graphene engineering: an ab initio study of the
TRANSCRIPT
Graphene EngineeringAn ab initio Study of the Thermodynamic Stability ofEpitaxial Graphene and the Surface Reconstructions
of Silicon Carbide
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium(Dr rer nat)
im Fach Physik
Spezialisierung Theoretische Physik
eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultaumlt
der Humboldt-Universitaumlt zu Berlin
von
Diplom Physikerin Lydia Nemec
Praumlsidentin Praumlsident der Humboldt-Universitaumlt zu BerlinProf Dr Jan-Hendrik Olbertz
Dekanin Dekan der Mathematisch-Naturwissenschaftlichen FakultaumltProf Dr Elmar Kulke
mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash-
Gutachterinnen 1 Prof Dr Claudia Draxl
2 Prof Dr Patrick Rinke
3 Prof Dr Ludger Wirtz
Tag der muumlndlichen Pruumlfung 7 Juli 2015
Abstract
Graphene with its unique properties spurred the design of nanoscale electronicdevices Graphene films grown by Si sublimation on SiC surfaces are promisingmaterial combinations for future graphene applications based on existing semi-conductor technologies Obviously the exact material properties of graphenedepend on the its interaction with the substrate Understanding the atomic andelectronic structure of the SiC-graphene interface is an important step to refine thegrowth quality In this work computational ab initio methods based on density-functional theory (DFT) are used to simulate the SiC-graphene interface on anatomistic level without empirical parameters We apply state-of-the-art density-functional approximation (DFA) in particular the Heyd-Scuseria-Ernzerhof hybridfunctional including van-der-Waals dispersion corrections to address the weakbonding between the substrate and graphene layers DFA simulations allow to in-terpret and complement experimental results and are able to predict the behaviourof complex interface system
Experimental work has shown that on the Si face of SiC a partially covalentlybonded carbon layer the zero-layer graphene (ZLG) grows as (6
radic3times 6
radic3)-R30
commensurate periodic film On top of the ZLG layer forms mono-layer graphene(MLG) as large ordered areas and then few-layer graphene By constructing anab initio surface phase diagram we show that ZLG and MLG are at least nearequilibrium phases Our results imply the existence of temperature and pressureconditions for self-limiting growth of MLG key to the large-scale graphene pro-duction At the interface to the substrate the Si atoms can be passivated by Hresulting in quasi-free-standing mono-layer graphene (QFMLG) We show that byH intercalation both the corrugation and doping are reduced significantly Ourcalculations demonstrate that the electronic structure of graphene is influenced byunsaturated Si atoms in the ZLG and therefore confirm that H intercalation is apromising route towards the preparation of high-quality graphene films
The situation on the C face of SiC is very different In experiment the growth oflarge areas of graphene with well defined layer thickness is difficult At the onsetof graphene formation a phase mixture of different surface phases is observedWe will address the stability of the surface phases that occur on the C side of SiCHowever the atomic structure of some of the competing surface phases as well asof the SiC-graphene interface is unknown We present a new model for the (3times3)reconstruction ndash the Si twist model The surface energies of this Si twist model theknown (2times2)C adatom phase and a graphene covered (2times2)C phase cross at thechemical potential limit of graphite which explains the observed phase mixtureWe argue that on the C face the formation of a well-controlled interface structurelike the ZLG layer is hindered by Si-rich surface reconstructions
v
Zusammenfassung
Die auszligergewoumlhnlichen Eigenschaften einer einzelnen Graphenlage ermoumlglichendas Design von elektronischen Bauteilen im Nanometerbereich Graphen kannauf der Oberflaumlche von Siliziumkarbonat (SiC) durch das Ausdampfen von Si epi-taktisch gewachsen werden Die Materialkombination SiC und Graphen ist daherhervorragend fuumlr die zukuumlnftige Anwendung von Graphen basierten Technologiengeeignet Die genauen Eigenschaften von epitaktischem Graphen haumlngen von derStaumlrke und Art der Interaktion mit dem Substrat ab Ein detailliertes Verstaumlndnisder atomaren und elektronischen Struktur der Grenzschicht zwischen Graphen undSiC ist ein wichtiger Schritt um die Wachstumsqualitaumlt von epitaktischem Graphenzu verbessern Wir nutzen Dichtefunktionaltheorie (DFT) um das HybridsystemGraphen-SiC auf atomarer Ebene ohne empirische Parameter zu simulieren Dieschwache Bindung zwischen dem Substrat und der Graphenlage beschreiben wirmit van-der-Waals korrigierten Austausch-Korrelations-Funktionalen insbeson-deren mit dem Hybrid Funktional von Heyd Scuseria und Ernzerhof [138 175]Die Simulationen geben Aufschluszlig uumlber das Verhalten und die Eigenschaften derkomplexen Grenzschicht
Experimentelle Arbeiten auf der Si-terminierten Oberflaumlche von SiC haben gezeigtdass die Grenzschicht durch eine teilweise kovalent gebundene Kohlenstofflage als(6radic
3times 6radic
3)-R30 kommensurate periodische Struktur waumlchst Die Grenzschichtzwischen SiC und Graphen wird haumlufig als ZLG bezeichnet Uumlber dem ZLG bildetsich die erste groszligflaumlchig geordnete Graphenlage (MLG) Durch das Konstru-ieren eines ab initio Oberflaumlchenphasendiagrams zeigen wir dass sowohl ZLGals auch MLG Gleichgewichtsphasen sind Unsere Ergebnisse implizieren dassTemperatur- und Druckbedingungen fuumlr den selbstbegrenzenden Graphenwach-stum existieren Die Si Atome an der Grenzschicht koumlnnen durch Wasserstoffatomeabgesaumlttigt werden und den ZLG zu quasi-freistehendem Graphen (QFMLG) um-wandeln Wir zeigen dass durch die H-Interkalation das Doping und die Riffel-lung von epitaktischem Graphene reduziert werden Aus unseren Rechnungenfolgt dass die ungesaumlttigten Si Atome an der Grenzschicht die elektronischen Ei-genschaften von Graphen beinflussen Anhand unsere Ergebnisse zeigen wir dassdie H-Interkalation eine vielversprechende Methode ist um qualitativ hochwertigesGraphen zu wachsen
Das Graphenwachstum auf der C-terminierten Oberflaumlche von SiC verhaumllt sichim Experiment qualitativ anders als auf der Si Seite Zu Beginn des Graphen-wachstums wird eine Mischung verschiedener Oberflaumlchenphasen beobachtet Wirdiskutieren die Stabilitaumlt dieser konkurierenden Phasen Die atomaren Strukturenvon einigen dieser Phasen inklusive der Graphen-SiC Grenzschicht sind nichtbekannt wodurch die theoretische Beschreibung erschwert wird Wir praumlsentierenein neues Model fuumlr die bisher unbekannte (3times3) Rekonstruktion ndash das Si Twist
vi
Model Die Oberflaumlchenenergie vom Si Twist Model und von der bekannten (2times2)C
Oberflaumlchenrekonstruktion schneiden sich direkt an der Grenze zur Graphitb-ildung Dies erklaumlrt die experimentell beobachtete Phasenkoexistenz zu Beginndes Graphenwachstums Wir schlussfolgern dass die Bildung von einer wohl-definierten Grenzschichtstruktur wie der ZLG auf der Si Seite auf der C Seitedurch Si-reiche Oberflaumlchenrekonstruktionen blockiert wird
vii
I do not try to dance better than anyone elseI only try to dance better than myself
Mikhail Baryshnikov
Contents
Introduction 1
I Electronic Structure Theory 9
1 The Many-Body Problem 1111 The Many-Body Schroumldinger Equation 1112 The Born-Oppenheimer Approximation 13
2 Basic Concepts of Density-Functional Theory 1521 The Hohenberg-Kohn Theorems 1522 The Kohn-Sham Equations 1723 Approximations for the Exchange-Correlation Functional 20
231 Local-Density Approximation (LDA) 20232 Generalised Gradient Approximation (GGA) 21233 Hybrid Functionals 22
24 Van-der-Waals Contribution in DFT 2325 Computational Aspects of DFT The FHI-aims Package 25
3 Swinging Atoms The Harmonic Solid 29
4 Ab initio Atomistic Thermodynamics 3241 The Gibbs Free Energy 3242 The Surface Free Energy 34
5 The Kohn-Sham Band Structure 39
II Bulk Systems 41
6 Carbon-Based Structures 4361 The Ground State Properties of Diamond Graphite and Graphene 4562 The Challenge of Inter-planar Bonding in Graphite 5063 The Electronic Structure of Graphene 58
7 Silicon Carbide The Substrate 6171 Polymorphism in Silicon Carbide 6272 The Electronic Structure of Silicon Carbide Polymorphs 66
III The Silicon Carbide Surfaces and Epitaxial Graphene 71
8 The 3C-SiC(111) Surface phases 7381 The Silicon-Rich Reconstructions of the 3C-SiC(111) Surface 74
xi
811 3C-SiC(111) (radic
3timesradic
3)-R30 Si Adatom Structure 74812 The Silicon-Rich 3C-SiC(111) (3times 3) Reconstruction 77
82 The Carbon-Rich Reconstructions of the 3C-SiC(111) Surface 79
9 Strain in Epitaxial Graphene 8591 ZLG-Substrate Coupling and its Effect on the Lattice Parameter 8592 Interface Models for Epitaxial Graphene 8893 Defect Induced Strain in Graphene 9594 Summary 100
10 Thermodynamic Equilibrium Conditions of Graphene Films on SiC 103
11 Band Structure Unfolding 107111 The Brillouin Zone (BZ) 110
1111 The Wave Function 112112 The Unfolding Projector 113113 The Hydrogen Chain Practical Aspects of the Band Structure Unfolding 117
12 The Electronic Structure of Epitaxial Graphene on 3C-SiC(111) 122121 The Electronic Structure of Epitaxial Graphene 123122 The Influence of Doping and Corrugation on the Electronic Structure 127123 The Silicon Dangling Bonds and their Effect on the Electronic Structure 133124 Summary 135
13 The Decoupling of Epitaxial Graphene on SiC by Hydrogen Intercalation 137131 The Vertical Graphene-Substrate Distance 138132 The Electron Density Maps 140133 Summary 142
14 Epitaxial Graphene on the C Face of SiC 143141 The 3C-SiC (111) Surface Reconstructions 143
1411 The (2 x 2) Si Adatom Phase 1431412 The (3 x 3) Surface Reconstruction An Open Puzzle 1451413 The Si Twist Model 150
142 Assessing the Graphene and SiC Interface 152143 Summary 156
Conclusions 157
Appendices 161
A Thermal Expansion and Phase Coexistence I
B Numerical Convergence IXB1 The Basis Set IXB2 Slab Thickness XIB3 k-Space Integration Grids XIIB4 The Heyd-Scuseria-Ernzerhof Hybrid Functional Family for 3C-SiC XIII
C Band Structure Plots of SiC XVC1 Cubic Silicon Carbide (3C-SiC) XVC2 Hexagonal Silicon Carbide 4H-SiC and 6H-SiC XVI
C21 The HSE Band Structure of 4H-SiC XVIC22 The HSE Band Structure of 6H-SiC XVII
xii
D Phonon Band Structure XVIIID1 The Phonon Band Structure of the Carbon Structures Graphene Graphite
and Diamond XVIIID11 Graphene XVIIID12 Graphite XXD13 Diamond XXI
D2 Phonon Band Structure of Silicon Carbide XXII
E Counting Dangling Bonds XXIII
F Comparing the PBE and HSE06 Density of States of the MLG Phase XXVF1 Influence of the DFT Functional XXV
G Bulk Stacking Order The Si-terminated 6H-SiC(0001) Surface XXVIII
H Influence of the Polytype XXIX
I The 3C-SiC(111)-(3x3) Phase Details on the Geometry XXX
J The History of Graphite Interlayer Binding Energy XXXII
Acronyms XXXV
List of Figures XLI
List of Tables LIII
Bibliography XCVII
Acknowledgements XCIX
Publication List CI
Selbstaumlndigkeitserklaumlrung CIII
xiii
Introduction
Introduction
The technological progress in the second half of the twentieth century wasdominated by the development of microelectronics based on the semicon-ductor silicon In the development and production of new electronic devicessilicon undeniably will continue to play a central role Today we aim to de-velop electronic devices at the nanoscale To meet the challenges that comewith advances in building components as small as a few nanometre newmaterials are necessary A promising material candidate to lead the fieldof device design is graphene Graphene is a two-dimensional atom-thicklayer of carbon atoms ordered in a honeycomb structure For many yearsgraphene was considered a theoreticianrsquos toy model and was assumed notto exist in a free state [338] Novoselov et al [235] succeeded to isolate asingle layer of graphene from a pencil stroke using Scotch tape Ironicallytheir low-tech approach [235] paved the way to the production of nano-scale high-tech devices [102 161 344 197 122 91 287] The discovery ofgraphenersquos outstanding electronic and structural properties was met withvast interest by the solid-state community [236 351 160 50] In 2010 An-dre K Geim and Konstantin S Novoselov at the University of Manchesterwere awarded the Nobel Prize in Physics for groundbreaking experimentsregarding the two-dimensional material graphene
The method of graphene exfoliation using Scotch tape is not practical forlarge-scale device production For practical applications of graphene-basedelectronics wafer-size high-quality graphene films are needed The groupof W A de Heer [18 19] demonstrated that epitaxial graphene on siliconcarbide (SiC) could be a successful alternative route towards wafer-sizegraphene growth Instead of graphene exfoliation they faciliate the surfacegraphitisation of high-temperature-treated SiC eg Refs [332 95 82 26367 302 346] In SiC the vapor pressure of Si is higher than that of carbonWhen heated Si sublimates and leaves a graphene-covered surface behind[332 95] This makes graphene growth on SiC special in the sense thatinstead of offering one or more of the components from the gas phasegraphene areas are formed by excess carbon at the surface [332 95] Asof today epitaxial growth of graphene on SiC by thermal decompositionis one of the most promising material combinations for future grapheneapplications based on established semiconductor technologies [18 19 82263 197 67 137 65 302 122 346] Indeed graphene-based devices andeven integrated circuits [161 344 287 197 137 122 91] were already createdemploying epitaxial graphene grown on SiC substrate
2
Figure 01 Schematic illustration of the graphene growth on the Si-side of SiC On theSi-side epitaxial graphene growth starts with a clean Si-terminated surface At temper-atures above 1000 K first a carbon interface layer forms [263] the so-called zero-layergraphene (ZLG) Increased temperature and growth time lead to the formation of mono-layer graphene (MLG) and bilayer graphene (BLG)
Graphene films grow on both polar surfaces of SiC the C-terminated [1819 334 64 302 141 83 21 337 213] and the Si-terminated surface [332 95264 82 263] The growth mechanism and the resulting graphene film differconsiderably between the two different SiC faces
Figure 01 illustrates schematically the growth process of epitaxial graphenefilms on the Si side In experiment the growth process starts from a cleanSi-terminated SiC surface With increased temperature the surface phasesgo from a Si-rich to a C-rich regime [332 95 263] First a (3times3) Si-richreconstruction [310] can be prepared then upon further annealing more Sisublimates and a simpler (
radic3timesradic
3)-R30 bulk-terminated Si rich surfaceforms [158 308] At temperatures above 1000 K a carbon interface layerforms [263] ndash the zero-layer graphene (ZLG) also called rsquobuffer-layerrsquo TheZLG is a carbon nanomesh with a honeycomb atomic structure similarto graphene but it is bonded to the substrate by strong covalent bonds[109] as indicated in Fig 01 With increased temperature and growth timeunderneath the already formed ZLG grows a new interface layer Thenew interface layer lifts of the previous ZLG layer forming a mono-layergraphene (MLG) The system now consists of the ZLG layer covered by theMLG layer (see Fig 01) Only the second carbon layer ndash the MLG ndash actslike a graphene layer displaying the linear π bands typical for the grapheneband structure [35] With increased layer thickness multi-layer graphenebehaves more like graphite than graphene
During annealing the nucleation of excess carbon atoms starts at the stepedge allowing for a layer-by-layer growth process On the Si side the aim isto control the graphene layer thickness and reduce the coexistence of MLG
3
and bi-layer graphene (BLG) [18 82 92 67] Across MLG-BLG graphenejunctions on the same Si-terminated surface a recent joint experimental-theoretical study finds particularly high local resistances [152] This mightbe a possible contributing factor to low carrier mobilities in graphene onthe Si face of SiC [82 152] Graphene films grown under ultrahigh vacuum(UHV) conditions are typically inhomogeneous [82 67] Improved MLGfilm homogeneity is achieved by increased growth temperature in an argonbackground buffer gas [82] de Heer et al [67] reported excellent wafer-sizeMLG films grown in a confined cavity that may retain a finite well-definedSi background pressure as Si evaporates from the surface Improving thequality of epitaxial graphene is a major and ongoing experimental goal [8267]
In the past the appearance of different phases on the Si-terminated surfacewas often interpreted [83 217 67] as successive intermediates formed byan outgoing Si flux that ultimately leads to the growth of bulk-like graph-ite layers Tromp and Hannon [329] demonstrated that the C-rich ZLGlayer [332 95 264 83] (not yet graphene) on the Si face is a reversible ther-modynamic equilibrium phase at high T and a controlled disilane (Si2H6)background pressure Reversibility is much harder to demonstrate once acomplete graphene plane has formed [124] What is still not clear howeveris whether MLG itself is an equilibrium phase under certain conditions
Graphene growth on the C-terminated surface differs greatly resultingin multilayer graphene During annealing a series of different surfacestructures were observed [332 304 145 192 24 292 208 140 307 139] Inan Si-rich environment graphene growth starts with a Si-rich (2times2) phase[24 192] or with an oxidic (
radic3timesradic
3) reconstruction [25 192] In the absenceof a Si background gas like disilane (Si2H6) a disordered oxidic layer with a(1times1) periodicity of bulk SiC is observed [304] Continued heating leads toa (3times3) phase
Using low-energy electron diffraction (LEED) Bernhardt et al [24] showedthat the (3times3) reconstructions originating from different starting structuresand environments are equivalent [24] Further annealing leads to a (2times2)Si adatom phase referred to as (2times2)C (notation taken from Ref [24])Just before graphene forms on the surface a coexistence of the two surfacephases the (3times3) and the (2times2)C is observed [24 337] The atomic structureof the (2times2)C was resolved by quantitative LEED [292] while the (3times3)reconstruction and the SiC-graphene interface remains a puzzle [192 145139 71]
Here two very different scenarios have been invoked for the structural
4
properties of the SiC-graphene interface
(a) The first carbon layer is strongly bound to the substrate [334 213 302]Here the Si sublimation rate during graphene growth is controlledby either using a confined geometry [129] by working in an inert gasatmosphere [82] or by providing an external Si gas phase [302 94] forexample disilane (Si2H6) background gas
(b) For samples prepared under UHV conditions the first carbon layer isweakly bound to the substrate and shows the characteristic behavior ofthe π-band at the K-point of the Brillouine zone Here an inhomogen-eous interface is present since the (3times3) and the (2times2)C reconstructionsare observed underneath the graphene layers [141 83 307 337]
For the weakly bound interface structure a (2times2) and (3times3) LEED patternwas observed underneath the graphene layer [141 83 307] An additionaltypical feature of this LEED pattern is a ring like structure originatingfrom rotational disordered graphene films [130 131 141 337 141] Therotational disorder originates from the growth process where graphenelayers nucleate on a terrace and grow in all directions on the surface [231]Due to the rotational disorder the electronic structure of the single graphenesheet decouples from the underlying graphene layer and exhibits single-layer-like electronic properties even for multilayer graphene films withvery high electron mobilities [131 301 21]
A recent study on graphene grown by molecular beam epitaxy (MBE) onthe C side exhibited the same structural characteristics as graphene grownby high-temperature annealing [218] This is a strong indication that indeedthe (2times2)C as well as the (3times3) reconstruction prevails below the graphenefilms
While some groups report the successful growth of large-scale MLG [146269] other reports suggest that the pure monolayer growth regime is dif-ficult to achieve on the C side [213] The exact material properties ofgraphene depend on the growth conditions and on the interaction betweenthe graphene layer and the substrate
The ultimate goal is to move the production of devices based on SiC-graphene hetero-structures from the laboratory to large scale productionsimilar to todayrsquos semiconductor device fabrication An important step torefine the growth process is to gain a deeper understanding of the atomicand electronic structure of the SiC-graphene interface We will see belowthat there are at least narrow thermodynamic equilibrium conditions for ZLG
5
MLG and even BLG grown on the Si-side of SiC However we found thata controlled graphene growth on the C-terminated surface is hindered bySi-rich surface reconstructions
This thesis presents first principles studies of the SiC phases on the Si and Cface laying the groundwork for a detailed atomistic understanding of thephase equilibria and resulting electronic properties of the surfaces involvedIn particular we will perform a density-functional theory (DFT) study onthe Si side and C side of the polar SiC surfaces using the all-electron numericatom-centered basis function code FHI-aims Due to the lattice mismatchbetween the SiC and graphene large commensurate surface structures formconsisting of up to sim 2800 atoms DFT is challenging for these systemsbecause of the system sizes and van-der-Waals (vdW) interactions notaccounted for by most standard density functionals We applied the ab initioatomistic thermodynamic formalism [340 279 280 259 260] to evaluatethe interface structure of epitaxial graphene and its competing surfacereconstructions The formation energies of different reconstructions andwidely-used model systems are presented as a function of the chemicalpotential of C
We show below that graphene films on the Si face of SiC grow at leastas near-equilibrium phases Therefore by tuning the chemical potentialwhich can be accomplished in experiment by varying the temperature andbackground pressure of C or Si it should be possible to grow high-qualityinterfaces as well as graphene structures on the Si side This makes theSi-terminated surface an ideal substrate for graphene growth We foundthat the interface layer ndash ZLG interface structure ndash plays a central role fora layer-by-layer graphene growth We demonstrate that the presence ofthe ZLG interface structure leads to a corrugated graphene film Not all Siatoms at the SiC-graphene interface are saturated some Si atoms remainunsaturated ndash the Si dangling bonds We show that the ZLG MLG and BLGare doped by the Si dangling bond states at the Fermi level Using DFTPerdew-Burke-Ernzerhof generalised gradient approximation [246] (PBE)including van-der-Waals effects [326] (PBE+vdW) and Heyd-Scuseria-Ern-zerhof hybrid functional [138 175] (HSE06+vdW) calculations we evaluatethe influence of the Si dangling bonds on the electronic structure of thegraphene films A different route to improve the electronic properties ofepitaxial graphene is to saturate the Si bonds at the SiC-graphene interfacewith hydrogen Hydrogen intercalation decouples the ZLG layer fromthe substrate forming quasi-free-standing mono-layer graphene (QFMLG)The QFMLG is flat and almost undoped featuring a homogeneous chargedensity at the interface The intercalation process improves the electronicproperties of the graphene film [97 315]
6
In experiment the same mechanism which leads to graphene growth onthe Si face fails on the C face Here controlling the layer thickness of thegraphene films remains a challenge [213] Just at the onset of surface graph-itisation a phase mixture of different surface phases is observed Studyingthe thermodynamic stability range of graphene films on the C side is diffi-cult since the atomic structure of some of the competing surface phases aswell as of the SiC-graphene interface are unknown We introduced a newmodel for the unknown 3times3-SiC(111) reconstruction ndash the Si-rich Si twistmodel inspired by the reconstruction known from the Si side [310 276] TheSi twist model captures the experimentally observed characteristics Com-paring the formation energy of the Si twist model and different interfacemodels indicates that the formation of a regulating interface structure likethe ZLG layer is hindered by Si-rich surface reconstructions
This thesis is structured in three parts
The first part introduces the basic concepts of electronic structure calcula-tions Chapter 1 familiarises the reader with the many-body problem in thecontext of condensed matter physics Throughout this work results areobtained by solving the many-body Schroumldinger equation (Eq 11) using DFTfor the electronic part The fundamental concepts of DFT and the derivationof the Kohn-Sham equations as well as the basic ideas behind the practicalapplication of DFT are given in Chapter 2 The joint vibrational motion ofnuclei in a lattice at a specific frequency are called phonons Material proper-ties such as the heat capacity thermal conduction or expansion depend onphonons In Chapter 3 the calculation of phonons in the DFT framework isexplained The central question throughout this work is whether thermody-namic equilibrium conditions like the temperature T and partial pressuresp can be found to control the growth of certain structures To tackle thisquestion we apply the ab initio atomistic thermodynamic formalism de-scribed in Chapter 4 In Chapter 5 we discuss the Kohn-Sham band structurewhich can provide a first insight into the electronic structure of a surface ifinterpreted with care
In the second part the reference bulk systems diamond graphite grapheneand SiC are introduced In Chapter 6 the structural and thermodynamicproperties of diamond graphite and graphene are characterised by ab initioelectronic structure calculations The three most relevant SiC polytypescubic silicon carbide (3C-SiC) and hexagonal 4H-SiC and 6H-SiC and theirphysical and electronic properties are discussed in Chapter 7
The third part discusses epitaxial graphene growth on the polar surfaces ofSiC In Chapter8 we introduce the different Si- and C-rich surface phases
7
observed during graphene growth on the Si-terminated SiC surface Inepitaxial graphene there are two aspects to strain Artificially induced strainby the choice of the coincidence lattice between the substrate and grapheneand strain caused by the bonding between the ZLG and the substrate Bothaspects of strain are discussed in Chapter 9 published in Ref [284] InChapter 10 we present our work on thermodynamic equilibrium conditionsof graphene films on SiC published in Ref [227] To evaluate the electronicstructure of epitaxial graphene and the interface grown on the Si face ofSiC we developed a method for the band structure unfolding exploiting theBloch-theorem in Chapter 12 In Chapter 13 we argue that the quality of epi-taxial graphene can be improved by hydrogen intercalation the experimetaland theoretical collaborative study was recently published in Ref [294]So far the focus was on epitaxial graphene on the Si-terminated surfaceIn Chapter 14 we investigate the relative phase stability of the competingsurface phases on the C-terminated surface in the thermodynamic rangeof graphitisation accepted for publication at Phys Rev B Rapid Comm(Ref [228])
8
Part IElectronic Structure Theory
9
This chapter introduces the theoretical toolbox applied in this thesis Webriefly retrace the underlying theory leading to the pragmatic aspects of den-sity-functional theory (DFT) The starting point is the quantum mechanicalmany-body problem Ch 1 and the Born-Oppenheimer (BO) approxim-ation Sec 12 The BO approximation formally separates the electroniccoordinates from the nuclear ones The electronic sub-problem is thensolved for a fixed set of nuclear coordinates greatly simplifying the many-body problem Then the electronic Schroumldinger equation is tackled in theDFT framework (Ch 2) The foundation of DFT are the Hohenberg-Kohntheorems Sec 21 Using the Hohenberg-Kohn theorems the electronicmany-body Hamiltonian is rewritten into a set of single particle Schroumldingerequations the Kohn-Sham equations To finally turn DFT into practice forsystem sizes up to several thousand atoms approximations for the un-known terms in the Kohn-Sham equations have to be introduced For areliable description of solids surfaces or weakly bonded layered materialssuch as graphite it is essential to include van-der-Waals (vdW) contribu-tions (Sec 24) Once the Kohn-Sham equations are solved in this workusing the FHI-aims package (Ch 25) the resulting total energy can be usedto determine the stability of structures and surface reconstructions (Ch 4)So-called phonons (Ch 3) the movement of the nuclei in a solid within theharmonic limit can also be included The Kohn-Sham eigenvalues can beused to obtain a first impression of the electronic structure of the systemThe discussion in this chapter follows the books by Kohanoff [170] Martin[211] Capelle [46] and the lecture given by Patrick Rinke in 20131
1Electronic Structure Theory - Technical University of Berlin Winter Term 2013httpwwwfhi-berlinmpgde~rinke
10
1 The Many-Body Problem
A piece of matter is a collection of interacting atoms The atoms can bearranged periodically into a bulk solid surface or wire They can also formmolecules and clusters or a mixture like a molecule absorbed on a surfaceHowever the description of materials on a quantum mechanical level ischallenging - known as the many-body problem For example a 1 cm3
large cube of 3C-SiC contains Ne 1025 electrons and Nn 1024 nucleiHere the Ne electrons interact with each other and with Nn nuclei In thefollowing we discuss the approach used throughout this work to gaininsight into material properties by finding approximations to the many-body problem
11 The Many-Body Schroumldinger Equation
The non-relativistic quantum-mechanical description of an N-particle sys-tem such as bulk 3C-SiC is governed by the many-body Schroumldingerequation In this work we focus on the time-independent Schroumldingerequation
HΨ = EΨ (11)
where H is the many-body Hamiltonian of the system E the energy andΨ the many-particle wave function Although the mathematical form ofthe Schroumldinger equation (Eq 11) is known an analytic solution is onlypossible in certain special cases eg the hydrogen atom [eg see 285Ch 63]
The many-body Hamiltonian (H) is a coupled electronic and nuclear prob-lem Throughout this thesis we will use atomic units this means
me = e = h =1
4πε0= 1
Then the general H in our system is given by Eq 12
11
bull Coulomb interactions between electrons Vee
bull Kinetic energy of the electronswith pi rarr minusinablari Te
H =Ne
sumi=1
minusp2i
2+
Ne
sumi=1
Ne
sumjgti
1∣∣ri minus rj∣∣
Nn
sumI=1
minusP2I
2MI+
Nn
sumI=1
Nn
sumJgtI
ZIZJ∣∣RI minus RJ∣∣ +
Ne
sumi=1
Nn
sumJ=1
ZJ∣∣ri minus RJ∣∣
(12)
bull Kinetic energy of the
nuclei with PI rarr minusihnablaRI Tn
bull Coulomb interactions between nuclei Vnn
bull Coulomb interactions between nuclei and electrons Vne
Ne number of electrons Nn number of nuclei
me electron mass Mn nuclear mass
e electronic charge Z nuclear charge
r electronic coordinate R nuclear coordinate
p electronic momentum P nuclear momentum
The electronic contributions in Eq 12 Te and Vee and the nuclear contribu-tions Tn and Vnn are coupled through the Coulomb interaction betweennuclei and electrons Vne The Schroumldinger equation (Eq 11) thus takes theform (
Te + Vee)+ Vne +(Tn + Vnn)Ψη = EηΨη (13)
where Ψη equiv Ψη(RI ri) is the many-body wave function that depends onthe coordinates of all electons ri and nuclei RI The ground state wave func-
12
tion (Ψ0) is a (Ne + Nn) middot 3 dimensional object and E0 is the correspondingground state energy All higher (ngt0) states correspond to (neutral) excitedstatesPreviously we estimated the total number of particles in a 3C-SiC cubeto be sim1025 Therefore the input to the ground state wave function is asim 1025 dimensional object The wave function Ψ0 has to be stored for manycombinations of coordinate values Usually Ψ0 is represented on a grid Tostore Ψ0 of this cube in a single grid point we would need a sim1013 Terabytedisc 1 Evidently we have to find approximations to solve the many-bodySchroumldinger equation
12 The Born-Oppenheimer Approximation
The first approximation we apply is the Born-Oppenheimer (BO) approxim-ation [see 285 Ch 152] The underlying idea is to reduce the complexity ofthe many-body problem by separating the electronic problem from the nuc-lear one In the Born-Oppenheimer (BO) approximation the electrons arecalculated in an external potential (Vext) generated by a fixed nuclear con-figuration The electronic Hamiltonian (He) within the BO approximationhas the form
He = Te + Vee + Vext (14)
The external potential (Vext) is generated by the Coulomb attraction betweenelectrons and nuclei given by (Vne) in Eq 12 with the ion positions R = RJkept fixed The electronic Schroumldinger equation then reads
He(ri R)φν(ri R) = E eνφν(ri R) (15)
where φν are the eigenfunctions of the electronic Hamiltonian and E eν its
eigenvalues The electronic spectrum parametrically depends on R Forany fixed nuclear configuration (R) the eigenfunctions φν(ri R) of He
form a complete basis set In the BO approximation the wave function Ψη
of the many-body Schroumldinger equation (Eq 13) can be approximated byan expansion in terms of the electronic eigenfunctions
Ψη(R ri) = sumν
cνη(R)φν(ri R) (16)
1In the case of a 1 cm3 SiC cube Ψ0 is a (Ne + Nn) middot 3 sim 33 middot 1025 dimensional object Ifwe store every entry of Ψ0 for a single grid point as a floating point number with singleprecision (4 byte per entry) sim 132 middot 1025 byte sim 1013 Terabyte are needed
13
where the expansion coefficients cνη(R) depend on the ionic configurationThe BO approximation holds when we can assume that the electrons adjustto the nuclear positions almost instantaneously This is a valid assumptionfor systems with well-separated adiabatic electronic states in which themovement of the nuclei (eg because of phonons or vibrations) does notlead to electronic transition All results shown in this work use the BOapproximation
14
2 Basic Concepts of Density-FunctionalTheory
In solid state physics material science or quantum chemistry electronicstructure theory is applied to gain information about material propertiesOne of the most successful and popular quantum mechanical approachesis density-functional theory (DFT) It is routinely applied for calculatingground-state properties such as cohesive energies phase stabilities or theelectronic structure of solids and surfaces
21 The Hohenberg-Kohn Theorems
The Hohenberg-Kohn theorems [144] are the foundation of density-functio-nal theory (DFT) which formally introduces the electron density as basicvariable In essence the electronic many-body problem (Eq 15) is reformu-lated in terms of the ground state density n0(r) instead of the ground statewave function Ψ0(ri) The advantage is that n0(r) depends on only 3 andnot 3Ne spatial coordinates but still contains all the informations needed todetermine the ground state The ground state density is defined as
Definition 211 Ground State Density
n0(r) equiv Ne
intd3r2
intd3rNe |Ψ0(r r2 rNe)|
2
Ne is the number of electrons and Ψ0 is the normalised ground state wave function
For simplicity it is assumed that the ground state is non-degenerate Thefirst Hohenberg-Kohn theorem states that there is a one to one mappingbetween n0(r) Ψ0(ri) and the external potential (Vext) [see Eq 15] Inother words the Vext uniquely defines the density
The second Hohenberg-Kohn theorem [144 Part I2] introduces the vari-ational principle The electron energy E of a system can be formulated as afunctional of the electron density n(r)
E [n(r)] =int
d3r n(r)vext︸ ︷︷ ︸specific
+F [n(r)]︸ ︷︷ ︸universal
(21)
15
F [n(r)] is called the Hohenberg-Kohn functional It contains the electronkinetic energy and the electron-electron interaction as functionals of n(r)but it is independent of the external potential (Vext)
Definition 212 Hohenberg-Kohn functional
F [n(r)] equiv Te[n(r)] + Eee[n(r)]
The second Hohenberg-Kohn theorem states that for any Vext the groundstate density (n0(r)) minimises the energy functional E [n(r)] The minim-ising density of Eq 21 is n0(r)
E0 = E [n0(r)] le E [n(r)] (22)
The Hohenberg Kohn theorems allow for an iteratively improvement of aninitially chosen trial density by minimising E [n(r)] However the majorchallenge is that F [n] is unknown In the original work by Hohenberget al the density variation was limited to a non-degenerate ground statesand v-representable densities These are densities that can be generated by aunique external potential The minimisation can be generalised by Levyrsquosconstrained search approach [191]
16
22 The Kohn-Sham Equations
The Hohenberg-Kohn theorems in Sec 21 provide the mathematical basisof density-functional theory (DFT) However guidance is not providedon how to built the Hohenberg-Kohn functional (F [n]) (Def 212) whoseminimisation gives the ground state energy and density One challenge isthe kinetic energy Te[n(r)] of the electrons because an explicit expressionin terms of the electron density is not known Kohn and Sham cast the cal-culation of the ground state energy into a set of single particle Schroumldingerequations Their approach is based on a non-interacting system of electronswhich can be described by a Slater determinant of one-particle orbitalsalso-called Kohn-Sham orbitals (ϕi(r)s)
-
-
--
-
-
-
--
- veff
interacting systemH auxiliary system hs =minusnabla2
2 + veff
Figure 21 The interacting system is mapped to a non-interacting auxiliary system
The central idea of the Kohn-Sham approach is to map the real physicalsystem onto a non-interacting reference system as sketched in Fig 21 Theinteraction of the electrons is described by the many-body Hamiltonian inEq 15 Here all electrons interact with each other (indicated by arrows inFig 21) The system can be converted into a system of non-interacting elec-trons governed by an effective potential veff veff is an effective interactionthat guarantees that the orbitals corresponding to the non-interacting elec-trons give the same density as the density of the interacting system Sincethe density is uniquely determined by the external potential (see Sec 21) thedensity of the Kohn-Sham system has to equal that of the fully interactingsystem However the wave function of the fully interacting and the non-interacting system are different In Sec 21 the Hohenberg-Kohn functional(Def 212) was introduced It contains the electron-electron interactionEee[n(r)] and the kinetic energy Te[n(r)] of the electrons Unfortunatelythe functional form of this two terms is unknown The electron-electroninteraction term can be separated into the classical Coulomb repulsion EH
17
and a non-classical part Encl[n(r)] containing all other contributions
Eee[n(r)] =e2
2
intintd3r d3rprime
n(r)n(rprime)|rminus rprime|︸ ︷︷ ︸
EH [n(r)]
+Encl[n(r)] (23)
The kinetic energy term can be separated in a known part the kinetic energyof the non-interacting reference system and in an unknown part
Te[n(r)] = minus h2m
N
sumi=1
intd3r ϕlowast(ri)nabla2ϕ(ri)︸ ︷︷ ︸Ts[n(r)]
+Tc[n(r)] (24)
The Hohenberg-Kohn functional can then be rewritten as
F [n(r)] = Ts[n(r)] + EH[n(r)] + Encl[n(r)] + Tc[n(r)]︸ ︷︷ ︸EXC[n(r)]
(25)
The exchange-correlation functional (EXC) contains the difference betweenthe real interacting system and the non-interacting single particle system
Definition 221 Exchange-correlation energy
EXC[n(r)] equiv Te[n(r)]minus Ts[n(r)] + Eee[n(r)]minus EH[n(r)]
The two dominant terms Ts[n(r)] and EH[n(r)] can be calculated and areoften much higher in energy than EXC Although EXC is not easier toapproximate than F [n(r)] the error relative to the total energy made byapproximations is smaller The energy functional E [n(r)] (Eq 21) cannow be rewritten containing all known terms and the unknown exchange-correlation energy as
unknown
E [n(r)] = Ts[n(r)] + EH[n(r)] +int
d3r n(r) vext(r) + EXC[n(r)](26)
known
All terms in Eq 26 are functionals of the density except for Ts which is
18
explicitly expressed as a functional of the non-interacting wave functionand is unknown as an explicitfunction of n(r) The HamiltonianHs can bewritten as a sum of effective one particle Hamiltonians hs(ri)
Hs(r) =Ne
sumi=1
hs(ri) = minus12
Ne
sumi=1nabla2 +
Ne
sumi=1
veff(ri) (27)
ϕi(r) are the eigenfunctions of the one-electron Hamiltonian hs(r) They areobtained by solving the one-electron Schroumldinger equation
Figure 22 The effective poten-tial veff of the many-body systemdepends on Kohn-Sham density(nKS(r)) which we are search-ing which depends on the Kohn-Sham orbital (ϕi(r)) which inturn depends on veff
hs(ri)ϕi = εi ϕi(minus1
2nabla2 + veff(ri)
)ϕi = εi ϕi
(28)
Equation 28 defines the Kohn-Sham equa-tions The Kohn-Sham density nKS(r) is givenby the occupied ϕi(r)
nKS(r) =Nocc
sumi=1|ϕi(r)|2 (29)
The minimisation problem of the interacting many-body system is replacedby the Kohn-Sham equations (28) Using the Rayleigh-Ritz principle andvarying the wave function under an additional orthonormalisation con-straint one finds the effective potential as
veff =δEXC[n(r)]
δn(r)+ VH + Vext (210)
The effective potential veff[n(r)] depends on the density therefore the Kohn-Sham equations have to be solved self-consistently As illustrated in Fig 22the self-consistency cycle starts with an initial guess for nKS(r) next thecorresponding effective potential veff can be obtained Then Eq 28 is solvedfor ϕi(r) From these orbitals a new density is calculated and used as a trialdensity to calculate the new veff and start again The process is repeateduntil the trial density and veff equal the new density and the newly obtainedveff However the exact forms of EXC is unknown The DFT total energyEtot is typically calculated using the sum over the Kohn-Sham eigenvalues
19
εi
Etot[n(r)] =Nel
sumi=1
εi minus EH[n(r)] + EXC[n(r)]minusint
d3rδEXC[n(r)]
δn(r)n(r) (211)
The Kohn-Sham orbitals are introduced to simplify the problem but theirmeaning is purely mathematical Neither the orbitals nor their correspond-ing eigenvalues εi have a strict physical meaning except for the eigenvalueof the highest occupied orbital εVBM [150] The energy level εVBM equalsthe ionisation energy of the system This is the minimum amount of energyrequired to remove an electron of an atom molecule or solid to infinity
23 Approximations for the Exchange-CorrelationFunctional
In the following sections we will discuss approximations for the exchange-correlation functional (EXC) Def 221 necessary in any real application sincethe true functional is not available The Kohn-Sham approach maps themany-body problem exactly onto a one-electron problem It is possible tosolve the problem if approximations for EXC can be found Exact solutionsare available for very few systems but good approximations can be appliedto a large class of problems The search for better functionals is an activefield of research
231 Local-Density Approximation (LDA)
In the pioneering paper by Kohn and Sham in 1965 an approximation forthe exchange-correlation functional (EXC) was proposed - the local-densityapproximation (LDA) Kohn and Shamrsquos idea was to approximate the ex-change-correlation functional (EXC) of an heterogenous electronic system aslocally homogenous and use EXC corresponding to the homogenous electrongas at point r in space Assuming a slowly varying electron density (n(r))they expressed
EXC[n(r)] =int
d3r n(r)εxc-LDA[n(r)] (212)
20
where εxc-LDA[n(r)] denotes the exchange-correlation energy per electron ofthe homogenous electron gas εxc-LDA[n(r)] can be rewritten as a sum of theexchange and correlation contributions
εxc-LDA[n(r))] = εx-LDA[n(r)] + εc-LDA[n(r)] (213)
The exchange part εx-LDA[n(r)] of the homogeneous electron gas was evalu-ated analytically by Dirac [73]
εx-LDA[n(r)] =34
3
radic3n(r)
πasymp 0458
rs(214)
where rs is the mean inter-electronic distance expressed in atomic units [73]For the correlation part the low- and high-density limits are known but ananalytic expression for the range between the limits is not known Howeverfor the homogeneous electron gas excellent parametrisations of accuratequantum Monte Carlo (QMC) calculations exist [52] The LDA appears tobe oversimplified in particular for strongly varying densities as one mightexpect in real materials Nonetheless LDA forms the basis for most of thewidely used approximations to EXC
232 Generalised Gradient Approximation (GGA)
The LDA only takes the density at a given point r into account One ofthe first extension to the LDA was the generalised gradient approximation(GGA) In the GGA the treatment of inhomogeneities in the electron densityis refined by a first order Taylor series with respect to density gradients
EXC asymp EXC-GGA[n(r)] =int
d3r n(r)εxc-GGA[n(r) |nablan(r)|]
equivint
d3r n(r)εx-hom FXC(n(r) |nablan(r)|)(215)
where FXC is a dimensionless enhancement factor As before the exchange-correlation energy per electron εxc-GGA is split into an exchange and correl-ation contribution The exchange contribution is the same as in the LDA(Eq 214) Unlike the LDA the functional form of GGAs and thus FXC isnot unique As a result a large number of GGAs have been proposed Acomprehensive comparison of different GGAs have been given for exampleby Filippi et al [90 chap 8] or Korth and Grimme [172]
The GGA suggested by Perdew et al is used throughout this work Theaim of the Perdew-Burke-Ernzerhof generalised gradient approximation
21
[246] (PBE) is to satisfies as many formal constraints and known limitsas possible sacrificing only those being energetically less important [246]PBE improves the description of bulk properties like cohesive energies andlattice constants as compared to LDA [311 312 328]
233 Hybrid Functionals
In the approximations for the exchange-correlation functional (EXC) dis-cussed so far a spurious self-interaction error remains The self-interactionerror as the name suggests is the spurious interaction of the electron dens-ity with itself For example the Hamiltonian of an one-electron systemdepends only on the kinetic energy and the potential due to the nucleiIn the Kohn-Sham framework the energy of such a system is given byEq 26 and contains two additional terms the Hartree term (Eq 23) andthe exchange-correlation functional (EXC) (Def 221) The exact exchange-correlation functional (EXC) would cancel the self-interaction introduced bythe Hartree term However in DFT most approximations to the exchange-correlation functional (EXC) leave a spurious self-interaction error Thisis different to the Hartree-Fock approximation where the self-interactionerror is cancelled by the exchange term
EHFx = minus1
2 sumij
intintd3r d3rprime
e2
|rminus rprime|ϕlowasti (r)ϕi(rprime)ϕlowastj (r
prime)ϕj(r) (216)
However Hartree-Fock theory neglects all correlation except those requiredby the Pauli exclusion principle This leads to sizeable errors in the de-scription of chemical bonding Combining DFT with Hartree-Fock exact-exchange is a pragmatic approach to deal with the problem These so-calledhybrid functionals were first introduced by Becke[16] His main idea was toreplace a fraction of the DFT exchange EDFT
X energy by the exact exchangeEHF
X energy In this work we use the Heyd-Scuseria-Ernzerhof hybrid func-tional [138 175] (HSE) therefore the discussion focuses on HSE For hybridfunctionals EXC is reformulated as
EhybXC = αEHF
X + (1minus α) EDFTX + EDFT
C (217)
where α specifies the fraction of exact exchange EHFX This approach works
well for a varity of systems reaching from semiconductors to moleculesand mitigates the effects of the self-interaction error reasonably well Heydet al introduced a screening parameter ω to separate the Coulomb operator
22
1|rminusrprime| =
1r into a short- and long-range part using the error function
1r=
1minus erf(ωr)r︸ ︷︷ ︸
SR
+erf(ωr)
r︸ ︷︷ ︸LR
(218)
That way only the short-range part of the Hartree-Fock exchange is included[138 175] Combining Eq 217 and Eq 218
EHSEXC (α ω) = αEHFSR
X (ω) + (1minus α) EPBESRX (ω) + EPBELR
X (ω) + EPBEC(219)
gives the so-called HSE functional Equation 219 contains two parametersα and ω one controls the amount of exact-exchange α and the second separ-ates the short- and long-range regime ω In the case of α = 0 and ω = 0 wefully recover PBE α = 025 and ω = 0 gives the PBE0 exchange-correlationfunctional [248] In the 2006 version of HSE α is set to 025 and the range-separation parameter ω to 011 bohrminus1 [175] so-called HSE06 The authorschose the value of ω by testing the performance of the functional for differ-ent test sets of atoms molecules and solids covering insulators semicon-ductors and metals For bulk systems they showed that band gaps are verysensitive to variations of the screening parameter ω while bulk propertieslike the bulk modulus or lattice parameters are less affected In addition toan improved description of the underlying physics the range-separationreduces the computational effort if compared to hybrid functionals withoutrange-separation mainly because the Hartree-Fock exchange decays slowlywith distance (1r-decay)
24 Van-der-Waals Contribution in DFT
Materials as well as molecules are stabilised by primary interatomic bondssuch as covalent ionic or metallic bonds In this section we discuss thesecondary bonding by van-der-Waals (vdW) interactions Van-der-Waalsforces are usually weaker than for example covalent interatomic forces Ofparticular importance for this work is their role for the interlayer bindingof graphite and few-layer graphene However all the exchange-corre-lation functionals introduced in Sec 23 do not include the long rangevdW tail Different scientific communities define vdW forces differentlythroughout this work the term vdW energy or dispersion energy will be used todescribe the energy contributions originating from induced dipole - induceddipole interactions The dispersion correction that we use in this thesis isbased on a non-empirical method that includs a sum over pairwise C6R6
AB
23
interactions between atoms A and B [343 116 117 326] It reads
EvdW = minus12 sum
ABfdamp(RA RB)
C6AB
|RA minus RB|6(220)
where |RA minus RB| = RAB is the distance between atom A and B and C6ABis the corresponding C6 coefficient The last term fdamp is the dampingfunction In practice the functional form of fdamp is chosen in such a waythat it eliminates the 1
R6AB
singularity The challenge is to find a good estimate
for the C6AB coefficient in Eq 220 Using the so-called Casimir-Polderintegral the C6AB coefficients are heteromolecular and can be written as
C6AB =3π
int infin
0dω αA(iω)αB(iω) (221)
where αAB(iω) is the frequency-dependent polarisability of atom A and BThis expression can be approximated by homonuclear parameters (C6AAC6BB) and their static polarisabilities (α0
A and α0B) only
C6AB =2C6AAC6BB(
α0B
α0A
C6AA +α0
Aα0
BC6BB
) (222)
In this work we use the scheme proposed by Tkatchenko and Scheffler (TS-vdW) In this approach the homonuclear C6 coefficients become density-dependent To this end an effective volume for an atom in a crystal ormolecule is defined by
VA =int
d3r wA(r)n(r) (223)
where n(r) is the total electron density and wA(r) is the Hirshfeld atomicpartitioning weight [142] It is defined as
wA(r) =nfree
A (r)
sumB nfreeB (r)
(224)
nfreeA (r) is the electron density of the free atom A and the sum goes over
all atoms in the system The electron densities for the free atom as wellas for the full system are calculated within DFT Finally the effective C6coefficients (Ceff
6AA) can be expressed in terms of the Cfree6AA coefficient of the
24
free atom (obtained from a database) and the Hirshfeld volume
Ceff6AA =
(VA
VfreeA
)2
Cfree6AA (225)
The C6R6AB terms of the TS-vdW scheme are derived from the self consist-
ent electronic density and thus adapt to their environment
As a final remark the nature of long-range dispersion is inherently a many-body effect The induced dipole moment of an atom in a solid (or molecule)depends on the induced dipole moments of all surrounding atoms Theseadditional screening contributions due to many-body effects cannot beaccounted for in a pairwise summation of vdW corrections Tkatchenkoet al proposed a refinement to the above scheme that corrects for the miss-ing screening effects by modeling the dispersion interactions as coupledquantum harmonic oscillators located at individual atomic sites That wayit additionally captures nonadditive contributions originating from simul-taneous dipole fluctuations at different atomic sites leading to an overallimproved screening and accounts for long-range anisotropic effects Thisapproach includes long-range many-body dispersion (MBD) effects employ-ing a range-separated (rs) self-consistent screening (SCS) of polarisabilities[324 5] and is therfore called MBDrsSCS [325 324 5] 1
25 Computational Aspects of DFT The FHI-aimsPackage
The Kohn-Sham equations 28 are solved self-consistently as described inSec 22 Usually the first step is to expand the Kohn-Sham orbitals ϕi(r)into a set of basis functions φj(r)
ϕi(r) = sumj
cijφj(r) (226)
Many basis function choices have been explored to list a few commonlyused basis functions and a selection of electronic structure codes usingthem2
1We use the MBDrsSCS formalism as implemented in the FHI-aims code since 06-20142An extensive list of quantum chemistry and solid-state physics software including
information about the basis set used can be found at W I K I P E D I A
25
Planewavesabinit [110]CASTEP [58]VASP [173]Quantum Espresso [104]
GaussiansGaussian [98]Crystal [78]
Full-potential (linearised) augmented planewaves (fp-LAPW)Wien2k [30 286]Exciting [121]
numeric atom-centered orbitals (NAOs)SIESTA [155]Dmol3 [69 70]FHI-aims [32]
In this work we use the FHI-aims package [32 133] FHI-aims is an all-electronfull-potential code that is computationally efficient without com-promising accuracy In order to achieve this for bulk solids surfaces or otherlow-dimensional systems and molecules the choice of basis functions iscrucial FHI-aims is based on numerically tabulated atom-centered orbitals(NAOs) of the following form
φj(r) =uj(r)
rYlm(Θ Φ) (227)
where Ylm(Θ Φ) are spherical harmonics and uj(r) is a radial part functionThe choice of uj(r) is flexible and is obtained by solving a radial Schroumldinger-like equation[
minus12
d2
dr2 +l(l + 1)
r2 + vj(r) + vcut(r)]
uj(r) = ε juj(r) (228)
The potential vj(r) defines the main behaviour of uj(r) Different potentialscan be chosen However the most common choices in FHI-aims are a self-consistent free-atom (or ion) radial potential or a hydrogen-like potentialThe second potential in Eq 228 is a confining potential vcut(r) It ensuresthat the radial function is strictly zero beyond the confining radius rcut anddecays smoothly In Fig 23 the silicon 3s basis function is shown withthe two potentials vj(r) and vcut(r) The basis functions are obtained bysolving Eq 228 on a logarithmic grid An advantage of NAO is that they
26
Figure 23 The radial function uj(r) of the 3s orbital for a free silicon atom is plotted alongradius r Also shown are the free-atom like potential vj(r) and the steeply increasingconfining potential vcut(r) The dotted line indicates the confining radius rcut
are localised around the nucleus which is essential to achieve linear scalingand calculating systems with 1000s of atoms The oscillatory behaviour ofthe electronic wave function close to the nucleus can be captured with justa few basis function as the oscillations are included by construction NAOsare non-orthogonal therefore the overlap sij of two basis functions localisedat two different atoms i and jint
d3r φlowastj (r)φj(r) = sij (229)
is taken into account An analytic treatment of the NAOs is not possibleinstead FHI-aims uses accurate and efficient numeric tools The main drawback is that the implementation of such algorithms is often more difficultHowever a major advantage of NAOs is that there are only basis functionsin regions with atoms eg the vacuum in a slab calculation contains nobasis functions
While plane waves allow a systematic improvement of the basis set byincreasing the number of plane waves NAOs cannot systematically beimproved by a single parameter To allow a systematic convergence ofthe calculation with respect of the basis size FHI-aims provides a libraryof pre-defined settings for all species which govern the key parametersregarding the numerical accuracy The library contains three different levelsof accuracy light tight and really tight (see Appendix B1) These three levelsspecify the accuracy of the real space grids Hartree potential and the basisset size The basis sets are organised in Tiers ordered by increasing accuracyEach Tier contains several basis functions of different angular momentaThe basis functions are determined by selecting them based on test sets ofnon-selfconsistent symmetric dimers in LDA That way the accuracy andperformance of calculations can be improved systematically by enabling
27
additional tiers one after another Further details on the construction of theFHI-aims basis set and the species default library can be found in Blum et al[32]
We use light settings to gain a first insight into a systemrsquos properties or forstructural to find a good first guess of the equilibrium atomic positionsThe next level the tight species defaults are rather safe for a variety ofsystems The really tight settings are over-converged for most purposes Theconvergence with respect to numerical and basis settings used for a specificproperty or system was ensured (see Appendix B
For structural relaxations and phonons total energy gradients with respectto nuclear positions are calculated In Section 22 the total energy of a systemis obtained by solving the Kohn-Sham equations self-consistently Forcesare then given by the first derivative of the total energy Etot (Eq 211) withrespect to the nuclear coordinate RI
FI = minusdEtot
dRI
= minuspartEtot
partRIminussum
ij
partEtot
partcij︸ ︷︷ ︸=0
partcij
partRI (230)
The derivative partEtotpartcij
= 0 because the self-constant energy Etot is variationalwith respect to the molecular orbital coefficients cij [101] The forces Eq 230are derived by calculating the derivative of the total energy Eq 211 termby term resulting in additional correcting terms (a full derivation anddiscussion is given in Gehrke [101])
bull The Pulay forces [255] originating from the dependency of the basisfunctions on the nuclear coordinates
bull The Hellman-Feynman forces [89 136] correspond to the classical forcesarising from the embedding of each nucleus in a field generated bythe electronic charge density of all other nuclei
bull The multipole-correction term originates from the derivative of theHartree energy Eq 23 because in FHI-aims the Hartree potentialis based upon a multipole expansion of the density instead of theelectron density [32]
28
3 Swinging Atoms The Harmonic Solid
In this chapter we discuss the collective vibrational motion of nuclei in alattice at a specific frequency so-called phonons (a detailed discussion canbe found eg in the book by Dove Ref [77]) Many physical propertiesof materials such as the heat capacity conduction or thermal expansiondepend on phonons
So far the nuclei were fixed at their nuclear positions RI and decoupled fromthe motion of the electrons by the Born-Oppenheimer (BO) approximationSec 12 The many-body Hamiltonian (Eq 12) separates into an electronicand a nuclear part In this sections we are dealing with the Hamiltonian ofthe nuclear system
H = Hnuc + E e = Tn + E e +12 sum
IJI 6=JV(RI minus RJ) = Tn + U (R) (31)
where E en are the eigenvalues of the electronic Schroumldinger Eq 15 1
2 sumIJI 6=J V(RIminusRJ) is the interatomic potential Vnn and Tn is the kinetic energy of the nucleigiven in Eq 12 In a crystalline solid the instantaneous atomic positionsRI = R0
I + uI(t) is given by the atomic equilibrium positions R0I and a time-
dependent displacement uI(t) For small displacements from the atomicequilibrium positions we can expand U (R) from Eq 31 in a Taylor series
U (RI) = U (R0I ) + sum
I
partU (RI)
partRI
∣∣∣∣R0
I
(RI minus R0
I
)︸ ︷︷ ︸
=0
+12 sum
I J
(RI minus R0
I
) part2U (RI)
partRIpartRJ
∣∣∣∣R0
I︸ ︷︷ ︸CI J
(RJ minus R0
J
)+O3 = Vharm(RI)
(32)
up to the second order this is known as the harmonic approximation Thefirst derivative is zero because all atoms are in their equilibrium positionThe second derivative with respect to the displacement uI(t) defines theinteratomic force constant matrix C in real space The Fourier transformedinteratomic force constant matrix CI J(q) is used to built the dynamicalmatrix DI J(q)
DI J(q) =CI J(q)radic
MI MJ (33)
29
The full solution for all vibrational states is given by the 3Nattimes 3Nat determ-inant equation to be solved for any wave vector q within the first Brillouinzone (BZ) with vibrational frequencies ωηq
det∣∣∣ macrD(q)minusω21
∣∣∣ = 0 (34)
with the identity matrix 1 The dependency of the vibrational frequenciesωη(q) on the wave vector q is known as the phonon dispersion and ηis the band index If there are Nat atoms per unit cell Eq 34 gives 3Nateigenvalues ndash vibrational frequencies ndash per reciprocal lattice vector q Thesecorrespond to the 3Nat branches of the dispersion relation In a one-atomicunit cell there are only three phonon bands for which ωη(q = 0) = 0they are called the acoustic modes They split in one longitudinal and twotransverse acoustic modes In crystals with more than one atom per unit cell(Nat gt 1) in addition to the three acoustic modes are higher branches so-called optical modes For a crystal with more than one atom in the primitivecell there are 3Nat minus 3 optical modes
The harmonic oscillator energy levels or eigenvalues for the mode ωq aregiven by
En(ωq) = hωq
(n +
12
) with n = 1 2 3 middot middot middot (35)
The full knowledge of the detailed dispersion relation ωη(q) is not alwaysnecessary For expectation values of thermodynamic potentials or theirderivatives the information about the number of phonon modes per volumeat a certain frequency or frequency range is sufficient the so-called phonondensity of states (phonon-DOS) The phonon-DOS gphon(ω η) of eachbranch η and the total phonon-DOS gphon(ω) are given by counting therespective number of phonons with a particular frequency ωη(q) withinthe BZ
gphon(ω) = sumη
1
(2π)3
intBZ
d3q δ(ωminusωη(q)) (36)
Phonons from first-principles The finite difference method
The interatomic force constant matrix C can be computed using ab initioquantum-mechanical techniques such as density-functional theory (DFT)Two different methods are commonly used to compute phonons
1 The finite difference method This scheme is based on numerical differ-entiation of forces acting on the nuclei when atoms are displaced bya small amount from their equilibrium positions In this work we
30
use the phonopy software package to calculate phonons applying thefinite difference method [327]
2 The linear response function or Greenrsquos function method Derivatives ofthe energy are calculated perturbatively up to third order A detaileddiscussion is given for example in [8 111]
In the following we focus on the discussion of the finite difference method asthis is the methodology used in this work In practice a single atom in asupercell (SC) is displaced by uIn and the induced forces on the displacedatom I and all other atoms in the SC are calculated The forces are
F(n I) = minussumnprime
CnInprimeJ middot uJn=0 (37)
where CnInprimeJ is the interatomic force constant matrix relating atoms I inthe unit cell n and J in unit cell nprime (Eq 32) Then the dynamical matrix inreciprocal space ˜D(q) (Eq 34) becomes
˜D(q) = sumnprimeJ
C0InprimeJradicMI MJ
times exp(minus2πi q middot (RI0 minus RJnprime)
)(38)
MI MJ and RI0 RJnprime are masses and positions of each atom plus anadditional phase factor because the atom I is located in the primitive unitcell and J in the nprime unit cell of the supercell Effects due to periodic boundaryconditions are included by displacing only atoms in the central unit cell(n = 0) and calculating the effect of the displacement on all the atoms in theSC At discrete wave vectors q the finite difference method is exact if thecondition
exp (2πi q middot L) = 1 (39)
is fulfilled (L is the lattice vector of the SC) Better accuracy of the phonondispersion is achieved by choosing a larger SC
The great advantage is that the calculation uses the same computationalsetup as other electronic structure calculations To describe the phononmodes at a reciprocal-lattice vector q the linear dimension of the SC is ofthe order of 2π
q Calculating a detailed phonon dispersion is demandingbecause the cost of an interatomic-force constant calculation will scale as3Nat times NUC where NUC is the number of unit cells in the SC and the factor3 accounts for the three generally independent phonon polarisations For-tunately making use of symmetry relations in the crystal can reduce thecomputational effort and phonons are often well behaved in regular solidsand can be interpolated reducing the SC size needed [245]
31
4 Ab initio Atomistic Thermodynamics
So far density-functional theory (DFT) was used to gain insight into ma-terial properties To include the effects of temperature and pressure onthe macroscopic system ab initio atomistic thermodynamics is applied Itcombines the results from DFT with concepts from thermodynamics andstatistical mechanics [340 279 280 259 260] Of particular interest in thiswork are phase diagrams to show conditions at which thermodynamicallydistinct phases can occur at equilibrium
In this chapter the ab initio atomistic thermodynamic formalism are de-scribed and its connection to DFT calculations In particular two questionare relevant for this work
1 How to determine the coexistence of materials in different phases
2 How can the energetics of different surface phases be compared
41 The Gibbs Free Energy
The coexistence of different phases as well as the stability of different single-component surface phases are best represented in a p-T-diagram Such adiagram is described by an isothermal-isobaric ensemble In this ensemblethe number of particles N the pressure p and the temperature T are inde-pendent variables The corresponding characteristic state function is theGibbs free energy
G(T p) = U minus T middot Scon f + p middotV (41)
where U is the internal energy Scon f is the configurational entropy and V isthe volume In the harmonic approximation we can rewrite U as a sum ofthe Born-Oppenheimer (BO) energy EBO (obtained from electronic structurecalculations) and the vibrational free energy Fvib
U = EBO + Fvib (42)
32
The vibrational free energy Fvib then can be written as
Fvib = Evib minus TSvib (43)
where Evib is the vibrational energy and Svib the vibrational entropy
The vibrational free energies In the previous chapter the phonon disper-sion in the harmonic approximation ωη(q) and the phonon density of states(phonon-DOS) were calculated using the finite difference method The micro-scopic partition function Z(T ω) can be calculated from the phonon-DOSand then be used to determine the thermodynamic state functions Evib andSvib of a solid
Evib = minus part
partβln (Z(T ω))
Svib = kB
(ln (Z(T ω)) + βEvib
)
(44)
Inserting Eq 44 into Eq 43 leads to
Fvib = minus 1β
ln (Z(T ω))
=int
dω gphon(ω)
[hω
2+
1β
ln(
1minus eminusβhω)] (45)
In some cases it is sufficient to include only the vibrational free energycontribution that originates from zero point vibrations In this case thetemperature goes to zero T rarr 0 and Eq 45 becomes
FZPE =int
dω gphon(ω)hω
2 (46)
Phase stabilities and the quasiharmonic approximation
Materials of the same chemical composition can occur in different phasessuch as graphite and diamond or the different polytypes observed for siliconcarbide The most stable phase for a given temperature and pressure is theone with the lowest Gibbs free energy (Eq 41)
The volume of a material changes with the temperature at constant pressureThis phenomenon is referred to as thermal expansion The basic physicalidea is that as the temperature rises the amplitude of the lattice vibrationsincreases so that due to anharmonicities the average value of the nuclear
33
displacement changes The nuclei then spend more time at distancesgreater or smaller than the original interatomic spacing and on average theatomic bond length changes The whole crystal changes so that the volumedependent free energy is minimised
In this work we only consider free energy contributions from phononsas they dominate the thermal properties in the systems of interest Theequilibrium structure at any temperature is always that with the lowest freeenergy
F(T V) = E(V) + Fvib(T V) (47)
where E(V) is the volume dependent total energy Fvib is the vibrational freeenergy (Eq 45) We use a phonon-based model the so-called quasiharmonicapproximation (QHA) to describe volume-dependency of the vibrationalfree energy (Eq 45) thermal effects In the QHA the unit cell volumeis an adjustable parameter The basic assumption is that the harmonicapproximation holds separately for a range of different volumes so thatthe temperature dependence of the phonon frequencies arises only fromthe dependence on the crystal volume Since by expanding the volume thelocal minimum of the BO surface changes so does the phonon dispersionleading to a change in the phonon-DOS and therefore in the vibrational freeenergy
42 The Surface Free Energy
In this section we focus on different surface phases We are interested inthe surface phase in equilibrium with its environment (eg backgroundgas or the silicon carbide (SiC) bulk) This means that the environmentacts as a reservoir because it can give or take any amount of atoms toor from the sample without changing the temperature or pressure TheGibbs free energy (Sec 41) is the appropriate thermodynamic potential todescribe such a system The Gibbs free energy G(T p Ni) depends on thetemperature and pressure and on the number of atoms Ni per species i inthe sample The most stable surface phase and geometry is then the onethat minimises the surface free energy For a surface system that is modeledby a slab with two equivalent surfaces the surface free energy is definedas
34
Definition 421 The surface free energy (γ)
γ =1
2A
[Gslab(T p Ni)minus
Nc
sumi
Nimicroi
]
=1
2A
[EBO + Fvib minus T middot Scon f + p middotV minus
Nc
sumi
Nimicroi
]
(48)
where A is the area of the surface unit cell G(T p Ni) is the Gibbs free energy ofthe surface under consideration Ni is the number of atoms of element i and microi theelementrsquos chemical potential and Nc the total number os different chemical speciesThe Gibbs free energy is given by Eq 41 with Eq 42 as an approximation to theinternal energy U
In this work we are interested in surface phases in equilibrium with the bulkphase In a solid like SiC the compressibility is extremely low 1 Thereforethe contribution of the p middotV term in Def 421 to the surface free energy willbe neglected In general vibrational energy contributions can be includedand may lead to small shifts However the necessary phonon calculationsof a large surfaces including more than 1000 atoms are computationallydemanding and not always feasible We neglect vibrational free energy Fvib
and configurational entropy contribution T middot Scon f to the surface free energyOnly the total energy term EBO is left as the predominant term This allowsus to rewrite Def 421 to
γ =1
2A
[EBO minus
Nc
sumi
Nimicroi
] (49)
For the input energies EBO we use total energies obtained from DFT calcu-lations DFT total energies are related to a thermodynamical quantity onlyin a restricted way They correspond to the internal energy U Eq 42 at zerotemperature and neglecting zero-point vibrations
Range of the chemical potential
The limits of the chemical potentials are given by the formation enthalpy of
1The volume of cubic silicon carbide (3C-SiC) changes by a factor of VV0 = 0927 ata pressure of 216 GPa and temperature of T = 298 K [300 313] In this work we areinterested in a low-pressure regime (lt 1MPa) In this pressure regime the change ofthe 3C-SiC bulk volume is negligible
35
SiC The enthalpy of a system is
H(T p) = G(T p) + T middot Scon f = U + p middotV (410)
The enthalpy of formation ∆H f is the energy released or needed to form asubstance from its elemental constituents in their most stable phases
Definition 422 The enthalpy of formation ∆H f
The enthalpy of formation ∆H f is the difference between the standard enthalpies offormation of the reactants and of the products
∆H f (T p) =Np
sumi
NiHi minusNr
sumj
NjHj (411)
where Np is the total number of products involved in the process Hi is the enthalpyof the product i and its respective stoichiometric coefficient Ni and Nr is the totalnumber of reactants Hj is the enthalpy of the reactant j and Nj its stoichiometriccoefficient
∆H f (SiC)(T p)
lt 0 products form
= 0 reactants and products are in equilibrium
gt 0 products precipitate into its reactants
As before for the surface free energy (Def 421) we will neglect the p middotVand temperature and vibrational contributions to the internal energy U inEq 410 Then we can express the enthalpy of formation ∆H f in terms oftotal energies obtained from DFT calculations
Throughout this work we address SiC surfaces grown by high temperaturesilicon (Si) sublimation Here the SiC bulk is in equilibrium with thesurrounding gas phase The gas phase is atomic carbon (C) and Si reservoirswhich supplies or takes away atoms from the surface The stability of theSiC bulk dictates microSi + microC = Ebulk
SiC where EbulkSiC is the total energy of a SiC
bulk 2-atom unit cell
As a next step the limits of the chemical potential have to be defined Theallowable range of the chemical potentials microSi and microC under equilibriumconditions are fixed by the elemental crystal phases of Si and C The dia-mond structure for Si is the appropriate bulk phase but for C there is a close
36
competition between diamond and graphite [23 347 348] Both referencesystems will be included in our analysis In Ch 6 and Appendix A a detaileddiscussion on the competition between the two C phases is presented
Finding the maximum of the chemical potentials
The bulk phases define the upper chemical potential limits Above theselimits the SiC crystal decomposes into Si or C
max(microSi) =12
HbulkSi (T p)
max(microC) =12
HbulkC (T p)
(412)
Applying the approximations to the enthalpy of the products and reactantsin Def 422 discussed above we will use total energies E obtained fromDFT calculations in the following derivation of the limits of the chemicalpotentials leading to
max(microSi) =12
EbulkSi max(microC) =
12
EbulkC (413)
for the maximum chemical potential of Si and C
Finding the minimum of the chemical potentials
Using the equilibrium condition of Def 422 microC + microSi = EbulkSiC we find for
the minimum chemical potential for C
min(microC) = EbulkSiC minus
12
ESi (414)
At the end we find for the limits of the chemical potential microC
EbulkSiC minus
12
ESi le microC le12
EbulkC (415)
Within the approximations to the enthalpy Eq 410 the limits can be rewrit-ten as
12
EbulkC + ∆H f (SiC) le microC le
12
EbulkC (416)
with ∆H f (SiC) = EbulkSiC minus
12 Ebulk
C minus 12 Ebulk
Si
The chemical potentials microC and microSi can be experimentally manipulated for
37
example through the substrate temperature and background pressure ofgases that supply Si or C [259 329 67 302] A precise control of the reser-voirs provided by the background gases (for instance disilane (Si2H8) [329])is desirable but calibration variations [202] may require exact (T p) rangesto be adjusted separately for a given growth chamber
It is important to note that small changes in the chemical potentials donot necessarily correspond to small changes in the experimental conditions(temperature and pressure) For example a drastic change in the number ofSi (NSi) and C (NC) atoms can correspond to a small change of the corres-ponding micro To cross beyond the carbon rich limit of graphite (or diamond)in equilibrium all Si has to be removed from the SiC crystal
In this chapter a bridge between DFT calculations and macroscopic phasestability was built The ab initio atomistic thermodynamics approach is a toolto compare different systems bulk or surfaces that are in thermodynamicequilibrium However it cannot give any information about the processof the phase formation or kinetic effects during the growth process Non-etheless a first very important step when dealing with systems in differentphases is to identify the thermodynamic equilibrium state
38
5 The Kohn-Sham Band Structure
Chapter 2 introduced the basic concepts of density-functional theory (DFT)Kohn and Sham turned the Hohenberg-Kohn theorems (Sec 21) into acomputationally feasible scheme (Sec 22) by mapping the system of inter-acting particles onto a fictitious system of non-interacting particles thatreproduces the same electron density as the many-body problem of interestThe central quantities obtained in DFT are the total energy and the electrondensity
In this section and throughout this work we will look at the Kohn-Shamband structure by plotting the Kohn-Sham eigenvalues εi(k) along high-symmetry lines in the Brillouin zone (BZ) Also DFT is a ground-statetheory the Kohn-Sham band structure can provide a first insight into theelectronic structure of a surface or crystal if interpreted with care
The Bulk Projected Band Structure In an electronic structure calculationthe transition from a three dimensional crystal with periodic boundaryconditions in all three directions to a two dimensional surface breaks thesymmetry of the direction perpendicular to the surface For a surface theperiodicity in the x-y-plane is conserved but broken along the z-directionFigure 51 illustrates how the periodicity of a periodic solid is broken alongthe z-axis In practice the surface is calculated within periodic boundaryconditions Surface calculations are realised by inserting a vacuum regionbetween the surface slabs as illustrated in Fig 51b The slab then is re-
a b c
Figure 51 (a) shows a periodic cubic silicon carbide (3C-SiC) crystal In (b) the periodicityis broken by inserting a vacuum region along the z-axis The bottom C atoms of the slabare terminated by H atoms In all surface calculations periodic boundary conditions areapplied Hence the surface slab is periodically repeated along the z-direction as shown in(c)
39
peated periodically along the z-direction (Fig 51c) Hydrogen (H) saturatesthe dangling bonds of the bottom carbon (C) atoms in order to avoid spuri-ous interactions or electron transfer As a consequence of the symmetrybreaking the wave number kz of the Bloch waves is no longer a goodquantum number In the lateral directions the crystal surface is still periodicTherefore the wave vector k are still good quantum numbers
Figure 52 A projected bulk bandstructure is illustrated schematicallyFor wave vectors kperp perpendicu-lar to the surface bulk states appearFor different kperp the band energiesbetween the lowest and highest val-ues correspond to regions of bulkstates These bulk states are shadedin grey in the band structure alongwave vectors parallel to the surfacek Outside the shaded bulk regionssurface states can occur (Figure cour-tesy of Johan M Carlsson Interna-tional Max-Planck Research School
ldquoTheoretical Methods for Surface SciencePart Irdquo (Presentation) )
The surface band structure can be characterised by a two dimensionalsurface BZ1 A projected band structure as shown in Fig 52 is a veryhelpful tool to identify and analyse electronic surface states Any wavevector k in the BZ splits into a component perpendicular to the surface kperpand one parallel k
k = k + kperp (51)
For each k-point in the surface plane there is a range of k-vectors with aperpendicular component To identify the bulk states in the 2-dimensionalsurface BZ the bands are calculated along a path in the surface BZ fordifferent kperp Due to the broken symmetry the band energies for differentkperp change The region between the minimum and maximum band energyis shaded like illustrated in Fig 52 The shaded regions correspond to bulkstates Outside the bulk regions surface bands can appear They correspondto states localised at the surface for example due to unsaturated bonds atthe surface
1A detailed discussion of the projection of 3 dimensional onto 2 dimensional BrillouinZone can be found in Bechstedt [13]
40
Part IIBulk Systems
41
Before investigating the hybrid SiC-graphene structures we will focus onthe pure bulk structures of crystalline carbon (C) and the substrate SiC Boththe substrate SiC and crystalline carbon phases exhibit a close competitionof different polymorphs
We introduced the ab initio atomistic thermodynamics formalism in Ch 4This formalism allows us to compare the stability of different phases in achemical potentials range The reference bulk phases SiC and crystalline Cand Si determine the limits of the chemical potential (Sec 42) In this partwe will discuss bulk SiC crystalline C in detail
SiC is observed in more than 250 different crystalline forms [57] Thebasic building block of any SiC polymorph is a Si-C-bilayer ordered in ahexagonal lattice The different polytypes of SiC differ in the stacking orderof these Si-C-bilayers Only the technologically most relevant polytypesare introduced in this chapter 3C-SiC and the hexagonal structures 4H-SiCand 6H-SiC
The situation of the carbon polymorphs is very different in the sense thatthe bonding characteristics change from a strong covalent tetrahedral - sp3
hybridisation in the case of C-diamond to planes of carbon atoms covalentlybonded by planar sp2 hybrid orbitals A single plane is called graphenegraphene layers stacked on top of each other and bonded by van-der-Waals(vdW) forces form graphite It is a challenge for electronic structure theoryto correctly describe the interlayer bonding in graphite In the followingwe discuss the accuracy we can achieve with the methodology used in thiswork The energetic competition between graphite and diamond at roomtemperature is very close In Appendix A we discuss the phase coexistenceof graphite and diamond
42
6 Carbon-Based Structures
Carbon is arguably the most versatile element in the periodic table of chem-ical elements due to its ability to form up to four covalent bonds by s-porbital hybridisation in various ways tetrahedral - sp3 planar - sp2 or linear- sp bonds If we take oxygen and hydrogen also into consideration carbonbuilds the structural backbone of an enormous variety of organic moleculesproviding the basic building blocks of life itself Even if we only considermaterials solely built from carbon atoms the structural diversity is largeThe two most commonly known carbon allotropes are the three dimensionalcrystalline structures diamond and graphite
In diamond carbon atoms arrange in a face-centered cubic lattice with twoatoms in the unit cell called diamond structure (see Fig 61 a) The bondnetwork is build from four identical sp3-tetrahedral hybrid orbitals Thestrong covalent C-C-bonds make diamond an extremely strong and rigidmaterial with an exceptionally high atom-number density Diamond is aninsulator with an optical band gap corresponding to ultraviolet wavelengthresulting in its clear and colourless appearance
Graphite on the other hand is a semimetal with good in-plane conductancegiving it its opaque grey-black appearance with a slight metallic shimmerIts structure differs from the diamond structure as it is a layered material (seeFig 61 b) Each layer is formed by carbon atoms arranged in a honeycomblattice The s px and py orbitals hybridise to form planar sp2 bonds Thenon-hybridised pz orbitals are oriented perpendicular to the plane andform weak π bonds with each of its neighbours [250] The carbon planesare stacked on top of each other and bonded by weak vdW forces Thisweak inter-plane bonding makes graphite soft enough to draw a line on apaper The very different nature of the in-plane and out-of-plane bondingleads to highly anisotropic electronic mechanical thermal and acousticproperties because electrons as well as phonons propagate quickly parallelto the graphite planes but they travel much slower from one plane to theother
43
Figure 61 Shows five different carbon structures The two commonly known threedimensional carbon crystals a diamond and b graphite A single layer of graphite formsgraphene a sp2 bonded two-dimensional structure labeled c d shows an example of acarbon nanotube and e the Buckminster-Fullerene - C60 also-called Buckyball
A single graphite sheet is called graphene (see Fig 61 c) In one of theearliest graphene related studies graphene was used as a theoretical modelto gain a deeper understanding of the graphite band structure [338] Wal-lace [338] first showed the unusual band structure with linear dispersingelectronic excitations characterising graphene as a semimetal At that timetwo-dimensional crystals such as graphene were expected to be thermo-dynamically unstable and therefore served as theoretical model systemsonly [50] 57 years after Wallacersquos publication [338] the modern era of thegraphene ldquogold-rushrdquo started when Novoselov Geim et al succeeded inisolating a single layer of graphene from a pencil stroke using Scotch tape[235] Since 2004 the interest in graphene has boomed and the publicationtrends in this area have been nearly exponential The extent of the ldquographenehyperdquo is visualised in Fig 62 For comparison the number of publicationsper year for SiC and for the hybrid SiC-graphene system are included aswell Figure 62 clearly demonstrates the rapid growth of interest in thisextraordinary two-dimensional material while the number of publicationsfor SiC has been almost constant for the last 20 years In 2010 Andre KGeim and Konstantin S Novoselov at the University of Manchester werearwarded the Nobel Prize in Physics for groundbreaking experiments re-garding the two-dimensional material graphene This atomically thin andhighly robust carbon layer is a promising material for future nano-electroniccomponents [333 102 287 233 234]
A single sheet of graphene rolled along a given direction and reconnectedthe C-C bonds at the end gives carbon nanotubes shown in Fig 61 d [149]For many physical aspects they show the behaviour of a one-dimensionalobject
Introducing pentagons in a single sheet of graphene creates a positivecurvature In this way the carbon atoms can be arranged spherically form-
44
Figure 62 Number of publications per year as given in a Web-of-Science search for thekeywords rsquographenersquo rsquosilicon carbidersquo and both keywords together [321]
ing so-called fullerenes molecules [174] (see Fig 61 e) They have discreteenergy states and can be thought of as zero-dimensional objects
In summary we introduced zero- one- two- and three-dimensional carbonstructures In this work we are mainly interested in the three- and two-dimensional carbon allotropes graphene graphite and diamond
61 The Ground State Properties of DiamondGraphite and Graphene
The aim of this Section is to characterise the structural properties and thethermodynamic stability of diamond graphite and graphene from ab initioelectronic structure calculations (Part I) In the description of thermody-namic properties such as the thermal expansion of the bulk phases vi-brational properties play a crucial role The crystalline carbon structureshave been subject to extensive theoretical and experimental studies (eg[349 164 103 221 250] and references therein)
We employ density-functional theory (DFT) using the FHI-aims all-electron
45
code [32 133] with rsquotightrsquo numerical settings (Appendix B) We compareresults obtained from different approximation to the exchange-correlationfunctional using the Perdew-Burke-Ernzerhof generalised gradient approx-imation [246] (PBE) the local-density approximation (LDA) [249] and theHSE hybrid functional with α=025 and ω=011 bohrminus1 [175] (HSE06) Foran accurate description of the different crystalline phases in particular theinterlayer bonding in graphite we include the effect of long-range electroncorrelations in an effective way using a vdW correction (Ch 24) forPBEand Heyd-Scuseria-Ernzerhof hybrid functional [138 175] (HSE)
To calculate equilibrium lattice parameters two different methods are usedin this work The first method minimises the energy with the help ofanalytic forces and stress tensor as implemented in the FHI-aims-code[169] For finding a minimum of the DFT based potential energy surface(PES) corresponding to the equilibrium lattice parameters a minimisationbased on analytic forces and stress tensor gives the most accurate resultsThe second method is based on numerically fitting computated energiesfor different volumes to the Birch-Murnaghan (B-M) equation of statesDef 611 which is used to account for temperature effects
Table 61 lists the structural parameter and the cohesive energy for diamondgraphite and graphene for different exchange-correlation functionals (LDAPBE HSE06 and vdW corrected PBE+vdW and HSE06+vdW) based onminimising the Born-Oppenheimer potential energy surface with the helpof analytic forces and stress tensor
The experimentally reported energy difference between diamond and graph-ite at T=0 K is 25 meVatom [23] and 150 meVatom [40] Based onthe potential energy minima (no zero-point vibrational correction (ZPC))Tab 61 graphite is found to be more stable than diamond in PBE+vdWby 60 meVatom This is qualitatively consistent with the extrapolatedexperimental phase hierarchy In plain PBE graphite is overstabilised by130 meVatom In LDA both phases are similarly stable Considering onlythe potential energy surface diamond is slightly more stable (by 12 meV)but already the inclusion of ZPC[347] neutralises this balance (graphitemore stable by 3 meVatom)
As a next step we include temperature effects by accounting for zero-pointvibrational effects At constant pressure the equilibrium volume of a mater-ial changes with temperature In Sec 41 we saw that the thermal expansioncan be understood in terms of lattice vibrations We introduced the quasihar-monic approximation (QHA) to calculate the volume dependent potentialenergy surface The basic assumption of the QHA is that the anharmonicity
46
Diamond
Functional a0 [Aring] Vatom [Aring3] Ecoh [eVatom]
LDA PES 3533 551 -895
PBE PES 3572 570 -774
PBE+vdW PES 3551 560 -794
HSE06 PES 3548 558 -756
HSE06+vdW PES 3531 551 -776
Exp 3567 [330] 567 [330] -7356+0007minus0039 [40]
Graphite
Functional a0 [Aring] c0 [Aring] Vatom [Aring3] Ecoh [eVatom]
LDA PES 2445 6643 8598 -894
PBE PES 2467 8811 11606 -787
PBE+vdW PES 2459 6669 8762 -800
HSE06 PES 2450 8357 9017 -765
HSE06+vdW PES 2444 6641 8591 -778
Exp 28815K [226] 2456plusmn1eminus4 [226] 6694plusmn7eminus4 [226] 8744 -7371+0007minus0039 [40]
Graphene
Functional a0 [Aring] Aatom [Aring2] Ecoh [eVatom]
LDA PES 2445 2589 -892
PBE PES 2467 2634 -787
PBE+vdW PES 2463 2627 -791
HSE06 PES 2450 2599 -765
HSE06+vdW PES 2447 2596 -769
Table 61 The structural and cohesive properties of diamond graphite and grapheneare listed for different exchange-correlation functionals (LDA PBE HSE06 and vdWcorrected PBE+vdW and HSE06+vdW) The lattice parameters a0 [Aring] and for graphite c0[Aring] calculated by minimising the energy with the help of analytic forces and stress tensoras implemented in the FHI-aims-code [169] The cohesive energy Ecoh [eV] as obtained inthis work Reference data from experiment is included ldquoPESrdquo refers to results computedbased on the Born-Oppenheimer potential energy surface without any corrections
47
is restricted to the change of the volume without any further anharmon-icities so that the lattice dynamics can still be treated within the harmonicapproximation (Ch 3) The equilibrium volume for every temperature isgiven by minimising F(T V) = E(V) + Fvib(T V) (Eq 47) with respect tothe volume at a fixed T For F(T V) analytical gradients are not availableIn this case the free energy potential surface is sampled by calculating theHelmholtz free energy (Eq 47) for different volumes Then the equilibriumvolume is calculated by fitting the F(T V) pairs to the B-M equation ofstates Def 611 for every temperature T [223 27]
The B-M equation of state describes the change of energy with the volumeat constant temperature
Definition 611 Birch-Murnaghan equation of state
E(V) = E0 +B0VBprime0
[(V0V)Bprime0
Bprime0 minus 1+ 1
]minus B0V0
Bprime0 minus 1
with
E0 equilibrium en-ergy
B0 = minusV(
partPpartV
)T
Bulk modulus
V0 equilibriumvolume
Bprime0 Bulk modulus derivativewith respect to pressure atp = 0
Table 62 gives the ground state lattice parameter for graphite and diamondcalculated without any temperature effects by fitting to Def 611 We alsoinclude the ZPC lattice parameter calculated with PBE+vdW The ZPClattice parameter were calculated by Florian Lazarevic and were presentedas part of his Master thesis [184] Exemplary phonon band structures andthe details of the underlying phonon calculations are given in Appendix D1Temperature effects beyond the ZPC and the diamond graphite phase coex-istence line are presented in the Master thesis by Florian Lazarevic [184] orin Appendix A
For diamond only one lattice parameter has to be determined so the fittingroutine is straightforward to apply In graphite two parameters have tobe obtained the in-plane lattice parameter a and the out-of-plane latticeparameter c In this case first the ratio a
c that minimises the energy for afixed volume is determined by a polynomial fit Then the minimum energyobtained from the polynomial fit enters the B-M-fit
48
Diamond Graphite
Functional a0 [Aring] Vatom [Aring3] Ecoh [eV] a0 [Aring] c0 [Aring] Vatom [Aring3] Ecoh [eV]
LDA PES 3533 5512 -895 2445 6647 8603 -894
ZPC 3547 558
PBE PES 3573 5702 -786 2446 8653 1139 -800
PBE+vdW PES 3554 5611 -806 2464 6677 8777 -813
ZPC 3581 574 2465 6674 8780
Table 62 Calculated ground state lattice parameter a0 and c0 unit cell volume per atomand cohesive energies for diamond and graphite using Def 611 ldquoPESrdquo refers to resultscomputed based on the Born-Oppenheimer potential energy surface without any correc-tions We also include ZPC lattice parameters (Data taken from Master thesis by FlorianLazarevic [184])
The lattice parameters obtained by fitting show good agreement with thelattice parameter obtained by lattice relaxation Tab 61 The only exceptionis the out-of-plane lattice parameter c0 of graphite calculated using the PBEexchange-correlation functional We present a detailed discussion of theinterlayer bonding of graphite for different exchange-correlation functionalin Sec 62 It shows that using PBE gives a very weak interlayer bondingresulting in large interlayer spacing This means that small changes in theenergy can correspond to large changes in the lattice parameter c0 makingan accurate polynomial fit as well as a stress minimisation difficult Apartfrom this special complication the minimisation procedure is able to givereliable and accurate equilibrium volumes
We also calculated the lattice paramter including ZPC for graphene usingPBE+vdW The phonon band structure of graphene and the details of theunderlying phonon calculation used are given in Appendix D1 For thezero-point corrected lattice parameter in graphene we found 2471Aring
The difference between the cohesive energies obtained for the stress tensorbased lattice relaxation (Tab 61) and the B-M-fit (Tab 62) are more pro-nounced The largest cohesive energy difference was found for graphiteFor the PBE and PBE+vdW exchange-correlation functionals the differenceis 013 eV The cohesive energies in Tab 61 are obtained from self-consistentDFT total energies while the energies in Tab 62 were obtained by a fittingprocedure For graphite two different energy fits are included in the cohes-ive energy first a polynomial fit to obtain the ratio between the minimumenergy with respect to the ration between the lattice parameter a and c Thisenergy then enters the B-M-fit
49
62 The Challenge of Inter-planar Bonding inGraphite
In Section 61 and Appendix A we are confronted with the challenge ofdetermining the equilibrium crystal structure of graphite The difficultyoriginates from the strongly anisotropic bonding in graphite While thebond length of the strong covalent in-plane bonds are well described fora variaty of exchange-correlation functionals (see Tab 61 and 62) theinteraction between the layers is very weak and is generally attributed tovdW forces Closely related to the question of the interlayer bond distance isthe interlayer energy Unfortunately a computational investigation withinDFT of the interlayer bonding in graphite is a challenge The problem arisesfrom the local and semi-local approximation to the exchange-correlationfunctional commonly used in DFT and the intrinsically long-range natureof vdW-bonding For an accurate description of the interlayer bondingin graphite long range electron correlation effects have to be included inDFT calculations (Ch 24) The inclusion of vdW effects is a very activescientific area of its own (eg see [271 116 182 117 188 326 325 3245] and references therein) Bjoumlrkman et al [28] and Graziano et al [115]gave a detailed discussion of incorporating vdW-effects in the theoreticaldescription of layered materials
In this section we focus on the comparison of two different vdW correctionschemes the well established Tkatchenko-Scheffler (TS) scheme [326] andthe latest progress incorporating many-body effects [325 324 5] Both vdWschemes depend on the electronic density Recapping briefly the summaryof the vdW section (Sec 24) the TS approach is based on a non-empiricalmethod that includs a sum over pairwise interactions The many bodyapproach includes long-range many-body dispersion (MBD) effects employ-ing a range-separated (rs) self-consistent screening (SCS) of polarisabilities[324 5] (MBDrsSCS) It captures non-additive contributions originatingfrom simultaneous dipole fluctuations at different atomic sites leadingto an improved overall screening and accounts for long-range anisotropiceffects
However the determination of the graphite interlayer binding energy isnot just a challenge for theory To our knowledge four different experie-ments have been published reporting binding energies in the order of30-55 meVatom [106 17 350 201] In the literature three different ener-gies are used to describe the interlayer binding energy exfoliation energybinding energy and cleavage energy The differences between these three termsare subtle Bjoumlrkman et al [29] gave an illustative description of this three
50
Figure 63 The interlayer binding energy sketch for three different types of binding a)exfoliation energy b) binding energy c) cleavage energy In (a) and (b) the layer of interest ismarked by a box The scissor and dashed line for the exfoliation (a) and cleavage (c) casesillustrates how the crystal is cut
energies in the supplemental material of Ref [29] We here will follow theirline of argumentation For completeness we will briefly derive the threeenergies
Figure 63 shows a sketch of the different terms for the binding energyThe interaction energies between two layers are labelled as ε1 for nearestneighbours ε2 for second nearest neighbours etc The exfoliation energy(EXF) is the energy to remove a single carbon layer (marked by a box inFig 63 (a)) from the graphite bulk structure We assume that energy con-tributions originating from in-plane relaxation are neglectible which is agood assumption for AB stacked graphite 1 From Fig 63 (a) we see thatEXF is a sum of the interaction energies
EXF = ε1 + ε2 + =infin
sumn=1
εn (61)
We determine EXF by calculating the total energy of a graphite slab withn layers (En) of a graphite slab with n minus 1 layers (Enminus1) and an isolatedgraphene layer E1 EXF then is given as
EXF(n) =1
Natom(En minus (Enminus1 + E1)) (62)
where E stands for calculated DFT total energies and Natom is the numberof atoms in the graphite slab
1 The assumption does not necessarily hold for graphite layers rotated by an arbitrayangle because due to the rotation there might be regions close to AA and other close toAB stacking AA and AB stacked graphite have different interlayer distances [220] Thisdifference results in a layer corrugation and consequently in in-plane relaxation whichleads to sizeable energy contributions
51
Another relevant energy is the energy needed to separate all layers fromeach other by increasing the interlayer distance ndash the binding energy (EB)(Fig 63 (b)) When we stretch the crystal to separate the individual layersfrom each other we break two bonds but since the bonds are shared thesum of interaction energies (εn) is halved
EB =12
infin
sumn=1
2 middot εn (63)
A comparison of Eq 61 and Eq 63 shows that the exfoliation energy isequal to the binding energy per layer We calculate the binding energy ofan infinite graphite crystal separated into isolated graphene layers
EB =1
Natom
(Egraphite minus NlayerEgraphene
)(64)
where Egraphite (Egraphene) is the total energy of graphite (graphene) andNlayer is the number of layers In the case of an infinite large slab Eq 62gives the same energy as Eq 64 In practice Eq 64 is used to calculateinterlayer binding energies
The cleavage energy is the energy required to split a graphite crystal in twohalves (see Fig 63 (c)) or in other terms it is the energy to create two unitsof surface The cleavage energy expressed in terms of εn is
Ecleav = ε1 + 2 middot ε2 + middot middot middot+ n middot εn =infin
sumn=1
n middot εn = 2 middot Esur f (65)
Comparing Eq 63 and Eq 65 shows that Eq 65 is greater than the bindingenergy and can only serve as an upper bound of the binding energy Thedefinitions of the exfoliation energy binding energy and cleavage energywill be helpful when interpreting experimentally obtained graphene inter-planar binding energies
52
Figu
re6
4H
isto
ryof
the
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hite
inte
rlay
erbi
ndin
gen
ergy
(meV
at
om)
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eth
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rst
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ent
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54[1
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enta
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[106
17
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201
113]
are
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ds
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epu
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atio
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eth
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ergy
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ato
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gree
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ergi
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App
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xJ
53
Finding an accurate description of the graphite interlayer binding energy isan ongoing quest in experiment and theory alike We first summarise thehistory of the search for the graphite interlayer binding energy (Fig 64)The four experiments [106 17 350 201] and the improved interpretationof the 2012 experiment [113] are shown as crosses Girifalco and Lad [106]gave their measured binding energy in units of ergscm2 the value of260 ergscm2 was erroneously converted to 23 meVatom (included inFig 64 as a dashed line) instead of the correct 42 meVatom In the literatureboth values were used reference The different theoretical values are shownas filled circles The colour of each circle depends on the experimentalwork cited A full list of theoretical and experimental values of the graphitebinding energy is given in Appendix J
Figure 65 Binding energy EB of graphite as a function of the interplanar distance dusing LDA PBE HSE06 and two different vdW correction schemes (TS [326] and MBD[325 324 5]) The symbols mark EB at the equilibrium interlayer distance The vertical linemarks the experimental lattice spacing of c0 = 6694Aring [226] For comparison EB valuesfrom RPAPBE calculations ( (1) Lebegravegue et al [185] (2) Olsen et al [239]) and fromexperiment ( (3) Girifalco et al [106] (4) Benedict et al [17] (5) Zacharia et al [350] (6) and(7) Liu et al [201 113]) are included
We first discuss the experimental data included in Fig 64 In 1954 Girifalcoand Lad provided the first experimental value EXF = (425plusmn 5) meVatomfor the graphite exfoliation energy obtained indirectly from heat of wettingmeasurements [106] In a unique experiment Benedict et al determined thebinding energy of twisted graphite based on transmission electron micro-scopy (TEM) measurements of the radial deformation of collapsed multiwall
54
carbon nanotubes Carbon nanotubes with large radii collapse and losetheir circular cross-section because of an interplay between the intersheetattraction and the resistance to elastical deformation of the curvature of thecarbon nanotube (mean curvature modulus) The TEM data was evaluatedby continuum elasticity theory and a Lennard-Jones (LJ) potential for thedescription of the graphite sheet attraction They found a binding energyof EB = (35+15
minus10) meVatom [17] Zacharia et al investigated the graphiteinterlayer bonding by thermal desorption of polyaromatic hydrocarbonsincluding benzene naphthalene coronene and ovalene from a basal planeof graphite To determine the interlayer binding energy of graphite Zachariaet al used the thermal desorption data They assumed the interaction ofcarbon and hydrogen atoms to be pairwise additive to the total bindingenergy of the molecules For the graphite cleavage energy they foundEcleav = (6165plusmn 5) meVatom and EB = (5265plusmn 5) meVatom for thebinding energy [350] Liu et al performed the first direct measurement ofthe graphite cleavage energy based on displacement measurements facili-ating mechanical properties of graphite [353] In the experiment a smallgraphite flake (a few micrometer) was sheared from a graphite island Theprofile of the cantilever graphite flake was measured using atomic forcemicroscopy (AFM) They derived a relationship between the cleaving forceand the displacement from a parameterised LJ potential They estimatedthe binding energy to be 85 of the cleavage energy and obtained a valueof EB = (31plusmn 2) meVatom In a recent work Gould et al suggested thatthe binding energy in the latter experiment might be substantially underes-timated because the experimental data were analysed using a LJ potentialAt large separation the LJ potential gives qualitatively incorrect interlayerbinding energies [75] The other experimental results in the literature mightalso be affected by the same difficulty
From a theoretical point of view the description of the interlayer bind-ing energy is equally challenging Various methods were employed ran-ging from semi-empirical approaches to advanced first-principles calcu-lations The obtained values for the graphite interlayer binding energyscatter widely (see Fig 64) In Sec 24 we already pointed out that stand-ard local and semi-local approximations such as the LDA and the PBEapproximation to the exchange-correlation functional cannot capture theinherently long range nature of vdW interactions A common techniqueto include the long-range tail of dispersion interactions is to add ener-gies obtained using pairwise interatomic potential to DFT total energies[116 117 128 326 118] This approach led to binding energies between43-82 meVatom [119 128 7 114 200 38 42 45] Another approach isthe seamless inclusion of vdW interaction in the exchange-correlation func-tional [271 72 182 53 322 207 272] resulting in binding energies between
55
24-66 meVatom [271 167 59 207 115 272 323 22] Spanu et al [299]applied a quantum Monte Carlo (QMC) method (variational Monte Carloand lattice regularized diffusion Monte Carlo [51]) to determine the layerenergetics (EB = (56plusmn 5) meVatom) and equilibrium lattice parameter(d = 3426 Aring) of graphite The calculations were performed in very smallsimulation supercells (2times 2times 2) and finite-size effects may limit the accur-acy of their results Two groups applied DFT (PBE) based random phaseapproximation (RPA) calculations of the correlation RPAPBE calculationsprovide a more sophisticated method for treating vdW interactions Bothgroups found an equilibrium interplanar distance d = 334 Aring Howevertheir binding energies differ by 14 meVatom (EB = 48 meVatom [185]and EB = 62 meVatom [239]) Lebegravegue et al confirmed the EB(d)sim c3d3
behavior of the correlation energy at large distances as predicted by ana-lytic theory for the attraction between planar π-conjugated systems [75]Olsen and Thygesen [239] also calculated the binding energy with the samek-point sampling as Lebegravegue et al [185] and found a binding energy of47 meVatom The remaining difference might originate from a differentsetup of the projector augmented wave method Using a very dense k-pointsampling of (26times 26times 8) to obtain the Hartree-Fock wave function theyfound 62 meVatom for the binding energy
We calculated the graphite interlayer binding energy as a function of theinterplanar distance d = c02 for different exchange-correlation function-als (LDA PBE HSE06) and vdW correction schemes (TS [326] and MBD[325 324 5]) shown in Fig 65 LDA is a purely local approximation to theexchange-correlation functional Therefore it is not capable of capturingthe long-range dispersion interactions However it gives a surprisinglygood equilibrium lattice parameter of 6643 Aring (Tab 61) but it clearly un-derestimates the binding energy in comparison to available experimentaland RPAPBE values [275 54 318 127 185 239] The PBE curve showsa very weak bonding (c0 = 8811 Aring) More interesting is the performanceof the vdW corrected exchange-correlation functionals We first comparethe two different vdW correction schemes For the pairwise TS schemewe find an equilibrium lattice parameter of c0 = 6669 Aring for PBE+vdWand c0 = 6641 Aring for HSE06+vdW in very good agreement with the ex-perimental value of 6694 Aring [226] The binding energies are overestimatedwith 840 meVatom (871 meVatom) for PBE+vdW (HSE06+vdW) re-spectively Combining the MBDrsSCS scheme with exchange-correlationfunctionals (PBE and HSE06) should improve the overall description of thegraphite inter-planar bonding because it additionally captures non-additivecontributions and accounts for long-range anisotropic effects Indeed forthe binding energy we find 476 meVatom and 525 meVatom for PBE in-cluding van-der-Waals effects with full many-body treatment (MBDrsSCS)
56
[325 324 5] (PBE+MBD) and HSE06+MBD respectively In Fig 65 in partic-ular the PBE+MBD but also the HSE06+MBD curve show a good agreementwith the RPA data of Lebegravegue et al [185] close to the minimum
As discussed above the interpretation of the experimental data is usuallybased on the LJ potential which might lead to an underestimation of thebinding energy in graphite [113] Here high quality theoretical values ofthe graphite binding energy are a challenge because electronic structurecalculations are difficult to converge [239] In the case of graphite interlayerbinding energies reliable benchmark data is still missing However a recenthigh quality QMC study provides a benchmark value for bilayer graphene[220] In the work of Mostaani et al [220] finite size effects were controlledby using large supercells (6times 6) They obtained a binding energy for ABstacked bilayer graphene of EB= 177 meVatom and an inter-planar latticeconstant of 3384 Aring [220] In comparison we find binding energies of bilayergraphene of EB= 220 meVatom using PBE+MBD and of EB= 365 usingPBE+vdW For bilayer graphene we find good agreement for the bindingenergy between PBE+MBD and high level benchmark data (QMC) and anoverestimation for PBE+vdW
The MBDrsSCS method is a very recent development At the current stateof the implementation it is not yet optimised well enough to treat largesystems containg 1000s of atoms Therefore we will use the TS scheme formost of the results presented in this work The TS scheme is an accurate non-empirical method to obtain the C6 coefficients from ground-state electrondensity and reference values for the free atoms allowing for an equallyaccurate treating of atoms in different chemical environments On theother hand most methods based on the pairwise summation of C6R6
coefficients rely on empirical or at least semiempirical determination of theC6 coefficients [343 116 117] In these methods the accuracy depends on thedetailed determination of the C6 coefficients However all methods basedon the pairwise summation of C6R6 coefficients cannot capture the non-additive contributions originating from simultaneous dipole fluctuations atdifferent atomic sites
In summary a reliable benchmark value for the graphite interlayer bindingenergy is still missing both experimentally and theoretically The mainobstacle is the correct description of the binding energy at large separa-tions In computational studies a sum over pairwise C6R6 interactionsis often used leading to a EB(d)sim c5d5 behaviour which is incorrect forgraphite [75] The same problem arises when interpreting experimentaldata [113] The best result currently available is given by RPA calculationsbecause RPA does not rely on assumptions of locality additivity nor C6R6
57
contributions [185 239] However a very dense sampling of the Brillouinzone (BZ) is needed for converged RPA calculations on graphite due to itssemimetallic behavior at the Fermi level which is computationally chal-lenging In this section we demonstrated clearly the importance but at thesame time problematic description of the anisotropic long-range dispersioneffects in graphite The best method currently available with FHI-aims isthe MBDrsSCS correction scheme which accounts for non-additive contri-butions However as this is very recent development most of the resultspresented in this work were calculated using the pairwise TS correctionscheme
63 The Electronic Structure of Graphene
The electronic properties of graphene originate from its very special crystalstructure The graphene honeycomb lattice can be seen as two triangu-lar sublattices lattice A and lattice B with one C atom in each The twosublattices are bonded by sp2 hybridisation of the 2s the 2px and 2py Corbitals forming strong in-plane σ bonds The remaining 2pz orbital is ori-ented perpedicular to the graphene x-y-plane The neighbouring 2pz orbitalform delocalised states across the graphene plane so-called π-bonds Elec-trons move easily in these π states resulting in the exceptionally electricalconductivity of graphene
Figure 66 a shows the hexagonal graphene Brillouin zone (shaded in grey)The graphene reciprocal lattice vectors b1 b2 connect the Γ-point of the firstBrillouin zone to the neighbouring Γ-points (indicated as blue arrows) Thegraphene Brillouin zone contains three high symmetry points
Γ-point in the zone center
M-point at the mid point of the Brillouin zone edge
K-point at the corners of the Brillouin zone
The graphene band structure of pritstine graphene on the level of PBE isshown in Fig 66 b We used the PBE+vdW lattice parameter given inTab 61 The graphene unit cell contains two π orbitals forming two bandsThese bands are referred to as π band the bonding lower energy valenceband (indicated by an arrow in Fig 66 b) and πlowast band the anti-bondinghigher energy conduction band The π bands touch at the K-point andclose the band gap which is why graphene is often referred to as a zero-gap
58
Figure 66 Subfigure (a) shows the graphene Brillouin zone (shaded in grey) and the highsymmetry points Γ M and K (marked by blue points) (b) The band dispersion of grapheneon the level of PBE The k-path is chosen along the high symmetry lines of the grapheneBrillouin zone as indicated by arrows in Subfigure (a)
semiconductor The energy-momentum dispersion of the π and πlowast bandsaround the K-point is approximately linear The energy where the two πbands touch is called the Dirac point (ED) (see Fig 66 b) The π bands of anundoped graphene sheet are half-filled ED equals the Fermi energy
The doping of the graphene layer shifts the Dirac point with respect to theFermi level In a neutral graphene layer ED equals the Fermi energy Theposition of the Dirac point is very sensitive to small doping concentrationsThis is a direct consequence of the linear dispersion of the bands in theclose vicinity of the K-point in the graphene BZ In reciprocal space theelectronic structure of graphene in the vicinity of the Dirac point is formedlike a cone The amount of dopants is proportional to the area enclosedby the Fermi surface which can be approximated by a circle with the areakF
2π In this simple model we find that the position of the Dirac point(ED) is proportional to the amount of dopants (δq) in the graphene layer asEDprop
radic|δq|
However as can be seen in Fig 66 b the π bands are not fully symmetricin the vicinity of the k-point [35 50 79] In the following we examinethe shift of the Dirac point with respect to the Fermi level for differentdoping concentrations The position of the Dirac point with respect tothe Fermi level is influenced by hole and electron doping of the graphenelayer We simulate the doping of the graphene layer using the virtual-crystalapproximation (VCA) [336 278 261] The VCA ensures that the simulationcell remains neutral and therefore allows to calculate a doped crystal usingperiodic boundary conditions This is achieved by removing a fraction of
59
the positive nuclear charge δq of all carbon atoms and adding the sameamount of negative charge to the conduction electrons This means that afractional electron of the charge minusδq is transferred to the conduction bandwhich gives the Fermi level in the VCA
Figure 67 The position of the grapheneDirac point with respect to the Fermi levelfor doping in the range ofplusmn 0005 eminusatomon the level of PBE PBE+vdW lattice para-meter were used (Tab 61)
We used the two atomic graphenecell introduced in Sec 61 with thelattice parameter optimised on thelevel of PBE+vdW as given in Tab 61The Fermi surface for different dopingchanges only in a small region aroundthe K-point in the graphene Brillouinzone In order to accurately accountfor these small changes we chose avery dense off-Γ k-grid with 48 pointsin the in-plane x and y direction Afull band structure similar to Fig 66 bwas calculated for 28 different dopingconcentrations using PBE We chosea doping range of plusmn0005 eminusatomwhich corresponds to a doping con-centrations of 008 Within the VCAhigher doping concentrations could beincluded However for doping con-centrations much higher than the onesconsidered here at the order of a fewtenth of the atomic charge local effectsin the electronic structure may not becaptured by the VCA [261]
Figure 67 shows the Dirac energy (ED)with respect to the Fermi level In the range between minus0005 eminusatom andzero the graphene layer is hole doped so that the Dirac point lies abovethe Fermi level In this case the Dirac cone is not fully occupied resultingin a p-type doped graphene layer (see diagram in Fig 67) In the rangebetween zero and 0005 eminusatom the graphene layer is n-type doped andthe Dirac point is shifted below the Fermi level In Chapter 12 we will usethe here presented analysis of doping in pristine graphene to aid a betterunderstanding of the doping of epitaxially grown graphene
60
7 Silicon Carbide The Substrate
The wide band gap semiconductor SiC is extremly rare in nature EdwardG Acheson found a wide-scale production method to synthesise SiC inthe early 1890rsquos [2] The new material quickly found industrial applicationas an abrasive because of its hardness1 Since then the interest in SiC hascontinued In the last 15 years there has been a continuous interest inSiC with an average number of publications of 1450plusmn 350 per year (seeFig 62)
SiC has been applied in various fields in particular for electronic devicessuch as Schottky diodes and metal-oxide-semiconductor field-effect tran-sistors for temperatures up to 973 K [26 49] The first prototype of a lightemitting diode (LED) was built using SiC in 1907 [282] Later SiC was com-mercially used for blue LEDs [216] until the much more efficient galliumnitride LEDs replaced it SiC is also a promising material for nanoelec-tromechanical systems such as ultrafast high-resolution sensors and highfrequency signal processing components [345] For high-temperature ap-plications such as temperature sensors or turbine engine combustion mon-itoring SiC is commonly used [49] due to its comparablely high thermalconductivity of 490 W
m K for 6H-SiC compared to 149 Wm K for Si [295 26] and
its low thermal expansion coefficient [195 194 159] However the indus-trial application of SiC is limited as the quality of the substrate material isstill below that of Si wafers although advances and improvements in thegrowth process have been made [224] Recently SiC has received muchattention because of its use as a substrate for epitaxial graphene growth byhigh-temperature Si sublimation [18 238 20 64 82 263] It is regarded asthe most promising route to large scale growth of high quality graphene forpractical applications
One of the most intriguing properties of SiC is the structural diversity withmore than 250 different crystalline forms [57] In this chapter we introducethe three SiC polytypes 3C-SiC and hexagonal 4H-SiC and 6H-SiC mostrelevant for this work and their physical and electronic properties
1On Mohs scale of mineral hardness SiC scales between 9-95 for comparison diamondscales as 10 and talc as 1
61
71 Polymorphism in Silicon Carbide
Figure 71 The white spheres represent a hexagonal struc-ture The possible stacking positions of a closed hexagonalstructure are marked as A B and C A SiC-bilayer overlay-ers the hexagonal structure The primitive unit cell of theSiC-bilayer is shown
The silicon carbide (SiC)crystal is composed of anequal number of silicon(Si) and carbon (C) atomsforming strong covalenttetrahedral bonds by sp3
hybridisation Four Catoms surround a Siatom and vice versa Fig-ure 71 shows a SiC-bilayer The central Siatom (marked A) formsthree in-plane bonds withneighbouring C atoms(marked B) A single SiC-bilayer can be picturedas a sheet of Si-C pairsarranged in a hexagonalpoint lattice with an Si-Cpair at each point The out-of-plane bonds connect the SiC-bilayers witheach other In a closed hexagonal structure there are three possible stackingpositions labeled A B and C in Fig 71 The SiC-bilayers can be stacked ontop of each other by shifting the central Si atom In Fig 71 the central Siatom is labeled A The position of the central Si atom of the next SiC-bilayeris shifted either to position B or C
We use the notation proposed by Ramsdell [258] to identify different SiCpolytypes The notation consists of a letter and a number The numbergives the number of SiC-bilayers stacked along the z-axis to complete theunit cell The letter C H or R describes the Bravais lattice type cubic Chexagonal H or rhombohedral R Every SiC polytype can be constructed byfive basic polytypes with small stacking periods 2H 3C 4H 6H and 15R[180] The 2H-SiC has a wurtzite structure It consists of two SiC bilayersin AB stacking 3C-SiC also known as β-SiC has a cubic Bravais latticewith two atoms in the unit cell (zinc blende structure) This structure canalso be understood as three SiC-bilayers stacked along the (111)-axis withABC stacking Figure 72 shows the atomic structure of the three differentpolytypes 3C-SiC 4H-SiC and 6H-SiC
We list the structural and cohesive properties of 3C-SiC 4H-SiC and 6H-SiC
62
Figure 72 The atomic structure of three different SiC polytypes are shown cubic siliconcarbide (3C-SiC) (left) and two hexagonal structures 4H-SiC (middle) and 6H-SiC (right)The atoms in the black box mark the unit cell The stacking sequence of the SiC-bilayersis shown for every polytype and indicated by a grey line in the structure 3C-SiC is alsoshown in its zinc blende structure (bottom left) The distance dnnminus1 between two bilayersand the distance Dn between a Si and C atoms within a SiC bilayer are shown
for different exchange-correlation functionals (LDA PBE HSE06 and vdWcorrected PBE+vdW and HSE06+vdW) in Tab 71 and Tab 72 We usedthe stress tensor based lattice relaxation to optimise the crystal structureThe choice of approximation to the exchange-correlation functional changesthe lattice parameter by less than 005 Aring for all three polytypes
PBE PBE+vdW LDA HSE06 HSE06+vdW
dnnminus1[Aring] Dn[Aring] dnnminus1[Aring] Dn[Aring] dnnminus1[Aring] Dn[Aring] dnnminus1[Aring] Dn[Aring] dnnminus1[Aring] Dn[Aring]
3C-SiC 1904 0635 1890 0630 1876 0625 1886 0629 1878 0626
4H-SiC 1901 0625 1891 0618 1878 0617
6H-SiC 1901 0631 1892 0626 1879 0623 1880 0622
Table 71 The distance dnnminus1 between two SiC-bilayer and the distance Dn between the Siand C sublayer with in SiC-bilayer are listed for 3C-SiC 4H-SiC and 6H-SiC calculated fordifferent exchange-correlation functionals (LDA PBE HSE and vdW corrected PBE+vdWand HSE06+vdW)
In Table 72 we also include the lattice parameters and bulk moduli includ-ing zero-point vibrational correction (ZPC) To calculate the ZPC latticeparameters we applied the QHA (see Sec 41) We calculated the Helmholtzfree energy (Eq 47) for different volumes by full phonon calculations ina finite-difference approach (details about the underlying phonon calcula-tions are given in Appendix D2) For 3C-SiC we used a 5times 5times 5 supercellfor 5 volumes We optimised the hexagonal structure in the same way as forgraphite (Sec A) Here we performed phonon calculations in a 3times 3times 2 su-
63
percell for 7 volumes and for every volume we use 9 different ratios betweenthe lattice parameters a and c The minimum-energy lattice parameter forT=0 K was obtained by fitting to the Birch-Murnaghan equation of state(Def 611) The inclusion of ZPCs changes the bullk cohesive propertiesonly slightly
The numerical convergence with respect to the integration grid in reciprocaland real space as well as with the number of basis functions included in thecalculation is discussed in Appendix B Differences in the lattice parametersobtained by B-M-fitting and the stress tensor based lattice relaxation weresmaller than 10minus3 Aring
For 3C-SiC we also tested density functionals that account for long-rangedispersion In Table 72 we include the TS scheme [326] (PBE+vdW andHSE06+vdW) and the recently developed many body vdW treatment [325324 5] (PBE+MBD) The effect of the many body vdW treatment on thelattice parameter cohesive energy and enthalpy of formation ∆H f is smallHowever using the PBE+MBD instead of the PBE+vdW method improvesthe bulk modulus by 01 Mbar for the groundstate calculation and by007 Mbar for the bulk modulus including ZPC The experimental value is248 Mbar [313]
Table 71 lists the distance dnnminus1 between two SiC-bilayers and the distanceDn between the Si and C sublayer where n denotes the number of theSiC-bilayer Dn and dnnminus1 are illustrated in Fig 72 In SiC the intralayerdistance is given by the Si-C bond length and the tetrahedral angle In aperfectly symmetric tetrahedron the central angle between any two verticesis the arccos (minus13) (7052) However in the hexagonal structures thebond tetrahedrons deviate from the symmetric tetrahedral angle by 036for 4H-SiC and 014 for 6H-SiC in PBE+vdW
The enthalpy of formation ∆H f of SiC is calculated using the referencephases bulk silicon and carbon in the diamond structure ∆H f is betweenminus053 eV and minus059 eV depending on the density-functional and polytypeused and shows good agreement with a calculated literature value ofminus058 eV using the pseudopotential plane wave (PSP-PW) method andLDA [355 126] The calculated enthalpy of formation is notably smaller(less energy gain) than the experimental room temperature value obtainedby an electromotive force (emf) measurement by Kleykamp [168] but in fairagreement with the value tabulated in the Handbook of Chemistry amp PhysicsHaynes [134]
The enthalpies of formation ∆H f of the three SiC polytypes are very similar
64
3C-SiCa0 [Aring] B0 [Mbar] Ecoh [eV] ∆H f [eV]
PES 436 210 -676 -056PBE+vdW
ZPC 438 209 -665 -058
PES 438 210 -652 -053PBE
ZPC 440 206 -641
PES 433 230 -741 -056LDA
ZPC 434 223 -730
PES 436 220 -678 -058PBE+MBD[5]
ZPC 438 216 -667
HSE06 PES 436 231 -635 -058
HSE06+vdW PES 434 231 -659 -059
T=300K 4358 [195] 248 [313] -634 [225] -077 [168]Experiment
T=0K 4357 [317] -065 [134]
Theory (LDA) PES 429 [157] 222 [157] -846 [157] -056 [126]
PSP-PW (LDA) PES 433 [355] 232 [12] -058 [355]
4H-SiCa0 [Aring] c0n [Aring] B0 [Mbar] Ecoh [eV] ∆H f [eV]
PES 308 251 215 -669 -057PBE+vdW
ZPC 309 253 212 -678 -059
PBE PES 310 253 -643 -054
LDA PES 307 249 -742 -057
Experiment T=0K 307 [108] 251 [108] -062 [134]
PSP-G (LDA) PES 304 [12] 249 [12]
6H-SiCa0 [Aring] c0n [Aring] B0 [Mbar] Ecoh [eV] ∆H f [eV]
PES 308 252 215 -669 -057PBE+vdW
ZPC 309 253 210 -658 -059
PBE PES 310 253 -643 -054
LDA PES 306 250 -742 -057
HSE06+vdW PES 306 250 -659 -061
Experiment 308 [1] 252[1] 230 [10] -077[168]
Theory (LDA) PES 308 [15] 252[15]
PES 307 [244] 252[244] 204 [244]
Table 72 The structural and cohesive properties of 3C-SiC 4H-SiC and 6H-SiC are lis-ted for different exchange-correlation functionals (LDA PBE HSE06 and vdW correctedPBE+vdW and HSE06+vdW) The lattice parameters a0 [Aring] and for the two hexagonalpolytypes c0 [Aringn] where n is the number of SiC bilayers the bulk modulus B0 [Mbar]cohesive energy Ecoh [eV] and enthalpy of formation ∆H f [eV] as obtained in this workReference data from experiment and theory is included The theoretical reference data wascalulated using DFT codes based on pseudopotentials with a plane-wave basis (PSP-PW)[15 244 157 355 126] or Gaussian basis (PSP-G) [12] in the LDA ldquoPESrdquo refers to resultscomputed based on the Born-Oppenheimer potential energy surface without any correc-tions We also include zero-point vibrational correction (ZPC) lattice parameters and bulkmoduli
65
Davis et al [63] measured ∆H f for 3C-SiC (∆H f = minus065 eV) and hexagonalSiC (∆H f = minus065 eV) under normal conditions and found the same valuefor both polytypes The energetic close competion between the differentpolymorphs is the main reason why it is difficult to grow high-quality SiCof a specific polytype [224] In a recent experiment Haynes [134] found aenergy difference of 003 eV between 3C-SiC and 4H-SiC
However once a specific polytype is grown the transformation from one tothe other occurs at temperatures as high as 2400 K by a reconstructive trans-formation proceeding through a vapour transport mechanism involvinggaseous silicon [342]
72 The Electronic Structure of Silicon CarbidePolymorphs
In Section 71 we introduced the atomic structure of SiC polytypes Thenearest neighbour configuration is the same for all polytypes because of thetetrahedral sp3 hybridisation but the polytypes differ in their long rangeorder These changes have a profound influence on the electronic structureFor example the band gap depends on the polytype with the smallest gapin 3C-SiC (22 eV) and the largest band gap in 2H-SiC (35 eV) [204] If onecould control the stacking order during the SiC growth process the bandgap could be tuned in this range In this section we investigate the electronicstructure of 3C-SiC 4H-SiC and 6H-SiC
Figure 73 and Figure 75 show the calculated density of states (DOS) of3C-SiC using the HSE06 and PBE exchange-correlation functional in com-parison with experimental soft x-ray spectroscopy data In HSE06 theamount of exact exchange is set to α=025 and the range-separation para-meter ω=02 Aringminus1 (for details about HSE see Sec 233 or [175]) Ramprasadet al [257] proposed to judge the quality of the exchange-correlation func-tionals with respect to the valence band width instead the Kohn-Sham (KS)band gap They observed a linear relationship between the valence bandwidth and formation energies for point defects over the (ω α) parameterspace of the HSE family We tested the influence of α on the band gap andvalence band width Appendix B4
Luumlning et al [204] performed synchrotron radiation excited soft x-ray emis-sion (SXE) and absorption (SXA) spectroscopy to measure the band gap ofbulk SiC The measured transition probability is to a good approximation
66
Figure 73 Density of states (DOS) around the band gap for 3C-SiC The top panel showsthe experimental species projected DOS (C solid line and Si dashed line) measured by softx-ray spectroscopy with a band gap of 22 eV [204] The middle panel shows the calculatedspecies projected DOS using the HSE06 exchange-correlation functional with a band gapof 233 eV In the bottom panel the calculated total Kohn-Sham DOS using the HSE06exchange-correlation functional is compared to the total DOS calculated with PBE Thetheoretical magnitude of the peaks in the spectra depends on the number of basis functionsincluded in the calculation
proportional to the local partial DOS The method is orbital and symmetryselective due to the selection rules and because of the core excitations itallows for element and site selectivity We included their spectra for 3C-SiCin the top panel of Fig 73
We calculated a species projected density of states (with a Gaussian broad-ening of 01 eV) using the HSE06 exchange-correlation functional and thevdW corrected lattice parameter from Tab 72 The species projected DOSaround the band gap is plotted in Fig 73 For HSE06 we find a band gap of233 eV in good agreement with experiment For comparison we includethe total DOS for HSE06 and PBE (bottom panel Fig 73)
In Figure 74 the calculated HSE06 band structure of 3C-SiC is compared to
67
experimental photoemission data [143] The calculated bands agree well inshape with the experimental band dispersion in Ref [143] The X5v energylevel in our calculation is at minus351 eV compared to minus268 eV in Ref [143]By optical reflectivity measurements Lambrecht et al [181] found for theX5v energy level minus36 eV which is in good agreement with our results andprevious theoretical studies X5v=minus347 eV (HSE06) [237] X5v=minus349 eV(GoWoLDA) [341] Appendix C shows the full band structure calculatedwith HSE06 for 3C-SiC 4H-SiC and 6H-SiC
Figure 74 Plot of the band structureof 3C-SiC from Γ to X The bandstructure (solid line) is calculated usingHSE06 exchange-correlation functionaland the vdW corrected lattice parameter(HSE06+vdW from Tab 72) The shadedarea (in grey) marks the band gap of233 eV In addition experimental photoe-mission data is included (filled cicle) andparameterised bands (dashed line) takenfrom Ref [143]
Figure 75 shows the density ofstates (DOS) of the valence band for3C-SiC For comparison we includethe photoemission data taken from Ref[296] The photoemission data gives avalence band width of 113 eV The cal-culated DOS (with a Gaussian broad-ening of 01 eV) using HSE06 gives aband width of 93 eV and PBE 842 eVHSE06 underestimates the band widthby sim18 However in the experi-mental spectra it is not clear what thebest starting and end point would beto determine the band width We chosetwo characteristic peaks in the experi-mental and HSE06 spectra (dashed linein Fig 75) the first one at minus25 eV(HSE06 minus24 eV) and a second oneat minus95 eV (HSE06 minus85 eV) If thepeak-to-peak distance is used as anreference HSE06 underestimates it by11 eV The species projected DOS (midpanel Fig 73) identifies the first peakat minus25 eV to be dominated by C statesand the second peak atminus95 eV is dom-inated by Si states in agreement withexperiment Luumlning et al [204]
In Table 73 we listed the direct and in-direct band gap and the band widthfor different exchange-correlation func-
tionals (PBE LDA HSE06) PBE and LDA underestimate the band gap bysim30 for 4H-SiC and 6H-SiC The band gap of 3C-SiC is underestimated bysim35 in PBE and sim40 in LDA Including a fraction of exact-exchange in
68
Figure 75 Shows the Density of states (DOS) of the valence band for 3C-SiC The toppanel shows the experimental DOS measured by photoemission spectroscopy with aband width of 113 eV [296] The calculated DOS using the HSE06 exchange-correlationfunctional with a band width of 93 eV is shown in the middle panel and using PBE with aband width of 84 eV in the bottom panel
HSE06 improves the description of the band gap and the band width overPBE but it still underestimates the band width if compared to the experi-mental spectra The agreement between the experimental band gaps andthe HSE06 calculated one is good for the polytypes 3C- 4H- and 6H-SiC
We demonstrated that the polymorphs of SiC relevant for this work are welldescribed by the level of theory applied In particular we achieved a verygood agreement for the structural properties Sec 71 independent of theapproximation to the exchange-correlation functional The band gap Tab 73is underestimated by PBE LDA However a good agreement with high-level theoretical calculations (GoWoLDA) and experiment can be achievedwhen a fraction of exact-exchange α=025 is included in the calculationby using HSE06 with a range-separation parameter set to ω=02 Aringminus1 Wefound that the default HSE06 value of α=025 captures both the band gapand the band width well and we therefore adopt it for our calculations
69
PBELD
AH
SE06R
ef(theory)Exp
gX
gΓminus
Xw
LminusΓ
gX
gΓminus
Xw
LminusΓ
gX
gΓminus
Xw
LminusΓ
gΓminus
X[eV
]g
ΓminusX
w
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
3C-SiC
457144
842455
132864
585233
93137
(PBE)235(H
SE06)[199]22
[204]113[ 296]
269(G
o Wo
LDA
)[341]242
[148]
gM
gΓminus
Mw
Γg
Mg
ΓminusM
wΓ
gM
gΓminus
Mw
Γg
ΓminusM
[eV]
gΓminus
Mw
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
[eV]
4H-SiC
340226
855335
222867
452326
933223
(PBE) 328(H
SE)[237]32
[204 ]112
[204 ]
356(G
o Wo
LDA
)[341]3285
[212]
6H-SiC
315205
856310
199869
292940
202(PBE)302
(HSE)[237]
29[204]
119[154]
325(G
o Wo
LDA
)[341]3101
[212]
Table73
The
direct
andind
irectband
gapand
thevalence
bandw
idth
arelisted
for3C
-SiC4H
-SiCand
6H-SiC
calculated
ford
ifferentex-
change-correlationfu
nctionals(
PBELDA
HSE06)
We
used
thegeom
etriesgiven
inTab72
forPBEH
SE06the
vdWcorrected
values
were
chosen
70
Part IIIThe Silicon Carbide
Surfaces and EpitaxialGraphene
71
In Chapter 7 we introduced the wide-band-gap semiconductor SiC whichcrystallises in various polytypes The technologically most relevant poly-types are the zinc-blende polytype 3C and the hexagonal polytypes 4H and6H shown in Fig 72
The natural growth direction of SiC is along the stacking axis of the SiC-bilayers that is the [111] direction for 3C-SiC and [0001] for the hexagonalpolytypes SiC is a polar material the two opposite faces the 3C-SiC-111and the 6H-SiC-0001 differ by their surface termination The 3C-SiC-(111)(6H-SiC-(0001)) face consists of Si atoms and the opposite face the 3C-SiC-(111) (6H-SiC-(0001)) of C-atoms Throughout this work we focus on thesurface perpendicular to the natural growth direction
Epitaxial graphene can be grown by high temperature Si sublimation onthe Si-terminated [332 95 264 82 263] and on the C-terminated surface[18 19 334 64 302 141 83 21 337 213] of SiC
In this part we investigate the surface reconstructions observed during SiCannealing and the hybrid SiC-graphene structures for the two polar SiCsurfaces
72
8 The 3C-SiC(111) Surface phases
Figure 81 The surface termination of 3C-SiC and 6H-SiC is shown for a clean surface In6H-SiC every third Si-C bilayer is rotated by 30 indicated by a kink in the grey line Thisrotation in the stacking order leads to different surface terminations Sn6H and Slowastn6H wheren runs from one to three
For the 3C-SiC(111) surface the bulk like stacking order close to the surfaceis independent of where the surface is cut but for specific surface recon-struction the stacking can change at the surface However for 6H-SiC(0001)this is different as here the SiC bilayers are rotated by 30 every third layerThe position of the rotation is indicated in Fig 81 by a kink in the greyline Due to these rotations the 6H-SiC offers more-or-less favourable stepterminations introducing an additional degree of freedom ndash the choice ofthe surface termination The six different surface terminations are oftenlabeled with Sn6H where n runs from one to three The symmetry equivalentsurfaces are marked by an additional asterix Slowastn6H (see Appendix G)
Independent of the polytype during graphene growth a series of surfacereconstructions is observed starting with a Si-rich (3times3) reconstruction [310]With increased temperature more Si sublimates and a simpler (
radic3timesradic
3)-R30 bulk-terminated Si rich surface forms [158 308] followed by the C-richsurface reconstruction the so-called zero-layer graphene (ZLG) or ldquobufferlayerrdquo followed by mono-layer graphene (MLG) bi-layer graphene (BLG)and few-layer graphene [332 95 264 83]
We performed DFT calculations for the Si-rich (radic
3timesradic
3)-R30 and (3times3) reconstruction (Sec 81) and the C-rich phases ZLG MLG and BLG(Sec 82) The calculations were carried out using the all-electron localisedbasis set code FHI-aims Sec 25 (rsquotightrsquo settings see Appendix B) For theapproximation to the exchange-correlation functional we use LDA PBEWe include long-range dispersion effects by using the TS scheme [326](PBE+vdW) The bulk lattice parameters of SiC are listed in Tab 72 theimpact of different standard functionals is well understood and systematic
73
(see Ch7) Predicted lattice parameters for diamond C and graphite arelisted in Tab 61 For the interplanar lattice parameter c of graphite vdWeffects must be included into the PBE functional (see discussion in Sec 62)To simulate the substrate in the surface calculations we use six SiC-bilayersstacked along the z-axis (see Fig 81 and Appendix B2) and a sufficientlythick vacuum region We fully relaxed the top three SiC-bilayers and allatoms above (residual energy gradients 8middot10minus3 eVAring or below)
81 The Silicon-Rich Reconstructions of the3C-SiC(111) Surface
811 3C-SiC(111) (radic
3timesradic
3)-R30 Si Adatom Structure
A Si-rich reconstruction withradic
3timesradic
3 translational symmetry was firstreported by Van Bommel et al [332] The structure was determined by DFTcalculations [232 273 156] and scanning tunnelling microscopy (STM) [241]and later confirmed by quantitative low-energy electron diffraction (LEED)measurements [305] The structure consists of a Si adatom on top of a bulk-like Si layer (shown in Fig 82) The adatom is at position B in Fig 71also known as T4-site At this position the adatom forms bonds with thethree Si atoms of the top SiC-bilayer reducing the number of dangling bondorbitals at the surface from 3 to 1 The new pz-like dangling bond orbitalcontains one unpaired electron per unit cell The Si adatom pushes the Catom underneath into the substrate leading to a corrugation δ1C in the topC-layer indicated by an arrow in Fig 82 [232 273 305 276]
The electronic structure of the 6H-SiC(0001)-(radic
3timesradic
3) surface was stud-ied by angle-resolved photoemission spectroscopy (ARPES) and k-vector-resolved inverse photoemission spectroscopy [154 319] The one-electronband corresponding to a localised state at the Si-adatom splits into twobands resulting in a semiconducting surface with a bandgap of 20 eV [319]The band splitting was discussed in the context of GoWoLDA calcula-tions [99 267] Their calculations showed that the band splitting is drivenby electron correlation effects of the top Si dangling bond In conventionalDFT the spin-unpolarised surface is by necessity metallic with a half-filledband at the Fermi level Introducing spin-polarisation opens a gap at thesurface
74
Figure 82 Geometry of the3C-SiC(111) (
radic3 timesradic
3)-R30 Siadatom structure The bondlength of the adatom with thetop Si atom (LSi-Si) the distancefrom the surface (D1SiSi) and thecorrugation of the top C layer(δ1C ) are shown
Alternativly the strong correlation betweenthe Si adatoms was addressed within theLDA+U framework by Furthmuumlller et al[99]
We use theradic
3timesradic
3 reconstruction to evalu-ate the influence of the exchange-correlationfunctional and the polytype by comparing res-ults for the 3C-SiC and 6H-SiC We took theSiC bulk lattice parameter from Tab 72 Wealso have to take into account the stackingorder for hexagonal SiC (see Fig 81) Wetested the influence of the stacking order onthe (radic
3timesradic
3)-R30 Si adatom structure Wefound the lowest surface energy (Eq 49) forABCACB and the symmetry equivalent ACB-ABC stacking order the so-called S36H andS3lowast6H terminations (see Appendix G) LEEDdata also indicates a S36H surface termination[291]
In Tab 81 we list the three characteristic para-meters of the Si adatom structure the bondlength between the adatom and the top layerSi atoms LSi-Si the distance between the ad-atom and top layer Si atoms along the z-axisD1SiSi and the corrugation of the top C-layerδ1C (see Fig 82) We include results from spin-polarised and unpolarisedcalculations for the 3C-SiC(111) and 6H-SiC(0001) surface The surfaceenergetics will be discussed in Ch 10
Interestingly the first Si-layer shows no corrugation with δ1Silt1 middot 10minus4 Aring forall exchange-correlation functionals used The Si-Si bond length is slightlylarger than in bulk Si (PBE+vdW 236 Aring) Although changing from a spin-unpolarised calculation to a spin-polarised one has a profound effect on theelectronic structure by opening a band gap the atomic structure is hardlyaffected
75
3C-SiC
(111)L
Si-SiD
1Si Si
δ1C
LDA
243175
026
PBE246
176027
spin-unpolarised
PBE+vdW245
177028
LDA
244175
026
PBE246
176027
PBE+vdW245
177028
spin-polarised
HSE06+vdW
244175
027
6H-SiC
(0001)L
Si-SiD
1Si Si
δ1C
spin-polarisedPBE+vdW
245177
027
References
LSi-Si
D1
Si Siδ1
C
6H-SiC
(0001)[305]expLEED
244175
033
4H-SiC
(0001)[276 ]expLEED
247177
034
3C-SiC
(111)[232]LD
A242
175022
6H-SiC
(0001)[273]LD
A241
171025
Table81A
tomic
positionofthe
Siadatomrelative
tothe
surfacefor
differentexchange-correlationfunctionalsThe
bulklattice
parameter
were
takenfrom
Tab72For6H
-SiCw
echose
aA
BC
AC
Bstacking
order
(seeA
ppendix
G)L
Si-Si isthe
bondlength
between
thead
atomand
thetop
layerSiatom
sD1
Si Si isthe
distancebetw
eenthe
adatomand
toplayer
Siatoms
alongthe
z-axisandδ1
Cis
thecorrugation
ofthetop
C-layer
(seeFig82)(A
llvaluesin
Aring)
76
Figure 83 The twist model of the 3C-SiC(111)-(3times 3) reconstruction On the left Thetwist model from a side view The layer distance between the substrate and the Si-ad-layerD1SiSi3 as well as the distance between the ad-layer and the Si-trimer DSi3Si2 and trimer-Si-adatom DSi2Si1 are indicated by arrows On the right The twist model from a top viewThe unit cell is shown in black
Comparing the structural data in Tab 81 we find that the change in theadatom geometry between the 3C-SiC(111) and 6H-SiC(0001) surface isnegligible for PBE+vdW In good agreement with a LEED analysis of the3C-SiC(111) and hexagonal (0001) surfaces [305] It seems that the surfaceSi adatom reconstruction on Si-terminated SiC surface is not affected by thepolytype [156 100 277 243]
In contrast to 6H-SiC only one type of stacking order is present at the surfaceof 3C-SiC [304] In fact the stacking sequence for the 6H-SiC and 3C-SiC arethe same down to at least four SiC-bilayers below the surface (see Fig 81)However the growth process [179] and the electronic structure [243] differamong these two polytypes (see Sec 72)
812 The Silicon-Rich 3C-SiC(111) (3times 3) Reconstruction
On the Si-terminated surface a (3times 3) structure forms during annealingunder simultaneous exposure to Si [158] The reconstruction was identifiedas a Si-rich structure by Auger electron spectroscopy (AES) and electron-energy loss spectroscopy (ELS) [158] The (3times 3) reconstruction attractedgreat attention after its stabilising role during SiC thin film growth bymolecular beam epitaxy (MBE) [316 93] was identified However thedetailed atomic structure of the (3times 3) reconstruction remained a mysteryfor many years
77
We briefly review the unsuccessful models of the 3C-SiC(111)-(3times 3) re-construction on the Si terminated SiC surface Later we will use thesemodels as structure models for the unknown (3times 3) reconstruction on theC terminated SiC surface (Ch 14) The first structure was suggested by Ka-plan [158] Based on AES and ELS measurements they rationalise that thestructure contains a Si ad-layer They suggested an adaption of the dimerad-atom stacking fault (DAS) model in analogy to the (7times 7) reconstructionof the Si(111) surface In this model the substrate is covered by a layer ofeight Si atoms On top of the ad-layer is a second layer consisting of six Siatoms occuping three-fold surface sites Finally a Si dimer saturates thedangling bonds of the underlying Si atoms However STM images did notsupport the DAS model because they showed only one protrusion per unitcell [178 192 309] Kulakov et al [178] suggested a model consisting of a Siad-layer and a single ad-cluster Li and Tsong [192] proposed a tetrahedralad-cluster structure without a Si ad-layer
DSi2Si1 DSi3Si2 D1SiSi3
LDA 143 112 233
PBE+vdW 144 114 236
PhD JSchardt [276] 145 097 237
Table 82 Layer distances of the Siadatom structure relative to the sur-face for different exchange-correla-tion functionals (LDA and PBE+vdW)of the 3C-SiC(111)-(3times 3) reconstruc-tion For comparison the layerdistances obtained from quantitat-ive LEED measurements are in-cluded [276] (All values in Aring)
In 1998 Starke et al proposed the now widely accepted structure of the3C-SiC(111)-(3 times 3) reconstruction in a combined STM LEED and DFTstudy [310] The twist model shown in Fig 83 consists of three layers Firsta Si-ad-layer then a ad-layer formed by a tetrahedral Si-ad-cluster The Si-ad-cluster is built from a Si-trimer and a single adatom on top The trimer istwisted with respect to the bulk position by a twist angle of 9 [306] and 93
(PBE+vdW) in this work The six atoms in the Si-ad-layer connected to thetrimer perform a rotation and expansion movement leading to the formationof five- and seven-membered rings arranged in a cloverleaf pattern (topview in Fig 83) In Appendix I we list the structural details calculatedusing PBE+vdW and compare it with quantitative LEED data from Schardt[276] finding an overall good agreement
78
82 The Carbon-Rich Reconstructions of the3C-SiC(111) Surface
In experiment the first stage of graphitisation is the carbon rich surfacestructure the so-called zero-layer graphene (ZLG) or rsquobuffer-layerrsquo Thedetailed atomic structure was debated for many years as there is conflictingexperimental evidence in the literature In STM a (6times6) pattern is imaged[192 210 55 307] On the other hand LEED measurements indicate a(6radic
3times6radic
3)-R30 periodicity [332 264] Also the detailed atomic structureof the carbon nano-mesh has been under debate until recently Chen et al[55] suggested a nano-mesh structure with isolated carbon islands and Qiet al [256] suggested an arrangement of C-rich hexagon-pentagon-heptagon(H567) defects in the ZLG (the defects are discussed in more detail inSec 92) In 2009 Riedl et al [262] demonstrated that the ZLG can beconverted into sp2 bonded graphene layer the so-called quasi-free-standingmono-layer graphene (QFMLG) (see Ch 13) by intercalation of hydrogenat the SiC-graphene interface This was interpreted as a strong argumentin favour of the hexagonal atomic arrangement in the ZLG [109] Goleret al [109] clarified the atomic structure of the ZLG by atomically-resolvedSTM imaging They demonstrated that the ZLG is indeed topologicallyidentical to a graphene monolayer and thus represents a true periodiccarbon honeycomb structure They could not observe any obvious atomicdefects in the QFMLG and concluded that also the defect concentrationin the ZLG is most likely very low Today the general consensus is thatthe C atoms arrange in a hexagonal atomic lattice such as in free-standinggraphene [264 83 335 166 183 263 109]
The (6radic
3times6radic
3)-R30 periodicity originates from the lattice mismatch be-tween the SiC and the graphene lattice parameter The large commensurateZLG unit cell consists of a (6
radic3times6radic
3) SiC supercell (108 Si and 108 C atomsper bilayer) covered by a (13times13) honeycomb graphene-like supercell (338C atoms) [264 83 335 166 263 109 227] The lattice match is almoststrain-free compared to a graphene plane in graphite (experiment 02 at T=0 K [195 9] PBE+vdW 01 Tab 72 see Ch 9) The C atoms inthe ZLG-layer are partially covalently bonded to the top Si atoms of thesubstrate
Continued heating and extended growth times detaches the C-plane of theZLG from the substrate to form MLG and a new ZLG-layer underneath [293125] The MLG-layer is a sp2 bonded graphene layer Further heating leadsto a successive formation of BLG [238] and few-layer graphene films In
79
the following we address the C-rich surface phases in their experimentallyobserved large commensurate (6
radic3times 6
radic3)-R30 supercells using slabs
containing six SiC-bilayers under each reconstructed phase (1742 up to 2756atoms for ZLG up to three-layer graphene (3LG) respectively)
Figure 84 Top view of the relaxed ZLG structure using PBE+vdW The unit cell in thex-y-plane is indicate by a grey rhombus The C atoms in the ZLG layer are colouredaccording to their z-coordinate (the colour scale reaches from yellow for atoms close tothe substrate to blue for atoms away from it) The top Si-C bilayer is shown as well herethe z-coordinate scales from black to white where black indicates that the atom is pushedtowards bulk SiC The ZLG hexagons furthest away from the substrate form a hexagonalpattern (marked in grey) The Si atom of the top Si-C bilayer in the middle of a ZLG carbonring has the minimum z-coordinate (in the middle of the unit cell marked in light green)
Figure 84 shows the geometric structure of the ZLG-layer and the top Si-C bilayer from a top view The unit cell in the x-y-plane is indicate by agrey rhombus The atoms in the structure are coloured depending on theirposition along the z-axis The atoms of the top Si-C bilayer are coloured fromblack to white where black indicates an atom position close to the substrateand white close to the ZLG-layer The colour scale for the atoms in theZLG-layer ranges from yellow for atom positions close to the substrate toblue for atoms away from it In the ZLG-layer the atoms furthest away fromthe substrate (shown in blue) form a hexagonal pattern (marked in grey)The corrugation of the ZLG-layer is also observed in STM experiments [166335 263 68 80] Riedl et al [263] used atomically resolved STM images toconstruct a model of the (6
radic3times 6
radic3)-R30 interface structure they found a
quasi (6times6) periodicity in good agreement with the pattern in Fig 84 (seeRef [263] Figure 3d) The Si atom of the top Si-C bilayer with the minimumz-coordinate is located in the middle of a ZLG carbon ring (in the center ofthe unit cell shown in Fig 84) The corrugation originates from an interplay
80
between C atoms covalently bonded to the substrate and sp2 hybridisedones
The next C-rich phase consists of the ZLG phase covered by a purely sp2
bonded graphene layer the MLG The MLG layer is a (13times13) graphenecell stacked on top of the ZLG-layer in graphite-like AB stacking TheBLG phase consists of the ZLG structure covered by two graphene layer ingraphite-like ABA stacking
Figures 85a-c show the ZLG MLG and BLG phases together with key geo-metry parameters predicted at the level of PBE+vdW To capture the extendof the corrugation shown in Fig 84 and analyse its effect on the substrateand the adsorbed graphene layer we plotted histograms for the atomicz-coordinates For illustration purposes we broadened the histogram linesusing a Gaussian with a width of 005 Aring
The corrugation from the highest C atom to the lowest C atom in the ZLG-layer is 083 Aring Emery et al [80] found a corrugation of 09 Aring by x-raystanding-wave-enhanced x-ray photoemission spectroscopy (XSW-XPS)and x-ray reflectivity (XRR) measurements [80] Few of the Si atoms of thetop Si-C bilayer are pushed into the carbon layer of the top Si-C bilayerWe highlighted the lowest Si atom in Fig 84 as pointed out above the Siatom is underneath the midpoint of a carbon hexagon in the ZLG-layer Thedangling bond of this Si atom pushes against the π-bonded parts of the Cinterface plane [334]
The addition of more graphene planes hardly affects the interface geometryThe corrugation of the ZLG-layer transfers to the MLG phase leadingto a significant buckling of the topmost graphene layer (041 Aring betweentop and bottom of the plane) This strong corrugation is qualitativelyconsistent with x-ray characterisation techniques (XSW-XPS XRR) [80] andSTM images [55 65 21] The corrugation is reduced in the BLG phase to032 Aring in the first graphene layer and 024 Aring in the second one This bucklingreflects some coupling to the covalently bonded interface C-rich planewhich is much more corrugated (08 Aring in our work similar to experimentalestimates [109 68]) The observed graphene interplanar distances near theinterface are slightly expanded compared to experimental bulk graphite(334 Aring [9]) and in good qualitative agreement with estimates from STM [21]and TEM [230]
81
Figure 85 Geometry and key geometric parameters determined by DFT-PBE+vdW for thethree phases (a) ZLG (b) MLG and (c) BLG on the Si face of 3C-SiC(111) and histograms ofthe number of atoms (Na) versus the atomic coordinates (z) relative to the topmost Si layer(Gaussian broadening 005 Aring) Na is normalised by NSiC the number of SiC unit cells Allvalues are given in Aring (Data published in Nemec et al [227])
82
We compared our findings to geometries for the PBE functional withoutvdW correction and for LDA (Fig 86) The LDA and PBE functional showa similar trend for the interplanar bonding of the MLG-layer as for graphite(see Ch 62) In PBE the interplanar distance between the ZLG and MLG-layer is unphysically expanded to 442 Aring In contrast the LDA geometryof the carbon planes agrees qualitatively with PBE+vdW although LDAincorporates no long-range vdW interactions The first qualitative geometrydifference between the PBE+vdW and LDA treatments appears in the cor-rugation of the Si atoms in the top Si-C bilayer In PBE+vdW the distancebetween top an bottom Si atom is 076 Aring but in LDA it is reduced to 053 Aring(PBE 008 Aring) The difference mainly originates from the central Si atom(shown in Fig 84) where the Si dangling bond pushs against the π-bondedparts of the C interface plane
Figure 86 Histogramm of the number of atoms Na versus the atomic coordinates (z)relative to the topmost Si layer (Gaussian broadening 005 Aring) The key geometric parameterof the MLG on the Si face of 3C-SiC(111) at the level of a) LDA and b) PBE All values aregiven in Aring
The atomic density map derived from combined XSW-XPS and XRR ana-lysis [80] also shows a significant broadening for the Si atoms in the top Si-Cbilayer However the corrugation in the top Si-C bilayer was not the focusof the experimental work [80] and therefore cannot clarify the observeddifference in the geometry obtained by using the LDA and PBE+vdW func-tional We will keep these subtle differences in mind when discussing ourresults
The Si-terminated surface of 6H-SiC is widely used to achieve a controlledformation of high quality epitaxial graphene monolayers [95 82 307 263]In the following we discuss the structural difference between 3C-SiC- and6H-SiC-MLG For the 6H-SiC-MLG we use the S36H terminated surface(see Fig 81) and the lattice parameter listed in Tab 72 For the structureoptimisation we used the PBE+vdW exchange-correlation functional
83
Figure 87 MLG on 6H-SiC(0001)and histogram of the number ofatoms Na versus the atomic co-ordinates (z) relative to the top-most Si layer (Gaussian broaden-ing 005 Aring) Na is normalised byNSiC the number of SiC unit cellsDnn+1 is the distance betweenthe layer n and n + 1 dn givesthe Si-C distance within the SiCbilayer n and δn the corrugationof the layer n All values aregiven in Aring (Figure published inRef [294])
Overall the atomic structure of 3C-SiC-MLG (Fig 85 b) and 6H-SiC-MLG(Fig 87 are very similar For 6H-SiC-MLG the corrugation of the MLG-layerand of the ZLG-layer is slightly increased to 045 Aring and 086 Aring respectivelyHowever the layer distances Dnn+1 are the same for 6H-SiC-MLG and3C-SiC-MLG Overall the changes with the polytype are minor which isnot surprising as the Si-C bilayer stacking order is identical for 3C-SiC andS36H terminated 6H-SiC
84
9 Strain in Epitaxial Graphene
There are two aspects to strain in epitaxial graphene The first one is specificto epitaxial graphene on the Si face of SiC caused by the coupling betweenthe ZLG and the epitaxial graphene layer (MLG) [229 87] In the followingwe discuss both aspects of strain in epitaxial graphene The second one iscaused by the choice of the coincidence lattice between the substrate andgraphene Depending on the interface structure used in the calculationsartificially induced residual strain may effect the energetics and the detailedbonding at the interface [290]
91 ZLG-Substrate Coupling and its Effect on theLattice Parameter
By means of Raman spectroscopy signatures of compressive strain wereobserved in epitaxial graphene on the Si face of SiC [229 87 266 281] Thestrain in the MLG is caused by the coupling between the ZLG and theMLG layer [88 109] Conversely in grazing-incidence x-ray diffraction(GID) an increase of the in-plane lattice parameter was measured for theZLG layer (2467 Aring) [283] if compared to lattice parameter of the MLG-layer(2456 Aring [283]) or the in-plane lattice parameter of graphite (2456 Aring [226])
In Section 82 we introduced the ZLG interface structure In Figure 91 (a)the ZLG layer and the top SiC bilayer is shown from atop The C atoms inthe ZLG are coloured depending on their vertical distance from the averageheight of the top Si layer from black to white The in-plane bonds betweenthe C atoms in the ZLG layer are coloured from blue to red dependingon their bond length In regions close to C atoms covalently bonded tothe substrate the C-C bond length is streched compared to free-standinggraphene For example the previously discussed Si atom in the middleof a carbon hexagon (marked in Fig 84 or at the center of Fig 91 (a)) issurrounded by 6 Si atoms covalently bonded to C atoms in the ZLG-layerThe Si-C bond length is 1995 Aring for comparison the Si-C bond length in3C-SiC is 1896 Aring The mixture of sp2 and sp3 hybridised carbon at thesubstrateZLG interface results in a corrugation of 083Aring (see Fig 85)
As alluded to above Schumann et al showed by GID that the lattice para-meter of the ZLG layer is larger than the in-plane lattice parameter of
85
graphite Figure 91 (b) shows the bond length distribution in the ZLG layerWe applied a Gaussian broadening of 0002 Aring for visualisation purposesonly The area under the curve is shaded with the same colour gradient usedin Fig 91 a) The bond length of free-standing graphene in the optimisedstructure on a PBE+vdW level and the average bond length in the ZLGlayer are indicated by two vertical lines
The bond length distribution curve shows two maximum a sharp one ata bond length of 1406 Aring and a second between 1468 Aring and 1504 Aring Thefirst maximum corresponds to a compression of the C-C bond length of11 compared to free-standing graphene On average the bond length inthe ZLG layer is 16 larger (1445 Aring) The double peak in the bond lengthdistribution can be understood when examining Fig 91 a) In regionswhere a C atom of the ZLG layer is covalently bonded to a Si atom inthe substrate the three in-plane C-C bonds are extended while in regionswith a sp2-like hybridisation in the ZLG layer the bond length is closerto free-standing graphene with a slight compression This result is notsurprising because an increase of the C-C bonds length has been observedwhen moving from sp to sp2 to sp3 hybridised C-C bonds [41] The presenceof inhomogeneous compressive strain was also suggested on the basis ofRaman spectroscopy by Roehrl et al [266] and Schmidt et al [281] Thecorrugation of the interface layer is reduced by an additional MLG layer ontop of the ZLG (see Fig 85 b) resulting in a slightly smaller average C-Cbond length
In GID measurements the lattice parameter is measured as a projectiononto the x-y-plane The changes in the bond length due to corrugation isnot captured by the experiment [283] This makes a direct comparison withour calculations difficult because the average C-C bond length projectedon the x-y-plane is fixed by the choice of lattice vectors used in the calcula-tion However the general trend is well captured in our calculations Theincreased lattice parameter can be explained by the mixture of the sp3 andsp2-like hybridisation of the C atoms in the ZLG layer
86
Figure 91 a) Top view of the ZLG layer and the top SiC bilayer fully relaxed usingPBE+vdW The carbon atoms of the ZLG layer are coloured according to their verticaldistance from the average height of the Si-layer ranging from black being close to the Silayer to white for C atoms away from the Si-layer The bonds between the ZLG C atomsare coloured from blue to red for increasing bond length b) Bond length distribution in theZLG layer A Gaussian broadening of 0002 Aring was applied for visualisation purposes onlyThe two vertical lines indicate the bond length of free-standing graphene in the optimisedstructure on a PBE+vdW level and the average bond length in the ZLG layer
87
92 Interface Models for Epitaxial Graphene
It would be computationally more efficient to use smaller-cell approximantphases to the true (6
radic3times6radic
3)-R30 supercell Indeed different smaller-cellapproximant phases have been used in literature [214 215 334 335 289243] However the residual artificial strain and incorrect bonding in thosephases hinder a meaningful surface energies comparison Sclauzero andPasquarello [288] compared three different structural models to the ZLGphase a SiC-(
radic3timesradic
3)-R30 supercell covered by an (2times 2) graphene cellan unrotated (4times 4) SiC cell covered by a (5times 5) graphene cell and theexperimentally observed SiC-6
radic3times 6
radic3-R30 ZLG structure They com-
pared the energy of the three SiC-graphene interface models and identifiedthe SiC-6
radic3times 6
radic3-R30 ZLG structure to be the most stable one
We constructed commensurate graphene-3C-SiC(111) interface structuresfrom rotated hexagonal supercells of 3C-SiC(111) and graphene We ad-dress artificially induced strain dangling bond saturation corrugationin the carbon nanomesh and the surface energy for different coincidencestructures
Figure 92 Comparison of the surface energies for six different smaller-cell approximantmodels of the 3C-SiC(111) ZLG phase relative to the bulk-terminated (1times1) phase as afunction of the C chemical potential within the allowed ranges using PBE+vdW The Sirich (3times 3) and (
radic3timesradic
3) reconstructions as well as the (6radic
3times 6radic
3)-R30 ZLG phasefrom Fig 101 are included for comparison The shaded areas indicate chemical potentialvalues outside the strict thermodynamic stability limits of Eq 415
88
By rotating a graphene layer on a SiC substrate Sclauzero and Pasquarello[290] found several model interfaces which give almost perfect commen-surability between the graphene and SiC supercells For a selection ofsuch interface models they calculated the change in energy of a plane free-standing graphene layer due to the strain necessary to built a commensurateSiC-graphene interface model [290] They found a change in energy to besmaller than 40 meV for all structures except the (
radic3timesradic
3)-R30 modelWe include six different smaller-cell approximant models of the 3C-SiC(111)ZLG phase in our analysis
(ZLG) SiC-(6radic
3times 6radic
3)-R30 amp (13times 13) [19 95 83 55]
(a) SiC-(5radic
3times 5radic
3)-R30 amp (11times 11)
(b) SiC-(5times 5) amp (3radic
13times 3radic
13)-R162 [242]
(c) SiC-(4times 4) amp (5times 5) [290]
(d) SiC-(6radic
3times 6radic
3) amp (13times 13)-R22 [131]
(e) SiC-(4times 4) amp (3radic
3times 3radic
3)-R30
(f) SiC-(radic
3timesradic
3)-R30 amp (2times 2) [334 214]
To our knowledge structure (a) and (e) have not yet been discussed inliterature We calculated a slightly rotated (5times5) approximant to the ZLGphase [242] (structure (b)) a periodicity sometimes seen in experiment [210264 307] Structure (b) has also been discussed in Ref [290] and Ref [289]Structure (c) was first introduced by Sclauzero and Pasquarello [290] Hasset al [131] used basic trigonomy to show that for the (6
radic3 times 6
radic3) SiC
supercell three rotational angle relative to SiC substrate can be found theexperimentally observed 30 (ZLG) and two additional rotational angleplusmn2204 The graphene layer rotated by 2204 can be constructed with thesupercell matrix (N ) introduced in Sec 11
N =
8 7 0
minus7 15 0
0 0 1
(91)
In addition to the ZLG structure we include the SiC-(6radic
3times 6radic
3) supercellwith a graphene layer rotated by 2204 (structure (d)) In Figure 92 wecompare the surface energies of the six different smaller-cell approximantmodels listed above We also included the Si rich (3times 3) and (
radic3timesradic
3)
89
SiC cell ZLG cell Angle α NsurfSiC NZLG NDB
Si δC[Aring] Strain [] γ [eVSiC(1x1)]
ZLG 6radic
3times 6radic
3 13times 13 30 108 338 19 083 014 minus044
(a) 5radic
3times 5radic
3 11times 11 30 75 220 19 095 minus065 minus036
(b) 5times 5 3radic
13times 3radic
13 162 25 78 3 083 027 minus035
(c) 4times 4 5times 5 0 16 50 3 065 019 minus029
(d) 6radic
3times 6radic
3 13times 13 22 108 338 19 074 014 minus029
(e) 4times 4 3radic
3times 3radic
3 30 16 54 10 145 36 minus015
(f)radic
3timesradic
3 2times 2 30 3 8 1 030 93 015
3times 3 2radic
3times 2radic
3 30 9 24 3 030 93 015
Table 91 Parameters of the simulated ZLG-3C-SiC(111) interface structures in differentcommensurate supercells the 3C-SiC and ZLG periodicity the rotation angle (α) the num-ber of surface Si atoms (Nsurf
SiC ) and C atoms in the ZLG (NZLG) the number of unsaturatedSi bonds (NDB
Si ) (see Appendix E) the corrugation of the ZLG layer (δC) the strain in thecarbon layer and the relative surface energy at the carbon rich graphite-limit (γ|microC=Ebulk
graphite)
reconstructions as well as the (6radic
3times 6radic
3)-R30 ZLG phase from Ch 8All structural models consist of six 3C-SiC bilayer covered by a graphenelattice the bottom C atoms of the SiC substrate have been saturated byhydrogen The top three SiC-bilayers and the carbon layer above are fullyrelaxed using PBE+vdW (residual energy gradients 8middot10minus3 eVAring or below)The surface energies (γ) were calculated using Eq 49 All surface energiesare given relative to the bulk-terminated (1times1) surface The range of theSi and C chemical potential are given in Sec 42 For instance the popular(radic
3timesradic
3)-R30 [334 214] approximant intersects the graphite stability lineat a relative surface energy in the carbon rich limit (ref Graphite) of 015 eVfar above the actually stable phases
In Table 91 we listed the parameters of the simulated ZLG-3C-SiC(111)interface structures for the different smaller-cell approximant models shownin Fig 92 the 3C-SiC and ZLG periodicity the rotation angle (α) thenumber of surface Si atoms (Nsurf
SiC ) and C atoms in the ZLG (NZLG) We alsolist the number of unsaturated Si bonds (NDB
Si ) The Si dangling bonds wereidentified on the basis of geometric position and bond length differences(for a detailed discussion see Appendix E) Table 91 further includes thecorrugation of the ZLG layer (δC) the strain in the carbon layer and therelative surface energy at the carbon rich graphite-limit (γ|microC=Ebulk
graphite)
In structure (b) the strain in the carbon ad-layer is small with 027 and theSi dangling bond saturation is improved if compared to the experimentallyobserved 6
radic3times 6
radic3-R30 However the relative surface energy intersects
90
with the chemical potential limit for graphite at minus035 eV still higher by006 eV than even the closest competing Si-rich phase the (
radic3timesradic
3)-R30
Si adatom phase The (5times5) phase is either a nonequilibrium phase or itsstructure is not the same as that assumed in Ref [242]
So far different mechanisms were assumed to stabilise the SiC-(6radic
3times 6radic
3)-R30 amp (13times 13)-ZLG structure First the surface energy is reduced byminimising the strain between the SiC substrate and the ZLG layer [290]Sclauzero and Pasquarello [288] pointed out a correlation between the bind-ing energy and the corrugation of the ZLG-layer In their study the bindingenergy was reduced with a reduction of the corrugation It also has beendiscussed that the structural properties are driven by saturation of danglingSi bonds of the top SiC-bilayer through bond formation with the ZLG Catoms [83 290] Structure (d) and the ZLG experience the same lattice mis-match Both phases saturate 89 Si atoms at the SiC-ZLG interface leaving19 Si dangling bonds (Tab 91) The corrugation of the ZLG-phase is higherby 009 Aring If we follow the line of argument and compare the two structuremodels the ZLG and structure (d) we would assume that the bindingenergy of strucutre (d) should be lower because the lattice mismatch andthe Si dangling bond saturation is the same but the corrugation of structure(d) is lower However this is not observed The ZLG phase is 015 eV lowerin surface energy Tab 91 and in LEED measurement show a 30 rotationbetween the substrate and the ZLG-layer eg in Ref [263]
Certainly the lattice mismatch the corrugation and the dangling bondsaturation play a crucial role in the stabilisation of the ZLG phase but thesethree factors cannot explain why the 30 rotation is preferred energeticallyover the 22 (Tab 91 and Fig 92) To shed some light on the origin of thedifference in surface energy between the two different rotation angles wecompare the bond length distribution of these two phases
Figure 93 shows the bond length distribution in the (6radic
3times 6radic
3) amp (13times13)-ZLG layer for two different rotational angles The bond length distri-bution of the 30 rotated ZLG layer is taken from Fig 91 (shown in greyin Fig 93) Structure (d) rotated by 2204 is shown in blue in Fig 93We applied a Gaussian broadening of 0002 Aring for visualisation purposesonly The average bond length of structure (d) is 1434 Aring which is 0012 Aringsmaller than for the ZLG-layer In Section 91 we demonstrated that thetwo peak structure can be understood as a result of a mixture of regionswith a sp2-like and sp3-like hybridisation in the ZLG layer The bond lengthof the sp3-like hybridised C atoms in the ZLG-layer range from 1468 Aring to1504 Aring If we compare the two curves in Fig 93 we see that in the bondlength range of the sp3-hybridised C atoms of the ZLG layer are only very
91
Figure 93 Bond length distribution in the (6radic
3times 6radic
3) amp (13times 13)-ZLG layer for twodifferent rotational angles The bond length distribution for the ZLG layer in the 30
rotated structure from Fig 91 is shown in grey and the ZLG structure rotated by 2204
in blue A Gaussian broadening of 0002 Aring was applied for visualisation purposes onlyThe two vertical lines indicate the bond length of free-standing graphene in the optimisedstructure on a PBE+vdW level and the average bond length in the ZLG layer rotated by2204
few C-C bonds of structure (d) Indeed the maximum C-C bond length instructure (d) is 1472 Aring indicating that the C-C bonds with sp3 characteristicare compressed Most bonds of structure (d) are between the two peaks ofthe ZLG structure From the analysis shown in Fig 93 we conclude thatthe surface energy of structure (d) is increased by the compression of thesp3-like hybridised C-C bonds and the stretching of the sp2-like hybridisedC-C bonds The difference in the bond length distribution originates in thedetailed bonding situation determined by the position of the Si danglingbonds However also the electronic structure of the specific arrangementwill play a role in the stabilisation of one structure over the other
Finally we discuss the influence of artificially strained approximant phaseson electronically relevant properties such as the formation energies ofdefects To calculate the defect formation energy we substract from thesurface energy (Eq 49) of the system including the defect (γdef) the surfaceenergy of the defect free surface (γZLG) and account for additional C atomsor C vacancies by the carbon chemical potential micro
GraphiteC (Eq 415)
∆G =1
Ndef
(γdef minus γZLG minus NCmicro
GraphiteC
) (92)
92
where NC is the number of substituted or vacant C atoms and Ndef the num-ber of defects in the slab For negative defect formation energies (∆G) thedefect is stable and for positive ∆G the formation of the defect is unlikely
As an example we consider a specific class of C-rich hexagon-pentagon-heptagon (H567) defects suggested as an equilibrium feature of the ZLGphase in Ref [256] The defects consist of three carbon heptagons andpentagons surrounding one carbon hexagon The central hexagon is rotatedby 30 with respect to the graphene lattice This defect incoporates twoadditional carbon atoms (NC =2) lowering the average C-C bond lengthin the ZLG Qi et al suggested two different defect positions ldquohollowrdquo andldquotoprdquo shown in Fig 94
In Figure 94 the H(567) defect is shown for both positions in the approxim-ated 3times 3 and in the (6
radic3times 6
radic3)-R30 ZLG phase The 3times 3 is a supercell
of the approximated (radic
3timesradic
3)-R30 ZLG phase with a massively strainedcarbon layer (see Tab 91) All four phases were fully relaxed using thesame procedure as described above Indeed both defects would be morestable than the hypothetical (
radic3timesradic
3)-R30 ZLG approximant when in-cluded in a (3times 3) arrangement as done in Ref [256] Using Eq 92 gives∆G = minus175 eV per defect for the hollow and ∆G = minus293 eV per de-fect for the top position However the same defects are unstable whenincluded into and compared to the correct (6
radic3times 6
radic3)-R30 ZLG phase
∆G = +528 eV per defect for the hollow and ∆G = +527 eV for the topsite again at the chemical potential limit for graphite The formation energyof the H(567) defect in the (6
radic3times 6
radic3)-R30 ZLG phase is too high to
make the formation of these defects likely The formation of a H(567) defectintroduces two additional C atoms lowering the overall strain of the C-Cbonds in the approximated (
radic3timesradic
3)-R30 ZLG phase which leads toa stabilisation of the defect This example illustrates the importance of acareful interpretation of any results obtained by using an approximatedinterface structure
93
Figure 94 The hexagon-pentagon-heptagon (H567) defect in the zero-layer grapheneshown in the approximated 3times 3 cell (insert a and b) and in the (6
radic3times 6
radic3)-R30 ZLG
phase The defect was placed in two different positions In inset a and c the defect is placedat the ldquohollowrdquo position with a silicon atom of the underlying SiC bilayer in the middle ofthe central hexagon and at the ldquotoprdquo position
94
93 Defect Induced Strain in Graphene
Schumann et al [284] demonstarted the feasibility of growing nano-crystallinegraphene films on (6
radic3times 6
radic3)-R30 ZLG phase using MBE as an altern-
ative method for graphene synthesis In a combined experimental andtheoretical study we investigated the strain observed in nano-crystallinegraphene grown by MBE published in Ref [284] Our experimental col-laborators investigated the structural quality of the MBE grown graphenefilms by AFM XPS Raman spectroscopy and GID XPS and Raman spec-troscopy indicate that the ZLG-layer persists throughout the growth processThey demonstrated that grown carbon films indeed consist of planar sp2-bonded carbon by XPS measurements The appearance of an intense Dpeak in Raman spectroscopy a characteristic feature in the Raman spectraof defects in graphene reveal that the MBE-grown carbon films possesses anano-crystalline and partially defective structure
From the Raman spectra they estimated the average lateral size of thegraphene domains to be at the order of 15-20 nm GID measurements showthat the graphene domains have the same in-plane orientation and arealigned to the substrate Interestingly the lattice parameter of the nano-crystalline graphene films is contracted by 045 in comparison to the latticeparameter of graphite (2461 Aring [132]) They measured the lattice parameterof the graphene films to be 2450 Aring The origin of the observed contractionis unclear We here list three potential reasons
i The linear thermal expansion coefficients of graphene and SiC are verydifferent in their low temperature behaviour [87 266] While the lat-tice parameter of graphene decreases [221] the volume of SiC expands[193 196 317] for increased temperatures These qualitatively differentbehaviour was related to the contraction of the MLG lattice parameter[87 266] Based on this difference Ferralis et al [87] estimated that acompressive strain in graphene of up to 08 can arise upon samplecooling down to room temperature
ii The strong corrugation in the ZLG as discussed in Sec 91 could contrib-ute to the apparent contraction of the (2D projected) lattice parameter ofthe uppermost nano-crystalline graphene despite the existent epitaxialrelation between them Indeed in Sec 82 we found a corrugation of theMLG films of 041Aring
iii The presence of defects and grain boundaries in the nano-crystallinefilm could also lead to a lateral contraction of graphene The strong
95
D peak in the Raman spectra would allow for the presence of zero- orone-dimensional defects in the films
Strain up to 04 has been inferred by Raman spectroscopy for the epitaxi-ally grown MLG [281] A similar contraction was directly linked to substratethermal contraction in a recent study of near-perfect CVD-grown grapheneon Ir(111) [132] However the contraction observed by the GID measure-ments extends to a smaller lattice parameter than that found in Ref [281]and Ref [132] For a perfect free-standing graphene layer the observedcontraction would most likely induce the formation of wrinklesnano-finsfor strain relief Schumann et al [284] performed an AFM analysis and didnot observe such wrinkles in the MBE grown graphene films
To estimate the possible contributions of (ii) and (iii) we performed DFT cal-culations of isolated graphene sheets In Table 61 we found for the infiniteperiodic flat graphene sheet a lattice parameter of a = 2463 Aring (PBE+vdW)The calculated value for MLG layer on SiC is practically the same (within01 see Tab 91) For straight PBE without vdW corrections the in-planelattice parameter of graphene is slightly affected leading to an increase of0004 Aring (see Tab 61) Beyond the original vdW correction of Tkatchenkoand Scheffler [326] the strength of effective interatomic C6 coefficients thatdescribe the vdW interaction in carbon-based nanostructures may varyconsiderably with the structure [107] and could significantly change out-ofplane interactions with a graphene sheet (Sec 62) However their effect willstill be small on the energy scale of interest for the in-plane lattice parameterwhich is dominated by the covalent interactions that are described by thePBE functional itself
First we address point (ii) the influence of the substrate-induced corruga-tion In Sec 82 we found that the corrugation of the ZLG-layer induces acorrugation in the MLG-layer of 041 Aring (Fig 85) The MBE-grown grapheneon the ZLG should show the same approximate corrugation If this corrug-ation led to significant stress in the plane a perfect graphene sheet with thesame (fixed) z corrugation should experience the same stress and shouldthus contract Our collaborator Volker Blum relaxed all in-plane coordinatesand lattice parameters of such a corrugated graphene plane with fixed zcorrugation However he found a calculated contraction of less than 005leading to a surface area that corresponds to an effective graphene latticeparameter of a = 2462 Aring
Finally we address the potential impact of different types of defects on thein-plane lattice parameter (point (iii) above) We first address one dimen-sional defect types (domain boundaries) Figure 95 shows the development
96
of the effective lattice parameter of finite-size hexagonal graphene flakes asa function of flake size The effective lattice parameter is calculated by fullyrelaxing the flat graphene flakes then calculating the average of all C-Cnearest-neighbour bond distances in the flake and converting this valueto the equivalent lattice parameter of a perfect honeycomb mesh We con-sidered two different types of flakes One type with a H-saturated boundary(inset in Fig 95) and those without H atoms at the boundary The effectivelattice parameter for small flakes is significantly contracted in either case Toapproach the GID-observed lattice parameter of 2450 Aring by this effect alonethe equivalent saturated flakes would have to be extremely small (less than08 nm in diameter) The equivalent unsaturated flakes however could besignificantly larger about 2 nm even if they were perfect otherwise Withincreasing flake size the net contraction decreases rapidly for larger sizesHowever the graphene domain size was estimated to be at the order of15-20 nm
Localised zero-dimensional (point-like) defects can also lead to strain andcorrugation in graphene sheets [47 274] In close collaboration with VolkerBlum we investigate the influence of point-like defects on the averagein-plane lattice parameter We used three different defect types mono-vacancies divacancies and Stone-Wales defects in three different periodicsupercell arrangements (for a recent review consider eg [47]) Figure 96shows the three defect types The monovacancy and 5-8-5 divacancy defectsare shown in a (10times 10) graphene unit cell and the Stone-Wales defect ina (7times 7 ) unit cell To give a quantitative prediction on how the graphenedefects influence strain corrugation and net area change of a periodicgraphene sheet (Tab 92) we fully relaxed the atomic structure and unit cell(residual forces and stresses below 0001 eVAring) of the three defect types indifferent supercell (SC) arrangements using PBE+vdW
In graphene the perhaps most studied point defect type are monovacancies(eg [274 265 6] and references therein) In a recent exhaustive studySantos et al [274] addressed the relation between theoretically imposedisotropic strain and monovacancy properties such as corrugation and spinpolarisation
The following anaysis of the different defect types in graphene was carriedout by Volker Blum published in a the joined publication Ref [284] Heconsidered only the fully relaxed stress-free local optimum structures ascalculated by DFT on the level of PBE+vdW Table 92 includes the calcu-lated data for monovacancies in three different supercell arrangements withand without spin polarisation For the structures without including spinpolarisation monovacancies would be associated with significant strain and
97
Defect Supercell aeff [Aring] ∆z [Aring] ∆Adefect [Aring2]
monovacancy 5times 5 2447 0741 -172
7times 7 2455 0800 -178
10times 10 2459 0856 -188
spin-polarised (ferro- or paramagnetic)
5times 5 2453 lt001 -106
7times 7 2458 0197 -111
10times 10 2461 0055 -098
divacancy 5times 5 2417 1023 -494
7times 7 2431 1566 -682
10times 10 2443 1888 -846
flat (meta stable)
7times 7 2447 lt002 -342
Stone-Wales defect 7times 7 2469 000 +054
Table 92 PBE+vdW calculated effective lateral lattice parameter aeff (in Aring) the top-to-bottom corrugation ∆z (in Aring) and the effective area lost (gained) per defect ∆Adefect ( inAring2) of different defect types and periodicities in an ideal free-standing graphene sheet(Table published in Ref [284])
corrugations of perfect graphene sheets (see Tab 92) The picture changesfor monovacancies calculated with the inclusion of spin polarisation InDFT on the level of PBE+vdW monovacancies are paramagnetic defectsthat carry a significant local moment This leads to a slight reduction ofthe compressive strain compared to a perfect graphene sheet and thusalso to a reduction of the overall distortion (buckling of the monovacancyand corrugation of the sheet) Even with the highest concentration con-sidered in Tab 92 (2 of monovacancies modeled by a (5times 5) supercell)monovacancies alone would not yet lead to the full strain seen in the GIDexperiments
The next defect type in graphene we considered is the divacancy In factdivacancies in graphene are thermodynamically more stable than mono-vacancies [47] a tendency that is even enhanced by compressive strain
98
[48] Volker Blum obtained the results for the divacancies with corruga-tion shown in Fig 96 by starting from the fully relaxed non-spinpolarisedmonovacancy geometries and removing the most strongly buckled atom(in z direction) in the monovacancy
As can be seen in the Tab 92 and the structure of the (10times 10) divacancydefect in Fig 96 this procedure leads to very significant strains and buck-ling in a free-standing graphene sheet The sheet curvatures seen at thedefect locations follow the trend described in the literature [47] It is alsoobvious that such defects could easily explain the GID-observed laterallattice parameter reduction even for relatively low defect concentrations((10times 10) case) In fact significant corrugations of this kind are seen inatomically resolved STM images of defects generated in Highly OrderedPyrolytic Graphite by ion implantation (eg Fig 1 in Ref [203]) Howevereven if a divacancy were completely flat the associated strain would still besignificant For comparison Tab 92 also includes the case of a flat (7times 7)periodic divacancy which is a local structure optimum about 01 eV higherin energy than the corrugated divacancy arrangement The compression ofthe lattice parameter for the defect densities of the divacancies included inTab 92 compresses the latttice parameter to much compared to the 2450 Aringmeasured by GID experiments
Finally we also include the case of the Stone-Wales defect which resultsfrom the rotation of a single C-C bond but the number of C atoms remainsunchanged Here a slight expansion not compression of the overall latticeparameter would result (Tab 92)
We here used idealised theoretical defects and boundaries as approxima-tions to the experimental reality of MBE-grown nano-crystalline graphenefilms Assuming that the growth process leads to defective graphene filmssuch that the defects influence the characteristic of such films the mor-phology of the MBE-grown graphene films would most likely consists of acombination of the defect types considered in this section as well as othersIt thus seems qualitatively plausible that the strain induced by defects mayindeed significantly contribute to the overall lattice parameter contractionthat was observed in GID However to achieve large-scale homogeneouselectronic properties of epitaxially MBE-grown graphene films it will beimportant to eliminate the potential for metastable defects
99
94 Summary
In summary a detailed analysis of the ZLG layer revealed the origin ofthe increased lattice parameter of the ZLG layer observed in GID measure-ments [283] The mixture of sp2 and sp3-like hybridisation of the carbonatoms in the ZLG layer leads to a double peak structure in the bond lengthdistribution The first peak corresponds to the sp2-like hybridised carbonatoms and the second one to the sp3-like hybridised carbon atoms leadingto an overall increase of the average in-plane C-C bond length
We compared different interface models for the ZLG structure Care hasto be taken when using smaller approximated interface structure to theexperimentally observed 6
radic3-ZLG interface By comparing the ZLG struc-
ture with a (6radic
3times 6radic
3) SiC supercell and a (13times 13)-R22 C layer wefound that the specific bonding situation at the interface plays a crucialrole in stabilising the SiC-graphene interface The differences in strain andthe detailed position of the Si-C bonds at the substrategraphene interfacecan for example misleadingly stabilise defects We calculated the hexagon-pentagon-heptagon (H567) defect as suggested by Qi et al [256] in the ZLGusing the approximated
radic3-ZLG cell and the 6
radic3-ZLG phase We found
that the defects calculated in theradic
3-ZLG would be energetically stable thesame defects calculated in the 6
radic3-ZLG are energetically unfavourable
To shed some light on the observed lattice contraction in MBE-grown nano-crystalline graphene we derived reference values for the lattice parametercontraction expected from a possible substrate-induced corrugation Toevaluate the influence of domain boundaries we calculated the C-C bondlength of finite carbon flakes (hydrogen-saturated or unsaturated) How-ever both the corrugation and the domain boundary effects were too smallto solely explain the observed lattice corrugation Finally we calculated thechange of the average in-plane lattice parameter for graphene sheets withmonovacancies divacancies and Stone-Wales defects in periodic supercellarrangements The calculations demonstrate that a lattice parameter con-traction will arise from all defects except for the Stone-Wales defect Thelargest contraction is associated with the divacancy which also induces asignificant buckling in free-standing graphene sheets A low concentrationof defects is thus one possible explanation for the observed contraction
100
Figure 95 (a) Calculated (PBE+vdW) effective lattice parameter (average C-C bondlength) in a series of fully relaxed graphene flakes of finite size with (squares) and without(circles) hydrogen termination at the edges The hydrogen terminated flakes are shownin the inset The lattice parameter of a flat strainfree periodic graphene sheet calculatedin PBE+vdW is indicated by a dashed line (b) Diameter (maximum C-C distance) of theflakes used in (a) (Figure published in Ref [284])
101
Figure 96 Top calculated fully relaxed (atomic postion and unit cell) structure of agraphene sheet with a (10times 10) periodic arrangement of a 5-8-5 divacancy defects (atomshighlighted in red) The in-plane unit cell is marked by black lines Bottom from left toright fully relaxed monovacancy and 5-8-5 divacancy defects in a (10times 10) unit cell andStone-Wales defect in a (7times 7 ) unit cell
102
10 Thermodynamic Equilibrium Conditionsof Graphene Films on SiC
In Chapter 8 we introduced the Si-rich (radic
3timesradic
3)-R30 and (3times 3) recon-structions (Sec 81) and the C-rich phases ZLG MLG and BLG (Sec 82) Inthis chapter we will examine the thermodynamic phase stability of thesedifferent phases using the ab initio atomistic thermodynamics formalism aspresented in Ch 4 The results presented in this chapter were published in2014 Nemec et al [227]
The surface energies (γ from Eq 49) of the known surface phases of SiC(Si face) are shown as a function of ∆microC = microC minus Ebulk
C in Fig 101a forPBE+vdW In the surface diagram in Fig 101a we included seven differentsurface reconstructions
(1times 1) SiC-(1times 1) calculated in a SiC-(radic
3timesradic
3)-R30 cell(see Fig 81)
(radic
3timesradic
3) SiC-(radic
3timesradic
3)-R30 amp Si adatom(see Sec 811)
(3times 3) SiC-(3times 3) amp 9 atomic Si ad-layer and 4 atomic Si ad-cluster(see Sec 812)
(ZLG) SiC-(6radic
3times 6radic
3)-R30 amp 1 (13times 13) C layer(see Sec 82)
(MLG) SiC-(6radic
3times 6radic
3)-R30 amp 2 AB-stacked (13times 13) C layer(see Sec 82)
(BLG) SiC-(6radic
3times 6radic
3)-R30 amp 3 ABA-stacked (13times 13) C layer(see Sec 82)
(3LG) SiC-(6radic
3times 6radic
3)-R30 amp 4 ABAB-stacked (13times 13) C layer
The most stable phase for a given value of ∆microC is that with the lowestsurface energy Going through Fig 101a from left to right we first find theexpected broad stability range of the very Si-rich (3times3) reconstruction (13atomic Si surface structure see Sec 812) followed by the (
radic3timesradic
3)-R30
Si adatom structure (see Sec 811) In the C-rich regime just before theC-rich limit (bulk graphite) is reached the C-rich phases stabilise First theZLG crosses the (
radic3timesradic
3) reconstruction and then the MLG and even for avery narrow stability range the BLG phase The 3LG crosses the BLG within
103
Figure 101 Comparison of the surface energies for five different reconstructions of the3C-SiC(111) Si side relative to the bulk-terminated (1times1) phase (always unstable) as afunction of the C chemical potential within the allowed ranges (given by diamond Sidiamond C or graphite C respectively) (a) PBE+vdW (b) PBE(c) LDA The shaded areasindicate chemical potential values outside the strict thermodynamic stability limits ofEq 415 (Data published in Ref Nemec et al [227])
1 meV of bulk graphite The respective stability ranges of the C chemicalpotential are very narrow for the different C-rich phases are small (inset ofFig 101a 4 meV 5 meV and lt1 meV for ZLG MLG and BLG respectivelyat the chemical potential axis) As pointed out in Sec 42 narrow chemicalpotential ranges do not necessarily correspond to narrow experimentalconditions
The differences in γ between the C-rich phases are also rather small (afew tens of meV per (1times1) SiC surface unit cell) which necessitates thecalculation of highly accurate total energies We ensured the convergencewith respect to the basis set size the sampling in real and reciprocal spaceand the number of SiC-bilayers included to represent the substrate (see
104
Appendix B)
As already discussed in Ch 4 one can further include the much smal-ler (p T) dependence of the solid phases by focusing on Gibbs free ener-gies G(T p) (see Eq 41) in the QHA instead of E to eg capture smalltemperature-dependent surface strain effects However quantifying thisT dependence precisely would necessitate accurate phonon calculationsfor structure sizes of the order of sim2000 atoms a task that is computation-ally prohibitive at present ZPC are small for the bulk phases (see Tab 72)However there is the possibility of small contributions of finite T for the sur-face phases [266] We keep them in mind when interpreting our calculatedresults below
The primary approximations that we cannot systematically improve inour calculations originate from the exchange-correlation functional usedStandard approximations to the exchange-correlation functionals cannotcapture correlation effects [14] rendering all discussed surface phases metal-lic However in experiment a small band gap of approximately 300 meVwas observed in scanning tunnelling spectroscopy (STS) measurements ofthe ZLG layer [270 109] A more advanced treatment might suppress themetallicity shifting the surface energetics towards lower energies mostlikely maintaining the relative energy differences between the graphenephases while the crossing point between the Si adatom phase and the ZLGphase could change
We evaluated the influence of different density functionals for the relatedbulk phases in Part II In Figures 101 b amp c) we show how the surfaceenergies would be affected by different density-functional treatments Wehave thus recomputed the surface phase diagram up to the MLG phase forthe PBE and LDA functionals
The results for the plain PBE functional which lacks long-range vdW inter-actions are shown in Fig 101 b) In Chapter 62 and Sec 82 we alreadydiscussed the importance of the inclusion of long-range vdW interactionsand demonstarted that plain PBE fails to describe the interplanar bondingbetween the graphene layers As expected the absence of vdW tails inthe plain PBE functional changes the phase diagram drastically Due tothe overstabilisation of graphite (130 meVatom Sec 61) its stability linemoves significantly further to the left as does the crossover point betweenZLG and MLG As a result neither ZLG nor MLG become stable over thecompeting Si-rich (
radic3timesradic
3)-R30 phase in PBE in outright contradictionto experiment [329]
105
In the LDA-derived phase diagram shown in Fig 101 c) the most significantchange compared to PBE+vdW is the apparent incorrect stability hierarchyof graphite vs diamond (see Ch 6 and Appendix A) If we focus on thegraphite line we find that the ZLG-MLG crossover point is almost exactlyon that line In LDA the net interlayer bonding between the first graphenelayer and the ZLG phase is thus the same as that in bulk graphite As a resultthe ZLG phase in LDA has a broader stability range than in PBE+vdW TheMLG phase would only be a (very) near-equilibrium phase in LDA Stilleven taking LDA at face value implies the existence of T-p conditions veryclose to equilibrium for MLG making the experimental search for suchconditions promising The key message of Fig 101 is thus that equilibriumor near-equilibrium chemical potential ranges for ZLG MLG and evenBLG corresponding to specific Tp conditions in experiment exist
The actual growth process proceeds by Si out-diffusion from underneathalready formed graphene planes Yet the external Si reservoir backgroundpressure still matters in equilibrium As long as the diffusion path to the out-side remains open so does the inward diffusion path and near-equilibriumwith the reservoir gas can be achieved During intermediate stages of theformation of a new graphene plane [230] such diffusion paths must beavailable This finding proves the potential for much better growth controlfor each phase than what could be expected if each phase were just a neces-sary (but not thermodynamically stable) kinetic intermediate Achievingtrue reversibility for actual MLG may be hard since the reverse growthprocess require to disassemble a fully formed graphene plane which is kin-etically difficult [329 124] However the active forward growth process fromMLG to BLG under Si out-diffusion should still be limitable by appropriateTp conditions A macroscopically homogeneous surface very close topure-phase MLG should thus be achievable in principle
Next we bring our results in the context of some specific growth-relatedobservations For epitaxial graphene growth in ultrahigh vacuum (UHV)the background pressures of Si and C are low and ill-defined leading to acoexistence phases with different number of graphene layer [82] Emtsevet al [82] achieved much more homogeneous graphene growth by introdu-cing an argon atmosphere although MLGBLG phase areas still coexistA uniform background partial pressure of Si however will not be strictlyguaranteed The observed thermodynamic ZLG stability and improvedgrowth of MLG by controlling a disilane (Si2H8) reservoir [329] is fullyconsistent with our findings The use of a confined cavity to control the Siflux away from the sample reportedly yields excellent wafer-size films [67]Maintaining a controlled Si partial pressure at constant temperature is mostlikely the important step for large-scale growth of high quality graphene
106
11 Band Structure Unfolding
The bulk projected band structure is a very helpful tool to analyse surfacesystems built from only a few unit cells However when studying theinterface between two materials with different lattice parameters such asSiC and graphene a coincidence structure has to be found These resultingstructures can easily become several hundreds of unit cells large Thecorresponding Brillouin zone of the supercell (SCBZ) then becomes verysmall and the resulting band structure hard to interpret
Figure 111 a Linear hydrogen chain with one hydrogen atom (H) in the unit cell andits band structure b supercell of the linear hydrogen chain with 5 times the one-atomicunit cell The band structure is folded at the edges of the reduced Brillouin zone of thesupercell The band structure corresponding to the actual one-atomic unit cell is plottedwith dashed lines c the hydrogen atoms in the 5 atomic chain are moved away from theirperiodic position The corresponding band structure is perturbed preventing the straightforward reversal of the band folding
In Figure 111 we demonstrated how a SC influences the band structureusing a linear hydrogen chain as an example We assume a periodic one-dimensional chain of hydrogen atoms shown in Fig 111 a) The bandstructure for the one-atomic unit cell is plotted from the zone edge (M) tothe Γ-point to the other M-point (Mrsquo) The bands of the linear hydrogen chainwith one atom in the unit cell show the typical parabolic behaviour Nextthe hydrogen atom is repeated five times in a perfect chain The supercell(SC) contains five hydrogen primitive cells (PCs) and the band structure isplotted in Fig 111 b) The bands of the SC are folded at the zone edges ofthe smaller SCBZ Comparing Fig 111 a and b) it becomes apparent thatthe Brillouin zone of the supercell (SCBZ) is reduced to one-fifth comparedto the Brillouin zone of the primitive cell (PCBZ) For example the bandstructure plot for the five-atomic hydrogen chain SC (Fig 111 b) showssix states at the Γ-point with two occupied states crossing at the Γ-point ofthe PCBZ(circled in red) A comparison of the band structure of the one-atomic and the perfectly periodic five-atomic hydrogen chain (Fig 111 a
107
and b) allows to lift the crossing by ordering the states of the folded SCband structure into the PCBZ as indicated by red circles in Fig 111 b) Inthe extended PCBZ it becomes clear that this crossing is an artifact of thezone folding because the eigenvalue is actually located at the neighbouringPCBZ In such a situation it is desirable to represent the band structure orspectral function of the SC in an extended BZ by exploiting the additionaltranslational symmetry In the third example the atoms in the five-atomicchain are moved away from their periodic position (see Fig 111 c) Theband crossing at the Γ-point of the SCBZ (circeled in red) is lifted preventingthe straight forward reversal of the folding operation Although the bandstructure of the perturbed hydrogen chain shows a close resemblance withthe unperturbed one it is unclear how to unfold the band structure intothe PCBZ of the one-atomic hydrogen chain In the following we present away to unfold the band structure of a supercell into that of a chosen unitcell
Many different approaches have been proposed to map the band structureof a SC onto a different unit cell and have been implemented in a range ofelectronic structure codes [189 37 36 176 253 254 4 187 84 147 3]
For the band structure of a periodically repeated primitive cell spanninga perfect supercell exact methods based on the Blochrsquos theorem [31] existto map the eigenstates from a SC calculation onto a unit cell For examplein the tight-binding community an exact method has been proposed forunfolding the band structure of a perfect SC into bulk dispersion curves[37] with an extension to imperfect SCs [36] or to imaginary bands [3]
Facing the challenge of investigating the electronic band structure of alloysPopescu and Zunger developed an unfolding method within the plane-wave DFT framework [253 254] To unfold the spectral function of func-tionalised carbon nanotubes [189] or bulk structures with impurities [176]the spectral function is represented in Wannier functions Farjam adaptedthe unfolding formalism from Lee et al [189] for the density-fuctional-basedtight-binding scheme to visualise the influence of point defects on the elec-tronic structure of graphene [84] Allen et al [4] and Huang et al [147]presented the unfolding formalism in a mathematical generalised notationand applied it to tight-binding band structures and phonon dispersions
Lee et al [187] unfolded the band structure using a basis of linear combin-ation of atomic orbitals (LCAO) It was developed independently of theunfolding routine presented in this work There is one major conceptualdifference between the unfolding formalism from Lee et al [187] and thiswork The formalism presented by Lee et al [187] (as well as the work by
108
Ku et al [176]) is built upon the concept of a connection between the SC anda conceptual normal cell In the case of a perfectly periodic SC the conceptualnormal cell is identical to the PC The relationship between the conceptualnormal cell and SC coordinates is given by what Ku et al refer to as ldquothemaprdquo This ldquomaprdquo consists of two pieces of information First a (3times 3)integer matrix the so-called N which expresses the SC lattice vectors interms of the PC lattice vectors Second a list of SC orbitals and their cor-responding PC lattice vectors and conceptual normal cell orbitals In aperfectly periodic SC the assignment of SC orbitals to their correspondingPC lattice vectors is trivial However as soon as the periodicity in the SCis broken for example by a defect substitution structural distortion or aninterface the identification of such a conceptual normal cell is not clear andtheir is no unique mapping As this ldquomaprdquo plays a fundamental role in theiralgorithm the unfolded band structure depends on the chosen conceptualnormal cell
Figure 112 a) shows 18 atoms arranged in aperturbed hexagonal lattice the unit cell andthe lattice vectors A1 and A2 b) the structurefrom a) is presented as a (3times 3) supercell Thenine primitive unit cells are indicated by bluelines and the primitive lattice vectors a1 anda2 by black arrows The blue atom in thecentral unit cell can be shifted to its periodicimage by a lattice vector translation
In this work we exploit Blochrsquos the-orem to unveil hidden translationalsymmetries in a SC calculation byspecifying an arbitrary supercellmatrix (N ) The formalism does notrequire the knowledge of any sym-metry in the system Before we in-troduce the formal unfolding form-alism we discuss the basic underly-ing idea
Any wave function can be unfoldedby projecting onto a system withshorter real space translation vec-tors corresponding to a higher trans-lational symmetry For any projec-tion we can determine the unfold-ing spectral weight The unfoldingspectral weight gives the probabil-ity of a state p to exist at K and takesa value between 0 and 1 For thespectral weight we can distinguishthree different cases First the as-sumed translational symmetry cor-responds exactly to the periodicityof the wave function Then the spec-tral weight will be one for one point
109
in the BZ (see example Fig 111 b) Second the wave function does notresembles the assumed translational symmetry Then the resulting projec-tion will be noisy and no point in the BZ accentuated The third case is therange in between Here the wave function is approximately periodic andthe spectral weight will be close to 1 or close to 0
Figure 112 a) shows 18 atoms forming a perturbed hexagonal lattice Thelattice in Fig 112 a) can also be seen as a 3times 3 SC with 2 atoms in the PCshown in Fig 112 b) The SC is spanned by the unit cell vectors A1 andA2 In the example sketched in Fig 112 the supercell matrix (N ) is a 2times 2matrix with the diagonal elements equal 3 In FHI-aims the wave functionof the system is expanded into numeric atom-centered orbitals (NAOs) Aset of NAOs is localised on every atom (see Sec 25) We can use Blochrsquostheorem to define an operator that projects out all the contributions of thewave function corresponding to the translational symmetry given by N Apart from the information available from the full system in Fig 112 a)the only additional requirement is the translation operator (marked in bluein Fig 112 b) defined by N The projector then translates a basis elementlocalised at an atom in the SC by the translation vector (eg a2 indicated byan arrow in Fig 112 b) To determine the projection weight in one k-pointthe basis element has to be translated to every PC determined by N
111 The Brillouin Zone (BZ)
Just for the moment we assume a perfect SC built from NC PCs and determ-ine the connection between the PC and the SC lattice of the same crystal inreal and reciprocal space
We consider a bulk crystal with a primitive cell (PC) described by real spacelattice vectors ai where the integer i denotes the spatial direction Thecorresponding unit cell in reciprocal space is shown in Fig 113 The BZ ofthe conventional unit cell contains a mesh of k-points (shaded in grey andthe k-points are shown in purple) and is spanned by the reciprocal latticevectors bj The real space lattice vectors and the reciprocal lattice vectorsare connected by the relation
ai middot bj = 2πδij (111)
The SC lattice vectors are generated by integer multiples of the primitive
110
lattice vectors aiAi = sum
jNijaj (112)
where Nij are the integer entries of the invertible (3times 3) supercell matrix(N ) The inverse of N is calculated using integer vector and matrix oper-ations only Using the transposed inverse of the supercell matrix (Nminus1)Tthe reciprocal lattice vectors can be obtained by
Bj = sumiNminus1
ji bi (113)
The Bj span the BZ of the SC (indicated in blue in Fig 113)
Figure 113 Sketch of a 2-dimensionalunit cell in reciprocal space The greyarea indicates the first BZ with the re-ciprocal lattice vectors bi of the conven-tional unit cell (in this example 64 k-points (k) (purple points)) and the SCBZof a 4times 4 supercell with lattice vectorsBi and four k-points (K) (white points)
All calculation in the BZ are evaluatedon a grid of k-points defined by Mj TheBZ of the SC contains Mj k-points in thedirection of the reciprocal lattice vectorBj (shown as white points in Fig 113)This gives Nk = prodj Mj k-points in totalThe K-vector pointing to one of thesepoints is
K = sumj=1
mj
MjBj (114)
where the integer mj = 0 Mj minus 1 isthe index of the k-point along the direc-tion j On the other hand the reciprocalrepresentation of the SC is periodicallyrepeated A K-vector in one BZ is con-
nected to the same point in a periodic repetition by a reciprocal latticevector
Gn =3
sumj=1
njBj (115)
Within the PCBZ there are NC different reciprocal lattice vector Gn Anyk-point in the PCBZ can be uniquely written as a sum of a SCBZ vector Kand lattice vector Gn
K + Gn =3
sumj=1
mjBj
Mj+ njBj =
3
sumij
mj + Mjnj
MjNminus1
ji bi = k (116)
111
For any k a vector of integers (pj = mj + Mjnj) can be found to express kin terms of reciprocal lattice vectors b 1
From these basic considerations we can draw some helpful conclusionsThe folded SCBZ can be repeated to span the unfolded PCBZ correspondingto the primitive unit cell Therefore every K in the folded SCBZ has a copyin all periodically repeated PCBZ With Eq 116 each k in the unfoldedPCBZ can be determined by an integer operation in terms of the reciprocallattice vectors B This facilitates a numerically accurate and easy assignmentof the k-path in the unfolded BZ
1111 The Wave Function
Figure 114 The real-spaceSC containing 3 times 3 PC isshaded in blue The latticevector Rj points to a periodiccopy of the SC The PC withlattice vectors a12 is shadedin grey The lattice vector ρlpoints to a periodic copy ofthe PC Each PC contains twoatoms (red circle) with atomicposition rmicro
Before we introduce the actual unfolding routinewe include some notational preliminaries
First we introduce the wave function of the su-percell (shaded in blue in Fig 114) The wavefunction (eigenvector) φKp of the eigenstate withband index (p) at SCBZ K-point (K) with energyεKp of a SC calculation is a Bloch function
φKp(r + R) = 〈r + R|φp(K)〉= eiKmiddotRφKp(r)
(117)
for any SC lattice vector R = sumi=1 niAi Theeigenvector |φp(K)〉 can be expanded in atomicbasis functions
|φp(K)〉 =Nbasis
summicro
Cpmicro(K)|K micro〉 (118)
where micro is the index of an orbital in the SC andNbasis the total number of basis functions |K micro〉
is the basis function in reciprocal space with
|K micro〉 = 1radicNS
NS
sumj
eiKmiddotRj |Rj micro〉 (119)
1For example applying Eq 116 to the BZ shown in Fig 113 the wave vector K becomesK = 1
2 B1 +12 B2 and k = 6
2middot4 b1 +4
2middot4 b2
112
where NS is the number of SCs included in the crystal Rj is the lattice vectorpointing to the jth periodic copy of the SC and |Rj micro〉 are the atomic basisfunctions in real space (Eq 227) in cell j and orbital micro
112 The Unfolding Projector
Unfolding the states |φp(K)〉 in Eq 118 into the PCBZ corresponds to aprojection onto Bloch states of higher periodicity given by the supercellmatrix (N ) We can thus project the component of a SC eigenvector at Kinto the nth reciprocal PC given by the reciprocal lattice vector Gn (Eq 115)by a projector Pn(K) This projection measures how close a state is to aBloch state with higher periodicity To express an unfolding projector werequire a basis of the same periodicity defined by N In general the SCis not an exact copy of NC PCs and the wave function Eq 118 does notnecessarily have the required periodicity We extend the basis |K micro〉 byadding periodic copies of each basis function with
T(ρl)|K micro〉 = |K micro l〉 (1110)
where T(ρl) is a translation operator that translates the basis function |K micro〉into the lth PC by the lattice vector ρl = sumi niai
The unfolding projector Pn(K) has to fulfil three constraints
1 Pn(K) has to be invariant under PC lattice translations ndash Pn(K) is aperiodic operator
2 once the projector acted on a state any additional application mustnot change the result beyond the initial application ndash Pn(K) is anidempotent operator
3 in a perfectly periodic SC Pn(K) projects a state fully in one reciprocalunit cell Gn only Any state with any Bloch symmetry has a uniquedecomposition into NC functions of higher PC translational symmetryThis has to be true for any arbitrary division in subcells that span theSC by lattice vector translations The sum of Pn(K) over all possibleNC SCBZ is one ndash Pn(K) sums to one
113
Definition 1121 The unfolding projector Pn(K) is defined by
Pn(K)|K micro〉 =1
NC
NC
suml
eminusiGnmiddotρl |K micro l〉
a Periodicity
Pn(K)|K micro l〉 = eiGnmiddotρl Pn(K)|K micro〉
where ρl is a PC lattice vector (seeFig 114)
b Idempotency
Pn(K)2 = Pn(K)
c Sum-to-one
sumn
Pn(K) = 1
In the following we show that the unfolding projector defined by Def 1121satisfies the three conditions
a Periodicity Pn(K)|K micro l〉 = eiGnmiddotρl Pn(K)|K micro〉
eiGnmiddotρlprimePn(K)|K micro〉 = 1NC
NC
suml
eiGnmiddot(ρlprimeminusρl)|K micro l〉
By construction |K micro l〉 is a Bloch function so that
eiGnmiddotρlprime |K micro l〉 = |K micro l + lprime〉
where ρlprime and ρl are both lattice vectors so that the sum of both isagain a lattice vector Inserting it into the Eq 112 gives
1NC
NC
suml
eminusiGnmiddotρl |K micro l + lprime〉 = Pn(K)|K micro lprime〉
114
b Idempotency Pn(K)2 = Pn(K)
Pn(K)Pn(K)|K micro〉 = 1NC
NC
suml
eminusiGnmiddotρl Pn(K)|K micro l〉
Inserting Eq 1121 for Pn(K)
=1
N2C
NC
sumllprime
eminusiGnmiddot(ρl+ρlprime )|K micro lprime〉
=1
N2C
NC
sumlprime
NCeminusiGnmiddotρlprime |K micro lprime〉
Wich is the definition of Pn(K) (Def 1121)= Pn(K)|K micro〉
c Sum-to-one sumNCn Pn(K) = 1
NC
sumn
Pn(K)|K micro〉 = 1NC
NC
suml
NC
sumn
eminusiGnmiddotρl |K l micro〉
applying the basic rules of Fourier transformationto the sum over n phase factors
=1
NC
NC
suml
NCδl0|K l micro〉
= |K micro〉
Here Eq 1110 was used for l = 0
|K l = 0 micro〉 = T(ρl=0)|K micro〉 = |K micro〉
The unfolding projector defined by Def 1121 satisfies the conditions a-cTherefore it can be used to uniquely determine the spectral weight in thePCBZ
As a final step the projector Pn(K) is applied to the eigenvector |φp(K〉 tocalculate the unfolded spectral weight wKp
n of the state p in the PCBZ n atK
wKpn = 〈φp(K)|Pn(K)|φp(K)〉 (1111)
115
Inserting Eq 118 for the SC eigenvector |φp(K〉 gives
wKpn =
Nbasis
summicromicroprime〈K microprime|Clowastp
microprime (K)Pn(K)Cpmicro(K)|K micro〉
=1
NC
NC
suml
Nbasis
summicromicroprime
Clowastpmicroprime (K)Cp
micro(K)eminusiGnmiddotρl〈K microprime|K micro l〉(1112)
In an orthonormal basis the object 〈K microprime|K micro l〉 would be a unity matrixδmicroprimemicrol and the work would be done However a numeric atom-centeredorbital-basis is non-orthogonal Using Eq 119 and Eq 1110 the matrixelement 〈K microprime|K micro l〉 becomes
〈K microprime|K micro l〉 = 1NS
NS
sumjjprime
eiK(RjprimeminusRj)〈Rjprimemicroprime|T(ρl)|Rjmicro〉
= [S(K)]microprimemicrol
(1113)
[S(K)]microprimemicrol is the overlap matrix element of the basis function microprime locatedin the SC jprime and the basis function micro translated by a PC unit cell vector ρlIn an exactly PC-periodic system the translation operator T(ρl) translatesthe basis function micro to an exact copy of micro In that case the (Nbasis times Nbasis)overlap matrix used in the DFT calculation already contains all matrixelements [S(K)]microprimemicrol The remaining task is to assign every micro to a primitivecell l However in the case of a perturbed system for example due tostructure relaxation or defects T(ρl) translates an orbital micro to an empty siteand the overlap matrix [S(K)] is an (Nbasis times NCNbasis) object The missingmatrix elements would have to be calculated However in practice weintroduce the first approximation by setting the overlap
[S(K)]micromicroprimelprime asymp [S(K)]micromicroprime (1114)
where in the overlap matrix [S(K)]micromicroprime microprime can be assigned to a PC lprime Thisapproximation allows us to use the overlap matrix as calculated in theFHI-aims-code For two atoms far apart the overlap is small and the changein the overlap matrix element due to changes in the atomic position shouldbe minor However for neighbouring PC the overlap between basis func-tions can still be sizeable Here the approximation only holds for smallchanges in the geometry
The unfolding spectral weight gives the probability of a state to exist at KThe sum of all spectral weights of a state p over the SCBZ is
116
Nc
sumn
wKpn = 1 (1115)
The normalisation is ensured by the sum-to-one constraint of the unfoldingprojector and the Eigenvector norm
Nbasis
summicromicroprime
Clowastpmicro (K) [S(K)]micromicroprime C
pmicroprime(K) = 1 (1116)
The unfolding formalism is exact and allows the mapping of any bandstructure onto a Brillouin zone By defining a supercell matrix one imposesan underlying periodicity on the system Only states corresponding tothe periodicity defined by N will show a band dispersion in the extendedBrillouin zone
113 The Hydrogen Chain Practical Aspects ofthe Band Structure Unfolding
In the following we discuss some practical details on the basis of a fewexamples shown in Fig 115
Example a Figure 115a shows a perfectly periodic 10-atom hydrogenchain In this example the band structure can be unfolded exactlyIn the PCBZ the spectral weight Eq 1112 is either zero or one
Example b Figure 115b shows a distorted geometry with broken trans-lational symmetry The overall shape of the band structure is verysimilar to that of example a but close to the Γ point the effect of thedistortion can be seen In this example the difference between an atomin the PC l at position ρl and an atom at position ρlprime in the lprimeth PC isnot a lattice vector of the PC The spectral weight reflects the strengthof the symmetry breaking by reaching a value between zero and oneHowever for very strong distorted geometry where the displacementis at the order of a lattice translation the approximation to the overlapmatrix Eq 1114 would break down In this case all overlap matrixelements [S(K)]micromicroprimelprime would need to be calculated explicitly
Example c In this example the hydrogen chain is perfectly periodic except
117
Figure 115 The unfolded band structures of four different 10-atom hydrogen chainsare plotted along the PCBZ path M to Γ to M Dashed lines indicate the SCBZ Thecorresponding atomic structure is shown at the bottom a A perfectly periodic hydrogenchain b a hydrogen chain with geometric distortions c a defect in the PC l = 6 in anotherwise perfectly periodic chain and d two chains with different periodicity
118
for one defect in the PC l = 6 Part of the original hydrogen bandcan still be observed but the symmetry breaking is clearly seen inFig 115c by small gap openings To evaluate Eq 1112 the overlapmatrix of atom microprime in PC lprime and atom micro in PC l = 6 is required but theoverlap is not calculated within FHI-aims because in the PC l = 6contains no basis functions This can be circumvent by introducing aghost atom that consists of the same basis functions but without anymass or charge
Example d The system contains two hydrogen chains a 10-atomic anda 2-atomic one separated by 25 Aring It is not clear how N should bechosen We are interested in the effect of the additional chain on the10-atomic hydrogen chain therefore
N =
10 0 0
0 1 0
0 0 1
(1117)
Figure 115d shows the unfolded band structure The band originatingfrom the 10-atomic hydrogen experiences a small gap opening atthe Fermi level The second 5-atomic chain is too far away for anychemical bonding between the chains which is reflected by the almostundisturbed band but introduces a periodic potential Additionalstates appear in particular some very weak states at the Fermi levelWhat cannot conclusively be determined from the plot is which statesoriginate from which chain In this system we face a similar problemas in example c We need to calculate the overlap of the two atoms inPC l and lprime as depicted in Fig 115d The PC lprime contains just one atomas a consequence the overlap between the basis function in l and lprime isnot fully available In this particular system ghost atoms can againbe used to unfold the combined system However in general the full(Nbasis times NCNbasis) overlap matrix defined in Eq 1113 is required tounfold an interface system formed by two subsystems with differentperiodicities
119
Figure 116 The unfolded bandstructure of the two chain systemintroduced in Example 113d Theband structure is projected on the10-atomic chain marked by a greybox
In summary the unfolding procedure intro-duced in this section facilitates the unfold-ing of the band structure of an arbitrarysystem by imposing a translational sym-metry However because of the overlapmatrix [S(K)]microprimemicrol Eq 1113 a first approx-imation was introduced see Example 113bFor small displacements of atoms in the PCthe changes of the respective overlap matrixelement are smalss such that the approx-imation holds As shown in Example 113cthere is a workaround by introducing mass-and chargeless basis functions at the posi-tion of ldquomissing atomsrdquo For interface sys-tems this workaround is only applicable ina few special cases
For an interface system formed by two sub-systems with different periodicities we aregenerally interested in the band structureprojected on the subsystem of interest Ina LCAO basis a straight forward projectionis given by a Mulliken partitioning [222]In Eq 1112 the sum over basis functions micro
should than run over all basis functions micros located in the subsystem and notover all basis functions In essence the projection splits the eigenvector Cp
in two parts
1 Eigenvector entries belonging to the subsytem Cpmicroprimes
2 Eigenvector entries belonging to the rest Cpmicroprimer
If the sum over basis functions in Eq 1116 is split into these two contribu-tions
Nprojbasis
summicrosmicroprimes
Clowastpmicros (K) [S(K)]microsmicroprimes Cp
microprimes(K) +
Nrestbasis
summicrormicroprimer
Clowastpmicror (K) [S(K)]micrormicroprimer Cp
microprimer(K) 6= 1
(1118)the eigenvector is no longer normalised because the contribution from theoff-diagonal blocks Clowastp
micros [S]microsmicror Cpmicror and Clowastp
microprimer[S]microprimermicroprimes Cp
microprimesis not included in the
sum As a result the eigenvector norm in Eq 1116 has to be introduced
120
explicitly by a rescaling factor Npsub such that
1Np
sub
Nprojbasis
summicrosmicroprimes
Clowastpmicros (K) [S(K)]microsmicroprimes Cp
microprimes(K) equiv 1 (1119)
the weight normalisation (Eq 1115) is ensured In analog to Eq 1119 anormalisation factor for the rest Np
rest can be calculated With
1minus (Npsub + Np
rest) = Jp (1120)
we can then determine how strongly the subsystem and the rest are coupledfor a state p For a combined system in which the two subsystems areinfinitely separated the overlap matrix becomes a block matrix and thecoupling coefficient Jp = 0 When interpreting Jp one has to keep in mindthat Jp strongly depends on the chosen basis set used for the projectionand there is no well defined upper limit of Jp in cases of strong couplingHowever it can be used to visualise general trends within one system
In Figure 116 the band structure of the system introduced in Example 113dis shown The two subsystems are well separated the overlap matrix is closeto a block matrix and the coupling coefficients are very small For the bandstructure plotted in Fig 116 the maximal coupling coefficient is Jp = 0038for the state p = 5 meV and the k-point at which the band touches the Fermilevel At this very point the projected band structure shows a very smallgap opening at the Fermi level in an otherwise undisturbed band
121
12 The Electronic Structure of EpitaxialGraphene on 3C-SiC(111)
In Section 82 we discussed the atomic structure of epitaxial grapheneMLG and the interface the ZLG structure on the Si side of SiC The atomiccomposition of theses two structures differ by the number of hexagonal car-bon layers We found that the ZLG consitsts of a mixture of sp3 hybridisedbonds to the substrate and in-plane sp2-like bonds while the MLG is a sp2
bonded graphene layer ARPES measurements Ref [83] and DFT (LDA)calculations of the (6
radic3times6radic
3)-R30 commensurate supercell Ref [166]showed qualitatively different electronic structure originating from thespecific bonding situation of the top carbon layer
Figure 121 Figure taken from Ref [263] Dispersion of the π-bands of graphene grown on6H-SiC(0001) measured with ARPES perpendicular to the ΓK-direction of the graphene Bril-louin zone for (a) an as-grown ZLG (Ref [263] Fig 15 a) (b) an as-grown MLG (Ref [263]Fig 15 e) (c) after hydrogen intercalation (Ref [263] Fig 15 b)
Figure 121 (taken from Ref [263]) shows the dispersion of the π-bandsof ZLG MLG and QFMLG grown on 6H-SiC(0001) measured with ARPESperpendicular to the ΓK-direction of the graphene Brillouin zone For theZLG layer the characteristic graphene π and πlowast states with their lineardispersion at the K-point in the BZ are not present (see Fig 121 a) [214 334263] However the in-plane σ bonds are already observed in ARPES [3581 83] Only with the MLG layer π and πlowast band appear Fig 121 b) [8334 263] In epitaxial graphene the Dirac point is shifted under the Fermilevel showing a n-type doping [83 35 238 263] When the ZLG surface isintercalated by hydrogen (H) QFMLG forms QFMLG is undoped if grownon 3C-SiC (see Fig 121 c) [96]
The origin of the doping and the effect of charge transfer is not yet fullyunderstood Improving the understanding of the SiC-graphene interface
122
and its influence on the electronic structure of MLG and few-layer grapheneis the aim of this chapter
In this chapter we discuss the electronic structure of epitaxial grapheneon the Si side of SiC For calculating the electronic structure we use theall-electron localised basis code FHI-aims For the approximation to the ex-change-correlation functional we use PBE and the hybrid functional HSE06(ω =0207Aringminus1 α =025) The atomic structure was introduced in Ch 8 (allstructures were optimised using PBE+vdW)
121 The Electronic Structure of EpitaxialGraphene
First we discuss the electronic structure of the competing C-rich surfacephases introduced in Sec 82 in the approximated (
radic3timesradic
3) 3C-SiC unitcell covered by a strained 2times 2 graphene cell (see Tab 91) [214 215 334 335243] There are three Si atoms in the top SiC layer of the 3C-SiC
radic3timesradic
3unit cell two of them are covalently bound to the 2times 2 ZLG layer andone Si atom is right underneath the midpoint of a carbon hexagon of theZLG layer This structure closely resembles the configuration at the centerof the experimentally observed (6
radic3times6radic
3)-R30 commensurate supercell(see Fig 84) In the following we will refere to the C-rich phases in the(radic
3timesradic
3) unit cell using the abbreviated formradic
3-ZLGradic
3-MLG andradic3-BLG
In Section 72 we found that HSE06 improves the description of the cohesiveproperties of SiC polytypes in specific the valence band width and band-gap over PBE so we used the hybrid functional HSE06 to calculate theelectronic structure A comparison between the HSE06 and PBE DOS oftheradic
3-MLG structure can be found in Appendix F For the bulk projectedstates the 3C-SiC unit cell was chosen such that the three SiC-bilayers arestacked along the z-axis In the x-y-plane a
radic3timesradic
3 cell was chosen Forthe bulk projected DOS the bulk states were multiplied by two to equal thesix SiC bilayers in the slab calculation
Figure 122 shows the calculated bulk projected band structure (see Sec 5)and DOS of the
radic3-ZLG
radic3-MLG and
radic3-BLG phases
123
Figure 122 The surface projected Kohn-Sham band structure and DOS of the (a) ZLG (b)MLG and (c) BLG calculated in the (
radic3times
radic(3)) approximate unit cell using the HSE06
hybrid functional
124
The bulk projected band structure DOS and the geometric structure of theradic3-ZLG phase are shown in Fig 122 a) The direct and indirect band gap
of bulk 3C-SiC are drawn into the band structure plot (for more detailsabout the 3C-SiC bulk properties see Sec 72) There is a single band atthe Fermi level giving rise to a peak in the DOS In the case of the 3C-SiCpolytype the conduction band minimum (CBm) reaches into the Fermi level(see Fig 122 a-c) This is different for hexagonal polytypes where the CBmis well above the Fermi level [243] The
radic3-ZLG layer contains a mixture
of sp2 and sp3 hybridised orbitals therefore the π bands with their lineardispersion close to the K-point characteristic for graphene are missing
However experimental studies of the ZLG suggest a small band gap aroundthe Fermi level of 300 meV in STS [270 109] We tested the influence ofspin-polarisation in separate calculations for
radic3-ZLG (not shown here) In
a non-spin-polarised calculation every state is occupied by two electronsIn a surface with one dangling bond the structure is by necessity metallicIntroducing spin-polarisation allows the surface to open a gap We com-pared the spin-polarised and non-spin-polarised band structure and DOSbut we could not find a change in the electronic structure or a gap openingwhich agrees well with a DFT study of the
radic3-ZLG structure using LDA
[215] For the SiCgraphene hybrid structures included in this section thenon-spin-polarised calculation gives the same result as the spin-polarisedone
Theradic
3-MLG layer forms on top of theradic
3-ZLG interface In the bandstructure Fig 122 b) the Dirac point is shifted by minus064 eV (minus062 eVusing LDA Pankratov et al [243]) leading to n-type doping similar to theexperimental value of minus04 eV [83 35 238 96] The energy differencebetween ED and the CBm is 063 eV The Fermi velocity is the slope of thegraphene π band at the Fermi level For the
radic3-MLG layer we found
100middot106 ms (PBE 084middot106 ms Experiment 090middot106 ms [96]) A flatsurface band appears at the Fermi level similar to the one observed in theband structure of the
radic3-ZLG phase
For the AB stackedradic
3-BLG (Fig 122 c) we find a n-type doping of theradic
3-BLG layer in agreement with experiment [238 263] Between the bilayergraphene related π bands a Kohn-Sham gap opens by 033 eV (PBE 025 eV012 eV in experiment using the 6H-SiC polytype as substrate [263]) As fortheradic
3-ZLG andradic
3-MLG there is a flat surface band at the Fermi level
A possible explanation for the surface band at the Fermi level could be thatthe absence of the band gap in the simulated band structure is due to theapproximated
radic3-ZLG structure In Figure 123 we compare the DOS of
125
theradic
3-ZLG and the 6radic
3-ZLG phase Because of the increased system sizewe will use the PBE exchange-correlation functional instead of the hybridfunctional HSE06 In addition we show in Fig 123 the projected DOS ofall Si atoms in the simulation cell (beige) and of the single Si dangling bond(pink) in the
radic3-ZLG and 19 dangling bonds in the 6
radic3-ZLG structure as
introduced in Sec 92
Figure 123 Comparison between the DOS of theradic
3-ZLG and 6radic
3-ZLG surface calcu-lated using PBE exchange-correlation functional The full DOS is shown in blue theprojected DOS of all Si atoms in the slab (beige) and of the Si dangling bonds (pink) Onthe right are shown images of the geometric structure optimised using PBE+vdW
We first discuss the DOS of theradic
3-ZLG structure in Fig 123 a) In theradic3-ZLG structure the surface band (see Fig 122 a) leads to a pronounced
peak in the DOS at the Fermi level The peak at the Fermi level cannotfully be contributed to the Si related states (shown in beige in Fig 123 a)However the projected DOS of the Si dangling bonds (shown in pink inFig 123 a) gives a clear peak at the Fermi level Between the peak at theFermi level and the SiC bulk states the DOS shows a gap (for comparisonsee Fig 122 a)
In the case of the 6radic
3-ZLG phase the picture changes Here also the DOSclose to the Fermi level increases The states at the Fermi level are domiatedby Si states (shown in beige in Fig 123 b) but the contribution to the Sistates originating from the Si dangling bonds is minor (shown in pink inFig 123 b) In the case of the 6
radic3-ZLG the projected DOS of the Si dangling
bonds does not show a clear peak like in theradic
3-ZLG structure
In a LDA based DFT study by Kim et al [166] the simulated ARPES spectra
126
of the ZLG layer on 4H-SiC shows several flat bands at the Fermi energyoriginating from the interface [166] However Kim et al [166] did not showa DOS to allow for a detailed comparison
However the LDA as well as PBE are known to severely underestimateband gaps [209] In the case of the ZLG layer the experimentally observedgap is of the order of 300 meV To clarify the electronic structure of theinterface close to the Fermi level a simulated band structure accounting forspin-polarisation on a dense k-grid at least on the level of HSE would bedesirable but at the moment prohibited by the computational costs forsystem sizes of sim2000 atoms
122 The Influence of Doping and Corrugation onthe Electronic Structure
For epitaxial graphene grown on SiC we found in Sec 82 that the MLGlayer is corrugated The corrugation is passed on to the MLG layer bythe substrategraphene interface structure In Section 121 we found byevaluating the band structure of
radic3-ZLG -MLG and -BLG structures that
epitaxial mono- and bi-layer graphene are n-type doped Electronic statesoriginating from the interface could affect device operation through electro-static screening even without directly contributing to electron transport Inaddition doping the graphene by charge transfer from the substrate or apossible symmetry breaking caused by the corrugation could affect electrontransport [219 19]
The interplay of saturated and unsaturated Si atoms at the interface causes acharge redistribution in the 6
radic3-ZLG carbon layer (Fig 128) We calculated
the change in the electron density at the interface between the ZLG surfaceand the MLG layer to evaluate how the charge redistribution in the ZLG-layer effects the MLG-layer
We calculated the electron density n(r) of a 3C-SiC slab with a reducedlayer thickness (4 SiC bilayers instead of 6) The electron density of theMLG layer nG(r) the substrate including the ZLG layer in the substratereference electron density nsub(r) and the full system nfull are calculated inisolation from each other The electron density is represented on an evenlydistributed grid (260times 260times 350) for all three systems The electron density
127
Figure 124 The electron density of the 6radic
3-MLG phase Panel (a) shows the full electrondensity nfull of the MLG surface integrated over the x-y-plane along the z-axis (solid blackline) the electron density of the substrate (filled in grey) nsub and the graphene layer nGThe average atomic layer position is indicated by dashed lines Panel (b) displays a cutoutof the atomic surface structure The boxes indicate the structures calculated in isolationblack the full slab grey the substrate and red the graphene layer Panel (c) shows theelectron density difference ∆n(z) (Eq 121 and Eq 122) The charge transfer is calculatedby integrating ∆n(z) from zprime to infin (integrated area filled in red) Panel (d) shows the chargedistribution in the x-y-plane at z = zprime
difference ∆n(r) is given by
∆n(r) = nfull(r)minus (nG(r) + nsub(r)) (121)
The change in the electron density along the z-axis is calculated by integrat-ing over the x-y-plane
n(z) =int
dx dy n(r) (122)
In Figure 124 a) the electron densities n(z) are shown for the full systemnfull (outline in black) the substrate nsub filled in grey and the MLG layernG filled in red To obtain information about a possible charge transfer fromthe substrate to the MLG layer we calculated the electron density difference∆n(r) applying Eq 121 Integrating ∆n(r) using Eq 122 gives the changein the electron density along the z-axis shown in Fig 124 c) Alreadythe second SiC bilayer shows only small changes in the electron densityAt the interface between the SiC bilayer and the ZLG layer the electrondensity redistribution is intensified We use ∆n(z) to calculate the chargetransfer by integrating ∆n(z) from the inflection point zprime into the vacuumregion above the MLG layer marked in red in Fig 124 c) It shows that theMLG layer is negatively charged by an average charge of 00019 eminusatomFigure 124 d) shows ∆n(r) in the x-y-plane at position z = zprime between thesubstrate and the graphene layer Figure 125 shows ∆n(r) in the x-y-planeat equidistant zi heights of 03 Aring between the MLG-layer and the substrateThe electron density pattern from Fig 124 d) is shown in Fig 125 8 Theresulting pattern is very similar for any chosen height The modulationsof the electron density in the MLG layer are governed by the interplay of
128
Figure 125 The difference in the electron density given by Eq 121 is shown for the 3C-SiCMLG phase Eq 122 gives ∆n averaged over the x-y-plane and plotted it along z Theposition of the Si C H and graphene-layer are indicated by dashed lines We show ∆n(zi)in the x-y-plane at equidistant zi heights between the MLG-layer and the substrate
saturated and unsaturated Si bonds to the ZLG layer
In Section 24 we introduced the Hirshfeld charge analysis as a tool to dividethe total electron density between different atoms of a system accordingto the electron density of the electron density of a free neutral atom In areal system we have an electron density in the field of positively chargednucleii which makes any definition of atomic charges arbitrary Hirshfeldcharges show the general tendency to underestimate the charge transfereg [190] We calculate the average Hirshfeld charge in the ZLG and MLGlayer (shown in Fig 126 a) The average Hirshfeld charge per C atom of theZLG layer amounts to 0024 eminusatom leading to a negatively charged layerHowever the charge of a C atom in the ZLG layer underneath the MLG layeris reduced to 0021 eminusatom Evaluating the Hirshfeld charges for the MLGlayer we found a negatively charge per C atom of 00015 eminusatom whichagrees well with the charge transfer of 00019 eminusatom calculated by chargedensity differences The charge transfer in the ZLG layer is approximately10 times larger than in the MLG layer This finding is not surprising as the
129
ZLG layer is covalently bonded to the substrate while the MLG layer isattached to the ZLG layer by weak vdW bonds
In experiment the MLG band structure shows a n-type doping ARPESmeasurements of the MLG layer show that the graphene Dirac point isshifted by approximately minus04 eV [83 35 238] From Fig 67 we canestimate that a charge transfer of sim0005 eminusatom would be necessaryto shift the Dirac point to minus04 eV An electron doping of 00019 eminusatom(00015 eminusatom) shifts the Dirac point tominus025 eV (minus022 eV) (see Sec 63)
The underestimation of the charge transfer could be an artefact of the ap-proximation to the exchange-correlation functional To test the influenceof the exchange-correlation functional on the charge transfer we used theradic
3-MLG structure introduced in Sec 121 (see Tab 121)
Hirshfeld charges in eminusatom
Layerradic
3-MLG 6radic
3-MLG
PBE HSE06 PBE
MLG 00045 00079 00015
ZLG 0019 0012 0021
Table 121 The Hirshfeld charge per C atom [eminusatom] in theradic
3-MLG calculated usingPBE and HSE06 and for comparison the Hirshfeld charges calculated in the 6
radic3-MLG
structure using PBE
In Table 121 we list the Hirshfeld charges per C atom calculated in theradic3-MLG structure using PBE and the hybrid functional HSE06 In addition
we include in Tab 121 the Hirshfeld charges for the 6radic
3-MLG structureusing PBE First we compare the difference in charge transfer betweenthe two different MLG structures The Hirshfeld charges listed in Tab 121for the
radic3-MLG-layer amounts to 00045 eminusatom three times higher than
the charge transfer for the 6radic
3-MLG-layer using PBE As a next step wecompare the charge transfer calculated using the two different exchange-correlation functionals PBE and HSE06 The charge transfer to the
radic3-
MLG-layer increases by almost a factor two from 00045 eminusatom for PBEto 00079 eminusatom for HSE06 We heere can conclude that the choice ofsimulation cell (6
radic3-MLG versus
radic3-MLG) has a larger influence on the
charge transfer than the functional at least for the functionals we tested
However the change in the charge transfer between HSE06 and PBE is
130
not large enough to explain the discrepancy between our calculations andthe experiment The difference might not solely originate from the ap-proximation to the exchange-correlation functional but originates fromthe experimental setup In experiment the SiC substrate is nitrogen dopedleading to a n-type doping [263] One can well imagine that during thegrowth process nitrogen migrates to the SiC surface enhancing the chargetransfer at the SiC-graphene interface
The 6radic
3-MLG-layer differs from free-standing graphene due to the inter-action with the substrate Here the graphene-substrate interaction on thelevel of PBE+vdW leads to a corrugation of 041 Aring (see Fig 85) and acharge transfer from the substrate to the graphene layer of 00019 eminusatom(00015 eminusatom respectively) In the following we will systematicallyevaluate the effect of the corrugation and the charge transfer on the electronband structure of the MLG layer
For a systematic study of the influence of doping and corrugation on thegraphene layer we calculated the Kohn-Sham band structure of an isolated13times 13 graphene cell including the corrugation as calculated in Sec 82 andthe charge transfere as given by the Hirshfeld analysis The band structureof the 13times 13 graphene supercell is then folded back into into the grapheneBZ applying the formalism introduced in Sec 11 The bands were unfoldedalong the high symmetry lines Γ - K - M - Γ in the graphene BZ First wecalculated the band structure of a perfectly symmetric 13times 13 graphene celland overlaid the unfolded band structure with the bands calculated in theprimitive graphene unit cell (Fig 126 b) As expected we find a perfectagreement for the perfectly periodic graphene cell
As a next step we were interested in the influence of the doping Theatom resolved Hirshfeld charges are shown in Fig 126 a) The Hirshfeldcharges are unevenly distributed and vary between 00038 eminusatom to00091 e+atom leading to an average Hirshfeld charge of 00015 eminusatomWe applied the VCA [336 278 261] every C atom was doped accordingto its Hirshfeld charge calculated in the experimentally observed 3C-SiC-6radic
3-MLG structure The unfolded Kohn-Sham band structure Fig 126 c)shows the graphene Dirac point shifted by -022 eV with respect to theFermi level However signs of symmetry breaking in the graphene bandstructure caused by the modulation of the Hirshfeld charge in the graphenelayer are not visible
131
Figure 126 The band structure of a 13times 13 graphene cell Subfigure a) shows the Hirsh-feld charge distribution of a MLG layer b) shows the unfolded band structure of a perfectlyperiodic 13times 13 graphene cell and overlayed in red is the band structure of free-standinggraphene in its 1times 1 unit cell c) unfolded band structure of a flat graphene layer dopedaccording to the Hirshfeld charges The charges lead to a shift of the Dirac point by 022 eVc) the unfolded band structure with the same corrugation as the MLG layer d) Bandstructure of a graphene layer which is both corrugated and doped The region around theDirac point is enlarged showing the shift of the Dirac point caused by the doping
132
Fig 126 d) shows the unfolded band structure of a corrugated graphenelayer We used the exact geometry of the MLG layer from the 3C-SiC-3C-SiC-6
radic3-MLG structure from Sec 82 without the substrate Likewise
the corrugation of graphene layer is to weak to break the symmetry ofgraphene enough to influence the graphene π bands
In this analysis we did not include the effects of the substrate directly Thereis still the possibility that electronic states originating from the interfacecould affect device operation through electrostatic screening However thecorrugation of the MLG-layer and the inhomogeneous charge redistributionalone do not show a significant change in the electronic band structure andtherefore should not affect device operation
123 The Silicon Dangling Bonds and their Effecton the Electronic Structure
3C-SiC has a band gap of 144 eV (PBE) and experimental gap of 22 eV[204] (see Tab 73) For ZLG the band gap is reduced to 300 meV (STSmeasurements) [270 109] In Section 121 we discussed the the DOS closeto the Fermi level However we could not reproduce the opening of a smallband gap
Figure 127 The bulk projected Kohn-Shamband structure and DOS of the
radic3-MLG
structure with the Si dangling bond satur-ated by an H atom The full band structureand DOS of the slab used in the calculationis shown in red and the 3C-SiC bulk states ingrey
In Chapter 9 we found 19 unsatur-ated Si atoms ndash the Si dangling bondsndash at the interface between the SiCsubstrate and the ZLG layer Theirinfluence on the electronic structureand how they effect the grapheneelectronic properties is not fully un-derstood It is speculated that thedifference in the electronic struc-ture between the ZLG- and MLGlayer is due to partial hybridisationof the C atoms in the ZLG layerand the substrate Si atoms [83 263]It is conceivable that for the ZLGthe observed low DOS in the SiCbulk band gap are caused by the un-saturated Si atoms in the top SiC
bilayer
133
Figure 128 Four times the ZLG layer coloured depending on the Hirshfeld charge of the Catom from blue for electron gain to red for electron loss The position of the unsaturated Siatoms is shown in pink The STM image taken from Qi et al [256] is shown for comparison
To evaluate the influence of the Si dangling bond we will use theradic
3-ZLGmodel as introduced in Sec 121 The advantage of this approximatedstructure over the true (6
radic3times 6
radic3)-R30 is that the Si dangling bond can
be clearly distinguished from the two covalently bounded Si atoms
In Figure 122 the Fermi level was pinned by a flat surface state in allthree phases the
radic3-ZLG
radic3-MLG and
radic3-BLG The main suspect is the
unsaturated Si dangling bond in the top SiC-bilayer at the interface Toevaluate the origin of the flat surface band we saturated the Si danglingbonds by one H atom After optimising the atomic structure we calculatedthe band structure and DOS In the band structure (Fig 127) the Diracpoint coincides with the Fermi level such that the surface is undopedThe 3C-SiC bulk CBm is shifted away from the Fermi level The energydifference between ED and the CBm is slightly reduced to 051 eV comparedto 063 eV in the
radic3-MLG (see Sec 121) The surface band at the Fermi
level disappeared From Fig 127 we can conclude that the doping of theMLG layer is induced by the Si dangling bonds at the interface and that itindeed give rise to the flat surface state at the Fermi level
Using theradic
3-ZLG model we found that the Fermi level is determinedby the interface structure most likely induced by charge transfer from thedangling Si bonds at the interface We marked the Si dangling bondsunderneath the ZLG layer in pink in Fig 128 On the basis of geometricposition and bond length differences we identified the Si dangling bonds
134
(for a detailed discussion see Appendix E) They form two different typesof cluster two triangular shaped clusters per unit cell and one hexagonalThe hexagonal cluster is centered around the Si atom positioned exactly inthe middle of a C hexagon of the ZLG layer (see Fig 128) The danglingbond cluster give rise to the characteristic corrugation of the ZLG layer asdiscussed in Sec 91
We can assign every C atom in the ZLG layer a charge by using the Hirshfeldpartitioning scheme [142] The calculated charge partitioning is shown inFig 128 On average a C atom in the ZLG layer is negatively chargedby 0024 eminusatom which is reduced to 0021 if a additional MLG-layer isadsorbed (see Tab 121) The charge transfer in the ZLG layer is maximal atC atoms with a Si atom right underneath forming a strong covalent bond(shown as blue C atoms in Fig 128) On the other hand for C hexagons ontop of a Si dangling bond the electrons disperse (indicated by red C atomsin Fig 128)
In 2010 Qi et al [256] visualised the ZLG surface using STM with a specialiron treated tip They found two differently shaped patterns in their STMimages one hexagonal and two triangular pattern (the STM image takenby Qi et al [256] is shown in Fig 128) They suggested the presence ofC-rich hexagon-pentagon-heptagon (H567) defects at two different defectpositions ldquohollowrdquo and ldquotoprdquo (see Fig 94) In Section 92 we demonstratedthat the formation energy for these defects make their presence at theinterface unlikely
Due to the two different types of cluster formed by the Si dangling bondsthe charge in the ZLG layer is redistributed The resulting pattern of theHirshfeld charge map of the ZLG layer looks similar to the pattern seen inSTM It is conceivable that Qi et al [256] actually visualised the Si danglingbonds in their measurements
124 Summary
In conclusion we used density-functional approximation (DFA) HSE06+vdWand PBE+vdW calculations to evaluate the origin of doping in epitaxialfew-layer graphene on the Si face of SiC The
radic3-approximated interface
structure features three Si atoms in the top layer of which two are covalentlybonded and one is unsaturated Because of the clear interface structure thedangling bond can be identified We evaluated the influence of the danglingbond on the ZLG-layer and few-layer graphene We found that indeed the
135
Si dangling bond dopes the ZLG MLG and BLG by a interface state at theFermi level The BLG structure has a Kohn-Sham band gap of 033 eV usingthe HSE06 functional Saturating the Si dangling bond by a hydrogen atomshifts the graphene Dirac point to the Fermi energy resulting in an undopedepitaxial graphene layer Evaluating the electron density of MLG we founda doping of 00019 eminusatom (experiment sim0005 eminusatom [83 35 238]) Thedoping shifts the Dirac point to minus022 eV (experiment minus04 eV) Howeverthe doping as well as the corrugation of the MLG layer hardly disturbthe graphene π bands In the case of the 6
radic3-ZLG structure we found
a very low DOS near the Fermi level which might be to low to be detec-ted experimentally Evaluating the electron density of the 6
radic3-ZLG we
found a pattern in the electron density of ZLG similar to the STM imagesfrom Qi et al [256] In the STM image as well as in the electron densityplot two differently shaped patterns are found one hexagonal and twotriangular pattern per unit cell Similar to the patterns visualised in STMmeasurements by Qi et al [256]
136
13 The Decoupling of Epitaxial Grapheneon SiC by Hydrogen Intercalation
In the previous Chapters we discussed how the mixture of covalent andnon-covalent bonding between the substrate (SiC) and the ZLG-layer influ-ences the atomic structure (see Ch 8) and the electronic structure (Ch 12)of the MLG-layer For future graphene applications it is desirable to find asubstrate for which the interactions are minimised and the extraordinaryproperties of a single graphene layer are preserved
In experiment Riedl et al [262] demonstrated the possibility to decouplegraphene from SiC by intercalation of H atoms forming QFMLG The inter-calation process removes the covalent Si-C bonds by saturating the Si atomsin the top SiC-bilayer The experimental band structure (see Fig 121 c)and core-levels of graphene indicate that indeed the intercalation processreduces the interaction with the substrate substantially [97]
For weakly interacting graphene the sensitivity of ARPES becomes insuf-ficient to assess the interaction with the substrate [153] An alternativecriterion to quantify the interaction strength of graphene with a substrate isits adsorption height However for H-intercalated graphene the adsorptionheight is not known experimentally Moreover it is not clear whether forsuch a weakly interacting system this height can be calculated reliably as itis entirely determined by vdW interactions which are difficult to treat (seeSec 24)
In this chapter we present the results from a theoretical and experimentalcollaborative study of the electronic and structural properties of QFMLG incomparison to MLG on 6H-SiC(0001) [294] The author of this thesis contrib-uted the calculations and theoretical analysis We present DFA calculationsof QFMLG in the large commensurate (6
radic3times 6
radic3)-R30 supercell consist-
ing of 6 SiC-bilayers using PBE+vdW We fully relaxed the top three SiCbilayers and all planes above (residual energy gradients lt 8 middot 10minus3 eVAring)Our experimental collaborators validate this calculations with an accurateexperimental height measurement by normal incidence x-ray standing wave(NIXSW) [294]
137
131 The Vertical Graphene-Substrate Distance
To test whether the structure of this predominantly vdW interacting inter-face can be predicted using DFA and to gain a detailed understanding ofhow H decouples the graphene layer from the substrate we performedDFA calculations for the QFMLG on the level of PBE+vdW The bulk lat-tice parameters are listed in Tab 72 In experiment and therefore in ourcalculations we use 6H-SiC as a substrate
Figure 131 (a) Vertical distances measured by NIXSW on QFMLG The position of theBragg planes around the surface are indicated by blue lines [294] PBE+vdW calculatedgeometry for (b) QFMLG on 6H-SiC(0001) and histograms of the number of atoms Naversus the atomic coordinates (z) relative to the topmost Si layer (Gaussian broadening005 Aring) Na is normalised by NSiC the number of SiC unit cells Dnn+1 is the distancebetween the layer n and n + 1 dn gives the Si-C distance within the SiC bilayer n and δnthe corrugation of the layer n All values are given in Aring (Figure published in Ref [294])
The NIXSW method combining dynamical x-ray diffraction and photoelec-tron spectroscopy is a powerful tool for determining the vertical adsorptiondistances at surfaces with sub-Aring accuracy and high chemical sensitivityApplying these techniques our collaborators determined the heights ofbulk C and bulk Si and graphene C with respect to the bulk-extrapolatedSiC(0006) atomic plane The CSiC
surf atoms are located at 061plusmn 004 Aring belowthe bulk-extrapolated Si plane and the SiSiC
surf 005plusmn 004 Aring above Thus weobtain an experimental Si-C distance of 066plusmn 006 Aring in agreement withthe SiC crystalline structure (Sec 71 Lee et al [186] Wang et al [339]) Inthe same way we find the adsorption height zads of the graphene layer withrespect to the topmost Si layer to be 422plusmn 006 Aring as shown in Fig 131 a
138
Figure 131 shows the measured (panel a) and the calculated (panel b)structure of the QFMLG on 6H-SiC predicted at the PBE+vdW level Inaddition we include a histogram of the atomic z coordinates relative to thetop Si layer normalised by the number of SiC unit cells For illustrationpurposes we broadened the histogram lines using a Gaussian with a widthof 005 Aring For the QFMLG we found a bulk-like distance of 189 Aring betweenthe SiC bilayers The Si-C distance within the top SiC bilayer (062 Aring) andthe remaining Si-C bilayer distances are practically bulk-like (063 Aring) ingood agreement with the experimental result (066plusmn 006 Aring) The distancebetween the top Si-layer and the graphene layer is zads = 416 Aring for 6H-SiCagain in good agreement with the measured 422plusmn 006 Aring Overall we findexcellent agreement between calculation and experiment [294]
Next we compare the QFMLG (Fig 131 b) with the MLG (Fig 87) structureThe most significant difference between the geometry of these two structuresis the significant reduction of the layer corrugation in the QFMLG For the6H-SiC-MLG we found a strong corrugation of the top Si-C bilayer (top Silayer 078 Aring and C layer 030 Aring) the ZLG-layer (086 Aring) and the MLG-layer(045 Aring) In comparison the corrugation of the QFMLG-layer is very small(002 Aring) and for the H layer and all layers underneath the corrugation is lt10minus2 Aring This is in agreement with STM results [109] showing no corrugationwithin the experimental accuracy
In Section 82 we demonstrated that the chosen exchange-correlation func-tional can have a significant impact on the geometric structure eg inplain PBE the interplanar layer distance between the graphene layer andthe ZLG-layer is unphysically expanded We also tested the influence ofthe exchange-correlation functional on the geometry and in particular theadsorption height for the QFMLG structure For our test calculations weused the approximated (
radic3timesradic
3)-R30 cell introduced in Sec 92 (see Ap-pendix F) For the adsorption height zads we found a small difference of001 Aring between PBE+vdW and HSE06+MBD Therefore we do not expectsignificant changes of the adsorption height in the 6
radic3-QFMLG for higher
level functionals such as HSE06+MBD
In addition we found that the influence of the SiC polytype on the atomicstructure of the SiC surface reconstructions is small [100 277 243] Wecompare the structural key parameters of the QFMLG (see Appendix H) andMLG (see Sec 82) on 3C-SiC and 6H-SiC The interface geometry hardlychanges with the polytype and the same qualitative difference between theQFMLG and MLG phases can be observed for both SiC polytypes
To evaluate how strong the QFMLG-layer is bound to the substrate we
139
calculated the graphene exfoliation energy (EXF) In Sec 62 we discussedthe challenge to reliably calculate the graphite interlayer binding energy Weintroduced the exfoliation energy (EXF) given by Eq 62 as the energy of anisolated graphene layer adsorbed on a graphite bulk structure On the levelof PBE+vdW we found a exfoliation energy for graphite of 81 meVatomWe rewrite Eq 62 to calculate the EXF of a graphene layer adsorbed on anysubstrate to
EXF =Efull minus (Esub + Egraphene)
Natom (131)
where Efull Egraphene and Esub are DFT total energies for the full system thegraphene layer and the substrate including the hydrogen layer for QFMLGand the ZLG-layer for MLG Natom is the total number of C atoms in thegraphene layer
Evaluating Eq 131 we found an exfoliation energy for the QFMLG-layer of59 meVatom and for the MLG-layer of 892 meVatom For the QFMLG-layer the exfoliation energy is significantly smaller than the correspondingvalues for MLG and graphite The vdW correction may not capture thecorrect absolut exfoliation energy however the error should be of the sameorder in the different structures compared here We consider only therelative change in the exfoliation energy therefore it is conceivable that therealtive energy differences are qualitatively correct
132 The Electron Density Maps
To evaluate how the presence of a flat intercalated layer translates intothe electronic structure of the graphene layer we calculate the change ofelectron density at the interface between the H-terminated SiC surfaceand the QFMLG-layer The calculations were performed with a 3C-SiCsubstrate as it allows us to use a smaller substrate thickness (4 SiC bilayersinstead of 6 for 6H-SiC) and renders the calculation more affordable Wefound the differences in geometry of the QFMLG to be small between the3C-SiC and 6H-SiC (see Sec 131 and Appendix H) We assume that thequalitative changes in the electron density would also be small between thetwo polytypes
Similar to the electron density investigation in Sec 122 for the MLG-layerwe calculated the electron density n(r) of a 4-bilayer 3C-SiC-QFMLG slabon an evenly distributed grid (260times 260times 350) for the full system nfull(r)the graphene layer alone nG(r) and the substrate alone nsub(r) including
140
Figure 132 The difference in the electron density given by Eq 121 is shown for the 3C-SiCQFMLG phase We integrated ∆n in the x-y-plane and plotted it along z (Eq 122) Theposition of the Si C H and graphene-layer are indicated by dashed lines We show ∆n(zi)in the x-y-plane at equidistant zi heights between the graphene layer and the substrate
the H layer
The electron density difference ∆n(r) was calculated using Eq 121 Weobtained the change in the electron density along the z-axis by integrating∆n(r) over the x-y-plane (Eq 122) In Figure 132 the change in the elec-tron density along the z-axis is shown for the QFMLG Figure 132 shows∆n(r) maps in the x-y-plane at equidistant heights of 03 Aring at the interfaceregion
In the QFMLG all Si atoms are saturated by H [33] resulting in negligiblevariations of the electron density within the x-y-plane independently of thechosen height as seen in Fig 132 On the other hand we found for the MLGphase that the in-plane electron density is influenced by the interplay ofsaturated and unsaturated Si bonds in the ZLG-layer (see Sec 122 Fig 125)The modulations in the charge distribution of the QFMLG-layer are very
141
small at any chosen position zi in the interface region This is an additionalindication for the improved decoupling of the graphene layer from thesubstrate and thus prevents its buckling
133 Summary
In conclusion we showed that our DFA calculations at the level of PBE+vdWprovide a valid description of the SiC-graphene interface In particular theyreproduce quantitatively the NIXSW-measured graphene-Si distances forthe QFMLG From a comparison of the corrugation between the QFMLG(002 Aring) and the MLG (045 Aring Fig 87) we can conclude that QFMLG isbetter suited for device applications than MLG This is also reflected in thelow exfoliation energy EXF calculated for QFMLG
The QFMLG is a very flat graphene layer with a very homogeneous electrondensity at the interface This significant difference between the MLG- andQFMLG-layer translates into a dramatic improvement of electronic devicesafter H intercalation [137 198]
142
14 Epitaxial Grapheneon the C Face of SiC
So far the focus of this work has been on epitaxial graphene on the Siterminated surface of SiC (Ch 8 - 13) On the Si side nearly perfect MLGfilms can be grown over large areas [82 66] This is very different from theC side where controlling the layer thickness of the graphene films remainsa challenge [213] In this chapter we will investigate the relative phasestability of the competing surface phases in the thermodynamic range ofgraphitisation published in Ref [228] Parts of the results presented in thischapter were obtained in close collaboration with Florian Lazarevic andhave been discussed in his master thesis Ref [184]
141 The 3C-SiC (111) Surface Reconstructions
On the C side during graphene growth a series of different surface struc-tures have been observed [332 304 145 192 24 292 208 140 307 139]The two most relevant surfcae phases for this work are the (2times2)C and the(3times3) reconstruction Seubert et al [292] resolved the atomic structure ofthe (2times2)C by quantitative LEED but the (3times3) reconstruction remains anopen puzzle
1411 The (2 x 2) Si Adatom Phase
The (2times2) cell contains four C atoms in the top Si-C bilayer so that the bulktruncated surface has four unsaturated bonds It is the only reconstructionon the C side for which a detailed atomic structure was identified by quant-itative LEED [292 291] The reconstructed surface consists of a Si adatomon top of a bulk-like C layer (shown in Fig 141) The adatom is at positionC in Fig 71 also known as H3-site At this position the adatom formsbonds with the three C atoms of the top Si-C-bilayer reducing the numberof dangling bond orbitals at the surface from 4 to 2 The two remainingpz-like dangling bond orbitals are located on the Si adatom and one on theunsaturated surface C atom
We performed DFA calculations using different exchange-correlation func-tionals The bulk lattice parameter were taken from Tab 72 As before weuse six SiC bilayers and the bottom silicon atoms are H terminated The
143
Figure 141 Geometry of the 3C-SiC(111) (2times2)C Si adatom structure The bond lengthof the adatom with the top C layer (LSi-C) the distance from the surface (DCSi) and thecorrugation of the top Si layer (δSi) are shown
top three SiC bilayers and all adatoms or planes above are fully relaxed(residual energy gradients 8 middot 10minus3 eVAring or below) In Table 141 we listthe structural key parameter of the 3C-SiC(111) (2times2)C Si adatom structurefor different exchange-correlation functionals
3C-SiC(111) LSi-C DCSi δC δSi
LDA 190 086 010 032
PBE 191 088 009 033Data from Ref [184]
PBE+vdW 191 087 009 033
HSE 187 086 010 033
HSE+vdW 186 086 009 034This work
HSE+MBD 186 085 010 034
6H-SiC(0001)
Exp data from Ref [291 292] 194 106 022 02
Table 141 Atomic position of the Si adatom relative to the surface for different exchange-correlation functionals The bulk lattice parameter were taken from Tab 72 LSi-C is thebond length between the adatom and the top layer C atoms DCSi is the distance betweenthe adatom and top layer C atoms along the z-axis and δSi is the corrugation of the topSi-layer (see Fig 141) (All values in Aring)
144
We include in Tab 141 experimental data from quantitative LEED meas-urements [292 291] The structure parameters on the level of LDA PBE andPBE+vdW were obtained by Florian Lazarevic and have been discussedin his master thesis Ref [184] The overall agreement of the structuralparameter between the different exchange-correlation functionals listed inthis work is very good However the theoretical prediction only agreesqualitatively with experiment [292 291] One has to keep in mind that theexperimental reference data from Ref [292 291] corresponds to the 6H-SiCpolytype More importantly Seubert et al [292] found that the (2times2)C re-construction preferes a S16H stacking order (CACBAB) (see Fig 81) LEEDdata indicates that the (2times2)C reconstruction on the 6H-SiC(0001) surfacestabilises hexagonal stacking at the surface (S16H) For the
radic3timesradic
3)-R30
adatom reconstruction on the Si terminated surface we found a preferredlinear stacking (S36H) (see Sec 811 and [304 308 309])
It remains to examine whether the differences in the geometry betweenexperiment and theory originate from the different choice of polytype Inthe following we will use the (2times2)C reconstruction as a reference phase
1412 The (3 x 3) Surface Reconstruction An Open Puzzle
Over the last two decades several structural models have been suggestedfor the (3times3) phase on the C face [192 145 139 71] however so far itremained an open puzzle We here summarise the experimentally observedcharacteristics of the (3times3) reconstruction
Si-richAES indicates that the stoichiometry of the surface recon-struction is Si-rich [145 24]
Triagonal adatom structure seen in STMThe corresponding filled-state STM image is consistentwith three adatoms residing at the same height [71 139]
SemiconductingSTS shows a semiconducting surface with a band gap of15 eV [140]
We performed DFT calculations using PBE+vdW and HSE06+vdW In ourcalculations we use six SiC bilayers and the bottom silicon atoms are H
145
Figure 142 All structural models based onthe model proposed by Hoster et al [145] forthe (3times3) reconstructions of the 3C-SiC(111)are shown in a side and top view In the topview the unit cell is marked in red We in-clude three different chemical compositions(a) from Hiebel et al [139] (b) from Deretzisand La Magna [71] (c) as a plain Si adatom
terminated The top three SiC bilay-ers and all adatoms or planes aboveare fully relaxed (residual energygradients 8 middot 10minus3 eVAring or be-low)
In the following we discuss thestructural model of the (3times3) recon-struction included in our analysisOn the basis of these characteristicsvarious alternative structural mod-els for the (3times3) reconstruction onthe C terminated surface have beenproposed in the literature [145 13971 192] We here briefly discuss thealternative surface reconstructionsof the 3C-SiC(111) surface
We start with a geometric configur-ation for the (3times3) reconstructionsuggested by Hoster Kulakov andBullemer [145]1 On the basis ofSTM measurements they construc-ted a model consisting of an ad-cluster containing 10 atoms On topof the truncated SiC surface forms3 dimers arrange in a hexagon withan additional atom in the middle ofthe hexagon The remaining 3 ad-atoms are adsorbed on top of the
hexagon (see Fig 142 a) The unit cell contains five dangling bonds Twodangling bonds originate from the C atoms of the substrate and three fromthe top adatoms Hoster et al [145] did not specify the chemical compositionof their model Several modifications of the chemical composition havebeen proposed (shown in Fig 142) [71 139]
Figure 142 a shows a variation of the model suggested by Ref [139] Thisstructure consists of a fractional bilayer with seven Si adatoms bonded tothe substrate and three C adatoms
1The structure was first calculated and discussed by Lazarevic [184] We took the geo-metries from Ref [184] We adapted the structures to the lattice parameters given inTab 72 and postrelaxed the structure with the numerical settings given in Appendix B
146
The next structure labeled b in Fig 142 is a carbon rich model suggestedby Deretzis and La Magna [71] It changes of the chemical compositionto six C adatoms forming a dimer ring and four Si adatoms We added amodification with all adatoms chosen to be Si Fig 142 c) (from Lazarevic[184])
Figure 143 A Si rich structural model for the(3times3) reconstructions shown in a side and topview from Ref [139] In the top view the unitcell is marked in red
A new model was suggested byHiebel et al [139] Their model con-sists of a Si-C-bilayer with a stack-ing fault of one half of the cell andtwo adatoms a Si adatom and onthe faulted side a C adatom shownin Fig 143 (This structure was notincluded in Ref [184])
For the (3times3) reconstruction on theSi- and C-terminated surface Li andTsong [192] proposed a Si or C-rich
tetrahedrally shaped cluster as adatoms We tested three different chemicalcombinations shown in Fig 144 (from Lazarevic [184]2)
Figure 144 Three different chemical compositions of the tetrahedral adcluster structuresuggested by Li and Tsong [192] In a) the tetrahedron is formed by a Si atom surroundedby three C atoms b) three C atoms surrounding a Si atom ( a) and b) shown in a side view)c) 4 Si atoms form a tetrahedron ( side and top view)
The first tetrahedron is formed by a Si atom surrounded by three C atoms(see Fig 144 a) The second one consists of three C atoms surrounding a Siatom (see Fig 144 b) and the last tetrahedron we tested is formed by fourSi atoms (see Fig 144 c) For further analysis we include the most stablecluster formed by four Si atoms shown in Fig 144 c)
2The structure was first calculated and discussed by Lazarevic [184] We took the geo-metries from Ref [184] We adapted the structures to the lattice parameters given inTab 72 and postrelaxed the structure with the numerical settings given in Appendix B
147
Next we include models for the (3times3) reconstructions which were proposedoriginally for the Si side of SiC
Figure 145 A Si rich structural model for the(3times3) reconstructions adapted from the Siface [177]
First the Si-rich structure for the6H-SiC(0001)-(3times3) reconstructionshown in Fig 145 by Kulakov et al[177]3 In this model dangling bondsaturation is optimal with only oneout of nine dangling bonds per(3times3) cell remaining
We base our last model on the Sitwist model [310 277]3 known fromthe 3C-SiC(111)-(3times3) reconstruc-
tion (see Sec 812) It is a silicon rich surface reconstruction Figure 146shows its geometry in a side view and from atop The top bulk C layeris covered by a Si ad-layer forming heterogeneous Si-C bonds Three Siadatoms form a triangle twisted by 77 with respect to the top SiC layer Incomparison the twist angle on the Si side amounts to 93 (see Sec 812))The topmost Si adatom is positioned on top of the triangle
Figure 146 The Si twist model adapted from the 3C-SiC(111)-(3 times 3) reconstructionSec 812 On the left The Si twist model from a side view On the right The Si twist modelfrom a top view The unit cell is shown in blue The Si adatoms are coloured depending ontheir distance to the top Si-C-bilayer The nine ad-layer Si atoms in dark blue the three Siadatoms on top of the ad-layer in blue and the top Si adatom in purple (Figure publishedin Ref [228])
In Chapter 4 we showed that for finding the most likely (3times3) reconstruc-tion a good indicator is a comparison of the respective surface free energiesas introduced in Chapter 4 As before we neglect vibrational and configura-tional entropy contribution to the free energy although in the coexistence
3The structure was first calculated and discussed by Lazarevic [184] We took the geo-metries from Ref [184] We adapted the structures to the lattice parameters given inTab 72 and postrelaxed the structure with the numerical settings given in Appendix B
148
region they might lead to small shifts
In Figure 147 we show the the relative surface energy γ as a function of∆microC = microC minus Ebulk
C (Eq 49) using PBE+vdW for the (2times2)C surface modelby Seubert et al [292] (Sec 1411) and the different models for the SiC-(3times3) reconstruction introduced above All surface energies are in eV perarea of a (1times1) SiC unit cell In Section 42 we found that the chemicalpotential limits of the C and Si reservoirs are fixed by the requirement thatthe underlying SiC bulk is stable against decomposition (see also Sec 42or Ref [227]) Because of the close competition between the diamond andgraphite structure for C (see Appendix A or [23 347 348]) we include bothlimiting phases in our analysis
Figure 147 Comparison of the surface energies relative to the bulk terminated (1times1) phaseas a function of the C chemical potential within the allowed ranges (given by diamondSi graphite C and for completeness diamond C respectively) The shaded areas indicatechemical potential values outside the strict thermodynamic stability limits Included inthe surface energy diagram are structure models as proposed for the C face [(b) [292]Sec 1411 (d) [192] Fig 144 c) (e) Fig 142 c) and (h) [145] with the chemical compositiongiven by [139] Fig 142 a) (f) [139] Fig 143 (g) [71] Fig 142 b] and models adapted fromthe Si face [(a) [310] Fig 146 and (c) [177] Fig 145] In the right panel the different modelsfor the (3times3) reconstructions of the 3C-SiC(111) are ordered by their surface energies inthe graphite C limit of the chemical potential with increasing stability from top to bottom(Figure published in Ref [228])
The structure with the lowest energy for a given ∆microC corresponds to themost stable phase The most stable phase in surface diagram (Fig 147) is the(3times3) Si twist The four most stable models in the surface diagram (Fig 147)
149
are structure (a) the (3times3) Si twist with 13 Si adatoms (see Fig 146) (c)(3times3) Kulakov from Ref [177] with 11 Si adatoms (see Fig 145) (e) (3times3) SiHoster with 10 Si adatoms (see Fig 142 c) and (d) (3times3) Li-Tsong fromRef [192] with 4 Si adatoms (Fig 144 c) These four structures have incommon that they only include Si atoms in their adatom structures
The three remaining less favourable structures Fig 147 (fg and h) include amixture of Si and C atoms in their reconstructions A comparison of the dif-ferent models shown in Fig 147 indicate that Si rich (3times3) reconstructionsare energetically preferred However we did not include enough structuresin our comparison to conclusively show this trend
1413 The Si Twist Model
In the surface diagram (Fig 147 of Sec 1412) the Si twist model adaptedfrom the 3C-SiC(111)-(3times 3) phase (Ref [310] and Sec 812) has the lowestenergy of all previously proposed (3times3) models (Sec 1412) Its formationenergy crosses that of the (2times2)C phase just at the C-rich limit (graphite)of the chemical potential with a surface energy difference of 047 meV infavour of the Si twist reconstruction To coexist with the (2times2)C phase andto be present at the onset of graphite formation the (3times3) phase has to crossthe graphite line very close to the crossing point between the graphite lineand the (2times2)C phase The Si twist model shown in Fig 148 satisfies thiscondition
In Figure 148 I we show the surface energetics of the coexisting phases(2times2)C and the Si twist model using the higher level HSE06+vdW func-tional with fully relaxed structures including spin-polarisation and unitcells (Tab 72) As can be seen the phase coexistence does not depend onthe chosen functional However to distinguish between a phase coexistenceor a close competition between the two phases the inclusion of entropyterms would be the next step [86]
In Section 1412 we summarised the experimentally observed characteristicsof the (3times3) reconstruction Next we will show that Si twist model forthe (3times3) reconstruction based on a Si-rich termination well-captures thevarious surface characteristics
Si-richIn agreement with the AES experiments [145 24] our model is Si-rich
150
Figure 148 I The a) and b) phases from Fig 147 calculated using the HSE06+vdWexchange-correlation functional with fully relaxed structures and unit cells (Tab 72) IISimulated constant current STM images for occupied- and empty-state of the Si twistmodel (unit cell shown in red) The three points of interest (A B C) are marked by arrowsand labeled according to Hiebel et al [139] III The DOS for the spin-up and spin-downstates clearly shows a band gap rendering the surface semiconducting (Figure publishedin Ref [228])
151
Triagonal adatom structure seen in STMThe atomic structure includes triangular Si adatom (see Fig 146)consistent with the filled-state STM image [71 139] Our simulatedSTM images Fig 148 (II) reproduce the measured height modulation[139] but the experimentally observed difference in intensity betweenoccupied and empty-state images is not captured by our simulatedimages
SemiconductingWe performed a spin-polarised calculation of the electronic structureusing HSE06 Figure 148 (III) shows the DOS for the spin-down (inred) and spin-up (in blue) channel The DOS clearly demonstrates thatthe surface is semiconducting in agreement with STS experiment [140]featuring a band gap of 112 eV (Exp 15 eV Ref [140]) In the DOSthe spin-down surface state gives rise to a peak in the SiC bulk bandgap above the Fermi level On the other hand the spin-up surface stateis in resonance with the SiC bulk states
The disagreement in the STM images might be an indication that a differentstructure is observed in the STM measurements An exhaustive structuresearch to find a surface model that is even lower in energy than the Sitwist model and that reproduces all experimental observations includingthe STM images is desirable However if an alternative model were tobe found its surface energy would have to be close to the Si twist modelat the graphite line to still coexist with the (2times2)C phase As a result itwould very likely coexist with the Si twist model The conceptual Si twistmodel - inspired by the Si side - appears to satisfy the existing experimentalconstraints well
142 Assessing the Graphene and SiC Interface
In the following we will use the Si twist model (Sec1413) as a repres-entative model to shed light on the SiC-graphene interface on the C faceDepending on the specific growth conditions two qualitatively different in-terface structures are observed In one case the first carbon layer is stronglybound to the interface and in the other case it is weakly bound
For the experimenatly observed strongly bound interface it is unclearwhether the different groups observe the same structures Based on theirLEED diffraction pattern Srivastava et al [302] proposed a (
radic43times
radic43)-
152
Figure 149 Comparison of the surface energies for four different interface structures of3C-SiC(111) relative to the bulk-truncated (1times1) phase as a function of the C chemicalpotential within the allowed ranges (given by diamond Si diamond C or graphite Crespectively) using the graphite limit as zero reference In addition the known (2times2)Creconstruction and the (3times3) Si twist model are shown (Figure published in Ref [228])
Rplusmn76 SiC substrate with a (8times8) carbon mesh rotated by 76 with respectto the substrate (rsquo
radic43-R76rsquo) [302]
Figure 149 shows a surface diagram including the (2times2)C (see Sec 1411)reconstruction and the (3times3) Si twist model (Sec1412) from Fig147 Inaddition we include 4 different SiC-graphene interface structures Twomodels represent the weakly bound C layer at the interface (Fig 149 c-d) and two models the strongly bound C layer at the interface shown inFig 149 e-f
As a first step we constructed the strongly bound interface structures Weconstruct an interface structure similar to the ZLG phase known from the Siface (see Sec 82)- a 6
radic3-R30-interface4 labeled e in Fig 149 As a second
4The structure was first calculated and discussed by Lazarevic [184] We took the geo-metries from Ref [184] We adapted the structures to the lattice parameters given inTab 72 and postrelaxed the structure with the numerical settings given in Appendix B
153
structure we included a purely C based model of theradic
43-R 76-interface5
as described above labeled f in Fig 149
As a next step we built the two weakly bound structures To model theinterface we limited our study to a 30 rotation between the substrate andthe graphene film This choice was motivated by the LEED study of Hasset al [131] who showed that graphene sheets on the C face appear mainlywith a 30 and a plusmn22 rotation A 30 rotation has also be seen in STMmeasurements for the graphene covered (2times2) and (3times3) phases [141] fromhere on called (2times2)G (Fig 149 c) and (3times3)G (Fig 149 d) We thereforechose a (6
radic3times 6
radic3) SiC supercell covered by a (13times 13) graphene cell
rotated by 30 with respect to the substrate
The (2times2)G-interface covers 27 unit cells of the (2times2)C reconstruction Ourmodel of the graphene covered (3times3) phase consist of the same (13times13)graphene supercell covering 12 units of the (3times3) Si twist model
The surface energies for the four different interface structures of the 3C-SiC(111)surface are shown in the phase diagram in Fig 149 All surface energies arecalculated using PBE+vdW for the fully relaxed interface structures Thesurface energies are given relative to the bulk-truncated (1times1) phase as afunction of the C chemical potential within the allowed ranges
The 6radic
3-R30-interface crosses the graphite line 046 eV above the crossingpoint of the (3times3) Si twist model rendering it unstable In our calcu-lations the
radic43-R 76-interface is even higher in energy than the 6
radic3-
R30-interface Theradic
43-R 76-interface is grown reproducibly and ob-served in LEED [206 302 303] however its detailed atomic structure isunknown [135] A recent LEED analysis indicates that the experiment-ally observed
radic43-R 76-interface involves a graphene-like layer witH-
intercalated Si atoms between this layer and a hydrogenated SiC C-facesurface [85]
The (2times2)G-interface crosses the (2times2)C reconstruction just to the right ofthe graphite limit at a chemical potential of 2 meV and a surface energy ofminus069 eV This finding demonstrates that the observed (2times 2) LEED patternunderneath the graphene layer is indeed consistent with the well known(2times2)C reconstruction The surface energy difference between (2times2)G- andthe (3times3)G-interface amounts to 013 eV at the graphite line favouring the(2times2)G-interface
5The structure was first calculated and discussed by Lazarevic [184] We took the geo-metries from Ref [184] We adapted the structures to the lattice parameters given inTab 72 and postrelaxed the structure with the numerical settings given in Appendix B
154
The two strongly bound interface models the 6radic
3-R30- andradic
43-R 76-interface have both a comparatively high surface energy compared to the(2times2)C reconstruction
Already in Sec 1412 for the different models of the (3times3) reconstructionit seemed that the formation of Si-C bonds with the surface C atoms isenergetically favourable This can be understood in terms of the enthalpy offormation ∆H f Def 422 The enthalpy of formation for SiC is quite large(minus077 eV in experiment [168] minus056 eV in DFT-PBE+vdW and minus059 eV inDFT-HSE06+vdW (Tab 72)) favouring the formation of SiC bonds Thisexplains why the (2times2)C and (3times3) Si twist model are more stable becausethey contain a large number of Si-C bonds Conversely the 6
radic3-R30- and
theradic
43-R 76-interface are made up of energetically less favourable C-Cbonds which increases the surface energy considerably
To put our results into context we revisit the growth process In experimentgrowth starts with a clean (3times3) reconstruction The sample is annealeduntil the surface is covered by graphene At this stage the (3times3) recon-struction is the dominant phase underneath graphene [139] However ashift from the SiC (3times3) to the (2times2)C surface reconstruction at the SiC-graphene interface can be stimulated by an additional annealing step at atemperature below graphitisation (950C - 1000C) leaving the graphenelayer unaffected [139] Graphene growth starts at a point where bulk SiCdecomposes The interface is determined by the momentary stoichiometryat which the sublimation stopped ndash a coexistences of different phases isobserved The final annealing step shifts the chemical potential into a re-gime in which SiC bulk decomposition stops and the graphene layer doesnot disintegrate but the SiC surface at the interface moves closer to localequilibrium in agreement with the phase diagram in Fig 149 forming aninterface structure between the (2times2)C reconstruction and graphene In ourview the main difference between the Si face and the C face is the fact thatSi-terminated phases are more stable on the C face due to the formation ofheterogeneous Si-C bonds The Si-rich phases are thus stable practically upto the graphite line allowing the surface to form graphene only at the pointwhere bulk SiC itself is no longer stable This makes MLG growth on the Cside more difficult than on the Si face
155
143 Summary
In summary we shed new light on the central aspects of the thermody-namically stable phases that govern the onset of graphene formation onSiC(111) We showed that a Si-rich model represented by the Si twist modelreproduces the experimental observations - Si-rich and semiconducting Wealso used the (2times2)C and the (3times3) Si twist model as interface structuresWe argue that the formation of C face surface MLG is blocked by Si-richphases in the same chemical potential range in which MLG formation canbe thermodynamically controlled on the Si face making the growth ofhomogeneous MLG much more difficult on the C face
156
Conclusions
Conclusions
Silicon carbide (SiC) is a widely used substrate for epitaxial growth ofmono- and few-layer graphene The exact material properties of graphenedepend on their interaction with the substrate In this work we appliedthe ab initio atomistic thermodynamic formalism (Ch 4) to evaluate theinterface structure of epitaxial graphene on the silicon (Si) and carbon (C)face of SiC applying all-electron density-functional theory (DFT) based onPerdew-Burke-Ernzerhof generalised gradient approximation [246] (PBE)and HSE hybrid functional with α=025 and ω=011 bohrminus1 [175] (HSE06)including van-der-Waals effects [326] The C-rich limit of the chemicalpotential is of particular interest for the thermodynamic evaluation of thecombined SiC-graphene surface structures The limiting structures in theC-rich limit are diamond and graphite However the very small differencein binding energy between diamond and graphite (see Ch 6) necessitatean accurate description of their energetics This is a challenge for theory inparticular the description of the interlayer bonding between graphite sheetsIn graphite the atomic bonding is very inhomogeneous The in-plane bondsare formed by strong sp2-hybridised C orbitals The bonding betweengraphene sheets on the other hand is formed by weak van-der-Waals (vdW)forces Accurately determining the inter-planar binding energy necessitatesthe inclusion of long-range vdW effects which is a major and ongoingexperimental and theoretical quest (Sec 62)
On the Si-terminated SiC surface wafer-size well-ordered areas of grapheneform as (6
radic3times 6
radic3)-R30 (6
radic3) commensurate periodic films [332 95 82
263 67 302 346] We address how the substrate affects the atomic structureCh 8 and the energetics of the different Si-rich reconstructions Ch 10 andthe C-rich surface structures from the first partially covalently bondedzero-layer graphene (ZLG) via mono-layer graphene (MLG) and bi-layergraphene (BLG) up to three-layer graphene (3LG) a challenge because ofthe system sizes up to sim 2800 atoms and vdW interactions not accountedfor by most standard density functionals We found narrow thermodynamicequilibrium conditions for the ZLG MLG and even BLG The equilibriumgeometry emerged as a direct prediction including the significant graphenelayer corrugation The remaining difficulty will be to find the precise Sipartial pressure and temperature in experiment an important step towardsequilibrium graphene growth
In Chapter 9 we compared different approximated interface models of theexperimentally observed 6
radic3-ZLG structure We also show that smaller-cell
158
approximants may miss some geometric features and could erroneouslystabilise hexagon-pentagon-heptagon (H567) defects due to artificiallyinduced strain The bonding in the 6
radic3-ZLG layer contains a mixture of
in-plane sp2-like and sp3-like hybridisation of the carbon atoms The Catoms with sp3-like hybridisation covalently bind the ZLG layer to theSiC substrate with an increased in-plane C-C bond length (compared tofree standing graphene) This explains the increased lattice parameterof the ZLG layer observed in grazing-incidence x-ray diffraction (GID)measurements [283]
Some of the Si atoms in the top SiC bilayer are covalently bonded to the ZLGlayer while others remain unsaturated the Si dangling bonds We used DFTPBE including van-der-Waals effects [326] (PBE+vdW) and HSE06+vdWcalculations to evaluate the influence of the Si dangling bonds on the elec-tronic structure of the C-rich surface phases on 3C-SiC(111) (Ch 12) Usingtheradic
3 approximated interface structure we found that the ZLG MLG andBLG is doped by the Si dangling bond state at the Fermi level The doping ofthe graphene layer can be removed by saturating the Si dangling bond withan H atom The situation in the (6
radic3times 6
radic3)-R30 is less clear In the dens-
ity of states (DOS) of the 6radic
3-ZLG structure we found a few states close tothe Fermi level in good agreement with scanning tunnelling spectroscopy(STS) measurements [270 109] A Hirshfeld atomic charge partitioninganalysis [142] revealed one hexagonal and two triangular shaped patternper unit cell similar to those found in scanning tunnelling microscopy (STM)images by Qi et al [256]
A possibility to eliminate the effects of the Si dangling bonds on the epitaxi-ally grown graphene layer is to saturate the Si atoms at the interface by Hatoms Indeed H intercalation of epitaxial graphene on the Si face of SiC de-couples the graphene layer from the substrate forming quasi-free-standingmono-layer graphene (QFMLG) [262] As described in Chapter 13 we foundthe adsorption height QFMLG layer calculated in the full (6
radic3times 6
radic3)-R30
unit cell to be in excellent agreement with normal incidence x-ray standingwave (NIXSW) experiments The corrugation is drastically reduced whencompared to the MLG layer and the electron density at the interface isvery homogeneous All these features improve the electronic properties ofQFMLG compared to epitaxial MLG
In contrast to graphene growth on the Si face experimentally controllingthe layer thickness on the C face is significantly more challenging [213] Torefine the growth quality of epitaxial graphene on the C side of SiC andimproving the resulting electronic character of these films it is importantto understand the atomic interface structure A phase mixture of different
159
surface phases is observed when surface graphitisation first sets in How-ever the atomic structure of some of the competing surface phases as wellas of the SiC-graphene interface are unknown We calculated the formationenergy of different reconstructions and model systems for the interface andcompared the different structures within the thermodynamically allowedrange We showed that the Si-rich Si twist model reproduces the experi-mental observations - Si-rich and semiconducting - of the hitherto unknown3times3-SiC(111) reconstruction Its formation energy crosses that of the (2times2)Cphase just at the carbon rich limit of the chemical potential which explainsthe observed phase mixture We used the (2times2)C and the (3times3) Si twistmodel as interface structures Our results indicate that the formation of aninterface structure like the ZLG on the Si-terminated surface is blocked bySi-rich phases This answers the key question why the graphene growthdiffers strongly between the C and Si-terminated surface
In conclusion graphene films on the Si face of SiC form at least as near-equilibrium phases The remaining challenge is to find the necessary Sipartial pressure and growth temperature and keep them constant close tothe surface This makes the Si-terminated surface an ideal substrate forgraphene growth The ZLG interface structure ndash which aids the graphenegrowth ndash on the other hand leads to a corrugation of the graphene filmand influences its electronic structure by states close to the Fermi leveloriginating from the Si dangling bonds By H intercalation the Si bonds atthe SiC-graphene interface are saturated lifting the ZLG layer up formingQFMLG The QFMLG is flat and undoped featuring a homogeneous chargedensity at the interface The same mechanism which leads to graphenegrowth on the Si face fails on the C face because here the formation ofa regulating interface structure like the ZLG layer is hindered by Si-richsurface reconstructions
160
Appendices
A Thermal Expansion and PhaseCoexistence
In 1938 the first phase diagram for carbon was constructed on the basisof thermochemical data for temperatures up to 1200 K by Rossini and Jes-sup [268] Although both forms of carbon diamond and graphite exist innature their phase diagram revealed that only graphite is thermodynam-ically stable at standard conditions for temperature and pressure (normalconditions)1 Fortunately the rate of reversion from diamond to graphiteis insignificant under normal conditions and only measurable at temper-atures above 1000 K On the basis of theoretical considerations Bermanand Simon [23] extended the phase line between diamond and graphiteto temperatures higher than 1200 K This linear extrapolation of the formp = a + bT where p is the pressure and T the temperature is knownas the Berman-Simon line Since then the interest in the phase diagramof carbon continued experimentally [44 162 43] as well as theoretically[205 164 103 349 120]
The following Sections are a brief outline of our results presented in theMaster thesis by Florian Lazarevic [184] about the phase coexistence linebetween graphite and diamond in the temperature range from 0 to 2000 KTemperature effects are included by calculating the Helmholtz free energyF(T V) = E(V)+ Fvib(T V) (Eq 47) where E(V) is the volume dependenttotal energy and Fvib is the vibrational free energy introduced in Sec 41
At constant pressure the equilibrium volume of a material changes withtemperature In Sec 41 we saw that the thermal expansion can be un-derstood in terms of lattice vibrations We introduced the quasiharmonicapproximation (QHA) to calculate the volume dependent potential energysurface The basic assumption of the QHA is that the anharmonicity isrestricted to the change of the volume without any further anharmonicitiesso that the lattice dynamics can still be treated within the harmonic approx-imation (Ch 3) The equilibrium volume for every temperature is given byminimising F(T V) with respect to the volume at a fixed T
The potential energy surface (PES) is sampled by calculating the Helmholtzfree energy (Eq 47) for different volumes Then the equilibrium volumeis calculated by fitting the F(T V) pairs to the Birch-Murnaghan (B-M)equation of states Def 611 for every temperature T [223 27]
1The National Institute of Standards and Technology (NIST) defines the standard condi-tions for temperature as 29315 K (20C) and the absolute pressure as 1 atm (101325 kPa)
I
Temperature [K] 288 423 573 723 873 973 1073
Vatom [Aring3]
PBE+vdW 8788 8802 8819 8839 8864 8883 8904
PBE-D [117] Ref[349]
8700 8722 8757 8791 8817 8837 8860
Exp Ref [226] 8744 8773 8808 8846 8885 8912 8939
Table A1 Calculated unit cell volume of graphite in the temperature range between 288 Kand 1073 K For comparison DFT data using vdW Grimme [117] corrected PBE-D (Ref Yuet al [349]) and experimental data (Ref Nelson and Riley [226])
As a next step the thermal expansion of graphite and diamond is calculatedwithin the QHA (Sec 41) To calculate the vibrational free energy we usedthe finite displacement method (Ch 3) We obtained the vibrational freeenergy for different volumes from phonon calculations for diamond andgraphite In the case of diamond we included 5 different volumes rangingfrom 528 Aring3atom to 595 Aring3atom For the phonon calculations we useda (3times 3times 3) supercell (SC) for diamond For graphite we calculated thevibrational free energy in a (5times 5times 2) SC for 4 different volumes rangingfrom 810 Aring3atom to 870 Aring3atom For each volume we included 5calculations to optimise the ratio between the lattice parameter a and cto find the minimum Helmholtz free energy
Figure A1 shows the fitting results at 4 different temperatures (0 K 300 K700 K and 1500 K) for graphite and diamond The energy for the chosenvolumes are shown in Fig A1 as crosses for diamond and squares forgraphite The volumes were chosen close to the ground state equilibriumvolume as given in Tab 62
In the range between 0 K and 2000 K the equilibrium volume and corres-ponding Helmholtz free energies was obtained by the fitting procedureexplained above For high temperatures gt 2000 K the inclusion of anhar-monic effects becomes relevant To include them we would need to gobeyond the QHA We are interested in the temperature range lt 2000 KTo ensure the quality of the data we only included data with a root meansquare fitting error of the B-M smaller than 7 meV
The thermal expansion of graphite and diamond is shown in Fig A2For diamond we included experimental x-ray data [320] Our computedthermal expansion shows good agreement with experiment At 29315 KThewlis and Davey measured the diamond lattice constant to be 3567 Aringour calculated lattice constant at 29315 K is 3568 Aring
II
Figu
reA
1
Cal
cula
ted
Hel
mho
ltzfr
eeen
ergi
esw
ithin
the
BO-a
ppro
xim
atio
nas
afu
nctio
nof
the
volu
me
per
atom
for
grap
hite
and
diam
ond
at4
dif
fere
ntte
mpe
ratu
res
(0K
700
K1
500
Kan
d30
00K
)usi
ngP
BE
+vd
WT
heB
-Mfi
tis
show
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red
solid
line
(blu
ed
ashe
dlin
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rd
iam
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(gra
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)Th
eca
lcul
ated
poin
tsus
edfo
rth
efit
ting
are
show
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cros
ses
(squ
ares
)for
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ond
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phite
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eph
ase
tran
sitio
nlin
e-a
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entt
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2-b
etw
een
grap
hite
and
diam
ond
issh
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[Plo
ttak
enfr
omLa
zare
vic
[184
]]
III
Figure A2 The unit cell volume for graphite (top panel) and diamond (bottom panel) fortemperatures ranging from 0 K to 1400 K calculated using PBE+vdW [Data taken fromLazarevic [184]] For comparison experimental x-ray measurements data from Thewlis andDavey [320] is included for diamond as green circles
For graphite we listed the volume per atom for temperatures between288 K and 1073 K in Tab A1 The temperatures were chosen to be the sameas listet in Ref Nelson and Riley [226] For comparison we included abinitio (ab initio) data from Yu et al [349] in the QHA and experimental datafrom Nelson and Riley [226] Yu et al used the abinit-code a plane-waveand pseudopotential code [110] to calculate the temperature dependentvolume of graphite using the vdW[117] corrected Perdew-Burke-Ernzer-hof generalised gradient approximation [246] (PBE) exchange-correlationfunctional (PBE-D) They fitted the equation of states to 11 E(V) data pointsAt each fixed volume they performed a constant-volume optimisation of thecell geometry to get the optimal lattice constant ratio ca Nelson and Rileyused a x-ray powder method to measure the temperature dependence of theunit-cell dimensions of hexagonal Ceylon graphite The stacking order ofhexagonal Ceylon graphite is ABC [11] while in the theoretical work thiswork and Yu et al [349] an AB stacking was used The largest disagreementbetween our results and the reference data is at 288 K here the unit cellvolume in this work is 10 larger than the value obtained by Yu et al [349]and 05 larger than the experimental value [226]
In thermodynamics the pressure for the canonical ensemble is defined as
p(V T) = minus(
partF(V T)partV
)T
(A1)
This is the negative of the slope of the F(T = const V) curve at a constanttemperature For a specific temperature both carbon phases - diamondand graphite - coexist at the pressure given by the common tangent of the
IV
graphite and diamond free energy
pcoex = pdia = pgra
minus(
partFpartV
)T
= minus(
partFdiapartV
)T
= minus(
partFgrapartV
)T
(A2)
For every temperature a coexistence pressure can be determined by cal-culating the common tangent The obtained pcoex(T) curve is called thecoexistence line The coexistence line gives the temperature pressure pairat which graphite and diamond coexist The common tangent is shownin Fig A1 for 4 different temperatures The volumes for graphite werechosen in such a way that the common tangent of graphite and diamondtouches the E(V) close to the calculated reference points Figure A3 showsthe coexistence line in a pressure-temperature phase diagram for differentexchange-correlation functionals At a fixed temperature for pressure lowerthan pcoex graphite is the most stable phase and for pressure higher thanpcoex diamond becomes more stable
Experimental data from three different experiments are included in Fig A3Bundy et al [44] measured the coexistence line in a temperature rangebetween 1500 to 2000 K and found that pcoex ranges from 49 GPa to 61 GPaThe second experiment was performed by Kennedy and Kennedy [162]who were able to reduce the error in the pressure determination In thelinear part of the coexistence line they found for the Berman-Simon lineP = 0194 GPa + T 32 middot 10minus3 GPaK There have been several theoreticalattempts to predict the graphite-diamond coexistence line In 2010 Khaliul-lin et al [164] studied the graphite-diamond coexistence by employing aneural-network (NN) mapping of the generalised gradient approximation(GGA)-derived ab initio potential energy surface They used the obtainedpotential to generate molecular-dynamics (MD) trajectories at the cost offorce field MD simulations (nanosecond-long trajectories are required tostudy thermodynamics and mechanism of phases transitions [164]) Thisallowed them to go beyond the QHA A comparison of their results with theexperimental data [43] shown in Fig A3 reveals that the NN overestimatesthe transition pressure by approximately 35 GPa However their slope ofthe Berman-Simon line agrees well with the experimental findings listedin Tab A2 Ghiringhelli et al [103] used a long-range carbon bond orderpotential (LCBOPI+) to sample the potential free energy surface using MDsimulations [103] In the 1000-2000 K range the coexistence line predictedby LCBOPI+ lies very close to the experimental line [43] at the onset cor-rectly estimating the 0 K transition pressure However the slope in thelow-temperature regime deviates from the experimental data [43]
V
FigureA
3The
graphite-diamond
coexistenceline
isshow
nin
apressure-tem
peraturephase
diagramfor
differentEX
C s(a)PBE(b)PBE+vdWand
(c)LD
AFor
comparison
previouscalculations
andexperim
entaldata
areinclud
edin
theplotLD
A(W
indl)[205](red
)LC
BO
PI+
data
(1)[103]neural-netw
orkingclassical(2)and
quantumnuclei(3)results
[164]Bermann-Sim
online
(4)[23]andexperim
entaldata(Exp1
(5)[44]Exp2
(6)[162]andExp3
(7)[43])[Plotadaptedfrom
Lazarevic[184]]
VI
Functional
LDA PBE PBE+vdW
a [GPa] minus037 56 37
b [GPaK] 27 middot 10minus3 31 middot 10minus3 27 middot 10minus3
pcoex(0K) [GPa] 02 66 42
References
theo(a) theo(b) exp(c) exp(d) exp(e)
a [GPa] 17 37 019 071 17
b [GPaK] 25 middot 10minus3 28 middot 10minus3 32 middot 10minus3 27 middot 10minus3 22 middot 10minus3
pcoex(0K) [GPa] 16 47 18
Table A2 For the exchange-correlation functional (LDA PBE PBE+vdW) the linearinterpolation of the p-T-data in the temperature range between 1000 K and 1500 K to theBerman-Simon line P = a + bT where b is the slope in GPaK In addition the coexistencepressure pcoex at 0 K is given for different functionals Reference data is given for twotheoretical works (a) Yu et al [349] (b) Khaliullin et al [164] and experimental data (c)Kennedy and Kennedy [162] (d) Bundy et al [44] and (e) Bundy et al [43]
First we compare our results obtained with the PBE functional to the experi-mental data For PBE the transition pressure pcoex at 1500 K is overestimatedby 55 GPa If we include vdW-effects the overestimation is reduced to19 GPa while LDA underestimates pcoex by 17 GPa The energy differencebetween graphite and diamond is of the order of a few meV For an accuratedescription of the transition pressure the exchange-correlation functionalhas to capture the E-V curve (Fig A1) for graphite and diamond equallygood It necessitates a highly accurate description of the ground state energyof these two systems which is a challenge for density-functional theory(DFT) The difficulty of DFT functionals to correctly describe the graph-itediamond energetics in particular the inter-planar bonding of graphite(see Sec 62) leads to the observed systematic shift of the coexistence linein Fig A3
We performed a linear interpolation of the p-T-data by fitting to the Berman-Simon line The Berman-Simon line is defined as
p = a + bT (A3)
where b is the slope in GPaK In experiment the data is usually fitted inthe temperature range from 1000 K to 2000 K We use the same temperature
VII
range for the fitting The results are given in Tab A2 for different functionalsand reference data The Berman-Simon line shown in Fig A3 was obtainedfrom experimental thermodynamic properties of diamond and graphite [23]Its overall shape is well-captured by our calculations The 0 K transitionpressure however varies depending of the used functional
Regardless of the energy errors introduced by the functional the onset of thecoexistence line at low temperatures is well described In the linear regimethe slope is well captured and in excellent agreement with experiment
VIII
B Numerical Convergence
B1 The Basis Set
The FHI-aims code employs numeric atom-centered basis sets basic descrip-tions of their mathematical form and properties as introduced in Sec 25or in Ref [32] The FHI-aims basis sets are defined by numerically determ-ined radial functions Eq 228 corresponding to different angula momentumchannels Each radial function is obtained by solving a radial Schroumldingerequation and is subject to a confinement potential vcut(r) in Eq 228 Itensures that the radial function is strictly zero beyond the confining radius
C Si
minimal [He]+2s2p [Ne]+3s3p
tier 1 H(2p 17) H(3d 42)
H(3d 60) H(2p 14)
H(2s49) H(4f 62)
Si2+(3s)
tier 2 H(4f 98) H(3d 90)
H(3p 52) H(5g 94)
H(3s 43) H(4p 40)
H(5g 144) H(1s 065)
H(3d 62)
Table B1 Radial functions used for C and Si The first line (ldquominimalrdquo) denotes theradial functions of the occupied orbitals of spherically symmetric free atoms as computedin DFT-LDA or -PBE (noble-gas configuration of the core and quantum numbers of theadditional valence radial functions) ldquoH(nl z)rdquo denotes a hydrogen-like basis function forthe bare Coulomb potential zr including its radial and angular momentum quantumnumbers n and l X2+(nl) denotes a n l radial function of a doubly positive free ion ofelement X See also Ref Blum et al [32] for notational details
IX
rcut and decays smoothly
What is important for the present purposes is to demonstrate the accurateconvergence up to a few meV at most of our calculated surface energieswith respect to the basis set used As is typical of atom-centered basissets (Gaussian-type Slater-type numerically tabulated etc) variationalflexibility is achieved by successively adding radial functions for individualangular momentum channels until convergence is achieved In practicethe basis functions for individual elements in FHI-aims are grouped in so-called ldquotiersrdquo or ldquolevelsrdquo tier 1 tier 2 and so forth In the present workbasis functions for Si and C up to tier 2 were used The pertinent radialfunctions are summarised in Table (B1) using the exact same notation thatwas established in Ref [32]
Figure B1 Effect of increasing the basis set size on the surface energy of theradic
3 approx-imant to the ZLG phase at the chemical potential limit of bulk graphite The PBE+vdWfunctional was used In the plot we use T1 (T2) as abbrevation of tier 1 (tier 2) respectively
The convergence of the calculated surface energies in this work with basissize is exemplified for the (
radic3 timesradic
3)-R30 small unit cell approximant[334 214] to the ZLG phase in Fig (B1) including 6 SiC-bilayer The bottomcarbon atoms were saturated by hydrogen Its geometry was first fullyrelaxed with FHI-aims tight grid settings a tier 1+dg basis set for Si anda tier 2 basis set for C basis settings This geometry was then kept fixed
X
for the convergence tests shown here What is shown in Fig (B1) is thedevelopment of the surface energy (Si face and H-terminated C face) withincreasing basis size for both C and Si were calculated using
γSi face + γC face =1A
(Eslab minus NSimicroSi minus NCmicroC
) (B1)
where NSi and NC denote the number of Si and C atoms in the slab respect-ively and A is the chosen area The computed surface energy is shown per1times1 surface area as in all surface energies given in the main text
The notation in the figure is as follows
bull ldquoT1rdquo and ldquoT2rdquo abbreviate the set of radial functions included in tier 1and tier 2 respectively (see Table B1)
bull ldquoSi T1- f rdquo denotes the Si tier 1 basis set but with the f radial functionomitted
bull ldquoC T2-g denotes the set of radial functions for C up to tier 2 butomitting the g-type radial function of tier 2
bull ldquoSi T1+dgrdquo denotes the radial functions included up to tier 1 of Si andadditionally the d and g radial functions that are part of tier 2 This isalso the predefined default basis set for FHI-aims rsquotightrsquo settings for Si
bull ldquoC T2rdquo denotes the radial functions of C up to tier 2 and is the defaultchoice for rsquotightrsquo settings in FHI-aims
In short the plot indicates the required convergence of the surface energyto a few meV(1times1) surface area if the default FHI-aims rsquotightrsquo settingsare used It is evident that the high-l g-type component for C contributesnoticeably to the surface energy
B2 Slab Thickness
The convergence of our surface calculations with respect to the number ofSiC bilayers is shown in Fig (B2) by considering surface energies for the(radic
3timesradic
3)-R30 small unit cell approximant [334 214] The zero reference
XI
energy is an unreconstructed six bilayer 1x1 SiC surface A six bilayer slabis sufficient to accurately represent bulk effects
Figure B2 Slab thickness dependence of the surface energy of the (radic
3timesradic
3)-R30 ap-proximant to the ZLG phase
B3 k-Space Integration Grids
We demonstrate the accuracy of the 2D Brillouin zone (BZ) integrals for ourgraphene-like surface phases by comparing different k-space integrationgrids in Table (B2) The ZLG and mono-layer graphene (MLG) surfaceenergies in the full (6
radic3times6radic
3)-R30 cell are compared using the Γ-pointonly and using a 2x2x1 k-space grid These k-mesh tests were performedin a four bilayer slab using the PBE including van-der-Waals effects [326](PBE+vdW) functional light real space integration grids a tier1 basis setwithout the f functions for Si and tier1 for C The geometries were keptfixed at the configuration relaxed with a Γ-point only k-space grid InTable (B2) the surface energies relative to the unreconstructed 1x1 surfacein the graphite limit are listed for the silicon rich
radic3xradic
3 reconstruction theZLG and MLG phases Table (B2) clearly shows that the surface energiesare well converged using the Γ-point only Hence in this work all surfaceenergies for the (6
radic3times6radic
3)-R30 phases were calculated using this k-spaceintegration grid For all bulk reference energies as well as for the two siliconrich surface reconstructions the convergence with respect to the k-meshsize has been tested and found to be well converged for grids equivalent tothe ZLG phase
XII
system k-grid
6x6x1 12x12x1 24x24x1radic
3xradic
3 -0439 -0435 -0436
1x1x1 2x2x1
ZLG -0426 -0426
MLG -0454 -0455
Table B2 Surface energies in [eVSiC(1x1)] relative to the unreconstructed 3C-SiC(1x1)surface for the chemical potential limit of graphite four-bilayer SiC slabs The silicon-richradic
3xradic
3 reconstruction ZLG and MLG phases using different k-grids are shown ThePBE+vdW exchange-correlation functional was used
B4 The Heyd-Scuseria-Ernzerhof HybridFunctional Family for 3C-SiC
In this work we used the HSE [175] for calculating the electronic structureand validating surface energies In HSE06 the amount of exact exchangeis set to α = 025 and the range-separation parameter ω = 02Aringminus1 Wecalculated the band gap and valence band width of 3C-SiC for differentvalues of α We used with rsquotightrsquo numerical settings as introduced in Sec B1with an 12times12times12 off-Γ k-grid
As can be seen in Fig B3 for fixed (ω) the band gap and valence bandwidth depend practically linearly on the exchange parameter α We testedthe HSE06 with respect to the band gap and band width the latter being ameasure for the cohesive properties of a crystal [257] The default HSE06value of α = 025 captures both the band gap and the band width well andwe therefore adopt it for our calculations
XIII
Figure B3 The Kohn-Sham band-gap (blue circles) and valence band width along Γ to X(red circles) of 3C-SiC as a function of α The HSE06 value α = 025 is marked by a verticalline The experimental valence band width is shown as horizontal line at 36 eV[143] andthe exp band gap at 242 eV[148]
XIV
C Band Structure Plots of SiC
C1 Cubic Silicon Carbide (3C-SiC)
Figure C1 The band structure of cubic silicon carbide (3C-SiC) is shown along the highsymmetry lines We used the Heyd-Scuseria-Ernzerhof hybrid functional [138 175] (HSE)The atomic structure was relaxed including vdW corrections (Tab 72) 3C-SiC has anindirect band gap between Γ and X of 233 eV
XV
C2 Hexagonal Silicon Carbide 4H-SiC and6H-SiC
C21 The HSE Band Structure of 4H-SiC
Figure C2 The band structure of 4H-SiC is shown along the high symmetry lines Weused the Heyd-Scuseria-Ernzerhof hybrid functional [138 175] (HSE) The atomic structurewas relaxed including vdW corrections (see Tab 72) 4H-SiC has an indirect band gapbetween Γ and M of 326 eV
XVI
C22 The HSE Band Structure of 6H-SiC
Figure C3 The band structure of 6H-SiC is shown along the high symmetry lines Weused the Heyd-Scuseria-Ernzerhof hybrid functional [138 175] (HSE) The atomic structurewas relaxed including vdW corrections (see Tab 72) 6H-SiC has an indirect band gapbetween Γ and M of 292 eV
XVII
D Phonon Band Structure
All phonon calculations are based on finite difference approach as describedin Sec 3 using the phonopy-code [327]
D1 The Phonon Band Structure of the CarbonStructures Graphene Graphite and Diamond
D11 Graphene
Figure D1 Calculated phonon dispersion relation of graphene along the high symmetrylines (PBE+vdW)
We used the finite difference approach to calculating phonon dispersionwith a supercell of 6times 6 and a finite displacement of 0008 Aring For an accuratedescription of the soft accoustic phonon mode near the Γ-point a very densereal space sampling (really tight Sec 25) was chosen
XVIII
Since the graphene unit cell contains two carbon (C) atoms A and B thereare six phonon dispersion bands Fig D1 three accoustic and three opticphonon branches The two atomic vibrations perpendicular to the grapheneplane correspond to one accoustic and one optic phonon mode The accous-tic one is the lowest phonon branch in the Γ-K direction and the optic onehas a phonon frequency of 873 cmminus1 at the Γ-point For the remaining 2accoustic and 2 optic phonon modes the vibrations are in-plane Near theBZ center (Γ-point) the two optic in-plane phonon modes correspond twovibrations of the two sublattices A and B against each other the so-calledE2g phonon mode The E2g mode is doubly degenerate in Γ at 1495 cmminus1
and is a first order Raman active phonon mode [331] However within BOapproximation (see Sec 12) the Raman G-peak position is independent ofFermi-level in the electronic band structure in contrast with experiments[251 62] where a stiffening (a shift to higher frequencies) of the E2g phononmode is observed with increased electron (hole) doping concentration TheE2g mode within the BO approximation is given by the perturbation ofthe electronic structure due to the atomic displacement (phonon) and therearrangement of the Fermi surface as a response Pisana et al showed thatin graphene these two contributions cancel out exactly because of the rigidmotion of the Dirac cones associated with the E2g phonon breaking the BOapproximation
XIX
D12 Graphite
Figure D2 Calculated phonon dispersion relation of graphite along the high symmetrylines (PBE+vdW)
We used the finite difference approach to calculating phonon dispersionwith a supercell of 5times 5times 2 and a finite displacement of 0005 Aring For anaccurate description of the soft accoustic phonon mode near the Γ-point adense real space sampling (tight Sec 25) was chosen
XX
D13 Diamond
Figure D3 Calculated phonon dispersion relation of diamond along the high symmetrylines (PBE+vdW)
We used the finite difference approach to calculating phonon dispersionwith a supercell of 3times 3times 3 and a finite displacement of 0001 Aring We chosea dense real space sampling (tight Sec 25)
XXI
D2 Phonon Band Structure of Silicon Carbide
Figure D4 Calculated phonon dispersion relation of SiC along the high symmetry lines(PBE+vdW)
We used the finite difference approach to calculating phonon dispersionwith a supercell of 3times 3times 3 and a finite displacement of 0001 Aring We chosea dense real space sampling (tight Sec 25)
XXII
E Counting Dangling Bonds
At the SiC-graphene interface on the Si face of SiC some Si atoms in the topSiC bilayer remain unsaturated the so-called Si dangling bonds To identifythe Si dangling bonds we computed the distance between a top layer Siatom and the closest C atom in the ZLG layer Figure E1 shows the Si-Cdistance for every Si atom in the top SiC bilayer ordered by bond lengthof the 3C-SiC-ZLG structure calcualted with PBE+vdW For all structureswe evaluated we found a similar picture The Si-C bond length of most Siatoms is between the bulk Si-C bond length of 189 Aringin the case of 3C-SiCand the average Si-ZLG layer distance The atoms well above the Si-ZLGlayer distance ar defined as Si dangling bonds
XXIII
FigureE1The
distancebetw
eena
Siatomofthe
topSiC
bilayerand
theclosestC
atomin
theZ
LGlayer
isused
toidentify
theSidangling
bonds
XXIV
F Comparing the PBE and HSE06 Densityof States of the MLG Phase
Figure F1 The Density of states of theradic
3-MLG structure calculated using the PBE andthe HSE06 exchange-correlation functionalShown in red is the DOS of the slab usedin the calculation and in grey is shown the3C-SiC-bulk projected DOS
In Chapter 12 we discussed theelectronic structure of epitaxialgraphene on the 3C-SiC(111) sur-face In a small approximated(radic
3 timesradic
3)-SiC unit cell with astrained (2 times 2) graphene layerwe calculated the electronic struc-ture using the hybrid functionalHSE06 However to gain insideinto the electronic structure of theexperimentally observed (6
radic3 times
6radic
3)R30 structure we use the PBEapproximation to the exchange-cor-relation functional to render the cal-culations affordable In Figure F1we compare the DOS of the the
radic3-
MLG structure calculated using thePBE and the HSE06 exchange-cor-relation functional We used a sixbilayer (
radic3timesradic
3)-SiC slab coveredby a (2times 2)-ZLG and (2times 2)-MLGlayer It shows in red the full DOS
of the slab used in the calculation and in grey the 3C-SiC-bulk projectedDOS PBE and HSE06 give qualitatively the same result however in HSE06the bulk band gap is larger and the overall spectrum seems to be strechedover a larger energy range
The doping of theradic
3-MLG layer increased with the fraction of exact ex-change included in the exchange-correlation functional from PBE to HSE06to PBE0 shifting the graphene Dirac point from -048 eV (PBE) -063 eV(HSE06) to -086 eV (PBE0)
F1 Influence of the DFT Functional
In Appendix B we included details on the numerical convergence of calcu-lations for similar structures with respect to the number of basis functions
XXV
and the grid density in real and reciprocal space For the (6radic
3times 6radic
3) in-terfaces we chose the Γ-point for accurate integrations in reciprocal spaceThe FHI-aims code employs numeric atom-centered basis sets Basic de-scriptions of their mathematical form and properties are published in inRef [32] or in Sec 25 The basis set and numerical real space grids are ofhigh quality as defined by the tight settings including a tier1+dg basis setfor Si and a tier2 basis set for C [32]
To test the influence of the exchange correlation functional on the geometrywe used three different exchange correlation functionals the LDA [249] thePerdew-Burke-Enzerhof generalised gradient approximation (PBE) [247]and the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06)[175] In HSE06the amount of exact exchange is set to α = 025 and a range-separationparameter ω = 02Aringminus1 is used
For an accurate description of the different surface phases in particu-lar hydrogen-graphene bonding in the quasi-free-standing mono-layergraphene (QFMLG) phase we include long range electron correlationsso-called van der Waals (vdW) effects Here we compare the results of twodifferent schemes The first scheme used in this work is the well establishedTkatchenko-Scheffler (TS) [326] method It is a pairwise approach wherethe effective C6 dispersion coefficients are derived from the self-consistentelectron density The second approach is a more recent refinement to the TSscheme incorporating many-body effects [325 324 5] In this scheme theatoms are modelled as spherical quantum harmonic oscillators which arecoupled through dipole-dipole interactions The corresponding many-bodyHamiltonian is diagonalised to calculate the many-body vdW energiesAs a result this approach accounts for long-range many-body dispersion(MBD) effects employing a range-separated (rs) self-consistent screening(SCS) of polarizabilities and is therefore called MBDrsSCS (for detailssee Ref [5]) To include the long-range tail of dispersion interaction wecouple each functional with either the pairwise TS scheme here referedto as PBE+vdW (HSE06+vdW) or the MBDrsSCS scheme in this workabbreviated as PBE+MBD (HSE06+MBD)
In Table F1 we list the layer distance (Dnn+1) the Si-C distance within a SiCbilayer (dn) and the layer corrugation (δn) the difference between the highestand lowest atom in the layer for QFMLG and MLG on 6H-SiC(0001) calcu-lated with different exchange correlation functionals and vdW correctionsWe conclude that PBE+vdW PBE+MBD HSE06+vdW and HSE06+MBDyield the same results for both phases
XXVI
6H-S
iCZ
LG
PBE+
vdW
PBE+
MBD
LDA
HSE
06+v
dWH
SE06
+MBD
nD
nn+
1d n
δ nSi
CD
nn+
1d n
δ nSi
CD
nn+
1d n
δ nSi
CD
nn+
1d n
δ nSi
CD
nn+
1d n
δ n
Z2
37mdash
mdash
030
235
mdashmdash
0
292
30mdash
mdash
037
238
mdashmdash
0
302
37mdash
mdash
030
11
920
480
42lt
10minus
21
920
510
40lt
10minus
21
910
530
31lt
10minus
21
920
480
48lt
10minus
21
920
480
47lt
10minus
2
21
880
61lt
10minus
2 lt
10minus
21
890
61lt
10minus
2 lt
10minus
21
880
60lt
10minus
2 lt
10minus
21
880
61lt
10minus
2 lt
10minus
21
880
61lt
10minus
2 lt
10minus
2
31
890
62lt
10minus
2 lt
10minus
21
900
63lt
10minus
2 lt
10minus
21
880
62lt
10minus
2 lt
10minus
21
880
62lt
10minus
2 lt
10minus
21
890
62lt
10minus
2 lt
10minus
2
6H-S
iCQ
FMLG
PBE+
vdW
PBE+
MBD
LDA
HSE
06+v
dWH
SE06
+MBD
nD
nn+
1d n
δ nSi
CD
nn+
1d n
δ nSi
CD
nn+
1d n
δ nSi
CD
nn+
1d n
δ nSi
CD
nn+
1d n
δ n
G2
75mdash
mdashlt
10minus
22
76mdash
mdashlt
10minus
22
71mdash
mdashlt
10minus
22
71mdash
mdashlt
10minus
22
77mdash
mdashlt
10minus
2
H1
50mdash
mdash
mdash1
50mdash
mdash
mdash1
51mdash
mdash
mdash1
49mdash
mdash
mdash1
49mdash
mdash
mdash
11
890
62lt
10minus
2 lt
10minus
21
890
62lt
10minus
2 lt
10minus
21
880
61lt
10minus
2 lt
10minus
21
880
62lt
10minus
2 lt
10minus
21
880
61lt
10minus
2 lt
10minus
2
21
890
63lt
10minus
2 lt
10minus
21
890
63lt
10minus
2 lt
10minus
21
880
63lt
10minus
2 lt
10minus
21
880
63lt
10minus
2 lt
10minus
21
880
63lt
10minus
2 lt
10minus
2
31
890
63lt
10minus
2 lt
10minus
21
890
63lt
10minus
2 lt
10minus
21
880
62lt
10minus
2 lt
10minus
21
880
62lt
10minus
2 lt
10minus
21
880
63lt
10minus
2 lt
10minus
2
Tabl
eF
1I
nflue
nce
ofth
efu
nctio
nalo
nth
ein
terf
ace
stru
ctur
eof
6H-S
iC(radic
3-Si
Cm
odel
cell)
Dn
n+1
isth
edi
stan
cebe
twee
nla
yer
nan
dn+
1d n
give
sth
edi
stan
cew
ithi
nSi
Cbi
laye
rn
and
δ nth
eco
rrug
atio
nof
laye
rn
All
dist
ance
sar
egi
ven
inAring
XXVII
G Bulk Stacking Order The Si-terminated6H-SiC(0001) Surface
Figure G1 The zero-layer graphene (ZLG) of the hexagonal SiC polytype (6H-SiC) usingan approximated
radic3timesradic
3-R30 unit cell and their surface energies are shown for differentSiC-bilayer stacking order calculated using PBE+vdW
For 3C-SiC(111) the stacking order close to the surface is independent ofwhere the surface is cut However for the 6H-SiC(0001) this is differentas the SiC bilayers are rotated every third layer by 30 degree in the unitcell The position of the rotation is indicated in Fig G1 by a kink in thegrey line We tested the influence of the stacking order for 6H-SiC usingan approximated
radic3timesradic
3-R30 unit cell The surface energies calculatedusing PBE+vdW of the 6H-SiC ZLG and the atomic structure are shownin Fig G1 We find the lowest surface energy for ABCACB and ACBABCstacked SiC We therefore use a ABCACB stacked 6H-SiC substrate
XXVIII
H Influence of the Polytype
QFMLG
6H-SiC 3C-SiC
n Dnn+1 dn δn SiC Dnn+1 dn δn SiC
G 266 mdash 002 268 mdash 001
H 150 mdash 000000 150 mdash 000000
1 189 062 000000 189 062 000000
2 189 063 000000 189 063 000000
3 189 063 000000 189 063 000000
MLG
6H-SiC 3C-SiC[227]
n Dnn+1 dn δn SiC Dnn+1 dn δn SiC
G 340 mdash 045 340 mdash 041
Z 236 mdash 086 236 mdash 082
1 192 055 078030 193 055 074031
2 190 061 021014 190 061 021013
3 189 062 008005 189 062 008005
Table H1 Geometry comparison of the layer distance (Dnn+1) the Si-C distance within aSiC bilayer (dn) and the layer corrugation the difference between the highest and lowestatom in the layer (δn) for two different SiC polytypes 6H-SiC(0001) and 3C-SiC(111)including the QFMLG and MLG surface structure calculated with PBE+vdW The datawas taken from [227]
To evaluate the influence of the SiC polytype on the geometry we list thekey geometry parameters for the QFMLG and MLG in Tab H1 phase fortwo different polytypes the 3C-SiC and 6H-SiC For the QFMLG phase theinterface geometries are practically identical Likewise the interface of theMLG phase shows polytype induced changes that are smaller than 004 AringTherefore 3C and 6H polytypes can be exchanged without qualitativelychanging the result of the calculation It should be noted that the advantageof 3C is that the number of bilayers one uses in the slab can be smaller thansix
XXIX
I The 3C-SiC(111)-(3x3) Phase Details onthe Geometry
Figure I1 Schematic representation of the 3C-SiC(111)-(3times 3) reconstruction The numer-ation of the Si adatoms in the unit cell (shown in black) is used to mark the atoms accordingto their appearance in Tab I1
In Sec 812 the 3C-SiC(111)-(3times 3) reconstruction was introduced Weperformed DFT calculations including structure realxation for the Si rich(3 times 3) reconstruction We here include details of the calculation usingthe generalised gradient approximation to the exchange-correlation func-tional including long-range dispersion effects using the well establishedTkatchenko-Scheffler [326] method (PBE+vdW) The bulk lattice parameterare listed in Tab72 Each slab consists of six SiC-bilayers and the bottomC bonds are saturated by hydrogen The top three SiC-bilayers and allatoms above are fully relaxed (residual energy gradients 8middot10minus3 eVAring orbelow)
Figure I1 shows a schematic top few of the 3C-SiC(111)-(3times 3) reconstruc-tion The structure consists of a silicon (Si) adatom (atom 1) a trimer (atoms2-4) and a Si-adlayer (atoms 5-13) The trimer is slightly rotated with respectto its bulk position by an twist angle of 93 in excellent agreement withRef [306]
In Table I1 we list the structural displacement of the Si adatoms withrespect to their symmetric bulk position and the total displacement in the
XXX
This work (PBE+vdW) J Schardt Ref [276]
Atom id x [Aring] y [Aring] ∆xy [Aring] x [Aring] y [Aring] ∆xy [Aring]
1 Top Atom 000 000 000 000 000 000
2 000 037 037 001 038 038
3 -033 -019 038 -033 -019 038
4 Trimer 033 -019 038 033 -019 038
5 Sublayer 070 -026 075 071 -024 074
6 -013 073 074 -014 073 074
7 -057 -048 075 -056 -049 074
8 031 -066 073 031 -067 074
9 042 060 073 043 061 074
10 -073 006 073 -074 007 074
11 000 000 000 000 000 000
12 000 000 000 000 000 000
13 000 000 000 000 000 000
Table I1 The displacement of the adatoms with respect to their symmetric bulk positionThe surface structure was calculated using PBE+vdW for the approximantion to the ex-change-correlation functional (EXC) The LEED optimised geometric structure from Schardt[276] is included for comparison
xy-plane (∆xy) The overall agreement between our calculation and theexperimental structure obtained from quantitative LEED measurements isvery good
XXXI
J The History of Graphite Interlayer BindingEnergy
The history of the graphite binding energy (Eb) for different experimentsand level of theory
Year Reference EbmeVatom
Method
1954 Girifalco [105] 42 exp Heat of wetting exper-iments
1956 Girifalco and Lad [106] 43plusmn5 exptheo
Lattice summationand experiment [105]
1998 Benedict et al [17] 35+15minus10 exp Collapsed carbon-
nanotube structure
2004 Zacharia et al [350] 52plusmn5 exp Thermal desorptionexperiments of pol-yaromatic hydrocar-bons
2012 Liu et al [201] 31plusmn2 exp Directly measure-ment of the interlayerbinding energy
2013 Gould et al [113] 44plusmn3 exptheo
[201] with improvedfitting
1952 Brennan [39] 173 theo LCAO+vdW
1983 DiVincenzo et al [74] 110 theo DFT+Thomas Fermi
1987 Jansen and Freeman [151] 82 theo All-electron augmen-ted plane wave
1992 Schabel and Martins [275] 24 theo DFT (LDA)
1994 Charlier et al [54] 20 theo DFT (LDA)
2003 Rydberg et al [271] 24 theo DFT PBE+ vdW cor-rection
2003 Telling and Heggie [318] 35 theo DFT (LDA)
2004 Hasegawa and Nishidate [127] 27 theo DFT (LDA)
XXXII
2005 Zhechkov et al [352] 385 theo Density-functional-based tight binding(DFTB)
2006 Dappe et al [60] 66plusmn6 theo DFT with perturb-ation theory basedvdWcorrection
2006 Donchev [76] 55 theo Quantum mechanicalpolarisable force field(QMPFF)
2006 Ortmann et al [240] 835 theo DFT PBE+ vdW cor-rection
2007 Grimme et al [119] 66 theo All-electron DFT(PBE-D)
2007 Hasegawa et al [128] 604 theo DFT+semi-empiricalcorrection
2007 Ziambaras et al [354] 53 theo DFT PBE+ vdW cor-rection
2008 Barone et al [7] 54 theo DFT PBE+semi-empirical addition ofdispersive forces
2008 Gould et al [114] 597 theo DFT PBE+ vdW cor-rection
2008 Kerber et al [163] 57 theo DFT PBE+D vdW cor-rected
2008 Kleis et al [167] 50 theo DFT PBE vdW-DF
2008 Silvestrelli [297] 61 theo DFT PBE+ vdW cor-rection
2009 Spanu et al [299] 56 theo QMC - not correctedfor finite size effects
2010 Cooper [59] 59 theo DFT vdW correctedexchange-correlationfunctional
2010 Grimme et al [118] 66 theo DFT+D vdW correc-ted
XXXIII
2010 Lebegravegue et al [185] 48 theo RPA-DFT
2010 Liu and Goddard [200] 43 theo DFT-low gradient(DFT-lg) formulaincluding ZPE
2010 Madsen et al [207] 41 theo DFT meta-GGA (M06-L)
2010 Podeszwa [252] 453 theo [SAPT(DFT)] ex-trapolated frombinding energy ofpolycyclic aromatichydrocarbons
2012 Dappe et al [61] 70plusmn5 theo Combined local or-bital DFT and secondorder perturbationtheory
2012 Graziano et al [115] 56 theo DFT optimised PBEvdW (optPBE-vdW)including ZPE
2012 Kim et al [165] 59 theo DFT PBE+ vdW cor-rection
2013 Brandenburg et al [38] 43 theo DFT-D3 basis set su-perposition error cor-rected
2013 Bucko et al [42] 55 theo DFT PBE (TS + SCS)vdW correction
2013 Chen et al [56] 5515 theo DFT PBE+vdWinterlayerpotential
2013 Gould et al [112] 48 theo DFT PBE+ vdW cor-rection
2013 Strutynski et al [314] 40 theo Parameterisated DFT-D
2013 Sabatini et al [272] 39 theo DFT rVV10 functional(vdW correction)
XXXIV
2013 Silvestrelli [298] 37 theo DFT PBE many bodydispersion correction
2013 Thrower et al [323] 58 theo DFT optB88-vdWfunctional
2013 Olsen and Thygesen [239] 62 theo RPA-DFT
2014 Ambrosetti et al [5] 50 theo DFTPBE0+MBDrsSCS
2014 Berland and Hyldgaard [22] 66 theo DFT vdW correctedexchange-correlationfunctional
2014 Bucko et al [45] 81 theo DFT PBE+TS withiterative Hirshfeldcharges
2014 Hamada [123] 60 theo DFT PBE+ vdW cor-rection
XXXV
Acronyms
Basic Theory
ab initio ab initio is derived from the two Latin words ab (ldquo from rdquo) andinitio (ldquo beginningrdquo) It can be translated to ldquofrom the beginningrdquo p 7
FHI-aims Fritz Haber Institute ab initio molecular simulations p 6
B-M Birch-Murnaghan p 46
BO Born-Oppenheimer p 10
BZ Brillouin zone p 30
CBm conduction band minimum p 125
DFA density-functional approximation p v
DFT density-functional theory p 6
DOS density of states p 66
EB binding energy The binding energy is the energy to stetch the graphitebulk structure along the axis perpendicular to the carbon planes untilthe individual layers are seperated to infinity p 50
ED Dirac point p 59
EXC exchange-correlation functional p 18
EXF exfoliation energy The exfoliation energy is the energy to remove asingle carbon layer from the graphite bulk structure p 51
GGA generalised gradient approximation p 21
HSE Heyd-Scuseria-Ernzerhof hybrid functional [138 175] p 6
HSE06 HSE hybrid functional with α=025 and ω=011 bohrminus1 [175] p 23
XXXVI
HSE06+MBD HSE06 including van-der-Waals effects with full many-bodytreatment (MBDrsSCS) [325 324 5] p 57
HSE06+vdW HSE06 including van-der-Waals effects [326] p 46
ϕi(r) Kohn-Sham orbital p 17
LCAO linear combination of atomic orbitals p 108
LDA local-density approximation p 20
LJ Lennard-Jones The Lennard-Jones potential is a mathematical modelused to approximate the interaction between a pair of neutral atomsor molecules It has the form VLJ = 4ε
[(σr)12 minus 2
(σr)6] where ε is the
depth of the potential well σ is the distance at which the intermolecu-lar potential between the two particles is zero and r is the distancebetween the two particles p 55
MBDrsSCS long-range many-body dispersion (MBD) effects employinga range-separated (rs) self-consistent screening (SCS) of polarisabilities[324 5] p 25
NAO numeric atom-centered orbital p 26
PBE Perdew-Burke-Ernzerhof generalised gradient approximation [246]p 6
PBE+MBD PBE including van-der-Waals effects with full many-body treat-ment (MBDrsSCS) [325 324 5] p 56
PBE+vdW PBE including van-der-Waals effects [326] p 6
PCBZ Brillouin zone of the primitive cell p 107
PC primitive cell p 107
PES potential energy surface p 46
phonon-DOS phonon density of states p 30
QHA quasiharmonic approximation p 34
XXXVII
QMC quantum Monte Carlo p 21
RPA random phase approximation p 56
SCBZ Brillouin zone of the supercell p 107
SC supercell p 31
N supercell matrix p 89
γ surface free energy p 35
VCA virtual-crystal approximation p 59
vdW van-der-Waals p 6
ZPC zero-point vibrational correction p 46
A Few Common Symbols
n(r) electron density p 20
n0(r) ground state density p 15
E0 ground state energy p 13
Ψ0(ri) ground state wave function p 15
Ψ0 ground state wave function p 12
H many-body Hamiltonian p 11
He electronic Hamiltonian p 13
nKS(r) Kohn-Sham density p 19
Vext external potential p 13
XXXVIII
Experimental Techniques
AES Auger electron spectroscopy p 77
AFM atomic force microscopy p 55
ARPES angle-resolved photoemission spectroscopy p 74
ELS electron-energy loss spectroscopy p 77
GID grazing-incidence x-ray diffraction p 85
LEED low-energy electron diffraction p 4
MBE molecular beam epitaxy p 5
NIXSW normal incidence x-ray standing wave p 137
STM scanning tunnelling microscopy p 74
STS scanning tunnelling spectroscopy p 105
TEM transmission electron microscopy p 54
XPS x-ray photoemission spectroscopy p 81
Materials and Surface Reconstructions
3C-SiC cubic silicon carbide p 7
3LG three-layer graphene p 80
4H-SiC hexagonal silicon carbide p 7
6H-SiC hexagonal silicon carbide p 7
BLG bi-layer graphene p 4
C carbon p 5
XXXIX
DAS dimer ad-atom stacking fault p 78
H hydrogen p 40
MLG mono-layer graphene p v
QFMLG quasi-free-standing mono-layer graphene p v
Si silicon p 36
SiC silicon carbide p 13
ZLG zero-layer graphene Also called ldquobuffer layerrdquo p v
Miscellaneous
LED light emitting diode p 61
UHV ultrahigh vacuum p 4
XL
List of Figures
01 Schematic illustration of the graphene growth on the Si-sideof SiC On the Si-side epitaxial graphene growth starts witha clean Si-terminated surface At temperatures above 1000 Kfirst a carbon interface layer forms [263] the so-called zero-layer graphene (ZLG) Increased temperature and growthtime lead to the formation of monolayer graphene (MLG)and bilayer graphene (BLG) 3
21 The interacting system is mapped to a non-interacting auxili-ary system 17
22 The effective potential veff of the many-body system dependson Kohn-Sham density (nKS(r)) which we are searchingwhich depends on the Kohn-Sham orbital (ϕi(r)) which inturn depends on veff 19
23 The radial function uj(r) of the 3s orbital for a free siliconatom is plotted along radius r Also shown are the free-atomlike potential vj(r) and the steeply increasing confining po-tential vcut(r) The dotted line indicates the confining radiusrcut 27
51 (a) shows a periodic cubic silicon carbide (3C-SiC) crystalIn (b) the periodicity is broken by inserting a vacuum re-gion along the z-axis The bottom C atoms of the slab areterminated by H atoms In all surface calculations periodicboundary conditions are applied Hence the surface slab isperiodically repeated along the z-direction as shown in (c) 39
52 A projected bulk band structure is illustrated schematicallyFor wave vectors kperp perpendicular to the surface bulk statesappear For different kperp the band energies between the low-est and highest values correspond to regions of bulk statesThese bulk states are shaded in grey in the band structurealong wave vectors parallel to the surface k Outside theshaded bulk regions surface states can occur (Figure cour-tesy of Johan M Carlsson International Max-Planck Research School
ldquoTheoretical Methods for Surface Science Part Irdquo (Presentation) ) 40
XLI
61 Shows five different carbon structures The two commonlyknown three dimensional carbon crystals a diamond andb graphite A single layer of graphite forms graphene asp2 bonded two-dimensional structure labeled c d showsan example of a carbon nanotube and e the Buckminster-Fullerene - C60 also-called Buckyball 44
62 Number of publications per year as given in a Web-of-Sciencesearch for the keywords rsquographenersquo rsquosilicon carbidersquo and bothkeywords together [321] 45
63 The interlayer binding energy sketch for three different typesof binding a) exfoliation energy b) binding energy c) cleavageenergy In (a) and (b) the layer of interest is marked by abox The scissor and dashed line for the exfoliation (a) andcleavage (c) cases illustrates how the crystal is cut 51
64 History of the graphite interlayer binding energy (meVatom)since the first experiment in 1954 [106] The experimentalvalues [106 17 350 201 113] are given as crosses and thetheoretical values as filled circles The colour of each circledepends on the experimental work cited in the publicationGirifalco and Lad [106] gave the binding energy in units ofergscm2 their value of 260 ergscm2 was erroneously con-verted to 23 meVatom (shown in light green) instead of thecorrect 42 meVatom (dark green) A full list of referencesand binding energies is given in Appendix J 53
65 Binding energy EB of graphite as a function of the inter-planar distance d using LDA PBE HSE06 and two differentvdW correction schemes (TS [326] and MBD [325 324 5])The symbols mark EB at the equilibrium interlayer distanceThe vertical line marks the experimental lattice spacing ofc0 = 6694Aring [226] For comparison EB values from RPAPBEcalculations ( (1) Lebegravegue et al [185] (2) Olsen et al [239])and from experiment ( (3) Girifalco et al [106] (4) Benedict etal [17] (5) Zacharia et al [350] (6) and (7) Liu et al [201 113])are included 54
66 Subfigure (a) shows the graphene Brillouin zone (shaded ingrey) and the high symmetry points Γ M and K (markedby blue points) (b) The band dispersion of graphene on thelevel of PBE The k-path is chosen along the high symmetrylines of the graphene Brillouin zone as indicated by arrowsin Subfigure (a) 59
67 The position of the graphene Dirac point with respect to theFermi level for doping in the range ofplusmn 0005 eminusatom on thelevel of PBE PBE+vdW lattice parameter were used (Tab 61) 60
XLII
71 The white spheres represent a hexagonal structure The pos-sible stacking positions of a closed hexagonal structure aremarked as A B and C A SiC-bilayer overlayers the hexagonalstructure The primitive unit cell of the SiC-bilayer is shown 62
72 The atomic structure of three different SiC polytypes areshown cubic silicon carbide (3C-SiC) (left) and two hexago-nal structures 4H-SiC (middle) and 6H-SiC (right) The atomsin the black box mark the unit cell The stacking sequence ofthe SiC-bilayers is shown for every polytype and indicatedby a grey line in the structure 3C-SiC is also shown in its zincblende structure (bottom left) The distance dnnminus1 betweentwo bilayers and the distance Dn between a Si and C atomswithin a SiC bilayer are shown 63
73 Density of states (DOS) around the band gap for 3C-SiC Thetop panel shows the experimental species projected DOS (Csolid line and Si dashed line) measured by soft x-ray spec-troscopy with a band gap of 22 eV [204] The middle panelshows the calculated species projected DOS using the HSE06exchange-correlation functional with a band gap of 233 eVIn the bottom panel the calculated total Kohn-Sham DOS us-ing the HSE06 exchange-correlation functional is comparedto the total DOS calculated with PBE The theoretical mag-nitude of the peaks in the spectra depends on the number ofbasis functions included in the calculation 67
74 Plot of the band structure of 3C-SiC from Γ to X The bandstructure (solid line) is calculated using HSE06 exchange-cor-relation functional and the vdW corrected lattice parameter(HSE06+vdW from Tab 72) The shaded area (in grey) marksthe band gap of 233 eV In addition experimental photoemis-sion data is included (filled cicle) and parameterised bands(dashed line) taken from Ref [143] 68
75 Shows the Density of states (DOS) of the valence band for3C-SiC The top panel shows the experimental DOS meas-ured by photoemission spectroscopy with a band width of113 eV [296] The calculated DOS using the HSE06 exchange-correlation functional with a band width of 93 eV is shown inthe middle panel and using PBE with a band width of 84 eVin the bottom panel 69
81 The surface termination of 3C-SiC and 6H-SiC is shown for aclean surface In 6H-SiC every third Si-C bilayer is rotated by30 indicated by a kink in the grey line This rotation in thestacking order leads to different surface terminations Sn6Hand Slowastn6H where n runs from one to three 73
XLIII
82 Geometry of the 3C-SiC(111) (radic
3timesradic
3)-R30 Si adatom struc-ture The bond length of the adatom with the top Si atom(LSi-Si) the distance from the surface (D1SiSi) and the corruga-tion of the top C layer (δ1C) are shown 75
83 The twist model of the 3C-SiC(111)-(3times 3) reconstruction Onthe left The twist model from a side view The layer distancebetween the substrate and the Si-ad-layer D1SiSi3 as well asthe distance between the ad-layer and the Si-trimer DSi3Si2and trimer-Si-adatom DSi2Si1 are indicated by arrows Onthe right The twist model from a top view The unit cell isshown in black 77
84 Top view of the relaxed ZLG structure using PBE+vdW Theunit cell in the x-y-plane is indicate by a grey rhombus TheC atoms in the ZLG layer are coloured according to theirz-coordinate (the colour scale reaches from yellow for atomsclose to the substrate to blue for atoms away from it) Thetop Si-C bilayer is shown as well here the z-coordinate scalesfrom black to white where black indicates that the atom ispushed towards bulk SiC The ZLG hexagons furthest awayfrom the substrate form a hexagonal pattern (marked in grey)The Si atom of the top Si-C bilayer in the middle of a ZLGcarbon ring has the minimum z-coordinate (in the middle ofthe unit cell marked in light green) 80
85 Geometry and key geometric parameters determined by DFT-PBE+vdW for the three phases (a) ZLG (b) MLG and (c)bi-layer graphene (BLG) on the Si face of 3C-SiC(111) andhistograms of the number of atoms (Na) versus the atomiccoordinates (z) relative to the topmost Si layer (Gaussianbroadening 005 Aring) Na is normalised by NSiC the numberof SiC unit cells All values are given in Aring (Data publishedin Nemec et al [227]) 82
86 Histogramm of the number of atoms Na versus the atomiccoordinates (z) relative to the topmost Si layer (Gaussianbroadening 005 Aring) The key geometric parameter of theMLG on the Si face of 3C-SiC(111) at the level of a) LDA andb) PBE All values are given in Aring 83
87 MLG on 6H-SiC(0001) and histogram of the number of atomsNa versus the atomic coordinates (z) relative to the topmostSi layer (Gaussian broadening 005 Aring) Na is normalised byNSiC the number of SiC unit cells Dnn+1 is the distancebetween the layer n and n + 1 dn gives the Si-C distancewithin the SiC bilayer n and δn the corrugation of the layer nAll values are given in Aring (Figure published in Ref [294]) 84
XLIV
91 a) Top view of the ZLG layer and the top SiC bilayer fullyrelaxed using PBE+vdW The carbon atoms of the ZLG layerare coloured according to their vertical distance from theaverage height of the Si-layer ranging from black being closeto the Si layer to white for C atoms away from the Si-layerThe bonds between the ZLG C atoms are coloured from blueto red for increasing bond length b) Bond length distributionin the ZLG layer A Gaussian broadening of 0002 Aring wasapplied for visualisation purposes only The two verticallines indicate the bond length of free-standing graphene inthe optimised structure on a PBE+vdW level and the averagebond length in the ZLG layer 87
92 Comparison of the surface energies for six different smaller-cell approximant models of the 3C-SiC(111) ZLG phase re-lative to the bulk-terminated (1times1) phase as a function ofthe C chemical potential within the allowed ranges usingPBE+vdW The Si rich (3times 3) and (
radic3timesradic
3) reconstructionsas well as the (6
radic3times 6
radic3)-R30 ZLG phase from Fig 101 are
included for comparison The shaded areas indicate chemicalpotential values outside the strict thermodynamic stabilitylimits of Eq 415 88
93 Bond length distribution in the (6radic
3times 6radic
3) amp (13times 13)-ZLGlayer for two different rotational angles The bond lengthdistribution for the ZLG layer in the 30 rotated structurefrom Fig 91 is shown in grey and the ZLG structure rotatedby 2204 in blue A Gaussian broadening of 0002 Aring wasapplied for visualisation purposes only The two verticallines indicate the bond length of free-standing graphene inthe optimised structure on a PBE+vdW level and the averagebond length in the ZLG layer rotated by 2204 92
94 The hexagon-pentagon-heptagon (H567) defect in the zero-layer graphene shown in the approximated 3times 3 cell (inserta and b) and in the (6
radic3times 6
radic3)-R30 ZLG phase The defect
was placed in two different positions In inset a and c thedefect is placed at the ldquohollowrdquo position with a silicon atomof the underlying SiC bilayer in the middle of the centralhexagon and at the ldquotoprdquo position 94
XLV
95 (a) Calculated (PBE+vdW) effective lattice parameter (aver-age C-C bond length) in a series of fully relaxed grapheneflakes of finite size with (squares) and without (circles) hy-drogen termination at the edges The hydrogen terminatedflakes are shown in the inset The lattice parameter of a flatstrainfree periodic graphene sheet calculated in PBE+vdWis indicated by a dashed line (b) Diameter (maximum C-C distance) of the flakes used in (a) (Figure published inRef [284]) 101
96 Top calculated fully relaxed (atomic postion and unit cell)structure of a graphene sheet with a (10times 10) periodic ar-rangement of a 5-8-5 divacancy defects (atoms highlighted inred) The in-plane unit cell is marked by black lines Bottomfrom left to right fully relaxed monovacancy and 5-8-5 diva-cancy defects in a (10times 10) unit cell and Stone-Wales defectin a (7times 7 ) unit cell 102
101 Comparison of the surface energies for five different recon-structions of the 3C-SiC(111) Si side relative to the bulk-terminated (1times1) phase (always unstable) as a function ofthe C chemical potential within the allowed ranges (givenby diamond Si diamond C or graphite C respectively) (a)PBE+vdW (b) PBE(c) LDA The shaded areas indicate chem-ical potential values outside the strict thermodynamic sta-bility limits of Eq 415 (Data published in Ref Nemec et al[227]) 104
111 a Linear hydrogen chain with one hydrogen atom (H) inthe unit cell and its band structure b supercell of the linearhydrogen chain with 5 times the one-atomic unit cell Theband structure is folded at the edges of the reduced Brillouinzone of the supercell The band structure corresponding tothe actual one-atomic unit cell is plotted with dashed lines cthe hydrogen atoms in the 5 atomic chain are moved awayfrom their periodic position The corresponding band struc-ture is perturbed preventing the straight forward reversal ofthe band folding 107
112 a) shows 18 atoms arranged in a perturbed hexagonal lat-tice the unit cell and the lattice vectors A1 and A2 b) thestructure from a) is presented as a (3times 3) supercell The nineprimitive unit cells are indicated by blue lines and the primit-ive lattice vectors a1 and a2 by black arrows The blue atomin the central unit cell can be shifted to its periodic image bya lattice vector translation 109
XLVI
113 Sketch of a 2-dimensional unit cell in reciprocal space Thegrey area indicates the first BZ with the reciprocal latticevectors bi of the conventional unit cell (in this example 64 k-points (k) (purple points)) and the SCBZ of a 4times 4 supercellwith lattice vectors Bi and four k-points (K) (white points) 111
114 The real-space SC containing 3 times 3 primitive cell (PC) isshaded in blue The lattice vector Rj points to a periodiccopy of the SC The PC with lattice vectors a12 is shaded ingrey The lattice vector ρl points to a periodic copy of the PCEach PC contains two atoms (red circle) with atomic positionrmicro 112
115 The unfolded band structures of four different 10-atom hy-drogen chains are plotted along the Brillouin zone of theprimitive cell (PCBZ) path M to Γ to M Dashed lines indic-ate the SCBZ The corresponding atomic structure is shownat the bottom a A perfectly periodic hydrogen chain b ahydrogen chain with geometric distortions c a defect in thePC l = 6 in an otherwise perfectly periodic chain and d twochains with different periodicity 118
116 The unfolded band structure of the two chain system intro-duced in Example 113d The band structure is projected onthe 10-atomic chain marked by a grey box 120
121 Figure taken from Ref [263] Dispersion of the π-bands ofgraphene grown on 6H-SiC(0001) measured with angle-re-solved photoemission spectroscopy (ARPES) perpendicularto the ΓK-direction of the graphene Brillouin zone for (a)an as-grown ZLG (Ref [263] Fig 15 a) (b) an as-grownMLG (Ref [263] Fig 15 e) (c) after hydrogen intercalation(Ref [263] Fig 15 b) 122
122 The surface projected Kohn-Sham band structure and DOSof the (a) ZLG (b) MLG and (c) BLG calculated in the (
radic3timesradic
(3)) approximate unit cell using the HSE06 hybrid functional124123 Comparison between the DOS of the
radic3-ZLG and 6
radic3-ZLG
surface calculated using PBE exchange-correlation functionalThe full DOS is shown in blue the projected DOS of all Siatoms in the slab (beige) and of the Si dangling bonds (pink)On the right are shown images of the geometric structureoptimised using PBE+vdW 126
XLVII
124 The electron density of the 6radic
3-MLG phase Panel (a) showsthe full electron density nfull of the MLG surface integratedover the x-y-plane along the z-axis (solid black line) theelectron density of the substrate (filled in grey) nsub and thegraphene layer nG The average atomic layer position is indic-ated by dashed lines Panel (b) displays a cutout of the atomicsurface structure The boxes indicate the structures calculatedin isolation black the full slab grey the substrate and redthe graphene layer Panel (c) shows the electron density dif-ference ∆n(z) (Eq 121 and Eq 122) The charge transfer iscalculated by integrating ∆n(z) from zprime to infin (integrated areafilled in red) Panel (d) shows the charge distribution in thex-y-plane at z = zprime 128
125 The difference in the electron density given by Eq 121 isshown for the 3C-SiC MLG phase Eq 122 gives ∆n averagedover the x-y-plane and plotted it along z The position of theSi C H and graphene-layer are indicated by dashed linesWe show ∆n(zi) in the x-y-plane at equidistant zi heightsbetween the MLG-layer and the substrate 129
126 The band structure of a 13times 13 graphene cell Subfigure a)shows the Hirshfeld charge distribution of a MLG layer b)shows the unfolded band structure of a perfectly periodic13times 13 graphene cell and overlayed in red is the band struc-ture of free-standing graphene in its 1times 1 unit cell c) unfol-ded band structure of a flat graphene layer doped accordingto the Hirshfeld charges The charges lead to a shift of theDirac point by 022 eV c) the unfolded band structure withthe same corrugation as the MLG layer d) Band structure ofa graphene layer which is both corrugated and doped Theregion around the Dirac point is enlarged showing the shiftof the Dirac point caused by the doping 132
127 The bulk projected Kohn-Sham band structure and DOS oftheradic
3-MLG structure with the Si dangling bond saturatedby an H atom The full band structure and DOS of the slabused in the calculation is shown in red and the 3C-SiC bulkstates in grey 133
128 Four times the ZLG layer coloured depending on the Hirsh-feld charge of the C atom from blue for electron gain to redfor electron loss The position of the unsaturated Si atoms isshown in pink The scanning tunnelling microscopy (STM)image taken from Qi et al [256] is shown for comparison 134
XLVIII
131 (a) Vertical distances measured by normal incidence x-raystanding wave (NIXSW) on QFMLG The position of theBragg planes around the surface are indicated by blue lines [294]PBE+vdW calculated geometry for (b) QFMLG on 6H-SiC(0001)and histograms of the number of atoms Na versus the atomiccoordinates (z) relative to the topmost Si layer (Gaussianbroadening 005 Aring) Na is normalised by NSiC the numberof SiC unit cells Dnn+1 is the distance between the layer nand n + 1 dn gives the Si-C distance within the SiC bilayer nand δn the corrugation of the layer n All values are given inAring (Figure published in Ref [294]) 138
132 The difference in the electron density given by Eq 121 isshown for the 3C-SiC QFMLG phase We integrated ∆n inthe x-y-plane and plotted it along z (Eq 122) The position ofthe Si C H and graphene-layer are indicated by dashed linesWe show ∆n(zi) in the x-y-plane at equidistant zi heightsbetween the graphene layer and the substrate 141
141 Geometry of the 3C-SiC(111) (2times2)C Si adatom structure Thebond length of the adatom with the top C layer (LSi-C) thedistance from the surface (DCSi) and the corrugation of thetop Si layer (δSi) are shown 144
142 All structural models based on the model proposed by Hosteret al [145] for the (3times3) reconstructions of the 3C-SiC(111)are shown in a side and top view In the top view the unitcell is marked in red We include three different chemicalcompositions (a) from Hiebel et al [139] (b) from Deretzisand La Magna [71] (c) as a plain Si adatom 146
143 A Si rich structural model for the (3times3) reconstructions shownin a side and top view from Ref [139] In the top view theunit cell is marked in red 147
144 Three different chemical compositions of the tetrahedral ad-cluster structure suggested by Li and Tsong [192] In a) thetetrahedron is formed by a Si atom surrounded by three Catoms b) three C atoms surrounding a Si atom ( a) and b)shown in a side view) c) 4 Si atoms form a tetrahedron ( sideand top view) 147
145 A Si rich structural model for the (3times3) reconstructions adap-ted from the Si face [177] 148
XLIX
146 The Si twist model adapted from the 3C-SiC(111)-(3times 3) re-construction Sec 812 On the left The Si twist model from aside view On the right The Si twist model from a top viewThe unit cell is shown in blue The Si adatoms are coloureddepending on their distance to the top Si-C-bilayer The ninead-layer Si atoms in dark blue the three Si adatoms on top ofthe ad-layer in blue and the top Si adatom in purple (Figurepublished in Ref [228]) 148
147 Comparison of the surface energies relative to the bulk ter-minated (1times1) phase as a function of the C chemical potentialwithin the allowed ranges (given by diamond Si graphite Cand for completeness diamond C respectively) The shadedareas indicate chemical potential values outside the strictthermodynamic stability limits Included in the surface en-ergy diagram are structure models as proposed for the Cface [(b) [292] Sec 1411 (d) [192] Fig 144 c) (e) Fig 142 c)and (h) [145] with the chemical composition given by [139]Fig 142 a) (f) [139] Fig 143 (g) [71] Fig 142 b] and mod-els adapted from the Si face [(a) [310] Fig 146 and (c) [177]Fig 145] In the right panel the different models for the(3times3) reconstructions of the 3C-SiC(111) are ordered by theirsurface energies in the graphite C limit of the chemical po-tential with increasing stability from top to bottom (Figurepublished in Ref [228]) 149
148 I The a) and b) phases from Fig 147 calculated using theHSE06+vdW exchange-correlation functional with fully re-laxed structures and unit cells (Tab 72) II Simulated con-stant current STM images for occupied- and empty-state ofthe Si twist model (unit cell shown in red) The three pointsof interest (A B C) are marked by arrows and labeled ac-cording to Hiebel et al [139] III The DOS for the spin-upand spin-down states clearly shows a band gap rendering thesurface semiconducting (Figure published in Ref [228]) 151
149 Comparison of the surface energies for four different interfacestructures of 3C-SiC(111) relative to the bulk-truncated (1times1)phase as a function of the C chemical potential within theallowed ranges (given by diamond Si diamond C or graphiteC respectively) using the graphite limit as zero reference Inaddition the known (2times2)C reconstruction and the (3times3) Sitwist model are shown (Figure published in Ref [228]) 153
L
A1 Calculated Helmholtz free energies within the BO-approxi-mation as a function of the volume per atom for graphite anddiamond at 4 different temperatures (0 K 700 K 1500 K and3000 K) using PBE+vdW The B-M fit is shown as red solidline (blue dashed line) for diamond (graphite) The calculatedpoints used for the fitting are shown as crosses (squares) fordiamond (graphite) The phase transition line - a tangentto both curves Eq A2 - between graphite and diamond isshown [Plot taken from Lazarevic [184]] III
A2 The unit cell volume for graphite (top panel) and diamond(bottom panel) for temperatures ranging from 0 K to 1400 Kcalculated using PBE+vdW [Data taken from Lazarevic [184]]For comparison experimental x-ray measurements data fromThewlis and Davey [320] is included for diamond as greencircles IV
A3 The graphite-diamond coexistence line is shown in a pressure-temperature phase diagram for different EXCs(a) PBE (b)PBE+vdW and (c) LDA For comparison previous calcula-tions and experimental data are included in the plot LDA(Windl) [205] (red) LCBOPI+ data (1) [103] neural-network-ing classical (2) and quantum nuclei (3) results [164] Ber-mann-Simon line (4) [23] and experimental data (Exp 1 (5)[44] Exp 2 (6) [162] and Exp 3 (7) [43]) [Plot adapted fromLazarevic [184]] VI
B1 Effect of increasing the basis set size on the surface energyof the
radic3 approximant to the ZLG phase at the chemical
potential limit of bulk graphite The PBE+vdW functionalwas used In the plot we use T1 (T2) as abbrevation of tier 1(tier 2) respectively X
B2 Slab thickness dependence of the surface energy of the (radic
3timesradic3)-R30 approximant to the ZLG phase XII
B3 The Kohn-Sham band-gap (blue circles) and valence bandwidth along Γ to X (red circles) of 3C-SiC as a function of αThe HSE06 value α = 025 is marked by a vertical line Theexperimental valence band width is shown as horizontal lineat 36 eV[143] and the exp band gap at 242 eV[148] XIV
C1 The band structure of cubic silicon carbide (3C-SiC) is shownalong the high symmetry lines We used the Heyd-Scuseria-Ernzerhof hybrid functional [138 175] (HSE) The atomicstructure was relaxed including vdW corrections (Tab 72)3C-SiC has an indirect band gap between Γ and X of 233 eV XV
LI
C2 The band structure of 4H-SiC is shown along the high sym-metry lines We used the Heyd-Scuseria-Ernzerhof hybridfunctional [138 175] (HSE) The atomic structure was relaxedincluding vdW corrections (see Tab 72) 4H-SiC has an in-direct band gap between Γ and M of 326 eV XVI
C3 The band structure of 6H-SiC is shown along the high sym-metry lines We used the Heyd-Scuseria-Ernzerhof hybridfunctional [138 175] (HSE) The atomic structure was relaxedincluding vdW corrections (see Tab 72) 6H-SiC has an in-direct band gap between Γ and M of 292 eV XVII
D1 Calculated phonon dispersion relation of graphene along thehigh symmetry lines (PBE+vdW) XVIII
D2 Calculated phonon dispersion relation of graphite along thehigh symmetry lines (PBE+vdW) XX
D3 Calculated phonon dispersion relation of diamond along thehigh symmetry lines (PBE+vdW) XXI
D4 Calculated phonon dispersion relation of SiC along the highsymmetry lines (PBE+vdW) XXII
E1 The distance between a Si atom of the top SiC bilayer andthe closest C atom in the ZLG layer is used to identify the Sidangling bonds XXIV
F1 The Density of states of theradic
3-MLG structure calculatedusing the PBE and the HSE06 exchange-correlation functionalShown in red is the DOS of the slab used in the calculationand in grey is shown the 3C-SiC-bulk projected DOS XXV
G1 The zero-layer graphene (ZLG) of the hexagonal SiC polytype(6H-SiC) using an approximated
radic3timesradic
3-R30 unit cell andtheir surface energies are shown for different SiC-bilayerstacking order calculated using PBE+vdW XXVIII
I1 Schematic representation of the 3C-SiC(111)-(3 times 3) recon-struction The numeration of the Si adatoms in the unit cell(shown in black) is used to mark the atoms according to theirappearance in Tab I1 XXX
LII
List of Tables
61 The structural and cohesive properties of diamond graphiteand graphene are listed for different exchange-correlationfunctionals (LDA PBE HSE06 and vdW corrected PBE+vdWand HSE06+vdW) The lattice parameters a0 [Aring] and forgraphite c0 [Aring] calculated by minimising the energy withthe help of analytic forces and stress tensor as implementedin the FHI-aims-code [169] The cohesive energy Ecoh [eV]as obtained in this work Reference data from experimentis included ldquoPESrdquo refers to results computed based on theBorn-Oppenheimer potential energy surface without any cor-rections 47
62 Calculated ground state lattice parameter a0 and c0 unit cellvolume per atom and cohesive energies for diamond andgraphite using Def 611 ldquoPESrdquo refers to results computedbased on the Born-Oppenheimer potential energy surfacewithout any corrections We also include ZPC lattice para-meters (Data taken from Master thesis by Florian Lazarevic[184]) 49
71 The distance dnnminus1 between two SiC-bilayer and the distanceDn between the Si and C sublayer with in SiC-bilayer arelisted for 3C-SiC 4H-SiC and 6H-SiC calculated for differentexchange-correlation functionals (LDA PBE HSE and vdWcorrected PBE+vdW and HSE06+vdW) 63
LIII
72 The structural and cohesive properties of 3C-SiC 4H-SiC and6H-SiC are listed for different exchange-correlation function-als (LDA PBE HSE06 and vdW corrected PBE+vdW andHSE06+vdW) The lattice parameters a0 [Aring] and for the twohexagonal polytypes c0 [Aringn] where n is the number of SiCbilayers the bulk modulus B0 [Mbar] cohesive energy Ecoh[eV] and enthalpy of formation ∆H f [eV] as obtained in thiswork Reference data from experiment and theory is includedThe theoretical reference data was calulated using DFT codesbased on pseudopotentials with a plane-wave basis (PSP-PW) [15 244 157 355 126] or Gaussian basis (PSP-G) [12]in the LDA ldquoPESrdquo refers to results computed based on theBorn-Oppenheimer potential energy surface without any cor-rections We also include zero-point vibrational correction(ZPC) lattice parameters and bulk moduli 65
73 The direct and indirect band gap and the valence band widthare listed for 3C-SiC 4H-SiC and 6H-SiC calculated for dif-ferent exchange-correlation functionals ( PBE LDA HSE06)We used the geometries given in Tab 72 for PBE HSE06 thevdW corrected values were chosen 70
81 Atomic position of the Si adatom relative to the surface fordifferent exchange-correlation functionals The bulk latticeparameter were taken from Tab 72 For 6H-SiC we chose aABCACB stacking order (see Appendix G) LSi-Si is the bondlength between the adatom and the top layer Si atoms D1SiSiis the distance between the adatom and top layer Si atomsalong the z-axis and δ1C is the corrugation of the top C-layer(see Fig 82) (All values in Aring) 76
82 Layer distances of the Si adatom structure relative to thesurface for different exchange-correlation functionals (LDAand PBE+vdW) of the 3C-SiC(111)-(3times 3) reconstruction Forcomparison the layer distances obtained from quantitativeLEED measurements are included [276] (All values in Aring) 78
91 Parameters of the simulated ZLG-3C-SiC(111) interface struc-tures in different commensurate supercells the 3C-SiC andZLG periodicity the rotation angle (α) the number of surfaceSi atoms (Nsurf
SiC ) and C atoms in the ZLG (NZLG) the numberof unsaturated Si bonds (NDB
Si ) (see Appendix E) the corruga-tion of the ZLG layer (δC) the strain in the carbon layer andthe relative surface energy at the carbon rich graphite-limit(γ|microC=Ebulk
graphite) 90
LIV
92 PBE+vdW calculated effective lateral lattice parameter aeff(in Aring) the top-to-bottom corrugation ∆z (in Aring) and the effect-ive area lost (gained) per defect ∆Adefect ( in Aring2) of differ-ent defect types and periodicities in an ideal free-standinggraphene sheet (Table published in Ref [284]) 98
121 The Hirshfeld charge per C atom [eminusatom] in theradic
3-MLGcalculated using PBE and HSE06 and for comparison theHirshfeld charges calculated in the 6
radic3-MLG structure using
PBE 130
141 Atomic position of the Si adatom relative to the surface fordifferent exchange-correlation functionals The bulk latticeparameter were taken from Tab 72 LSi-C is the bond lengthbetween the adatom and the top layer C atoms DCSi is thedistance between the adatom and top layer C atoms alongthe z-axis and δSi is the corrugation of the top Si-layer (seeFig 141) (All values in Aring) 144
A1 Calculated unit cell volume of graphite in the temperaturerange between 288 K and 1073 K For comparison DFT datausing vdW Grimme [117] corrected PBE-D (Ref Yu et al[349]) and experimental data (Ref Nelson and Riley [226]) II
A2 For the exchange-correlation functional (LDA PBE PBE+vdW)the linear interpolation of the p-T-data in the temperaturerange between 1000 K and 1500 K to the Berman-Simon lineP = a + bT where b is the slope in GPaK In addition thecoexistence pressure pcoex at 0 K is given for different func-tionals Reference data is given for two theoretical works (a)Yu et al [349] (b) Khaliullin et al [164] and experimental data(c) Kennedy and Kennedy [162] (d) Bundy et al [44] and (e)Bundy et al [43] VII
B1 Radial functions used for C and Si The first line (ldquominimalrdquo)denotes the radial functions of the occupied orbitals of spher-ically symmetric free atoms as computed in DFT-LDA or-PBE (noble-gas configuration of the core and quantum num-bers of the additional valence radial functions) ldquoH(nl z)rdquodenotes a hydrogen-like basis function for the bare Coulombpotential zr including its radial and angular momentumquantum numbers n and l X2+(nl) denotes a n l radialfunction of a doubly positive free ion of element X See alsoRef Blum et al [32] for notational details IX
LV
B2 Surface energies in [eVSiC(1x1)] relative to the unrecon-structed 3C-SiC(1x1) surface for the chemical potential limitof graphite four-bilayer SiC slabs The silicon-rich
radic3xradic
3reconstruction ZLG and MLG phases using different k-gridsare shown The PBE+vdW exchange-correlation functionalwas used XIII
F1 Influence of the functional on the interface structure of 6H-SiC(radic
3-SiC model cell) Dnn+1 is the distance between layer nand n + 1 dn gives the distance within SiC bilayer n and δnthe corrugation of layer n All distances are given in Aring XXVII
H1 Geometry comparison of the layer distance (Dnn+1) the Si-Cdistance within a SiC bilayer (dn) and the layer corrugationthe difference between the highest and lowest atom in thelayer (δn) for two different SiC polytypes 6H-SiC(0001) and3C-SiC(111) including the QFMLG and MLG surface struc-ture calculated with PBE+vdW The data was taken from[227] XXIX
I1 The displacement of the adatoms with respect to their sym-metric bulk position The surface structure was calculatedusing PBE+vdW for the approximantion to the exchange-cor-relation functional (EXC) The LEED optimised geometricstructure from Schardt [276] is included for comparison XXXI
LVI
Bibliography
[1] Silicon carbide (SiC) lattice parameters In O Madelung U Roumlsslerand M Schulz editors Group IV Elements IV-IV and III-V CompoundsPart a - Lattice Properties vol 41A1a of Landolt-Boumlrnstein - Group IIICondensed Matter (p 1-13) Springer Berlin Heidelberg (2001)[ISBN 9783540640707][doi10100710551045_253] 71
[2] Acheson E G Production of Artificial Crystaline Garbonaceous Materials(February 1893) US Patent 492767httpwwwgooglecompatentsUS492767 7
[3] Ajoy A Murali K V R M and Karmalkar S Brillouin zone unfoldingof complex bands in a nearest neighbour tight binding scheme Journal ofPhysics Condensed Matter vol 24 (5) 055504 (2012)[doi1010880953-8984245055504] 11
[4] Allen P B Berlijn T Casavant D A and Soler J M Recoveringhidden Bloch character Unfolding electrons phonons and slabs PhysicalReviews B vol 87 085322 (February 2013)[doi101103PhysRevB87085322] 11
[5] Ambrosetti A Reilly A M DiStasio R A and Tkatchenko A Long-range correlation energy calculated from coupled atomic response functionsThe Journal of Chemical Physics vol 140 (18) 18A508 (2014)[doi10106314865104] 24 62 65 62 71 F1 J1 J
[6] Araujo P T Terrones M and Dresselhaus M S Defects and impurit-ies in graphene-like materials Materials Today vol 15 (3) 98 - 109 (2012)ISSN 1369-7021[doi101016S1369-7021(12)70045-7] 93
[7] Barone V Casarin M Forrer D Pavone M Sambi M and Vit-tadini A Role and effective treatment of dispersive forces in materialsPolyethylene and graphite crystals as test cases J Comput Chem vol 30 934 (2008)[doi101002jcc21112] 62 J1
[8] Baroni S Giannozzi P and Testa A Greenrsquos-Function Approach toLinear Response in Solids Phys Rev Lett vol 58 (18) 1861ndash1864 (May
LVII
1987)[doi101103PhysRevLett581861] 2
[9] Baskin Y and Meyer L Phys Rev vol 100 544 (1955) 82 82
[10] Bassett W A Weathers M S Wu T-C and Holmquist T Com-pressibility of SiC up to 684 GPa Journal of Applied Physics vol 74 (6) 3824-3826 (1993)[doi1010631354476] 71
[11] Batsanov S S and Batsanov A S Introduction to Structural ChemistrySpringerLink Buumlcher Springer (2012)[ISBN 9789400747715]httpbooksgoogledebooksid=YiHHkIA7K4MC A
[12] Baumeier B Kruumlger P and Pollmann J Self-interaction-correctedpseudopotentials for silicon carbide Phys Rev B vol 73 (19) 195205(May 2006)[doi101103PhysRevB73195205] 71 72 J
[13] Bechstedt F Principles of Surface Physics Advanced Texts in PhysicsSpringer Heidelberg 1 ed (2003)[ISBN 978-3-642-55466-7][doi101007978-3-642-55466-7] 1
[14] Bechstedt F and Furthmuumlller J Electron correlation effects on SiC(111)and SiC(0001) surfaces J Phys Condens Matter vol 16 (17) S1721ndashS1732 (2004)[doi1010880953-89841617014] 10
[15] Bechstedt F Kaumlckell P et al Polytypism and Properties of SiliconCarbide physica status solidi (b) vol 202 (1) 35ndash62 (1997) ISSN1521-3951[doi1010021521-3951(199707)2021lt35AID-PSSB35gt30CO2-8]71 72 J
[16] Becke A D A new mixing of Hartree-Fock and local density-functionaltheories The Journal of Chemical Physics vol 98 (2) 1372-1377 (1993)[doi1010631464304] 233
[17] Benedict L X Chopra N G Cohen M L Zettl A Louie S Gand Crespi V H Microscopic determination of the interlayer bindingenergy in graphite Chem Phys Lett vol 286 (5-6) 490ndash496 (April
LVIII
1998)[doi101016S0009-2614(97)01466-8] 62 64 62 65 62 J1J
[18] Berger C Song Z et al Ultrathin epitaxial graphite 2D electron gasproperties and a route toward graphene-based nanoelectronics J PhysChem B vol 108 (52) 19912ndash19916 (December 2004)[doi101021jp040650f] (document) 7 III
[19] Berger C Song Z et al Electronic confinement and coherence in pat-terned epitaxial graphene Science vol 312 (5777) 1191 (May 2006)[doi101126science1125925] (document) III 92 122
[20] Berger C Song Z et al Electronic confinement and coherence in pat-terned epitaxial graphene Science vol 312 (5777) 1191 (May 2006)[doi101126science1125925] 7
[21] Berger C Veuillen J et al Electronic properties of epitaxial grapheneInt J Nanotechnol vol 7 (4) 383ndash402 (2010) (document) III 82
[22] Berland K and Hyldgaard P Exchange functional that tests the ro-bustness of the plasmon description of the van der Waals density functionalPhysical Review B vol 89 035412 (January 2014)[doi101103PhysRevB89035412] 62 J1
[23] Berman R and Simon F On the Graphite-Diamond Equilibrium Z fElektrochem Ber Bunsenges Phys Chem vol 59 333-338 (1955) 4261 1412 A A3 A J
[24] Bernhardt J Nerding M Starke U and Heinz K Stable surfacereconstructions on 6H-SiC(0001 Materials Science and Engineering Bvol 61 ndash 62 (0) 207 ndash 211 (1999)[doi101016S0921-5107(98)00503-0] (document) 1411412 1413
[25] Bernhardt J Schardt J Starke U and Heinz K Epitaxially idealoxide-semiconductor interfaces Silicate adlayers on hexagonal (0001) and(000-1) SiC surfaces Applied Physics Letters vol 74 (8) 1084-1086(1999)[doi1010631123489] (document)
[26] Bhatnagar M and Baliga B Comparison of 6H-SiC 3C-SiC and Sifor power devices Electron Devices IEEE Transactions on vol 40 (3)
LIX
645-655 (March 1993)[doi10110916199372] 7
[27] Birch F Finite Elastic Strain of Cubic Crystals Physical Review vol 71 809ndash824 (June 1947)[doi101103PhysRev71809] 61 A
[28] Bjoumlrkman T Gulans A Krasheninnikov A V and Nieminen R MAre we van der Waals ready Journal of Physics Condensed Mattervol 24 (42) 424218 (2012)[doi1010880953-89842442424218] 62
[29] Bjoumlrkman T Gulans A Krasheninnikov A V and Nieminen R Mvan der Waals Bonding in Layered Compounds from Advanced Density-Functional First-Principles Calculations Physical Review Letters vol 108 235502 (June 2012)[doi101103PhysRevLett108235502] 62
[30] Blaha P Schwarz K Sorantin P and Trickey S Full-potential lin-earized augmented plane wave programs for crystalline systems ComputerPhysics Communications vol 59 (2) 399 - 415 (1990)[doi1010160010-4655(90)90187-6] 25
[31] Bloch F Uumlber die Quantenmechanik der Elektronen in KristallgitternZeitschrift fAtilde1
4r Physik vol 52 (7-8) 555-600 (1929) ISSN 0044-3328[doi101007BF01339455] 11
[32] Blum V Gehrke R et al Ab initio molecular simulations with numericatom-centered orbitals Comp Phys Commun vol 180 (11) 2175ndash2196(2009)[doi101016jcpc200906022] 25 25 25 61 B1 B1 F1J
[33] Bocquet F C Bisson R Themlin J-M Layet J-M and AngotT Reversible hydrogenation of deuterium-intercalated quasi-free-standinggraphene on SiC(0001) Physical Review B vol 85 (20) 201401 (2012)[doi101103PhysRevB85201401] 132
[34] Bostwick A McChesney J Ohta T Rotenberg E Seyller T andHorn K Experimental studies of the electronic structure of grapheneProgress in Surface Science vol 84 (11) 380ndash413 (2009)[doi101016jprogsurf200908002] 12
LX
[35] Bostwick A Ohta T Seyller T Horn K and Rotenberg E Quasi-particle dynamics in graphene Nature Physics vol 3 (1) 36ndash40 (2007)[doi101038nphys477] (document) 63 12 121 122 124
[36] Boykin T B Kharche N Klimeck G and Korkusinski M Approx-imate bandstructures of semiconductor alloys from tight-binding supercellcalculations Journal of Physics Condensed Matter vol 19 (3) 036203(2007)[doi1010880953-8984193036203] 11
[37] Boykin T B and Klimeck G Practical application of zone-foldingconcepts in tight-binding calculations Phys Rev B vol 71 115215(March 2005)[doi101103PhysRevB71115215]httplinkapsorgdoi101103PhysRevB71115215 11
[38] Brandenburg J G Alessio M Civalleri B Peintinger M F BredowT and Grimme S Geometrical Correction for the Inter- and Intramolecu-lar Basis Set Superposition Error in Periodic Density Functional TheoryCalculations The Journal of Physical Chemistry A vol 117 (38) 9282-9292 (2013)[doi101021jp406658y] 62 J1
[39] Brennan R O The Interlayer Binding in Graphite J Chem Phys vol 20 40 (1952)[doi10106311700193] J1
[40] Brewer L Cohesive energies of the elements (January 1975)[doi1021727187973] 61 61
[41] Brown M G Atom hybridization and bond properties Some carbon-containing bonds Trans Faraday Soc vol 55 694-701 (1959)[doi101039TF9595500694] 91
[42] Bucko T c v Lebegravegue S Hafner J and Aacutengyaacuten J G Tkatchenko-Scheffler van der Waals correction method with and without self-consistentscreening applied to solids Phys Rev B vol 87 064110 (February 2013)[doi101103PhysRevB87064110] 62 J1
[43] Bundy F Bassett W Weathers M Hemley R Mao H and Gon-charov A The pressure-temperature phase and transformation diagram forcarbon updated through 1994 Carbon vol 34 (2) 141 - 153 (1996)[doi1010160008-6223(96)00170-4] A A A3 A2 J
LXI
[44] Bundy F P Bovenkerk H P Strong H M and Wentorf Jr R HDiamond-Graphite Equilibrium Line from Growth and Graphitization ofDiamond The Journal of Chemical Physics vol 35 (2) 383 - 391 (August1961) ISSN 0021-9606[doi10106311731938] A A A3 A2 J
[45] Bucko T Lebegravegue S Aacutengyaacuten J G and Hafner J Extending theapplicability of the Tkatchenko-Scheffler dispersion correction via iterativeHirshfeld partitioning The Journal of Chemical Physics vol 141 (3) 034114 (2014)[doi10106314890003] 62 J1
[46] Capelle K A birdrsquos-eye view of density-functional theory eprint arXiv(November 2002)[arxivcond-mat0211443] I
[47] Carlsson J M Simulations of the Structural and Chemical Properties ofNanoporous Carbon vol 3 of Carbon Materials Chemistry and PhysicsSpringer Netherlands (2010)[ISBN 978-1-4020-9717-1][doi101007978-1-4020-9718-8_4] 93 93
[48] Carlsson J M (2013) Private communication 93
[49] Casady J B and Johnson R W Status of silicon carbide (SiC) as awide-bandgap semiconductor for high-temperature applications A reviewSolid-State Electronics vol 39 (10) 1409 - 1422 (October 1996)[doi1010160038-1101(96)00045-7] 7
[50] Castro Neto A H Guinea F Peres N M R Novoselov K S andGeim A K The electronic properties of graphene Rev Mod Phys vol 81 109ndash162 (January 2009)[doi101103RevModPhys81109] (document) 6 63
[51] Casula M Filippi C and Sorella S Diffusion Monte Carlo Methodwith Lattice Regularization Physical Review Letters vol 95 100201(September 2005)[doi101103PhysRevLett95100201] 62
[52] Ceperley D M and Alder B J Ground State of the Electron Gas by aStochastic Method Phys Rev Lett vol 45 (7) 566ndash569 (August 1980)[doi101103PhysRevLett45566] 231
LXII
[53] Chakarova-Kaumlck S D Schroumlder E Lundqvist B I and LangrethD C Application of van der Waals Density Functional to an ExtendedSystem Adsorption of Benzene and Naphthalene on Graphite Phys RevLett vol 96 146107 (April 2006)[doi101103PhysRevLett96146107] 62
[54] Charlier J C Gonze X and Michenaud J P First-principles study ofthe stacking effect on the electronic properties of graphite(s) Carbon vol 32 289ndash299 (1994)[doi1010160008-6223(94)90192-9] 62 J1
[55] Chen W Xu H et al Atomic structure of the 6H-SiC(0001) nanomeshSurf Sci vol 596 176 - 186 (2005)[doi101016jsusc200509013] 82 82 92
[56] Chen X Tian F Persson C Duan W and Chen N-x Interlayerinteractions in graphites Scientific Report vol 3 (November 2013)[doi101038srep03046] J1
[57] Cheung R Silicon Carbide Micro Electromechanical Systems for HarshEnvironments Imperial College Press (2006)[ISBN 9781860946240] II 7
[58] Clark S Segall M et al First principles methods using CASTEPZeitschrift fuumlr Kristallographie vol 220 (5-6) 567ndash570 (2005)[doi101524zkri220556765075] 25
[59] Cooper V R Van der Waals density functional An appropriate exchangefunctional Physical Review B vol 81 161104 (April 2010)[doi101103PhysRevB81161104] 62 J1
[60] Dappe Y J Basanta M A Flores F and Ortega J Weak chemicalinteraction and van der Waals forces between graphene layers A combineddensity functional and intermolecular perturbation theory approach Phys-ical Review B vol 74 205434 (2006)[doi101103PhysRevB74205434] J1
[61] Dappe Y J Bolcatto P G Ortega J and Flores F Dynamicalscreening of the van der Waals interaction between graphene layers Journalof Physics Condensed Matter vol 24 (42) 424208 (2012)[doi1010880953-89842442424208] J1
[62] Das A Pisana S et al Monitoring dopants by Raman scattering in an
LXIII
electrochemically top-gated graphene transistor Nature Nanotechnologyvol 3 (4) 210 - 215 (April 2008) ISSN 1748-3387[doi101038nnano200867] D11
[63] Davis S G Anthrop D F and Searcy A W Vapor Pressure of Siliconand the Dissociation Pressure of Silicon Carbide The Journal of ChemicalPhysics vol 34 (2) 659-664 (1961)[doi10106311701004] 71
[64] De Heer W Berger C and Wu X Epitaxial graphene Solid StateCommunications vol 143 92ndash100 (2007) (document) 7 III
[65] De Heer W Seyller T Berger C Stroscio J and Moon J EpitaxialGraphenes on Silicon Carbide MRS Bulletin vol 35 (4) 296 ndash 305(2010)[doi101557mrs2010552] (document) 82
[66] de Heer W A Epitaxial graphene A new electronic material for the 21stcentury MRS Bulletin vol 36 632ndash639 (August 2011)[doi101557mrs2011158] 14
[67] de Heer W A Berger C et al Large area and structured epitaxialgraphene produced by confinement controlled sublimation of silicon carbideProc Nat Acad Sci vol 108 16900 (2011) (document) 42 10 143
[68] de Lima L H de Siervo A et al Atomic surface structure of grapheneand its buffer layer on SiC(0001) A chemical-specific photoelectron diffrac-tion approach Physical Review B vol 87 081403(R) (February 2013)[doi101103PhysRevB87081403] 82
[69] Delley B An all-electron numerical method for solving the local densityfunctional for polyatomic molecules The Journal of Chemical Physicsvol 92 (1) 508-517 (1990)[doi1010631458452] 25
[70] Delley B From molecules to solids with the DMol3 approach The Journalof Chemical Physics vol 113 (18) 7756-7764 (2000)[doi10106311316015] 25
[71] Deretzis I and La Magna A A density functional theory study ofepitaxial graphene on the (3 times 3)-reconstructed C-face of SiC AppliedPhysics Letters vol 102 (2013)
LXIV
[doi10106314794176] (document) 1412 142 1412 1471413 J
[72] Dion M Rydberg H Schroder E Langreth D and Lundqvist B Van der Waals densityfunctional for general geometries Phys Rev Lett vol 92 (24) 246401(June 2004)[doi101103PhysRevLett92246401] 62
[73] Dirac P A M Note on Exchange Phenomena in the Thomas AtomMathematical Proceedings of the Cambridge Philosophical Society vol 26 376ndash385 (7 1930) ISSN 1469-8064[doi101017S0305004100016108] 231 231
[74] DiVincenzo D P Mele E J and Holzwarth N A W Density-functional study of interplanar binding in graphite Physical Review Bvol 27 (4) 2458ndash2469 (February 1983)[doi101103PhysRevB272458] J1
[75] Dobson J F White A and Rubio A Asymptotics of the DispersionInteraction Analytic Benchmarks for van der Waals Energy FunctionalsPhys Rev Lett vol 96 073201 (February 2006)[doi101103PhysRevLett96073201] 62
[76] Donchev A G Ab initio quantum force field for simulations of nanostruc-tures Phys Rev B vol 74 235401 (2006)[doi101103PhysRevB74235401] J1
[77] Dove M T Introduction to Lattice Dynamics Cambridge Topics inMineral Physics and Chemistry Cambridge University Press (1993)[ISBN 0-521-39293-4][doi1022770521392934] 3
[78] Dovesi R Orlando R et al CRYSTAL14 A program for the ab initioinvestigation of crystalline solids International Journal of Quantum Chem-istry vol 114 (19) 1287ndash1317 (2014)[doi101002qua24658] 25
[79] Elias D C Gorbachev R V et al Dirac cones reshaped by interactioneffects in suspended graphene Nature Physics vol 7 (9) 701ndash704 (July2011)[doidoi101038nphys2049] 63
LXV
[80] Emery J D Detlefs B et al Chemically Resolved Interface Structure ofEpitaxial Graphene on SiC(0001) Phys Rev Lett vol 111 (21) 215501(2013)[doi101103PhysRevLett111215501] 82 82
[81] Emtsev K Seyller T et al Initial stages of the graphite-SiC (0001)interface formation studied by photoelectron spectroscopy In MaterialsScience Forum vol 556 (p 525ndash528) Trans Tech Publ scientificnet(September 2007)[doi104028wwwscientificnetMSF556-557525] 12
[82] Emtsev K V Bostwick A et al Towards wafer-size graphene layers byatmospheric pressure graphitization of silicon carbide Nature Materialsvol 8 (3) 203ndash207 (February 2009)[doi101038nmat2382] (document) a 7 III 82 10 14 143
[83] Emtsev K V Speck F Seyller T Ley L and Riley J D Interactiongrowth and ordering of epitaxial graphene on SiC 0001 surfaces Acomparative photoelectron spectroscopy study Phys Rev B vol 77 (15) 155303 (2008)[doi101103PhysRevB77155303] (document) b III 8 8292 92 12 12 121 122 123 124
[84] Farjam M Visualizing the influence of point defects on the electronic bandstructure of graphene Journal of Physics Condensed Matter vol 26 (15) 155502 (2014)[doi1010880953-89842615155502] 11
[85] Feenstra R M (2014) Private communication 142
[86] Feibelman P J and Alavi A Entropy of H2O Wetting Layers TheJournal of Physical Chemistry B vol 108 (38) 14362-14367 (2004)[doi101021jp049934q] 1413
[87] Ferralis N Maboudian R and Carraro C Evidence of StructuralStrain in Epitaxial Graphene Layers on 6H-SiC(0001) Physical ReviewLetters vol 101 156801 (October 2008)[doi101103PhysRevLett101156801] 9 91 i
[88] Ferralis N Maboudian R and Carraro C Determination of substratepinning in epitaxial and supported graphene layers via Raman scatteringPhys Rev B vol 83 081410 (February 2011)[doi101103PhysRevB83081410] 91
LXVI
[89] Feynman R P Forces in Molecules Phys Rev vol 56 (4) 340ndash343(August 1939)[doi101103PhysRev56340] 25
[90] Filippi C Gonze X and Umrigar C J Generalized Gradient Approx-imations to Density Functional Theory Comparison with Exact Resultsvol 4 of Theoretical and Computational Chemistry chap 8 (p 295ndash326)Elsevier Science (1996)[ISBN 0-444-82404-9] 232
[91] Fiori G and Iannaccone G Multiscale Modeling for Graphene-BasedNanoscale Transistors Proceedings of the IEEE vol 101 (7) 1653ndash1669(2013)[doi101109JPROC20132259451] (document)
[92] First P De Heer W Seyller T Berger C Stroscio J and Moon JEpitaxial graphenes on silicon carbide MRS Bulletin vol 35 (4) 296ndash305(2010)[doi101557mrs2010552] (document)
[93] Fissel A Schroumlter B and Richter W Low-temperature growth of SiCthin films on Si and 6H-SiC by solid-source molecular beam epitaxy AppliedPhysics Letters vol 66 (23) 3182-3184 (1995)[doi1010631113716] 812
[94] Forbeaux I Themlin J M Charrier A Thibaudau F and DebeverJ M Solid-state graphitization mechanisms of silicon carbide 6H-SiC polarfaces Applied Surf Sci (p 406 - 412) (2000)[doiDOI 101016S0169-4332(00)00224-5] a
[95] Forbeaux I Themlin J-M and Debever J-M Heteroepitaxial graphiteon 6H minus SiC(0001) Interface formation through conduction-band elec-tronic structure Phys Rev B vol 58 16396ndash16406 (December 1998)[doi101103PhysRevB5816396] (document) III 8 82 92143
[96] Forti S Emtsev K V Coletti C Zakharov A A Riedl C andStarke U Large-area homogeneous quasifree standing epitaxial grapheneon SiC(0001) Electronic and structural characterization Physical ReviewB vol 84 125449 (September 2011)[doi101103PhysRevB84125449] 12 121
[97] Forti S and Starke U Epitaxial graphene on SiC from carrier density
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engineering to quasi-free standing graphene by atomic intercalation Journalof Physics D Applied Physics vol 47 (9) 094013 (March 2014)[doi1010880022-3727479094013] (document) 13
[98] Frisch M J Trucks G W et al Gaussian09 Revision D01 (2009)Gaussian Inc Wallingford CT 2009httpwwwgaussiancomindexhtm 25
[99] Furthmuumlller J Bechstedt F Husken H Schroter B and RichterW Si-rich SiC (111)(0001) 3 x 3 and sqrt (3) x sqrt (3) surfaces A Mott-Hubbard picture Physical Review B vol 58 (20) 13712 (November1998)[doi101103PhysRevB5813712] 811 811
[100] Furthmuumlller J Kaumlckell P et al Model of the epitaxial growth of SiC-polytypes under surface-stabilized conditons Journal of Electronic Materialsvol 27 (7) 848ndash852 (1998)[doi101007s11664-998-0108-1] 811 131
[101] Gehrke R First-principles basin-hopping for the structure determinationof atomic clusters PhD thesis () 25
[102] Geim A and Novoselov K The rise of graphene Nature materialsvol 6 (3) 183ndash191 (2007)[doi101038nmat1849] (document) 6
[103] Ghiringhelli L M Los J H Meijer E J Fasolino A and FrenkelD Modeling the Phase Diagram of Carbon Phys Rev Lett vol 94 145701 (April 2005)[doi101103PhysRevLett94145701] 61 A A A3 J
[104] Giannozzi P Baroni S et al QUANTUM ESPRESSO a modular andopen-source software project for quantum simulations of materials Journalof Physics Condensed Matter vol 21 (39) 395502 (2009)[doi1010880953-89842139395502] 25
[105] Girifalco L A (1954) (Girifalco PhD thesis 1954 cited in L AGirifalco and R A Lad J Chem Phys 25 693 (1956)) J1
[106] Girifalco L A and Lad R A Energy of Cohesion Compressibility andthe Potential Energy Functions of the Graphite System J Chem Physvol 25 693 (1956)[doi10106311743030] 62 64 62 65 62 J1 J
LXVIII
[107] Gobre V V and Tkatchenko A Scaling laws for van der Waals in-teractions in nanostructured materials Nature Communincation vol 4(August 2013)[doi101038ncomms3341] 93
[108] Goldberg Y Levinshtein M E and Rumyantsev S L Properties ofAdvanced Semiconductor Materials GaN AlN SiC BN SiC SiGe JohnWiley amp Sons Inc New York (2001)[ISBN 9780471358275] 71
[109] Goler S Coletti C et al Revealing the atomic structure of the bufferlayer between SiC(0001) and epitaxial graphene Carbon vol 51 249ndash254(2013)[doi101016jcarbon201208050] (document) 82 82 9110 121 123 131 143
[110] Gonze X Beuken J M et al First-Principles Computation of MaterialProperties the ABINIT Software Project Computational Materials Sciencevol 25 (3) 478ndash492 (2002)[doi101016S0927-0256(02)00325-7] 25 A
[111] Gonze X and Lee C Dynamical Matrices Born Effective Charges Dielec-tric Permittivity Tensors and Interatomic Force Constants from Density-Functional Perturbation Theory Phys Rev B vol 55 (16) 10355ndash10368(1997)[doi101103PhysRevB5510355] 2
[112] Gould T Lebeacutegue S and Dobson J F Dispersion corrections ingraphenic systems a simple and effective model of binding Journal ofPhysics Condensed Matter vol 25 (44) 445010 (2013)[doi1010880953-89842544445010] J1
[113] Gould T Liu Z Liu J Z Dobson J F Zheng Q and LebAtildeumlgue SBinding and interlayer force in the near-contact region of two graphite slabsExperiment and theory The Journal of Chemical Physics vol 139 (22) 224704 (2013)[doi10106314839615] 64 62 65 62 J1 J
[114] Gould T Simpkins K and Dobson J F A theoretical and semiempricalcorrection to the long-range dispersion power law of stretched graphitePhysical Review B vol 77 165134 (2008)[doi101103PhysRevB77165134][arxiv08014426] 62 J1
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[115] Graziano G Klimes J Fernandez-Alonso F and Michaelides AImproved description of soft layered materials with van der Waals densityfunctional theory Journal of Physics Condensed Matter vol 24 (42) 424216 (2012)[doi1010880953-89842442424216] 62 62 J1
[116] Grimme S Accurate Description of van der Waals Complexes by DensityFunctional Theory Including Empirical Corrections Journal of Computa-tional Chemistry vol 25 (12) 1463ndash1473 (2004)[doi101002jcc20078] 24 62 62
[117] Grimme S Semiempirical GGA-type density functional constructed witha long-range dispersion correction Journal of Computational Chemistryvol 27 (15) 1787ndash1799 (2006)[doi101002jcc20495] 24 62 62 A A1 A J
[118] Grimme S Antony J Ehrlich S and Krieg H A consistent andaccurate ab initio parametrization of density functional dispersion correction(DFT-D) for the 94 elements H-Pu The Journal of Chemical Physics vol132 (15) 154104 (2010)[doi10106313382344] 62 J1
[119] Grimme S Muumlck-Lichtenfeld C and Antony J Noncovalent Interac-tions between Graphene Sheets and in Multishell (Hyper)Fullerenes TheJournal of Physical Chemistry C vol 111 (30) 11199-11207 (2007)[doi101021jp0720791] 62 J1
[120] Grochala W Diamond Electronic Ground State of Carbon at Temperat-ures Approaching 0K ANGEWANDTE CHEMIE-INTERNATIONALEDITION vol 53 (14) 3680-3683 (APR 1 2014) ISSN 1433-7851[doi101002anie201400131] A
[121] Gulans A Kontur S et al EXCITING a full-potential all-electron pack-age implementing density-functional theory and many-body perturbationtheory Journal of Physics Condensed Matter vol 26 (36) 363202 (2014)[doi1010880953-89842636363202] 25
[122] Guo Z Dong R et al Record Maximum Oscillation Frequency in C-FaceEpitaxial Graphene Transistors Nano Letters vol 13 (3) 942-947 (2013)[doi101021nl303587r] (document)
[123] Hamada I van der Waals density functional made accurate Physical
LXX
Review B vol 89 121103 (Mar 2014)[doi101103PhysRevB89121103] J1
[124] Hannon J B (2012) Private communication (document) 10
[125] Hannon J B Copel M and Tromp R M Direct Measurement of theGrowth Mode of Graphene on SiC (0001) and SiCeth000 1THORN Phys Rev Lettvol 107 (16) 166101 (October 2011)[doi101103PhysRevLett107166101] 82
[126] Harl J and Kresse G Accurate Bulk Properties from Approximate Many-Body Techniques Phys Rev Lett vol 103 056401 (July 2009)[doi101103PhysRevLett103056401] 71 71 72 J
[127] Hasegawa M and Nishidate K Semiempirical approach to the energeticsof interlayer binding in graphite Physical Review B vol 70 205431(2004)[doi101103PhysRevB70205431] 62 J1
[128] Hasegawa M Nishidate K and Iyetomi H Energetics of interlayerbinding in graphite The semiempirical approach revisited Physical ReviewB vol 76 (11) 115424 (September 2007)[doi101103PhysRevB76115424] 62 J1
[129] Hass J de Heer W A and Conrad E H The growth and morphologyof epitaxial multilayer graphene J Phys Condens Matter vol 20 (32) 323202 (2008)[doi1010880953-89842032323202] a
[130] Hass J Feng R et al Highly ordered graphene for two dimensionalelectronics Applied Physics Letters vol 89 (14) 143106 (2006)[doi10106312358299] (document)
[131] Hass J Varchon F et al Why Multilayer Graphene on 4H-SiC(0001Acircmacr)Behaves Like a Single Sheet of Graphene Physical Review Letters vol100 (12) (March 2008)[doi101103PhysRevLett100125504] (document) 92 142
[132] Hattab H NrsquoDiaye A T et al Interplay of Wrinkles Strain and LatticeParameter in Graphene on Iridium Nano Letters vol 12 (2) 678-682(2012)[doi101021nl203530t] 93 93
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[133] Havu V Blum V Havu P and Scheffler M Efficient O (N) integ-ration for all-electron electronic structure calculation using numeric basisfunctions J Comp Phys vol 228 (22) 8367ndash8379 (2009)[doi101016jjcp200908008] 25 61
[134] Haynes W CRC Handbook of Chemistry and Physics 95th Edition CRCHandbook of Chemistry and Physics Taylor amp Francis 95 ed (2015)[ISBN 9781482208672]httpwwwhbcpnetbasecom 71 71
[135] He G Srivastava N and Feenstra R Formation of a Buffer Layer forGraphene on C-Face SiC0001 Journal of Electronic Materials vol 43 1-9(2013)[doi101007s11664-013-2901-8] 142
[136] Hellmann H Zur Rolle der kinetischen Elektronenenergie fAtilde14r die zwis-
chenatomaren Kraumlfte Zeitschrift fuumlr Physik vol 85 (3-4) 180-190 (1933)[doi101007BF01342053] 25
[137] Hertel S Waldmann D et al Tailoring the graphenesilicon carbideinterface for monolithic wafer-scale electronics Nature Comm vol 3 957(2012)[doi101038ncomms1955] (document) 133
[138] Heyd J Scuseria G E and Ernzerhof M Hybrid functionals basedon a screened Coulomb potential The Journal of Chemical Physics vol118 (18) 8207-8215 (2003)[doi10106311564060] (document) 233 233 233 61 C1C2 C3 J
[139] Hiebel F Magaud L Mallet P and Veuillen J-Y Structure andstability of the interface between graphene and 6H-SiC(0001) (3times 3) anSTM and ab initio study Journal of Physics D Applied Physics vol 45 (15) 154003 (2012)[doi1010880022-37274515154003] (document) 1411412 142 1412 143 1412 147 148 1413 142 J
[140] Hiebel F Mallet P Magaud L and Veuillen J-Y Atomic and elec-tronic structure of monolayer graphene on 6H-SiC(0001) A scanningtunneling microscopy study Phys Rev B vol 80 (23) 235429 (Decem-ber 2009)[doi101103PhysRevB80235429] (document) 141 14121413
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[141] Hiebel F Mallet P Varchon F Magaud L and Veuillen J-YGraphene-substrate interaction on 6H-SiC(0001) A scanning tunnelingmicroscopy study Phys Rev B vol 78 153412 (Oct 2008)[doi101103PhysRevB78153412] (document) b III 142
[142] Hirshfeld F Atom Fragments for Describing Molecular Charge-DensitiesTheoretica Chimica Acta vol 44 129 - 138 (1977)[doi101007BF00549096] 24 123 143
[143] Hoechst H Tang M Johnson B C Meese J M Zajac G W andFleisch T H The electronic structure of cubic SiC grown by chemicalvapor deposition on Si(100) Journal of Vacuum Science amp Technology Avol 5 (4) 1640-1643 (1987)[doi1011161574537] 72 74 B3 J
[144] Hohenberg P Kohn W Rajagopal A K and Callaway J Inhomogen-eous Electron Gas Phys Rev vol 136 B864-B871 (November 1964)[doi101103PhysRev136B864] 21 21 21
[145] Hoster H E Kulakov M A and Bullemer B Morphology and atomicstructure of the SiC(0001)3 x 3 surface reconstruction Surface Science vol382 (1ndash3) L658 ndash L665 (1997)[doi101016S0039-6028(97)00084-8] (document) 1411412 142 1412 147 1413 J
[146] Hu Y Ruan M et al Structured epitaxial graphene growth and proper-ties Journal of Physics D Applied Physics vol 45 (15) 154010 (2012)(document)
[147] Huang H Zheng F Zhang P Wu J Gu B-L and Duan W Ageneral group theoretical method to unfold band structures and its applica-tion New Journal of Physics vol 16 (3) 033034 (2014)[doi1010881367-2630163033034] 11
[148] Humphreys R Bimberg D and Choyke W Wavelength modulatedabsorption in SiC Solid State Communications vol 39 (1) 163 - 167(1981) ISSN 0038-1098[doi1010160038-1098(81)91070-X] 72 B3 J
[149] Iijima S Helical microtubules of graphitic carbon Nature vol 354 56 ndash58 (November 1991)[doi101038354056a0] 6
LXXIII
[150] Janak J F Proof that partEpartni
= ε in density-functional theory Physical ReviewB vol 18 7165ndash7168 (December 1978)[doi101103PhysRevB187165] 22
[151] Jansen H J F and Freeman A J Structural and electronic propertiesof graphite via an all-electron total-energy local-density approach PhysicalReview B vol 35 (15) 8207ndash8214 (May 1987)[doi101103PhysRevB358207] J1
[152] Ji S-H Hannon J B Tromp R M Perebeinos V Tersoff J andRoss F M Atomic-Scale transport in epitaxial graphene Nature Materials(2011)[doi101038NMAT3170] (document)
[153] Johannsen J C Ulstrup S et al Electron-phonon coupling in quasi-free-standing graphene Journal of Physics Condensed Matter vol 25 (9) 094001 (2013)[doi1010880953-8984259094001] 13
[154] Johansson L Owman F Maringrtensson P Persson C and LindefeltU Electronic structure of 6H-SiC (0001) Phys Rev B vol 53 (20) 13803ndash13807 (May 1996)[doi101103PhysRevB5313803] 72 811
[155] Junquera J Paz O Sagravenchez-Portal D and Artacho E NumericalAtomic Orbitals for Linear-Scaling Calculations Phys Rev B vol 64 (23) 235111 (November 2001)[doi101103PhysRevB64235111] 25
[156] Kaumlckell P Furthmuumlller J and Bechstedt F Polytypism and surfacestructure of SiC Diamond and Related Materials vol 6 (10) 1346 - 1348(1997) ISSN 0925-9635[doi101016S0925-9635(97)00089-7] Proceeding of the 1stEuropean Conference on Silicon Carbide and Related Materials (EC-SCRM 1996) 811 811
[157] Kaumlckell P Wenzien B and Bechstedt F Influence of atomic relaxationson the structural properties of SiC polytypes from ab initio calculationsPhys Rev B vol 50 17037ndash17046 (December 1994)[doi101103PhysRevB5017037] 71 72 J
[158] Kaplan R Surface structure and composition of β- and 6H-SiC Surf Scivol 215 (1-2) 111 - 134 (1989) ISSN 0039-6028
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[doi1010160039-6028(89)90704-8] (document) 8 812812
[159] Karch K Pavone P Windl W Strauch D and Bechstedt F Abinitio calculation of structural lattice dynamical and thermal propertiesof cubic silicon carbide International Journal of Quantum Chemistryvol 56 (6) 801ndash817 (December 1995)[doi101002qua560560617] 7
[160] Katsnelson M I Novoselov K S and Geim A K Chiral tunnellingand the Klein paradox in graphene Nature Physics vol 2 (9) 620ndash625(2006)[doi101038nphys384] (document)
[161] Kedzierski J Hsu P-L et al Epitaxial Graphene Transistors on SiCSubstrates Electron Devices IEEE Transactions on vol 55 (8) 2078-2085(Aug 2008) ISSN 0018-9383[doi101109TED2008926593] (document)
[162] Kennedy C S and Kennedy G C The equilibrium boundary betweengraphite and diamond Journal of Geophysical Research vol 81 (14) 2467ndash2470 (1976)[doi101029JB081i014p02467] A A A3 A2 J
[163] Kerber T Sierka M and Sauer J Application of semiempirical long-range dispersion corrections to periodic systems in density functional theoryJournal of Computational Chemistry vol 29 (13) 2088ndash2097 (2008)ISSN 1096-987X[doi101002jcc21069] J1
[164] Khaliullin R Z Eshet H Kuumlhne T D Behler J and ParrinelloM Graphite-diamond phase coexistence study employing a neural-networkmapping of the ab initio potential energy surface Physical Review B vol 81 100103 (March 2010)[doi101103PhysRevB81100103] 61 A A A3 A2 J
[165] Kim H Choi J-M and Goddard W A Universal Correction ofDensity Functional Theory to Include London Dispersion (up to Lr Element103) The Journal of Physical Chemistry Letters vol 3 (3) 360-363 (2012)[doi101021jz2016395] J1
[166] Kim S Ihm J Choi H J and Son Y W Origin of anomalous electronicstructures of epitaxial graphene on silicon carbide Phys Rev Lett vol
LXXV
100 (17) 176802 (May 2008)[doi101103PhysRevLett100176802] 82 82 12 121
[167] Kleis J Schroumlder E and Hyldgaard P Nature and strength of bondingin a crystal of semiconducting nanotubes van der Waals density functionalcalculations and analytical results Phys Rev B vol 77 205422 (May2008)[doi101103PhysRevB77205422] 62 J1
[168] Kleykamp H Gibbs energy of formation of SiC A contribution to thethermodynamic stability of the modifications Ber Bunsenges phys Chemvol 102 (9) 1231ndash1234 (1998)[doi101002bbpc19981020928] 71 71 142
[169] Knuth F Carbogno C Atalla V Blum V and Scheffler M All-electron formalism for total energy strain derivatives and stress tensor com-ponents for numeric atom-centered orbitals Computer Physics Communica-tions vol 190 (0) 33 - 50 (2015) ISSN 0010-4655[doi101016jcpc201501003] 61 61 J
[170] Kohanoff J Electronic Structure Calculations For Solids And MoleculesTheory And Computational Methods Cambridge University Press(2006)[ISBN 0-521-81591-6] I
[171] Kohn W and Sham L J Self-Consistent Equations Including ExchangeAnd Correlation Effects Phys Rev vol 140 1133ndash1138 (1965)[doi101103PhysRev140A1133] 22 231 5
[172] Korth M and Grimme S Mindless DFT Benchmarking Journal ofChemical Theory and Computation vol 5 (4) 993-1003 (2009)[doi101021ct800511q] 232
[173] Kresse G and Furthmuumlller J Efficient iterative schemes for ab initiototal-energy calculations using a plane-wave basis set Phys Rev B vol 54 11169ndash11186 (October 1996)[doi101103PhysRevB5411169] 25
[174] Kroto H W Heath J R OrsquoBrien S C Curl R F and SmalleyR E C60 Buckminsterfullerene Nature vol 318 162 ndash 163 (November1985)[doi101038318162a0] 6
LXXVI
[175] Krukau A V Vydrov O A Izmaylov A F and Scuseria G EInfluence of the exchange screening parameter on the performance of screenedhybrid functionals The Journal of Chemical Physics vol 125 (22) 224106(2006)[doi10106312404663] (document) 233 233 233 61 72143 B4 C1 C2 C3 F1 J
[176] Ku W Berlijn T and Lee C-C Unfolding First-Principles BandStructures Physical Reviews Letters vol 104 216401 (May 2010)[doi101103PhysRevLett104216401] 11
[177] Kulakov M Henn G and Bullemer B SiC(0001)3 x 3-Si surfacereconstruction a new insight with a STM Surface Science vol 346 (1ndash3) 49 ndash 54 (1996)[doi1010160039-6028(95)00919-1] 145 1412 147 1412 J
[178] Kulakov M A Henn G and Bullemer B SiC(0001)3 x 3-Si surfacereconstruction a new insight with a STM Surface Science vol 346 (1ndash3) 49 ndash 54 (February 1996)[doi1010160039-6028(95)00919-1] 812
[179] La Via F Camarda M and La Magna A Mechanisms of growthand defect properties of epitaxial SiC Applied Physics Reviews vol 1 (3) 031301 (2014)[doi10106314890974] 811
[180] Lambrecht W Limpijumnong S Rashkeev S and Segall BElectronic band structure of SiC polytypes a discussion of theory andexperiment phys stat sol (b) vol 202 (1) 5ndash33 (July 1997)[doi1010021521-3951(199707)2021lt5AID-PSSB5gt30CO2-L]71
[181] Lambrecht W R L Segall B et al Calculated and measured uv reflectiv-ity of SiC polytypes Physical Review B vol 50 10722ndash10726 (October1994)[doi101103PhysRevB5010722] 72
[182] Langreth D C Dion M Rydberg H Schroumlder E HyldgaardP and Lundqvist B I Van der Waals density functional theory withapplications International Journal of Quantum Chemistry vol 101 (5) 599ndash610 (October 2005)[doi101002qua20315] 62 62
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