arxiv:0907.0122v2 [cond-mat.mes-hall] 1 oct 2009

arX

iv:0

907.

0122

v2 [

cond

-mat

.mes

-hal

l] 1

Oct

200

9 Electronic properties ofgraphene antidot lattices

J A F�urst,1 J G Pedersen,2 C Flindt,3 N A M ortensen,2 M

B randbyge,1 T G Pedersen4 and A -P Jauho1;5

1 Departm entofM icro and Nanotechnology,TechnicalUniversity ofDenm ark,DTU

Nanotech,DTU-building 345 east,DK -2800 K ongensLyngby,Denm ark2 Departm entofPhotonicsEngineering,TechnicalUniversity ofDenm ark,DTU

Fotonik,DTU-building 343,DK -2800 K ongensLyngby,Denm ark3 Departm entofPhysics,Harvard University,17 O xford Street,Cam bridge,02138

M assachusetts,USA4 Departm entofPhysicsand Nanotechnology,Aalborg University,DK -9220 Aalborg

�,Denm ark5 Departm entofApplied Physics,HelsinkiUniversity ofTechnology,P.O .Box 1100,

FI-02015 TK K ,Finland

E-m ail:[email protected]

A bstract. G raphene antidot lattices constitute a novel class of nano-engineered

graphenedeviceswith controllableelectronicand opticalproperties.An antidotlattice

consistsofa periodic array ofholeswhich causesa band gap to open up around the

Ferm ilevel,turning graphene from a sem im etalinto a sem iconductor. W e calculate

the electronic band structure of graphene antidot lattices using three num erical

approacheswith di�erentlevelsofcom putationalcom plexity,e�ciency,and accuracy.

Fast�nite-elem entsolutionsofthe Dirac equation capture qualitative featuresofthe

band structure,whilefulltight-binding calculationsand density functionaltheory are

necessary form ore reliable predictionsofthe band structure. W e com pare the three

com putationalapproachesand investigatetheroleofhydrogen passivation within our

density functionaltheory schem e.

PACS num bers:71.15.M b,73.20.At,73.21.Cd

http://arxiv.org/abs/0907.0122v2

Electronic propertiesofgraphene antidotlattices 2

1. Introduction

Since itsdiscovery in 2004 [1,2],graphene hasbecom e a research �eld oftrem endous

interest within the solid state physics com m unity [3]. The interest stem s from the

particular electronic properties ofgraphene as wellas the prom ising perspectives for

future technologicalapplications[4].The electronic excitationsaround the Ferm ilevel

ofgrapheneresem blethoseofm assless,relativisticDiracferm ions,allowing predictions

from quantum electrodynam ics to be tested in a solid state system [5]. From a

technologicalpoint ofview,severalfuture applications have already been envisioned.

These include the use of graphene for single m olecule gas detection [6], graphene-

based �eld-e�ecttransistors[1],and quantum inform ation processingin nano-engineered

graphene sheets [7]. Additionally, graphene is the strongest m aterial ever tested,

suggesting theuseofcarbon-�berreinforcem entsin novelm aterialcom posites[8].

M etam aterialsconstituteanotherpopular�eld ofresearch in contem porary science.

Contrary to conventional, naturally occurring m aterials, m etam aterials derive their

properties from their arti�cial,m an-m ade,periodic sm all-scale structure rather than

their chem ical or atom ic com position [9]. W hen properly designed and fabricated,

m etam aterialso�eroptim ized and unusual,som etim eseven counter-intuitive,responses

to speci�cexcitations[10].Exam plesincludem etam aterialswith negativeperm ittivity

and perm eability [11],superlenses[12,13],and cloaking devices[14].Photonic[15,16]

and phononic [17] crystals are closely related to m etam aterials, although they are

typically designed to alter the response to electrom agnetic and acoustic excitations,

respectively, at wavelengths sim ilar to the dim ensions of the sm all-scale structure.

The realization ofarti�cialband structuresin two-dim ensionalelectron gassesm ay be

pursued with sim ilarapproaches[18,19],allowing the form ation ofe.g.Dirac conesin

conventionalantidotlattices[20,21].

Basedontheaboveideas,som eofushaverecentlyproposedtoalterinacontrollable

m anner the electronic and optical properties of graphene by fabricating a periodic

arrangem entofperforationsorholesin a graphene sheet[22]. W e referto thiskind of

structureasa grapheneantidotlatticeowing to itscloseresem blancewith conventional

antidot lattices de�ned on top ofa two-dim ensionalelectron gas in a sem iconductor

heterostructure [23,24]. Using tight-binding calculations we have shown that such

a periodic array ofholes in a graphene sheet causes a band gap to open up around

the Ferm ilevel[22], changing graphene from a sem im etalto a sem iconductor with

corresponding clearsignaturesin the opticalexcitation spectrum [25]. Soon afterour

proposal,grapheneantidotlatticeswererealized experim entally by Shen and co-workers

[26]and Erom s and W eiss [27] with lattice constants below 100 nm . The rapidly

im proving ability to pattern m onolayer �lm s with e-beam lithography suggests that

grapheneantidotlatticeswith typicaldim ensionstowardsthe10nm scalem aybewithin

reach [28,29]. Furtherm ore,Giritand co-workersrecently m onitored the dynam icsat

the edgesofa growing hole in realtim e using a transm ission electron m icroscope [30],

and Jia and co-workers dem onstrated a m ethod for producing graphitic nanoribbon


edges in a controlled m anner via Joule heating [31]. Very recently,Rodriguez-M anzo

and Banhart created individualvacancies in carbon nanotubes using a 1 �A diam eter

e-beam [32]. These advances suggest that fabrication ofnano-scale graphene antidot

latticeswith desired holegeom etriesm ay bepossiblein thenearfuture.

In the endeavors of m odeling these structures one is faced with a com prom ise

between com putational e�ciency and accuracy. Sm all-scale lattices with perfect

periodicity and identicalfew-nm sized holes can be treated accurately with density

functionaltheory (DFT),but this is a com putationally heavy and tim e consum ing

approach, which lim its the possibilities to perform large, system atic studies. For

exam ple,in order to m odellattice disorder,such as variations in the hole geom etry

and alignm ent,itm ay benecessary to form a supercellcontaining severalholesatthe

costofan increased com putationaltim e.In orderto circum ventthisproblem ,onecan

m ake use ofthe pseudo-relativistic behaviorofelectronsin bulk graphene close to the

Ferm ileveland solve the corresponding Diracequation using com putationally cheaper

m ethods,however,possibly atthecostofa decreased com putationalaccuracy.

The aim ofthispaperisto study the band structure ofgraphene antidotlattices

using threenum ericalapproachesofdi�erentcom putationalcom plexity,e�ciency,and

accuracy. W e �rstdevelop a com putationally cheap schem e based on a �nite-elem ent

solution ofthe Dirac equation. This m ethod gives reasonable predictions forthe size

ofthe band gap due to the antidotlattice,buthaslim ited accuracy in predicting the

fullband structure.Forbetterpredictionsoftheband structure,weem ploy a �-orbital

tight-binding schem e,which is stillnum erically cheap and capable oftreating larger

antidotlattices.Theresultsarecom pared with com putationallydem anding,full- edged

ab initio calculations,based on density functionaltheory,which we expect to predict

theband structurewith thehighestaccuracy.Thetight-binding calculationsagreewell

with qualitativefeaturesoftheband structurecalculationsbased on density functional

theory,although som edi�erencesarefound on a quantitative level.Finally,wediscuss

hydrogen passivation alongtheedgesoftheholesin agrapheneantidotlatticeand study

thein uenceon theelectronicpropertiesusing density functionaltheory.

The paper is organized as follows: In Section 2 we introduce graphene antidot

lattices and give a briefoverview ofthe existing literature on the topic. In Section 3

we describe ourthree com putationalapproaches;�nite-elem ent solutions ofthe Dirac

equation (DE),a �-orbitaltight-binding schem e (TB),and density functionaltheory

calculations(DFT).A com parison and discussion oftheresultsobtained usingthethree

m ethodsaregiven in Section 4.Finally,wediscussin Section 5thein uenceofhydrogen

passivation on theband structure,beforestating ourconclusionsin Section 6.

2. G raphene antidot lattices

A graphene antidot lattice consists ofa periodic arrangem ent ofholes in a graphene

sheet[22].In thefollowing,weconsidera hexagonallatticeofcircularholes,butother

latticestructures,e.g.squarelattices,with di�erentholesshapesareexpected toexhibit


{12, 3} {7, 3} {10, 6.4}

R

L

Figure 1. Unitcellsofthree hexagonalgraphene antidotlatticeswith di�erentside

lengthsL and hole radiiR. The structuresare denoted asfL;Rg with both lengths

m easured in unitsofthe graphenelatticeconstanta ’ 2:46 �A.Herewehaveassum ed

thatthe edgesofthe holeshave been hydrogen passivated (hydrogen shown aswhite

atom s).

sim ilarphysics.In particular,weanticipatean opening ofaband gap around theFerm i

levelforalargeclassofantidotlattices[33].Thehexagonalunitcellswith di�erenthole

sizesareshown in Fig.1.Thestructuresarecharacterized by thesidelengthsL ofthe

hexagonalunitcellsand theapproxim ateradiiR oftheholes,both m easured in unitsof

thegraphenelatticeconstanta =p3lC ’ 2:46 �A,wherelC = 1:42 �A isthebond length

between neighboring carbon atom s. In Fig.1,the holesare assum ed to be passivated

with hydrogen,using thebond length 1.1 �A between neighboring carbon and hydrogen

atom s. Throughout the paper,we denote a given structure asfL;Rg,where L is an

integer,butR notnecessarily.W ewillconsideronly very sm allstructureswith L � 10.

Although itm ay notbe conceivable to fabricate such sm allstructureswithin the near

future,thesm allunitcellsallow forsystem aticcom parisonsofourthreecom putational

schem es. In particular,with sm allunit cells we can perform com putationally heavy

DFT calculations.Im portantly,sim plescalingrelationshavebeen dem onstrated forthe

sizeoftheband gap in term softhetotalnum berofatom sand thenum berofrem oved

atom swithin a unitcell,m aking itpossible to extrapolate resultsto largergeom etries

[22]. Such scaling relations m ay be helpfulwhen m odeling on-going experim ents on

grapheneantidotlattices[26,27].

In ouroriginalproposalforgrapheneantidotlattices,wefocused on thepossibility

offabricating intentional‘defects’by leaving out one or m ore holes in the otherwise

periodic structure [22]. Aswe showed,such defects lead to the form ation oflocalized

electronicstatesatthelocationsofthedefectswith energiesinsidetheband gap.Several

such (possibly coupled)defectswould then form a platform forcoupled electronic spin

qubitsin a graphene-based quantum com puting architecture [22]. Sim ilarideasbased

on conventionalantidot lattices de�ned on a two-dim ensionalgas in a sem iconductor


heterostructurehavepreviously been studied bysom eofus[18,19].However,asalready

m entioned, the perfectly periodic graphene antidot lattice constitutes an interesting

structure on its own. In particular, the controllable opening of a band gap m ay

potentially pave the way forgraphene-based sem iconductordevices. In Ref.[25]som e

ofus studied the opticalproperties ofgraphene antidot lattices, showing that they

behaveasdipole-allowed directgap two-dim ensionalsem iconductorswith apronounced

optical absorption edge. Additional studies of the electronic properties have been

perform ed by Vanevi�c, Stojanovi�c, and Kinderm ann [34] as well as by som e of us

[33]. Vanevi�c and co-workers studied in detailthe occurrence of at bands due to

sublattice im balances and irregularities in the hole shapes at the atom ic level. In

ourstudy,we addressed the rolesofgeom etry relaxation and electron spin using DFT

calculations.Very recently,Rosalesand co-workersstudied the transportpropertiesof

antidotlatticesalong graphene nanoribbons[35]. Turning around the ideasofm aking

graphene sem iconducting using periodic superlattices,ithasrecently been shown that

periodic potentialm odulationsm ay create graphene-like electronic band structures of

two-dim ensionalgases in sem iconductor heterostructures [20,21]. In that case, the

possibility to controlthe slope ofthe linear bands and thus the velocity ofthe Dirac

ferm ionsisofgreatinterest.

3. C om putationalm ethods

In the following we outline the three com putationalm ethods em ployed in this work.

As a com putationally cheap approach we consider �rst �nite-elem ent solutions ofthe

Dirac equation (DE).W ithin this approach,large unit cells can be treated and the

com putations are fast. The m ethod relies on the linear bands of bulk graphene

around the Ferm ilevel. Asa m ore re�ned approach,we considernext�-orbitaltight-

binding calculations(TB).Thism ethod goesbeyond the assum ption ofa linearband

structureofbulk graphene,and theedgesoftheantidotholescan becarefully treated,

including possiblee�ectsduetovalley m ixing.Finally,weconsiderfull- edged abinitio

calculationsusing DFT.W hilethism ethod iscom putationally heavy,DFT isa widely

used standard fordoing�rstprinciplescalculationsand weexpectittoprovidethem ost

detailed description oftheelectronicstructure.

3.1.Dirac equation (DE)

W e�rstdescribeour�nite-elem entsolutionsoftheDiracequation.Them ethod isbased

on the band structure ofbulk graphene close to the two Dirac pointsbeing linearand

welldescribed by the Dirac equation [3]. W ithin this picture,the atom ic honeycom b

lattice structure ofgraphene isreplaced by an e�ective continuum description. Asan

exam ple,we show in Figs.2a and 2b,respectively,a graphene antidotlattice unitcell

and the corresponding continuum dom ain on which the Dirac equation issolved. The

hole in the unitcellism im icked with a m assterm M (r)in the Dirac equation atthe


��

��

��

��

��

��

M (r)

n

φ

a) b) c)

Figure 2. Unitcell,continuum dom ain,and �nite-elem entm esh.a) Hexagonalunit

cellofthe f7;3g graphene antidot lattice. b) Corresponding continuum dom ain on

which the Dirac equation issolved. The hole (hatched area)ism odeled with a m ass

term M (r)in theDiracequation.Thenorm alvectorto theholen,form ing theangle

� with thehorizontalaxis,isused to de�neappropriateboundary conditionsalongthe

edgeofthehole(seetext)c)Corresponding�nite-elem entm esh on which wesolvethe

Diracequation.Theedgeoftheholeisshown with red.PeriodicBloch conditionsare

im posed on the outerboundary ofthe unitcell.

location ofthe hole; see explanation following Eq.(2). For large m asses,the Dirac

ferm ionsare e�ectively excluded from the location ofthe hole and the m assterm can

be replaced by appropriate boundary conditionsalong the edge ofthe hole,indicated

with red in Fig.2c. In Fig.2c we also show an exam ple ofthe �nite-elem entm esh on

which theDiracequation isdiscretized and solved.PeriodicBloch boundary conditions

areim posed on theouteredgesoftheunitcell,m aking theproblem equivalentto that

ofan in�nitely largegrapheneantidotlattice.

Electronic states close to one ofthe two Dirac points ofbulk graphene can be

expressed in term sofenvelope wave functionscontained in the two-com ponentspinor

jiwith onecom ponentcorresponding to each ofthetwo sublatticesin thehoneycom b

structure ofgraphene [3]. Spinors corresponding to states close to one ofthe Dirac

pointssatisfy theDiracequation

H ji= [� F p�� + M (r)�z]j i= E ji; (1)

where�F ’ 106 m s�1 istheFerm ivelocity[2],p = [px;py]isthem om entum ,� = [�x;�y]

is the pseudo-spin corresponding to the two sublattices,and M (r) is the m ass that

couplesto �z and isnon-zero only inside the holes. Spinorsassociated with the other

Dirac point satisfy Eq.(1) with the replacem ent � ! � � = [�x;� �y]. W ithin this

description,statesclose to di�erentDiracpointsareassum ed notto couple.The real-

space representation ofthe spinorjiis(r)� hrji= [ 1(r); 2(r)]T,where 1 and

2 are the envelope functions corresponding to each ofthe two sublattices. Equation

(1)iscorrespondingly written"

M (r) �i~�F (@x � i@y)

�i~�F (@x + i@y) �M (r)

#"

1(r)

2(r)

#

= E

"

1(r)

2(r)

#

: (2)

W e now consider the situation,where Dirac ferm ionsare excluded from the holes,by


takingthelim itM (r)! 1 insidetheholes.In thatlim it,wecan derivetheappropriate

boundary conditions for the spinor along the edges ofa hole and solve the resulting

problem outside the holes. The boundary conditionsare derived by requiring thatno

particle current runs into a hole. The particle current operatoris j� r p H = �F �,

and the localparticle currentdensity in thestate (r)isj(r)= y(r)j(r).Im posing

n � j(r)= 0 along the edge ofa hole with n being the outward-pointing norm alvector

to thehole,onecan derivethecondition 1(r)= ie�i� 2(r)along theboundary,where

theangle� isde�ned in Fig.2b.Thisprocedurewasoriginally developed by Berry and

M ondragoninstudiesofneutrinobilliards[36]andm orerecentlyem ployed byTworzyd lo

and co-workersin thecontextofgraphene[37].Along theouterboundariesoftheunit

cellweim poseperiodicBloch boundary conditions,and wearethusleftwith a system

ofcoupled di�erentialequations on a �nite-size dom ain with wellde�ned boundary

conditions. Problem s of this type are wellsuited for com m ercially available �nite-

elem ent solvers, and the num ericalim plem entation is relatively straightforward and

fastusing thestandard �nite-elem entpackageCOM SOL M ultiphysics[38].The �nite-

elem entsolverdiscretizesand solvestheproblem on an optim ized m esh ofthe�nite-size

dom ain.Them esh shown in Fig.2cwasgenerated with COM SOL M ultiphysics.

3.2.Tight-binding (TB)

W e nextdescribe ourtight-binding schem e. The Dirac equation approach introduced

above is a continuum description of the electronic properties, ignoring the detailed

atom ic structure ofgraphene and the edgesofthe holes,which m ay lead to scattering

between the two Dirac points. It m oreover assum es linear bands ofbulk graphene.

To capture e�ects ofthe atom ic structure,including the in uence ofedge geom etry,

and in order to incorporate a realistic description of the band structure of bulk

graphene,weneed to go beyond thesim pleDiracferm ion picture.In ourtight-binding

schem e,thestarting pointistheSchr�odingerequation fora singleelectron in real-space

representation

HTB (r)=

�

�~2

2m e

r 2 + V (r)

�

(r)= � (r); (3)

whereV isan e�ectivepotentialand m e istheelectron m ass.Theunknown eigenstate

j iissubsequently expanded in a setoflocalized \atom ic" wave functionsj~R;liasa

superposition j i =P

C ~R ;lj~R;li with expansion coe�cients C ~R ;l

. Here,each atom ic

stateislabeled by theorbitalsym m etry (l=s;px;pz...) and theposition oftheatom ~R.

Thistransform stheSchr�odingerequation into a m atrix equation readingX

~R 0;l0

h~R;ljH TBj~R 0;l0iC ~R 0;l0

= �X

~R 0;l0

h~R;lj~R 0;l0iC ~R 0;l0

: (4)

At this point, several approxim ations can be adopted in order to sim plify the

calculations. First, the atom ic orbitals are usually taken to be orthogonal, i.e.,

h~R;lj~R 0;l0i= �~R ;~R 0�l;l0. This m eans thatthe m atrix problem becom es a sim ple rather

than ageneralized eigenvalueproblem .Second,them atrixelem entsofH TB areregarded


as em piricalparam eters �tted,usually,to experim entaldata. In the sim plest tight-

bindingdescription ofplanarcarbonstructurescontained inthe(x;y)-plane,justasingle

pz or�-orbitalon each siteisconsidered and only nearest-neighborm atrix elem entsare

retained. This \hopping integral" is denoted as ��,with � � 3:033 eV [39]. Other

valuesofthehoppingintegralcan alsobefoundin theliterature.Forexam ple,thechoice

� � 2:7 eV provides low-energy band structures for bulk graphene consistently with

density functionaltheory calculations[40]. However,the Ferm ivelocity isdeterm ined

by therelation �F =p3�a=2~ and by choosing � � 3:033eV,weobtain �F = 9:9� 105

m s�1 in good agreem entwith experim ents[2].

The reason for considering only �-orbitals is that �-orbitals with odd z-parity

decouplefrom the�-orbitalsspanned by s;px,and py statesthatallhaveeven z-parity.

M oreover,the bands in the vicinity ofthe band gap are allproduced by the loosely

bound �-orbitals.Hence,forallstructuresconsidered in thepresentwork,weneed only

include�-orbitalsexplicitly.Also,even though realisticstructureswillcontain hydrogen

term inated edges,the hydrogen atom scouple only to the �-orbitalsand are therefore

irrelevant for �-states. In a m ore sophisticated m odel,bare or hydrogen term inated

edgeslead to a sm allm odi�cation ofthe�-electron hopping integralsnearan edgedue

to relaxation ofthegeom etry.Thism odi�cation isignored asitsim ply leadsto a sm all

additionalopening oftheband gap [22].

3.3.Density functionaltheory (DFT)

Finally, we discuss our DFT calculations. This m ethod provides the m ost detailed

description ofgraphene antidot lattices,and we expect it to yield the m ost accurate

results.Theaccuracycom esatthecostofthem ethod beingnum ericallydem andingand

therequired com putationalresourcesexceed thosetypically availableon astandard PC.

Density functionaltheory isa widely used standard forelectronicstructurecalculations

and weshallhereonly brie y outlinetheunderlying theory [41].

Them ethod takesasstartingpointthefullinteractingm any-bodysystem involving

allelectrons and atom nucleim aking up the graphene antidot lattice. Diagonalizing

the corresponding m any-body Ham iltonian isa trem endoustask,butthe problem can

be brought to a som ewhat sim pler form using the Born-Oppenheim er approxim ation

in which the positions ofthe nucleiare �xed. W e are then considering a system of

interacting electrons m oving in an externalpotentialcreated by the nucleiat �xed

positions. This is stilla di�cult m any-body problem ,but further advances can be

m ade following Hohenberg and Kohn who showed that the ground state energy is

uniquely determ ined by the ground state electron density [42]. Kohn and Sham (KS)

later realized that this density can be obtained from a single-particle picture ofnon-

interacting electrons.Thecorresponding Ham iltonian forthesingle-particleKS orbitals

i isexpressed by theKS equationsas[43]

HK S i(r)=

�

�1

2r 2 + Ve�(r)

�

i(r)= �i i(r); (5)


wherethee�ectivepotential

Ve�(r)=

Z

dr0 �(r

0

)

jr� r0

j+ Va(r;fR iag)+

�Exc[�(r)]

��(r)(6)

dependsexplicitlyon thedensity�(r)=P

ioj i(r)j

2 with thesum runningoveroccupied

KS orbitals. Here,Va(r;fR iag)isthe externalpotentialdue to the atom satpositions

R ia.Theso-called exchange-correlation term E xc(r)accountsforallm any-body e�ects

and isnotknown exactly,butm ustbeappropriately approxim ated.Finally,theground

stateenergy oftheinteracting problem is

E [�(r)]= T[�(r)]+

Z

dr�(r)Va(r;fR iag)

+1

2

Z Z

drdr0 �(r)�(r

0

)

jr� r0

j+ E xc[�(r)]; (7)

whereT isthekineticenergy corresponding to thedensity �(r).

W earenow leftwith theproblem ofdeterm ining thedensity �(r).Thedensity isa

functionofonlythreecoordinates,unliketheN -particlewavefunction of3N coordinates.

ThesetofKS equationsissolved self-consistently:starting from an initialdensity,the

e�ective potentialis com puted together with the KS orbitals and the corresponding

density, and this procedure is repeated until convergence has been reached. The

band structure can then by calculated corresponding to the chosen coordinatesofthe

nucleiR ia. The totalenergy ofthe system can further be m inim ized with respect

to the coordinates of the nuclei. This is referred to as geom etry relaxation. The

m ethod can easily be extended to include spin as wellas di�erent species ofnuclei.

In this work, we use spin-polarized DFT as im plem ented in the Siesta code [44].

The structures are relaxed using com putationally cheaper DFT based tight-binding

m ethods [45]. Perform ing electronic structure calculations using DFT on geom etries

relaxed in thisway isknown to provide accurate results[33]. Ascom m only done,the

core electrons are replaced by pseudo-potentials and the rem aining valence-electrons

are described with localized atom ic orbitals. For the exchange-correlation potential

weem ploy thewidely used Perdew-Burke-Ernzerhofparam etrization ofthegeneralized

gradientapproxim ation [46].W em ainly usea so-called double-� polarized (DZP)basis

set size, consisting of13 functions per carbon atom . Contrary to the DE and TB

m ethods,theantidotedgesarehydrogen-passivated intheDFT calculations.Thee�ects

ofpassivation are discussed in Section 5. FurtherdetailsofourDFT calculationscan

befound in Ref.[33].

4. B and structures

W enow presentand com pareourresultsfortheelectronicband structureobtained using

thethreem ethodsdescribed in theprevioussection.Thisprovidesvaluableinsightinto

thephysicsdom inating theelectronic propertiesofgrapheneantidotlatticesaswellas

an indication ofthe range ofvalidity ofthe less com putationally expensive m ethods.


0

0.5

1

0

0.5

1

1.5

2

0

0.5

1

1.5

2

DETB

E(e

V)

K Γ M K K Γ M K K Γ M K

{12, 3} {7, 3} {10, 6.4}

DETB

Figure 3. Band structures ofthree representative graphene antidot lattices. Full

linesindicateresultsobtained by solving theDiracequation (DE),whiletight-binding

results(TB)areshown with red dashed lines.W ithin thesecom putationalapproaches

we have exact particle-hole sym m etry, and consequently only positive energies are

shown.Note thedi�erentenergy scaleon the leftm ost�gure.

Both the �nite-elem ent solutions of the Dirac equation (DE) and our tight-binding

calculations (TB) were carried out on a standard PC,and a single band structure

calculation could typically be perform ed in a few m inutesforthe relatively sm all-scale

graphene antidot lattices considered in the following. The density functionaltheory

calculations(DFT)werecarried outon 8 AM D Opteron CPUsin paralleland typically

lasted around 48 hours.UnliketheTB and theDFT m ethods,thecom putationaltim e

ofour DE schem e does not increase with the size ofthe unit cell,determ ined by L,

butonly dependson the ratio R=L. Forlarge unitcells,the DE schem e willtherefore

outperform both theTB and theDFT m ethodsin term sofcom putationaltim e.

In Fig.3 weshow band structureresultsforthreerepresentative grapheneantidot

lattices using DE and TB.Both m ethods predict band gaps ofa few hundred m eVs

fortheserelatively sm alldim ensionsofgrapheneantidotlattices.Forlow energies,DE

predictswellthequalitativefeaturesofthebandsobtained usingTB,butthedeviations

becom e pronounced athigherenergies. Thisisnotsurprising asthe Dirac equation is

only a valid description atlow energies,where the band structure ofbulk graphene is

linear. Roughly,thism eansenergiesbelow 0:1� ’ 0:3 eV.Additionally,the increased

kineticenergyduetothecon�nem entoftheparticlesrenderstheDE resultslessaccurate

forlarge antidotradiirelative to the dim ensions ofthe unitcell. This isapparent in

the �gure,where the bands at higher energies becom e increasingly inaccurate as the

antidotradiusisincreased. However,even forthe fL;Rg = f10;6:4g structure,there


f12;3g f7;3g f10;6:4g

eV � f12;3g eV � f12;3g eV � f12;3g

DE 0.54 (0.29) 1 1.27 (0.82) 2.35 (2.83) 1.53 (1.22) 2.83 (4.21)

TB 0.23 1 0.74 3.22 1.01 4.39

DFT 0.19 1 0.61 3.21 0.82 4.32

Table 1. Band gaps of three representative graphene antidot lattices. W e show

results obtained by solving the Dirac equation (DE),via tight-binding calculations

(TB),and using density functionaltheory (DFT).Valuesin parenthesesareobtained

using DE and corrected for the low-radius behavior (see text). The band gaps are

given in eV aswellasin dim ensionlessvaluesrelative to the size ofthe band gap for

the fL;Rg= f12;3g structure.

isa qualitativeagreem entbetween theshapesofthebandsatlow energiesfound using

thetwo m ethods.

W hile the shapes ofthe bands at low energies are approxim ately the sam e for

the DE and TB approaches, the sizes ofthe band gaps vary signi�cantly. For the

fL;Rg = f12;3g structure, the band gap predicted by DE is m ore than twice as

large as thatobtained using TB.These di�erences m ay be traced back to two ofthe

underlying assum ptionsofDE:linearbandsofbulk grapheneand absenceofscattering

between the two Dirac points. In orderto illum inate the discrepancy we considertwo

lim iting cases.W e�rstconsiderthelim itoflargeunitcells,i.e.,largevaluesofL.By

investigating a large sam ple ofdi�erentgraphene antidotlattice using TB,som e ofus

have dem onstrated a sim ple scaling-law between the hole size and the band gap E g,

showing thatE g /pN hole=N cellforsm allvaluesoftheratio R=L [22].Here,N hole / R 2

isthenum berofcarbon atom sthathavebeen rem oved from theintactunitcellin order

to createthehole,and N cell/ L2 isthetotalnum berofcarbon atom sin theintactunit

cell(before the hole ism ade). W e then �nd E g / (R=L)=L,showing thatfora �xed

value ofthe geom etric ratio R=L,the band gap E g fallso� as1=L with increasing L.

Forsu�ciently largeunitcellswethusexpecttheband gap tobewellwithin theenergy

window forwhich theelectronicbandsofbulk graphenein factarelinear,and theband

gapsobtained using DE should thusagreebetterwith TB.Thelim itofsm allholes,i.e.,

R going to 0,isanotherim portantcheck pointofourm ethods. In the DE approach,

theboundary condition along theedgeofa holeenforcesa phaseshiftbetween thetwo

spinorcom ponents,given by the angle � indicated in Fig.2b. W ith R going to 0,the

phaseshiftm ustoccuratasinglepointin space,resultingin acom pletely undeterm ined

phase relationship atthispoint. Consequently,there isno adiabatic transition from a

graphene antidotlattice to bulk graphene in the lim itofvanishing hole sizes. Indeed,

forsm allvaluesofR,we �nd a non-vanishing band gap using DE,and extrapolating

theresultsto R = 0 we�nd a band gap oftheapproxim atesize1:02�=L.Ifwecorrect

forthisby sim ply subtracting thisvalue from the band gapscalculated using DE,we

�nd better agreem ent with the results obtained using TB,as shown in Table 1. For


-1

-0.5

0

0.5

1

TBDZP

E(e

V)


{12, 3} {7, 3} {10, 6.4}

TBDFT

Figure 4.Band structuresofthreerepresentativegrapheneantidotlattices.Fulllines

indicate results obtained using density functionaltheory (DFT),while tight-binding

results(TB)are shown with red dashed lines. W ithin the DFT schem e,particle-hole

sym m etry isnotassum ed,and wethusshow resultsforenergiesboth aboveand below

the Ferm ienergy atzero.

largevaluesofL,thecorrection tendsto zero,aswewould expect.

In Table1 we also show resultsforthe band gapsusing density functionaltheory.

The band gapscalculated using DFT are within 30% ofthe corresponding TB results,

with DFT consistently reporting lower band gaps than TB.This follows the general

tendency thatenergy gaps are underestim ated in DFT [47]. Forthe three structures

shown here,the band gaps increase with increasing relative hole size. This trend is

captured wellbyallthreem ethods.Theband structurescalculated with DFT areshown

in Fig.4,togetherwith resultsobtained using TB forcom parison. Generally,there is

a reasonable qualitative agreem entbetween thetwo m ethodsin term softhe shapesof

the bands,in particular,at energies close to the Ferm ilevel. At larger energies,the

qualitative featuresstartto deviatesigni�cantly.Unlike theTB calculations,theDFT

approach doesnotim ply particle-holesym m etry,and thecorrespondingband structures

arenotsym m etric around theFerm ilevelatzero.Thisdi�erenceisclearly seen in the

�gure.

In contrast to the DE and TB calculations,our DFT schem e also includes the

spin degree offreedom and is thereby able to predict the m agnetic properties ofthe

grapheneantidotlattice.W ithin theTB description,grapheneisconsidered a bipartite

latticestructurewith twosublattices,A and B ,with non-zerohoppingelem entsbetween

di�erentsublatticesitesonly.In thatcase,thetotalm agneticm om entperunitcellM ,

canbedeterm ined from Lieb’stheorem [48],statingthatM = N A � N B withN A (B )being


-1

-0.5

0

0.5

1

DZP

E(e

V)


{12, 3} {7, 3} {10, 6.4}

DZPSZ

b

a

Figure 5.Band structuresofthreerepresentativegrapheneantidotlatticescalculated

with DFT.Fulllinesindicate resultsobtained using the DZP basisset,while dashed

linescorrespond to the sm allerSZ basisset(seetext).The real-spacerepresentations

ofthe statescorresponding to the pointsa and bareshown in Fig.6.

thenum berofsitesofsublattice A(B )in theunitcell.By inspection ofthestructures

in Fig.1,we see thatthey have zero sublattice im balance,i.e.,N A = N B ,and we thus

expecta zero totalm agneticm om entaccording to Lieb’stheorem .Although ourDFT

calculationsarenotbased on a description ofgraphenein term softwo sublatticeswith

onlynearest-neighborcoupling,westill�nd azerototalm agneticm om ent.Additionally,

we�nd thatnolocalm agneticm om entsareform ed in anyoftheinvestigated structures.

Lieb’s theorem does not concern the form ation oflocalm agnetic m om ents, but the

absence in the present casescan be understood from the circularshapes ofthe holes,

which inhibitthe form ation oflongerzig-zag shaped partsofthe edge. Thisissim ilar

to resultsobtained forgraphene akes,where relatively large zig-zag partsare needed

forlocalm agneticm om entsto form [49].W ethusconclude thatthebandsareallspin

degeneratein thecaseswehaveinvestigated.

W ithin ourDFT schem e,thecom putationaltim ecan bereduced by usingasm aller

basisset. W e thuscom pare resultsobtained with the double-� polarized (DZP)basis

set involving 13 basis functions per carbon atom ,used thus far,and results obtained

using a single-� (SZ) basis with only 4 basis functions per carbon atom . Results for

the band structures obtained using the two di�erent basis sets are shown in Fig. 5.

Theband structuresobtained using thesm allerSZ basisagreewellwith thoseobtained

usingtheDZP basis,and thecom putationaltim eissigni�cantlyreduced.An interesting

di�erence between the band structuresobtained using DFT com pared to DE and TB,

is the very low dispersion ofthe band roughly 0.5 eV below the Ferm ilevelfor the


Figure 6. Real-space representation ofelectronic states. The leftpanelcorresponds

to thepointon the atband in Fig.5 indicated by a.Forcom parison,therightpanel

showsthe state corresponding to the pointb in Fig.5. W e show the absolute square

ofthe wavefunctions.

fL;Rg = f10;6:4g structure. The absolute square ofthe wavefunction forone ofthe

spin degenerate statesatthe �-point,denoted by a in Fig.5,isshown in Fig. 6. For

com parison,we also show the state denoted by bin Fig.5.The state on the atband

isstrongly localized on the zig-zag partsofthe edge. The lowerdispersion com pared

to TB ispossibly dueto thegradually increasing totalelectronicpotentialwithin DFT,

when approaching the edge ofa hole. The increased on-site energy ofthe edge atom s

within DFT m ay thuscausestrongerlocalization.

5. Passivation

Finally,we discuss the in uence ofedge passivation ofthe holes with hydrogen. In

orderto addressthisquestion weem ploy ourDFT schem e.Detailsoftheedgesarenot

considered within our�nite-elem entsolutionsofthe Dirac equation (DE),and within

a tight-binding description (TB),passivation istypically included sim ply asa shiftof

thehopping integralbetween carbon atom salong theedgesdueto therelaxed carbon-

carbon bond length [50].Thiscorrection leadstoslightly increased energy gapsbuthas

been ignored in the TB calculations in the present work. In contrast,DFT carefully

treatsthepresence ofhydrogen along theedgeofa hole,and,im portantly,them ethod

includesthe spin degreesoffreedom ,which turnsoutto be crucialin determ ining the

in uence ofpassivation on the electronic properties. W e consider as an illustrative

exam ple the structure fL;Rg = f4;2g depicted in Fig.7,shown with and without

hydrogen passivation.W enotethattheholegeom etry in thiscaseishexagonal,leading

again to a vanishing m agneticm om entwithoutpassivation.

The resulting band structures, shown in Fig. 8, with and without passivation

are very di�erent. W ith fullhydrogen passivation,severalbandsare spin degenerate.

Thisdegeneracy islifted withoutpassivation,and low dispersion bandsstem m ing from

danglingbondsareclearly present.Thedispersionsofthesebandsariseduetocoupling


Figure 7.TheunitcellofthefL;Rg= f4;2g structure.Thestructureisshown with

(right) and without (left) com plete hydrogen passivation ofthe carbon atom s along

the edgeofthe hole.

between neighboring edge atom s. Each dangling bond introducesa calculated spin of

one Bohrm agneton,1.00�B ,giving a totalm agnetization of12.00�B percell,causing

the lifting ofthe spin degeneracy. This m agnetization involves only the sp2-orbitals

and is strongly localized at the sites ofthe dangling bonds. W e �nd that structural

relaxation hasno qualitativeim pacton theresultsin thesetwo cases.

Next, we investigate the e�ects of a single carbon vacancy at the edge. This

introducesa sublattice im balance ofjN A � N B j= 1,resulting in an expected non-zero

totalm agneticm om entaccording to Lieb’stheorem .Elaborating on Lieb’swork,Inui,

Trugm an and Abraham have shown that such a sublattice im balance is accom panied

by at least jN A � N B jm idgap states with zero energy for a perfect bipartite lattice

[51]. A recent discussion of sim ilar statem ents can be found in Ref.[52]. In Fig.

9 we show the geom etries of a single carbon vacancy at the edge, both with and

withouthydrogenpassivation,aswellaswithandwithouthavingrelaxed thegeom etries.

The corresponding band structures are shown in Fig.10. Generally,we �nd two low

dispersion bandscloseto theFerm ilevel.Thesearethem idgap stateswith an induced

spin splitting. The spin degeneracy is lifted for allbands due to the non-zero total

m agneticm om ent.Them ain �nding in thecasewithoutpassivation oftheatom sclose

to the vacancy are the two at bands stem m ing from the dangling bonds,indicated

with arrows in Fig.10. The dangling bonds are found to overlap in case ofwhich it

is energetically m ost favorable for the dangling bonds to have zero totalspin as our

calculations show. W e �nd a m agnetization of1:00�B per unit cellfor both system s

in the unrelaxed case. Thism agnetization isentirely due to the sublattice im balance,

and is,contrary to thecaseofdangling bonds,largely non-local,residing m ainly on the

�-orbitals.

W hereas relaxation has m inor e�ects when passivation is included,the opposite

is true for carbon vacancies without passivation. In that case,the two unpassivated

carbon atom s at the edge approach each other,form ing a pentagon as seen in Fig.


-2

-1

0

1

2

E(e

V)

K Γ M K K Γ M K

Unpassivated Passivated

Figure 8. Band structure ofthe fL;Rg = f4;2g graphene antidotlattice calculated

with DFT.Theleft(right)panelshowstheband structurewithout(with full)hydrogen

passivation,corresponding to the unit cells in Fig.7. The system s are fully relaxed

and the spin degree offreedom is included. M ajority (m inority) spin is shown with

black (green).The Ferm ilevelisatE = 0.

9d. A sim ilarphenom enon hasbeen observed theoretically forsingle carbon vacancies

in bulk graphene,where the spin ofsuch vacancies can usually be understood as an

unsaturated dangling bond on theneighboring carbon atom ,notform ing thepentagon

[53,54,55].Thisresultsin a calculated m agneticm om entofaround 1�B .In ourcase,

however,there are only two neighboring atom s. In fact,in both cases the m agnetic

m om entisbetterunderstood usingLieb’stheorem ,asdiscussed by Palacios,Fern�andez-

Rossier,and Brey,in thecaseofcarbon vacanciesin bulkgraphene[56].In thepentagon

geom etry,two sitesbelongingto thesam esublatticebond strongertoeach other,which

isre ected in thesm allerbond length of1.67 �A com pared to 2.46�A in thecasewithout

relaxation.Consequently,thedangling bondsarethen saturated and thecorresponding

at bands are not present. Additionally, the bipartite lattice sym m etry is broken,

causing a reduction in the m agnetic m om ent from 1:00�B to 0:50�B . Consequently,

thespin splitting ofthebandsisreduced.Them idgap statesarestillobserved,butin

thiscasewith m oredispersion.Thefeaturesofthebipartitelatticearethusm aintained

in a m oderated version,when pentagonsareform ed dueto a carbon vacancy along the

edge ofthe hole. W e stress thatwhile the m agnetization arising from Lieb’stheorem

is predictable,the m agnetization due to dangling bonds is m erely a result ofenergy

optim ization and thereforeharderto predict.


1.0µB 1.0µB

1.0µB 0.5µB

Figure 9. Single carbon vacancy at the edge ofthe hole in the fL;Rg = f4;2g

structure.In the left(right)panelsthe structureshavenotbeen (have been)relaxed.

In the upper (lower) panels,the carbon atom s next to the vacancy have (have not)

been passivated with hydrogen.Thecalculated m agneticm om entisindicated in each

panel.

6. C onclusions

W e have carried out a num ericalstudy of the band structures of graphene antidot

lattices,using threedi�erentcom putationalapproachesofvarying levelsofcom plexity

and accuracy.Finite-elem entsolutionsoftheDiracequation (DE)provideasim pleand

fast schem e,capturing essentialqualitative features ofthe band structures and band

gaps. For m ore reliable predictions ofthe band structures,we em ployed a �-orbital

tight-binding schem e (TB)aswellascom putationally heavy density functionaltheory

calculations(DFT).Thethreem ethodsallpredictan openingofaband gap on theorder

ofa few hundred m eVsforthe nano-scale structured graphene antidotlatticesstudied

in thiswork.Qualitativesim ilaritieswerefound fortheband structurescalculated with

the three di�erent m ethods. Finally,we discussed the e�ects ofhydrogen passivation

alongtheedgesoftheholes.Passivation wasfound tohaveasigni�cantin uenceon the

band structures,and thepresenceofcarbon vacanciesalong theholeedgeswereshown

to inducem idgap bands.

7. A cknow ledgm ents

W ethank B Trauzettel,J Li,M Vanevi�c,and V M Stojanovi�cforinsightfuldiscussions.

Thework by CF wassupported by theVillum Kann Rasm ussen Foundation.Financial

supportfrom Danish Research CouncilFTP grant‘Nanoengineered graphenedevices’is

gratefully acknowledged.Com putationalresourceswereprovided by theDanish Center

for Scienti�c Com putations (DCSC).APJ is gratefulto the FiDiPro program ofthe

Finnish Academ y.


E(e

V)

K Γ M K K Γ M K

Unpassivated Passivated

Figure 10. Band structures ofthe fL;Rg = f4;2g graphene antidot lattice with

a single carbon vacancy in the unit cell. Dashed lines indicate band structures for

the unrelaxed geom etry shown in Fig.9,panels (a) and (c),while fulllines are the

corresponding results for the relaxed structures, panels (b) and (d). The un�lled

bands of the dangling bonds are indicated in the left panel by horizontalarrows.

The corresponding �lled bandsatlowerenergiesare notshown in the plot. M ajority

(m inority)spin isshown with black (green).TheFerm ilevelisatE = 0.

R eferences

[1]Novoselov K S,G eim A K ,M orozov S V,Jiang D,Zhang Y,Dubonos S V,G rigorieva IV and

Firsov A A 2004 Science 306 666

[2]Novoselov K S,G eim A K ,M orozov S V,Jiang D,K atsnelson M I,G rigorieva IV,DubonosS V

and Firsov A A 2005 Nature 438 197

[3]Castro Neto A H,G uinea F,PeresN M R,Novoselov K S and G eim A K 2009 Rev.M od.Phys.

81 109

[4]G eim A K 2009 Science 324 1530

[5]K atsnelson M I,Novoselov K S and G eim A K 2006 Nature Physics 2 620

[6]Schedin F,G eim A K ,M orozov S V,HillE W ,BlakeP,K atsnelson M Iand Novoselov K S 2007

Nature M aterials 6 652

[7]TrauzettelB,Bulaev D V,LossD and Burkard G 2007 Nature Physics 3 192

[8]LeeC,W eiX,K ysarJ W and Hone J 2008 Science 321 385

[9]Pendry J B,Holden A J,StewartW J and YoungsI1996 Phys.Rev.Lett.76 4773

[10]Pendry J B,Schurig D and Sm ith D R 2006 Science 312 1780

[11]Veselago V G 1968 Sov.Phys.Usp.10 509

[12]Pendry J B 2000 Phys.Rev.Lett.85 3966

[13]Fang N,LeeH,Sun C and Zhang X 2005 Science 308 534

[14]Schurig D,M ock J J,Justice B J,Cum m er S A,Pendry J B,Starr A F and Sm ith D R 2006

Science 314 977

[15]Yablonovitch E 1987 Phys.Rev.Lett.58 2059

[16]John S 1987 Phys.Rev.Lett.58 2486

[17]VasseurJ O ,Deym ierP A,ChenniB,Djafari-RouhaniB,DobrzynskiL and PrevostD 2001Phys.

Rev.Lett.86 3012


[18]FlindtC,M ortensen N A and Jauho A P 2005 Nano Lett.5 2515

[19]Pedersen J,FlindtC,M ortensen N A and Jauho A P 2008 Phys.Rev.B 77 045325

[20]Park C H and Louie S G 2009 Nano Lett.9 1793

[21]G ibertiniM ,Singha A,PellegriniV,PoliniM ,VignaleG ,Pinczuk A,Pfei�erL N and W estK W

2009 Phys.Rev.B 79 241406

[22]Pedersen T G ,FlindtC,Pedersen J,M ortensen N A,Jauho A P and Pedersen K 2008 Phys.Rev.

Lett.100 136804,ibid.(2008)100 189905

[23]RoukesM L and SchererA 1989 Bull.Am .Phys.Soc.34 633

[24]Ensslin K and Petro� P M 1990 Phys.Rev.B 41 12307

[25]Pedersen T G ,FlindtC,Pedersen J,Jauho A P,M ortensen N A and Pedersen K 2008 Phys.Rev.

B 77 245431

[26]Shen T,W u Y Q ,Capano M A,Rokhinson L P,EngelL W and Ye P D 2008 Appl.Phys.Lett.

93 122102

[27]Erom sJ and W eissD 2009 New.J.Phys.11 093021

[28]Han M Y,�O zyilm azB,Zhang Y and K im P 2007 Phys.Rev.Lett.98 206805

[29]Fischbein M D and Drndi�cM 2008 Appl.Phys.Lett.93 113107

[30]G irit C O ,M eyer J C,ErniR,RossellM D,K isielowskiC,Yang L,Park C H,Crom m ie M F,

Cohen M L,Louie S G and ZettlA 2009 Science 323 1705

[31]Jia X,Hofm ann M ,M eunier V,Sum pter B G ,Cam pos-Delgado J,Rom o-Herrera J M ,Son H,

Hsieh Y P,Reina A,K ong J,TerronesM and DresselhausM S 2009 Science 323 1701

[32]Rodriguez-M anzo J A and BanhartF 2009 Nano Lett.9 2285

[33]F�urstJ A,Pedersen T G ,BrandbygeM and Jauho A P 2009 Phys.Rev.B 80 115117

[34]Vanevi�cM ,Stojanovi�cV M and K inderm ann M 2009 Phys.Rev.B 80 045410

[35]RosalesL,Pacheco M ,Barticevic Z,Le�on A,Latg�e A and O rellana P A 2009 Phys.Rev.B 80

073402

[36]Berry M V and M ondragon R J 1987 Proc.R.Soc.A 412 53

[37]Tworzyd lo J,TrauzettelB,Titov M ,Rycerz A and BeenakkerC W J 2006 Phys.Rev.Lett.96

246802

[38]www.com sol.com

[39]Saito R, Dresselhaus G and Dresselhaus M S 1998 Physical Properties of Carbon Nanotubes

(Im perialCollegePress)

[40]Reich S,M aultzsch J,Thom sen C and O rdej�on P 2002 Phys.Rev.B 66 035412

[41]M artin R M 2004 Electronic Structure - Basic Theory and Practical M ethods (Cam bridge

University Press)

[42]Hohenberg P and K ohn W 1964 Phys.Rev.136 B864

[43]K ohn W and Sham L J 1965 Phys.Rev.140 A1133

[44]SolerJ M ,Artacho E,G aleJ D,G arcia A,Junquera J,O rdejon P and Sanchez-PortalD 2002 J.

Phys.: Condens.M atter 14 2745

[45]Porezag D,Fraunheim T,K ohlerT,SeifertG and K aschnerR 1995 Phys.Rev.B 51 12947

[46]Perdew J P,BurkeK and ErnzerhofM 1996 Phys.Rev.Lett.77 3865

[47]O nida G ,Reining L and Rubio A 2002 Rev.M od.Phys.74 601

[48]Lieb E H 1989 Phys.Rev.Lett.62 1201

[49]Jiang D E,Sum pterB G and DaiS 2007 J.Chem .Phys.127 124703

[50]Son Y-W ,Cohen M L and LouieS G 2006 Phys.Rev.Lett.93 187202

[51]InuiM ,Trugm an S A and Abraham sE 1994 Phys.Rev.B 49 3190

[52]W ang W L,Yazyev O V,M eng S and K axirasE 2009 Phys.Rev.Lett.102 157201

[53]M a Y,Lehtinen P O ,FosterA S and Niem inen R M 2004 New.J.Phys.6 68

[54]Lehtinen P O ,FosterA S,M a Y,K rasheninnikov A V and Niem inen R M 2004 Phys.Rev.Lett.

93 187202

[55]Yazyev O V and Helm L 2007 Phys.Rev.B 75 125408

[56]PalaciosJ J,Fern�andez-RossierJ and Brey L 2008 Phys.Rev.B 77 195428

arxiv:0907.0122v2 [cond-mat.mes-hall] 1 oct 2009

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