Optical and vibrational excitations of clean and hydrogen covered Si(001)surfaces: RA and HREELS

C. H. Patterson1

School of Physics, Trinity College Dublin, Dublin 2, Ireland.a)

(Dated: 28 April 2012)

Reflectance anisotropies and surface loss functions are calculated using a hybrid density functional theorymethod for hydrogen covered and clean Si(001) surfaces. The reflectance anisotropy is calculated for (2x1),p(2x2) and c(4x2) dimer arrangements of the clean surface and the spectrum for the Si(001)-c(4x2) surface isin excellent agreement with the measured spectrum. Surface conductivities and loss functions are calculatedfor the (2x1), p(2x2) and c(4x2) clean surface phases and the (2x1) H and D covered surfaces. Surface con-ductivities perpendicular to the surface are significantly smaller than conductivities parallel to the surface.The surface loss function is compared to high resolution electron energy loss measurements. There is goodagreement between calculated loss functions and experiment for H and D covered surfaces. However, agree-ment between experimental data from different groups and between theory and experiment is poor for cleanSi(001) surfaces. Formalisms for calculating electron energy loss spectra are reviewed and the mechanism ofelectron energy losses to surface vibrations is discussed.

I. INTRODUCTION

Reflectance anisotropy (RA) and high resolution elec-tron energy loss spectroscopy (HREELS) are importanttechniques for monitoring changes in surface structureand electronic structure through optical or vibrationalexcitations. RA has important advantages over elec-tron spectroscopic techniques in that it can be appliedat ambient pressure and no charging occurs in insulat-ing samples, but it is a difference technique, measur-ing reflectance difference of normally incidence light withorthogonal polarizations1, rather than probing excita-tions directly. First principles calculations are generallyneeded to identify key features in RA spectra, and wehave shown recently that, for adsorbates on semiconduc-tor surfaces, theory is also needed to identify surface vi-brational losses, because of the breakdown of the surfacedipole selection rule2. Here we report calculations of RAand HREELS spectra of clean and H or D covered Si(001)surfaces. Dimer buckling arrangements of the clean sur-face with c(4x2), p(2x2) and (2x1) surface unit cells areconsidered, as well as the (2x1)-H and -D surfaces.

RA measurements on the clean Si(001) surface havebeen reported by Yasuda et al.3, Kipp et al.4, Shiodaand Van der Weide5 and Jaloviar et al.6 and first prin-ciples density functional theory (DFT) calculations ofRA on the Si(001) surface have been reported7,8,10–12.Calculations of optical excitations at semiconductor sur-faces present several challenges to computational meth-ods. While DFT is relatively inexpensive in computa-tional cost, it underestimates the band gap and thereforeoptical excitation energies which enter expressions forRA. Superior many-body methods such as GW and theBethe-Salpeter equation (BSE) have been applied to cal-culations of optical excitations at surfaces13,14 but they

a)Electronic mail: Charles.Patterson@tcd.ie.

are probably too expensive to apply routinely to RA cal-culations in thick silicon slabs. The usual approach tocorrecting the band gap problem in DFT calculations ofoptical excitations in semiconductors is to shift the con-duction bands upward so that there is a closer match be-tween predicted and measured optical absorption peakpositions - a so-called scissors shift. In this paper weapply an alternative correction to the band gap prob-lem in semiconductors using a hybrid DFT method. Hy-brid density functionals such as B3LYP15,16 contain aweighted Fock exchange term which increases the bandgap nearly linearly with the weight of Fock exchange.This method is relatively inexpensive and can be appliedin a self-consistent determination of the surface struc-ture, electronic structure and optical and vibrational ex-citations.

HREELS can measure vibrational excitations at cleansemiconductor surfaces via electric-dipole or impactmechanisms17. Strong losses in HREELS experimentsdone in a specular scattering geometry are believed tooccur only when a surface vibration has a strong dy-namic dipole moment perpendicular to the surface18. Ina classic paper on HREELS measurements on H adsorbedon W(100), Ho, Willis and Plummer19 showed that theonly mode which was observed in specular scattering wasthe totally symmetric stretching mode in which the Hatom moves perpendicular to the surface. The scatter-ing cross-section for that mode decreased sharply in off-specular scattering; two additional modes, which involveH atom motion parallel to the surface, were observedin off-specular scattering with a similar, weak cross-section. The strong, specular scattering cross-sectionwas attributed to an electric-dipole mechanism whereasthe weaker, off-specular scattering was attributed to theshort-range, impact mechanism19. Experimental obser-vation of vibrational modes with a strong dynamic dipolemoment perpendicular to the surface in specular scatter-ing is sometimes referred to as the surface dipole selectionrule19. In early theoretical work17,18, Mills and cowork-

2

FIG. 1. Real space and reciprocal space unit cells for theSi(001) c(4x2) and p(2x2) surfaces. Real (a) and reciprocal(b) space cells of the p(2x2) dimer arrangement. Real (c) andreciprocal (d) space cells of the c(4x2) dimer arrangement.Dimer rows are shown in red.

ers attributed the absence of scattering by modes withdynamic dipole moments oriented parallel to metal sur-faces (or narrow band gap semiconductor surfaces, suchas Si) to screening of those moments by the bulk responseof the substrate.

Using hybrid DFT calculations of vibrational modesat clean and H covered Si(001) surfaces, we find thatatoms in the first three atomic layers in the Si(001) sur-face have large Born charges. Born charge tensor com-ponents which determine charge displacement parallelto the surface in a vibrational mode exceed e in somecases while tensor components for charge displacementperpendicular to the surface are much smaller (of or-der 0.01e). Displacement of ions parallel to the surfaceresults in large electron transfer between ions in cova-lent bonds, especially in chemically unsaturated, cleanSi(001) surfaces, while displacement of ions perpendic-ular to the surface does not, in contradiction of themodel mentioned above. The relatively small chargetransfer associated with atomic motion perpendicular tothe surface, compared to motion parallel to the surface,makes it especially difficult to assign HREELS spectrafor complex surfaces. Knowledge of ion displacementsin phonon eigenvectors is insufficient to assign HREELSspectra of semiconductor surfaces reliably. Below weshow that atomic motions, both perpendicular and par-allel to the surface, contribute to the HREELS dipolescattering cross-section. We report HREELS scatteringcross-sections for the Si surfaces considered here and as-sign experimentally observed modes. Good agreement isfound between experimental and calculated spectra for Hand D covered surfaces.

There have been surprisingly few reports of HREELS

measurements at the clean Si(001) surface in thevibrational20,21 and interband22 loss regions. First prin-ciples calculations of electron energy losses due to in-terband transitions at the Si(001) surface were reportedrecently10. Takagi et al.20 and Eremtchenko et al.21 re-ported room temperature HREELS measurements of theclean Si(001) surface. The dearth of HREELS measure-ments at this important surface may be due to the lowelectron scattering cross-sections of phonons at the cleanSi(001) surface and the tendency for electron beams todisorder the surface at low temperature23. HREELSspectra for the Si(001)-(2x1)H and D surfaces were re-ported recently by Eremtchenko et al. in the samepaper21. These measurements followed earlier workon the Si(001)-(2x1)H surface24–27. Assignment of Hstretching and bending modes for the (2x1)H and D sur-faces is straightforward and therefore provides a goodtest of a first principles calculation of HREELS spectra.Allan and Mele calculated the phonon dispersion rela-tion for the Si(001)-(2x1)H surface28 with one H atomper surface Si atom. Phonon dispersion relations havebeen reported for the clean Si(001) surface by Mele andcoworkers29,30 using a tight-binding method, by Fritschand coworkers31,32 using a density functional pertur-bation theory method and by Tutuncu et al.33 usinga bond-charge model. Alerhand and Mele also calcu-lated the dipole activities of phonons at the Si(001)-(2x1)surface30.

Earliest scanning tunneling microscopy (STM) stud-ies of the Si(001) surface showed that the surface con-sisted of (2x1), p(2x2) and c(4x2) domains of buckled Sidimers34,35. More recent STM36–41 and atomic force mi-croscopy (AFM)42 studies have shown that the Si(001)surface at low temperature consists of both c(4x2) andp(2x2) domains, with c(4x2) predominating. The sur-face unit cell of each structure is determined by orderingof buckled dimers in rows. Along a particular row, thedirection of dimer tilt alternates along the row. c(4x2)domains result when tilting in neighboring rows is outof phase between rows and p(2x2) domains result whentilting is in phase between rows, as shown in Fig. 1.Total energies of dimerized Si(001) surfaces have beenwidely studied by first principles (DFT43–55, GW 13 andquantum Monte Carlo56) methods. These calculationsshow that tilted dimers are energetically favored oversymmetric dimers by around 0.2 eV per dimer54,56 andthat the c(4x2) dimer order is favored over the p(2x2)arrangement by ∼1 meV per dimer54,55. The latterstudy showed that the ground state of the Si(001) sur-face can be switched from the c(4x2) arrangement to thep(2x2) arrangement by doping with electrons or apply-ing an electric field. Subsequent STM and AFM studieshave shown that dimer-tip interactions affect the STMimage36–42, that dimer tilts can be flipped by the tip37

and that reversible switches between c(4x2) and p(2x2)structures can be induced by the tip37.

3

II. THEORY

A. Reflectance anisotropy

The RA is defined to be

∆r

r= 2

rx − ry

rx + ry

, (1)

where rx and ry are complex reflectivity amplitudesmeasured with the electric vector parallel to the x (=[110]) and y (= [110]) directions. We use the McIntyre-Aspnes 3-layer model57 to calculate the reflectivity of asurface from the susceptibility difference for a slab justover 30 Athick. The change, ∆R, in average reflectivity,R, induced by a surface layer with a nonzero, anisotropic,surface susceptibilty, ∆χxs, relative to the average reflec-tivity, is given by,

∆R

R= 2Re

∆r

r= 4kd

∆χixsχ

rb − ∆χr

xsχib

|χb|2, (2)

where,

∆χxs(ω) = χxs,x(ω) − χxs,y(ω), (3)

the superscripts r and i indicate real and imaginaryparts, k is the wavevector magnitude of the incidentlight and d is the surface layer thickness. The frequency-dependent surface excess susceptibility58, is defined by,

χxs,i(ω) = χs,i(ω) − χb(ω), (4)

where χs,i is the surface susceptibility in direction i,and χb is the bulk susceptibility. Surface and bulk sus-ceptibilities are calculated using the single-particle sus-ceptibilty expression,

χi(ω) =2e2

m2ǫoΩ~2ω2

∑

nn′k

(fo(Enk) − fo(En′k))

(Enn′k − ~ω − iδ)|pi

nn′k|2,

(5)where p

nn′ki and E

nn′k are momentum matrix ele-ments and transition energies, respectively, connectingstates n and n′ at k-point k. Ω is a unit cell volume.

B. HREELS

The theory of electron energy loss at surfaces was ini-tially developed by Lucas and Sunjic59 and by Evans andMills18,60. The formalism by Mills is given in detail inthe book by Ibach and Mills17. Lambin, Vigneron andLucas61 adopted an approach in which the energy losswas calculated from the work done by a charged parti-cle as it passes through a dielectric. Approaches based

on the first Born scattering approximation adopted byMills and coworkers18,60, Persson62 and others and clas-sical approaches by Lucas and coworkers61,63 come toessentially the same conclusions regarding the scatteringcross-section. It can be expressed as a product of a kine-matic term, A(Q||, ω), and a surface loss function term,P(Q||, ω),

d2S

dΩ(kS)d~ω= A(Q||, ω)P (Q||, ω), (6)

where

A(Q||, ω) =2m2e2v4

⊥

π~5cosθI

kS

kI

|RI |2[v2

⊥Q2|| + (ω − v||.Q||)2]2

. (7)

kI and kS are the incident and scattered wavevectormagnitudes, θI is the incidence angle, RI is the reflectionamplitude for the surface in the absence of any surfacevibration, m, e, ~ω and v are the scattering electronmass, charge, energy and velocity.

P (Q||, ω) =2~Q||

π[1 + n(ω)]Im

−1

ǫ(Q||, ω) + 1. (8)

1 + n(ω) is a Bose-Einstein factor, Q|| is the compo-nent of the scattering vector parallel to the surface andǫ(Q||, ω) is an effective dielectric function which containsboth surface and bulk responses to the incoming electron.The surface response function, ǫ, in Eq. 8 is derived froma 3-layer model containing a vacuum layer, a thin surfacelayer with dielectric function ǫs and bulk dielectric func-tion ǫb. It is given by17,

ǫ(Q||, ω) = ǫs(ω)1 + ∆(ω)e−2Q||d

1 − ∆(ω)e−2Q||d, (9)

where

∆(ω) =ǫb(ω) − ǫs(ω)

ǫb(ω) + ǫs(ω). (10)

ǫs contains the static, electronic contribution to thedielectric function as well as the surface phonon contri-bution. The phonon contribution to the bulk dielectricconstant is zero by symmetry in cubic Si, but at the sur-face there is an anisotropic contribution from surface lo-calized phonons.

C. Phonon calculations and Born charge tensors

The change in slab polarization, ∂Pi caused by a dis-placement, ∂uαj, of the αth atom, defines the atomicBorn charge tensor,

4

Z∗α,ij =

∂Pi

∂uαj

Ω

e, (11)

where Ω is the slab cell volume and charges are inunits of the fundamental charge, e. The change in polar-ization induced by displacements of ions along a phononcoordinate is required for the phonon contribution to thedielectric function and conductivity of the surface. Borncharge tensors in the normal (Z) and atomic, Cartesian(Z∗) coordinate systems are related by,

Zp,i =∑

α,j

tp,αjZ∗α,ij√

Mα

, (12)

where p labels the vibrational mode, tp,αi containscomponents of the pth phonon eigenvector and Mα is themass of the αth atom.

The bulk, static dielectric contribution to the surfacedielectric function is ǫb, and the phonon contribution isgiven in terms of Born charge tensors in the phonon basisand phonon frequencies,

ǫii(ω) = ǫb +e2

ǫoΩ

∑

p

Zp,iZp,i

ω2p − ω2 − iωγ

. (13)

ωp and γ are the frequency of the pth mode and aphenomenological damping parameter.

The phonon contribution to the (diagonal) surface con-ductivity tensor in S per is

σii(ω) =e2

ǫoA

∑

p

γω2Zp,iZp,i

ω2p − ω2 − iωγ

. (14)

A is the area of the surface unit cell.

D. Surface electric fields and the scattering potential

Electric fields inside and outside the surface of anideal dielectric, with dielectric constant ǫ, may be cal-culated by the method of images64. If a charge, Q, invacuum, is placed above an ideal dielectric, it inducesa circularly symmetric positive screening charge at thevacuum/dielectric interface. The total potential of theexternal plus induced charges, outside the dielectric, isequivalent to that produced by the external charge plusan image charge, -Q(ǫ - 1)/(ǫ +1), inside the dielectric.The total potential inside the dielectric is equivalent tothat produced by a single charge, 2Qǫ/(ǫ + 1), at the siteof the vacuum charge. Electric fields inside and outsidethe dielectric are therefore,

Ein⊥ =

2

ǫ + 1E⊥, (15)

Eout⊥ =

2ǫ

ǫ + 1E⊥, (16)

Ein|| = Eout

|| =2

ǫ + 1E||, (17)

where E⊥ and E|| are components of the field of thecharge, Q, in vacuum. The usual boundary conditionsfor the parallel component of an electric field and per-pendicular component of the displacement field at a vac-uum/dielectric interface are satisfied. Thus the perpen-dicular component of the electric field of an electronwhich is incident on the surface of an ideal dielectric isenhanced by a factor of 2ǫ/(ǫ + 1) just outside the dielec-tric, while the parallel component is reduced by a factor2/(ǫ + 1).

The role of local fields at surfaces in the 3-layer modelof the surface loss function can be understood further byexpanding it in powers of Q||d

17, as

Im−1

ǫ(Q||, ω) + 1= Im

−1

ǫb + 1+ [

ǫ2b(ǫb + 1)2

Im−1

ǫs

+1

(ǫb + 1)2Imǫs]Q||d + O(Q2

||d2). (18)

Electromagnetic modes in bulk solids, such as bulkplasmons, occur at the poles of the inverse bulk dielectricfunction, 1/ǫ. Surface excitations at a vacuum/dielectricinterface occur at poles of 1/(ǫ + 1)65. Electromagneticmode frequencies of bulk materials with a macroscopicslab geometry are determined by the macroscopic fieldand the bulk susceptibility, χ(ω). The macroscopic fieldcontains the external field, E⊥, and a depolarizing field,−P⊥/ǫo, created by surface polarization charges. Thepolarization perpendicular to a slab face is therefore,

P⊥ = ǫoχ(E⊥ − P⊥

ǫo

), (19)

P⊥ = ǫoχ(1 + χ)−1E⊥, (20)

and it is clear that excitations occur at zeros of (1 + χ)(or poles of ǫ−1). In the present context, 1 + χ is the sumof the phonon and static, bulk, electronic contributionsto ǫii. Zeros of 1 + χ occur at

5

ω2 = ω2p +

Ω2p

ǫb

, (21)

where ωp are phonon frequencies and Ωp is the surfacephonon plasma frequency. Electromagnetic modes polar-ized parallel to the surface occur at phonon frequencies,ω = ωp, i.e. without any shift due to surface local fields.Phonon calculations which are presented below are per-formed within a self-consistent field approximation whichincludes the depolarizing field created by surface polar-ization charges. Hence the shifts in frequencies of modes

polarized perpendicular to the surface, Ω2p/ǫb, are implic-

itly included in the phonon calculations and ǫ−1s in Eq.

18 is replaced by ǫzz in Eq. 22. Terms in Eq. 18 cantherefore be interpreted as follows: the term indepen-dent of Q||d is the surface response of the semi-infinitebulk layer, which is present even in the absence of a sur-face layer; the term containing Im ǫ−1

s is the responseof the entire surface layer perpendicular to the surface;the term containing Im ǫs is the response of the surfacelayer parallel to the surface. Prefactors containing thebulk dielectric function, ǫb are discussed in Sec. VI. Thesurface phonon contribution to the surface loss function,in terms of the dielectric function calculated from Borncharges and phonon frequencies is,

Im−1

ǫ(Q||, ω) + 1= [

1

2(ǫb + 1)2Imǫxx +

1

2(ǫb + 1)2Imǫyy +

ǫ2b(ǫb + 1)2

Imǫzz]Q||d + O(Q2||d

2). (22)

A factor 12

has been introduced in the xx and yy termsin Eq. 22 to account for equal proportions of either do-main orientation of Si dimers.

III. COMPUTATIONAL METHODS

All electronic structure calculations reported here wereperformed using the Crystal code66. Hybrid DFT calcu-lations used a Hamiltonian similar to that in the B3LYPfunctional15,16 while LDA calculations used the LDAexchange67 and von Barth-Hedin correlation68 poten-tials. The B3LYP functional contains Hartree-Fock ex-change with weight 0.2 and the local density and general-ized gradient approximations to exchange with combinedweight 0.8; this standard functional results in overestima-tion of the optical band gap of bulk Si. In this work, therelative weight for Hartree-Fock exchange was reduced to0.05, which results in good agreement with experimentfor the value of the static real part and the position ofthe E2 peak in the imaginary part of the dielectric func-tion of bulk Si. This modified hybrid density functionalis used throughout this work, except where LDA calcu-lations of RA are explicitly referred to.

The Crystal code used in this work employs a frozenphonon method69,70 to calculate the dynamical matrixfor Γ point phonons. Born effective charge tensors inCrystal are obtained using a Berry phase method71.Slabs containing 16 Si layers with (2x1), p(2x2) or c(4x2)surface unit cells and both surfaces terminated by Sidimers were used for RA and HREELS calculations ofclean Si(001) surfaces. Slabs containing 25 Si layers with(2x1)H or D terminations were used for HREELS calcu-lations of hydrogen covered surfaces. Lattice parametersand atomic coordinates were relaxed until all forces onatoms were below 10−4 hartree/bohr. A 6x6 Monkhorst-

Pack (MP) net72 was used for all self-consistent field andfrozen phonon calculations.

The Exciton code used to calculate RA spectrauses a tetrahedron method to perform Brillouin zoneintegrations73. A 24x24 MP net was used for RA cal-culations on c(4x2) and p(2x2) unit cells (correspondingto 2304 points in the (1x1) surface Brillouin zone). A16x32 MP net was used for RA calculations on (2x1)unit cells and a 24x24x24 MP net was used for the bulkdielectric function. The Gaussian orbital basis used forSi74 is described in Ref. [75].

IV. RESULTS

A. Electronic Structure

The electronic structures of the (2x1), p(2x2) andc(4x2) phases of Si(001) calculated using hybrid DFTare shown in Fig. 2. In order to facilitate comparison ofsurface state dispersion in the (2x1) and p(2x2) phases,the band structures of symmetric and asymmetric dimerswith a (2x1) perodicity were calculated in a (2x2) super-cell. Surface state dispersions are similar to previously re-ported surface state dispersions54, except that band gapsof around 0.5 eV have opened in the electronic struc-tures of the asymmetric (2x1), p(2x2) and c(4x2) phaseswhereas there is no gap or a very small gap in local den-sity approximation (LDA) DFT calculations54. The sym-metric (2x1) phase is metallic in both hybrid DFT andLDA-DFT.

6

−2−1

0123

(2x1) asym.

−2−1

012

(2x1) sym.

Ene

rgy

(eV

)

−2−1

012

Γ−

J’−

p(2x2)

K−

J−

Γ−

−2−1

0123

c(4x2)

Γ−

Y−

Y’−

X−

Γ−

Ene

rgy

(eV

)

FIG. 2. Band structures for (2x1) surface unit cells with en-ergy minimized asymmetric and symmetric dimers plotted in(2x2) surface Brillouin zones (top panels). Band structure forp(2x2) surface unit cell and c(4x2) unit cell (lower panels).

B. Reflectance Anisotropy Spectra

STM experiments mentioned in Sec. I clearly show thesensitivity of the surface structure of the Si(001) surfaceto preparation conditions. This sensitivity is also foundin RA measurements3–6. In order to obtain a surfacewith a nonzero RA signal, a surface must be preparedwith a majority of one or other domain orientation. Thiscan be achieved by cutting the Si crystal slightly off the[001] plane to obtain a stepped surface with narrow ter-races. Alternatively, a prefered domain orientation withmicron sized terraces can be prepared by electromigra-tion. Yasuda et al.3 and Kipp et al.4 used Si(001) crys-tals offcut by 3 and 5o, respectively, while Shioda andVan der Weide5 and Jaloviar et al.6 used the electromi-gration technique. RA data from either method are quitedifferent. Here we compare results of our RA calculationsto data from Shioda and Van der Weide.

As noted in Sec. I, one of the advantages of using amethod which correctly predicts the single-particle bandgap is that there is no need to perform a scissors-shift of

0

10

20

30

40

50

60

70

0 2 4 6 8 10

Energy (eV)

ε2ε2

Si bulk exptSi bulk hybrid DFT

E parallel to dimersE perpendicular to dimers

FIG. 3. Imaginary part of the dielectric function for bulk Sifrom experiment and hybrid DFT calculations and for Si(001)slabs with c(4x2) dimer terminations with the electric fieldapplied parallel or perpendicular to the dimers.

the conduction band states when optical spectra are tobe calculated. In Fig. 3 we show dielectric functions forbulk Si from experiment and from hybrid DFT calcula-tions. The leading and trailing edges of the calculatedbulk dielectric function are slightly lower and higher inenergy than the experimental dielectric function edges.The E2 peaks in the two spectra occur at nearly thesame energies. The E1 peak in the hybrid DFT calcu-lation is nearly absent when compared to the bulk di-electric function. This peak is caused by electron-holeattraction (excitonic) effects which are not included inthese calculations. In Fig. 4 we show RA spectra for thec(4x2) surface obtained using LDA-DFT and LDA-DFTwith the conduction bands of both the Si(001) slab andSi bulk used in the RA calculation shifted upward by 0.3eV, in order to achieve better agreement with the ex-perimental RA spectrum. In addition to a shift in peakpositions to higher energies, there is a marked decreasein the relative intensity of the surface state peak, whichis found in experiment around 1.5 eV. Fig. 4 also showsthe hybrid DFT spectrum and experimental RA spec-trum scaled by 10. The hybrid DFT spectrum closelyresembles the shifted LDA-DFT spectrum and peaks inthe hybrid DFT calculation closely match experimentalpeaks in relative intensity and position, with the excep-tion of the surface state peak at 1.5 eV, which occurs at1.1 eV and is more intense in the hybrid DFT calcula-tion. The absolute value of the measured intensity of theRA spectrum depends on the domain structure of the

7

sample used. If domains containing dimers with eitherorientation are present in equal abundance, then therewill be no RA signal. A sample containing a 55/45 ratioof dimers of either orientation would have an RA signal10 times smaller than a mono-domain surface.

-0.02

-0.01

0.00

0.01

0.02

0 1 2 3 4 5 6

-0.02

-0.01

0.00

0.01

0.02

Energy (eV)

∆R/R

Expt x10 DFT Hybrid LDA LDA shifted 0.3 eV

FIG. 4. RA spectra of the Si(001)-c(4x2) surface calculatedusing LDA-DFT, LDA-DFT with a scissors shift of 0.3 eVand hybrid DFT and from experimental data redrawn fromRef.[5].

RA spectra for various dimerized surfaces from hybridDFT calculations are shown in Fig. 5. The top part ofthe figure compares spectra from symmetric and asym-metric dimers in a (2x1) cell. The peak associated withthe dangling bond surface states changes markedly ondimer symmetry breaking; the surface state peak for thesymmetric dimer surface has a positive peak at 0.7 eVwhich shifts to positive and negative going peaks around1.3 eV in the asymmetric dimer (2x1) surface. Thesepeaks become strong negative peaks at 1.3 and 1.1 eV inthe p(2x2) and c(4x2) surfaces, in better agreement withthe experimental RA spectrum. The hybrid DFT RAspectra for the (2x1) surfaces do not agree with the ex-perimental data at all in the bulk absorption energy rangeabove 3 eV. The p(2x2) and c(4x2) spectra are fairly sim-ilar over most of the energy range shown here. This isnot surprising since the atomic structures are quite sim-ilar. The dip in the experimental spectrum around 3 eVis reproduced as two sharp peaks in the c(4x2) spectrumand the features with minima and maxima at 3.6 and 4.4eV in the experimental spectrum are closely reproducedonly for the c(4x2) surface. These calculations show thesensitivity of RA spectra to the fine detail of surface re-constructions at Si surfaces.

-0.02

-0.01

0.00

0.01

0.02

0 1 2 3 4 5 6

-0.02

-0.01

0.00

0.01

0.02

Energy (eV)

∆R/R

Expt x10 (2x1) sym. (2x1) asym.

p(2x2) c(4x2)

FIG. 5. RA spectra of the dimerized Si(001) surface calcu-lated using hybrid DFT for symmetric and asymmetric dimersin a (2x1) cell and asymmetric dimers in p(2x2) and c(4x2)cells, compared to experimental data redrawn from Ref.[5].

V. HREELS

In this section we present calculations of the phononcontribution to the surface conductivity, the HREELSscattering cross-section and analysis of the modes whichare responsible for the strongest scattering in HREELS.A limited amount of experimental HREELS data is avail-able for clean Si(001) surfaces. Different primary beamenergies are used by Takagi et al.20 and by Eremtchenkoet al.21 and near-specular scattering spectra reported bythem differ considerably. In order to demonstrate a testof the method used to calculate HREELS spectra, we re-port spectra for hydrogen covered surfaces and comparethem to experimental spectra. HREELS spectra for theSi(001)-(2x1)H and D surfaces have been measured byEremtchenko et al.21 and by Eggeling et al.27 There isless uncertainty in the structure of this surface when asample is prepared for an HREELS measurement com-pared to the clean Si(001) surface where (2x1), p(2x2)or c(4x2) domains may coexist. Experimental HREELSdata for Si(001)-(2x1)H from several groups are in goodagreement21,25–27. Our calculations of loss energies andintensities are in very good agreement with loss energiesand relative intensities from experiment for the hydro-gen covered surfaces. This approach can therefore be ex-pected to predict HREELS spectra for clean Si(001) sur-faces reliably. The conductivity (Eq. 14) gives greaterweight to high frequency modes than the loss function(Eq. 8). The latter contains a thermal factor which givesgreater weight to modes with ~ω ≤ kT and the formercontains an extra factor of ω in the numerator of Eq.14. Consequently, modes with relatively low intensitypeaks in the conductivity appear as significant losses inthe HREEL spectrum.

8

A. Si(001)-(2x1)H and D phonons and HREEL spectra

The Si-H(D) symmetric stretching mode is reported tooccur at 2090(1525) cm−1 by Eremtchenko et al.21 and2095 cm−1 by Eggeling et al.27 We find four modes inthe range 2147 to 2183 (1550 to 1576) cm−1 in our cal-culations for the H(D) covered surfaces. The modes withthe largest dipole activities are at 2179(1573) cm−1. Si-H(D) stretching frequencies are therefore overestimatedby around 3% when compared to experiment. Bonds toH atoms are strongly anharmonic and the harmonic ap-proximation used here is known to overestimate H atomstretching frequencies. In the case of brucite, (Mg(OH)2),Pascale and coworkers76 found the OH stretching fre-quency was overestimated by around 5% using a B3LYPHamiltonian and that the stretching frequency agreedwith the experimental value within 10 cm−1 when al-lowance was made for anharmonicity of the OH stretch-ing potential.

0

2

4

6

8

0 100 200 300 400 500 600 700

σ (x

10-1

0 S s

q.-1

)

Energy (cm-1)

zx10

0

1

2

σ (x

10-7

S s

q.-1

)

y

x10 0

1

2

σ (x

10-7

S s

q.-1

)

xx10

(2x1) (2x1)H

FIG. 6. Conductivities of Si(001)-(2x1) and Si(001)-(2x1)H.Si(001)-(2x1)H data have been scaled by 1/10 for energiesabove 550 cm−1. Note differences in scales of coordinate axes.

Surface conductivities up to 700 cm−1 for Si(001)-(2x1)H are compared to those for the clean Si(001)-(2x1)surface in Fig. 6. A line broadening parameter, γ, of 5cm−1 was used in conductivity calculations (Eq. 14). Sidimer bonds are parallel to the [110] (x) direction anddimer rows are parallel to the [110] (y) direction. Fre-

quencies of modes with prominent peaks in the conduc-tivity are given in Table I.

TABLE I. Mode frequencies in cm−1 and polarization in thesurface conductivity for the Si(001)-(2x1)H and D surfaces.

polarization H ω (cm−1) D ω (cm−1) characterx 226 224 dimer rockx 648 493 in-phase H/D bendy 635 407 in-phase H/D bendy 513 in-phase H/D bendz 344 330 anti-phase H/D bendz 373 420 anti-phase H/D bendz 424 468 anti-phase H/D bendz 620 484 anti-phase H/D bendz 2179 1573 Si-H/D stretch

The conductivity parallel to the Si dimer (x) direc-tion at the H covered surface shows a strong peak at226 cm−1 which is the dimer rocking (r) mode and anin-phase Si-H bending mode at 648 cm−1. Allan andMele report the r mode at 210 cm−1 at the K point ofthe Brillouin zone28. The conductivity parallel to dimerrows (y) shows a strong peak at 635 cm−1 which is Si-Hbending perpendicular to the Si dimer axis. The con-ductivity perpendicular to the surface plane (z) showspeaks at 344, 373, 424, 620 and 648 cm−1. Peaks in therange 300 to 470 cm−1 correspond to coupled Si dimerstretching/anti-phase Si-H bending along the Si dimeraxis.

Modes with a large conductivity parallel to Si dimerbonds (x) at the D covered surface are found at 224 cm−1

and in the range 450 to 500 cm−1, with the most intensemode at 493 cm−1(Fig. 7). The mode at 224 cm−1 is ther mode. The 493 cm−1 mode corresponds to the in-phasebending mode observed at 648 cm−1 in the H-coveredsurface. Two bending modes with a large conductivityperpendicular to the dimer axis (y) are found at 407 and513 cm−1 and correspond to the single mode at 635 cm−1

in the H-covered surface. Several Si-D bending/dimerstretching modes with conductivity perpendicular to thesurface plane (z) are found at 330, 420, 468 and 484cm−1. The mode at 620 cm−1 in the H covered sur-face lies above the highest frequency in the bulk Si DOS(which is around 500 cm−1) and is therefore strongly lo-calized on Si-H bonds. The corresponding modes in theD covered surface mix strongly with Si bulk modes sincethey fall below 500 cm−1.

Phonon eigenvectors of modes with a relatively largeconductivity perpendicular to the H and D covered sur-faces are shown in Fig. 8. The slab used for these calcu-lations does not have a mirror plane of symmetry perpen-dicular to the Si dimer axis and so there is no requirementfor modes to be odd or even about the dimer axis.

HREEL spectra are calculated using Eq. 22. Absolutevalues of experimental scattering cross-sections are notavailable and calculated cross-sections have been scaledarbitrarily. Weights of contributions from xx, yy andzz elements of the dielectric functions in Eq. 22 are

9

0

1

2

0 100 200 300 400 500 600 700

σ (x

10-9

S s

q.-1

)

Energy (cm-1)

z

0

1

σ (x

10-6

S s

q.-1

)

y

0

1

2

σ (x

10-7

S s

q.-1

)

x

(2x1) (2x1)D

FIG. 7. Conductivities of Si(001)-(2x1) and Si(001)-(2x1)D.Note differences in scale compared to Si(100)-(2x1)H conduc-tivities.

1/2(ǫb + 1)2, 1/2(ǫb + 1)2 and ǫ2b/(ǫb + 1)2, respectively,where ǫb has been chosen to be 12. Factors of 1/2 in thexx and yy weights are included as the surface is assumedto consist of equal proportions of the two possible orienta-tions of Si dimer domains. Experimental and computedHREEL spectra for the H and D-covered surfaces arecompared in Figs. 9 and 10. Contributions to the com-puted HREEL spectrum from polarization parallel andperpendicular to the surface is also shown in Figs. 9 and10. A Lorentzian function corresponding to the elasticpeak has been added to the calculated HREEL spectrain Figs. 9 and 10 to aid comparison to experiment. Thisfunction was not added to the decomposition into x, yand z contributions in these figures. A line broadeningparameter, γ, of 30 cm−1 was used in dielectric functions(Eq. 13) to match the experimental resolution.

Experimental HREELS data from Eremtchenko et

al.21 for the H covered surface show losses at 145 and220 cm−1, a broad hump between 310 and 390 cm−1 anda strong, asymmetric loss at 617-622 cm−1 . Eggeling et

al.27 report a loss at 625 cm−1 for the H covered surface.There are relatively weak losses at 140 and 215 cm−1 andthree distinct, intense losses in the experimental data forthe D covered surface at 340, 410 and 490 cm−1. Thecomputed HREEL spectra for the H(D) covered surfaces

FIG. 8. Phonon eigenvectors for modes at Si(001)-(2x1)D(left) and Si(001)-(2x1)H (right) surfaces with the greatestconductivities perpendicular to the surface plane.

0 100 200 300 400 500 600 700

145 220 310 390 480 622

Energy Loss cm-1

(2x1)-H

Expt.

Theory

x1/8

xyz

x+y+z

FIG. 9. HREELS loss function for the Si(001)-(2x1)H sur-face (red) and experimental HREELS data (black) fromEremtchenko et al.21. The primary beam energy in the ex-perimental work was 2.5 eV and the incidence angle was 64o.Vertical lines indicate experimental peak positions.

show weak losses at 124(126) cm−1 which are Si-H(D)

10

bending motion coupled to bulk Si motion and may cor-respond to the experimental losses at 145(140) cm−1.

0 100 200 300 400 500 600 700

140 215 340 410 490

Energy Loss cm-1

(2x1)-D

Expt.

Theory

xyz

x+y+z

FIG. 10. HREELS loss function for the Si(001)-(2x1)Dsurface (red) and experimental HREELS data (black) fromEremtchenko et al.21. The primary beam energy in the ex-perimental work was 2.5 eV and the incidence angle was 64o.Vertical lines indicate experimental peak positions.

The dimer rocking modes observed in experiment at220(215) cm−1 are calculated to be at 226(224) cm−1.The lower panels of Figs. 9 and 10 show that therocking mode is associated with polarization parallel tothe Si dimer axis only. It therefore breaks the surfacedipole selection rule for HREELS, whereby only modeswith polarization perpendicular to the surface plane areobserved. The computed spectrum for the H coveredsurface contains weak contributions from Si-H bend-ing/dimer stretching modes with contributions from bothx and z polarizations and must correspond to the broadhump seen in experiment between 310 and 390 cm−1.In contrast, the D covered surface has three strong lossfeatures in the range 300-500 cm−1. Here there is mix-ing of Si-D bending and surface Si atom motion. Thereare weak contributions to the D covered surface HREELspectrum from modes with x polarization and strongercontributions from modes with y and z polarization. Themodes with y polarization are Si-D bending perpendicu-lar to the Si dimer axis and the modes with z polarizationat 330, 420 and 468 cm−1 are shown in Fig. 8. There isvery good agreement between the calculated and experi-mental spectra. Since this depends on inclusion of largecontributions from polarization parallel to the surface,this provides a further demonstration that the surfacedipole selection rule is invalid for these surfaces.

Surface conductivities (Figs. 6 and 7) and HREELspectra (Figs. 9 and 10) of both H covered and cleanSi(001) surfaces show strong features in the range 450to 500 cm−1. There are no strongly scattering surfacemodes in this energy range for the H covered surface, butmany modes localized in the slab interior make contribu-tions to the scattering at this energy. The relative inten-sities of the losses around 500 cm−1 in our calculations,which we assign to surface resonances of bulk phonons,increased when a thicker slab was used for the phononcalculation. This energy corresponds to the maximumof a strong peak in the bulk phonon density of states33

and it is present for both the clean and H covered (2x1)surface conductivities along z (Fig. 6). Hence an assign-ment to resonances of bulk states at both clean and Hor D covered surfaces seems reasonable. Some of the lossintensity around 500 cm−1 in the Si(001)-(2x1)D surfaceis due to Si-D bending modes, as mentioned above.

B. Si(001) clean surface phonons and HREEL spectra

Allan and Mele29 report three surface phonons, la-belled r, s and sb, in gaps of the projected bulk phonondensity of states of the Si(001)-(2x1) surface. r is a dimerrocking mode in which the dimer tilt changes, s is a dimerswing motion in which dimers oscillate about axes per-pendicular to the surface and sb is a motion of the subsur-face bonds. Alerhand amd Mele also refer to a five foldring mode30 in which atoms in five membered rings whichinclude pairs of dimer atoms and atoms immediately be-low the dimer (subdimer atoms) are in motion. Thesesurface atom motions can couple to bulk Si atom motionin more than one way and we find that a particular kindof surface atom motion makes contributions to the sur-face conductivity and HREELS spectra in more than onemode. Therefore more than one peak in a spectrum canbe assigned to a dimer swing motion, s, etc. Reportedfrequencies for these modes, converted to cm−1, are givenin Table II.

The surface conductivity of the clean Si(001)-(2x1) sur-face is compared to that for the (2x1)H surface in Fig.6. The clean surface conductivity parallel to dimers (x)is dominated by a group of peaks up to 184 cm−1 whichare dimer rocking modes. The conductivity along dimerrows (y) is large in the clean surface in the range 400 -500 cm−1 and a peak appears at 134 cm−1 when H isremoved from the surface. The peaks at 134, 412 and477 cm−1 are s modes and a peak at 432 cm−1 is an sbmode. The s modes around 412 cm−1 are primarily local-ized on the upper dimer atom and the atom in the valleybetween dimers while the s mode at 477 cm−1 is pri-marily localized on the lower dimer atom and the valleyatom. The conductivity perpendicular to the surface (z)has an r mode at 265 cm−1 and an sb mode at 484 cm−1.The clean (2x1) surface has a surface localized mode at501 cm−1 which lies above the bulk phonon DOS andis localized on the lower dimer atom and the backbond

11

0

2

4

6

8

0 100 200 300 400 500 600 700

s (x

10-1

0 S s

q.-1

)

Energy (cm-1)

z

0

2

4

s (x

10-7

S s

q.-1

)

y

0

1

2

s (x

10-7

S s

q.-1

)

x

c(4x2)p(2x2)

FIG. 11. Conductivities of Si(001)-p(2x2) and Si(001)-c(4x2).

atom. This surface state was also reported by severalother groups (Table II). However, we do not find a sur-face localized mode above the bulk DOS in the p(2x2) orc(4x2) surfaces.

TABLE II. Mode frequencies in cm−1 for the Si(001)-(2x1)surface.Reference rocking mode r swing mode s subsurface bond sb

AM a 207 356 494FP b 158 337 494TJS c 168 348 527FP d 175 525TJS e 171 513This work f 184 412, 477 501

a Allan and Mele K point, Ref. [29]b Fritsch and Pavone K point, Ref. [31]c Tutuncu, Jenkins and Srivastava, K point, Ref. [33]d Fritsch and Pavone Γ point, Ref. [31]e Tutuncu, Jenkins and Srivastava, Γ point, Ref. [33]f Γ point

Conductivities of Si(001)-p(2x2) and c(4x2) surfacesare shown in Fig. 11. Conductivities of these surfacesare very similar along dimers or dimer rows. The conduc-tivity along dimers (x) is associated with dimer rockingat 194 and 295 cm−1 and sb modes at higher frequen-cies. The conductivity along dimer rows is dominated

by a mode at 382 cm−1, which is an s mode localizedon dimer atoms and valley atoms. The c(4x2) surfacehas strong contributions to the conductivity perpendicu-lar to the surface (z) from modes at 295 and 476 cm−1.These are r and sb modes, respectively. Phonon eigen-vectors for several key modes at the (2x1) and c(4x2)surfaces are shown in Fig. 12. The s mode at 382 cm−1

is quite different from those at 412 or 477 cm−1 in theclean (2x1) surface, since motion on the valley atoms isvertical rather than horizontal.

FIG. 12. Phonon eigenvectors for modes at Si(001)-(2x1)(left) and Si(001)-c(4x2) (right) surfaces. Phonon frequenciesare given in cm−1.

Alerhand and Mele30 reported conductivities of the(2x1), p(2x2) and c(4x2) Si(001) surfaces using a tightbinding method. They found that conductivities wereof the same order of magnitude parallel and perpendic-ular to the surface normal. We find similar magnitudesfor conductivities parallel to surfaces, but a magnitudearound 100 times smaller for the perpendicular direction.Polarization parallel and perpendicular to the surface isobtained self-consistently in our calculations whereas itmay not have been in the earlier tight-binding calcula-tions. This may explain the difference in the predictionsof the two calculations for polarization perpendicular tothe surfaces.

HREELS data for the clean Si(001) surface reported byTakagi et al.20 and by Eremtchenko et al.21 are redrawnin Fig. 13 and compared to the HREELS loss function(Eq. 8) for the Si(100)-(2x1) and c(4x2) surfaces. Databy Takagi et al. were recorded at room temperature usinga 2.5 eV primary beam energy20. Data by Eremtchenkoet al. were recorded from a surface which exhibited a(2x1) low energy electron diffraction (LEED) pattern21

12

TABLE III. Mode frequencies in cm−1 for the Si(001)-c(4x2)surface.Reference rocking mode r swing mode s subsurfaced bond sb

FP a 186 482This work b 194 382

a Fritsch and Pavone J point, Ref. [31]b Γ point

0 100 200 300 400 500 600 700

150 280395 480 530

Energy Loss cm-1

(2x1)

c(4x2)

Eremtchenko Ep = 2.5 eV

Takagi Ep = 6.2 eV

x+y+zxyz

FIG. 13. HREELS spectra for the clean Si(001)-(2x1) andc(4x2) surfaces obtained from the surface loss function andexperimental data for clean Si(001) surfaces redrawn fromEremtchenko et al.21 and Takagi et al.20 Primary beam en-ergies and incidence angles in the experimental work byEremtchenko et al. and by Takagi et al. were 2.5 eV and64o and 6.2 eV and 60o, respectively. Vertical lines indicateexperimental peak positions referred to in Refs. [20] and [21].

using a 6.2 eV primary beam energy. Eremtchenko et

al.21 report losses at 150, 280, 480 and 530 cm−1. Takagiet al.20 report losses at 97, 161, 242, 266, 395, 476 and516 cm−1. Both groups assigned modes on the basis ofbond charge model calculations for the Si(001)-(2x1) sur-face by Tutuncu et al.33 although no information on lossintensities was available from the bond charge model cal-

culations. The loss reported by Eremtchencko(Takagi) et

al. at 530(516) cm−1 could be the sb surface mode abovethe bulk phonon DOS, which we find at 501 cm−1 andother groups find between 494 and 527 cm−1 (Table II).The loss reported at 480(476) cm−1 may be caused by thehigh density of bulk modes in this region which have someamplitude on surface atoms which carry a nonzero Borncharge. Conductivities perpendicular to the H coveredand clean (2x1) and c(4x2) surfaces have peaks in thisregion. The loss reported by Takagi et al. at 395 cm−1

may be the s mode with a strong peak around 382 cm−1

in the c(4x2) surface, if there are c(4x2) domains present.The loss at 280 cm−1 may be an r mode which we findat 265 cm−1 in the (2x1) surface and at 295 cm−1 in thec(4x2) surface. The loss reported at 150(161) may be anr mode which we find at 184 cm−1 in the (2x1) surface.Clearly, there are major differences between the two setsof experimental data in Fig. 13 and between the exper-imental data and results of our calculations. Agreementbetween theory and experiment is significantly better forthe H and D covered Si(001) surfaces.

C. Atomic Born charge analysis

Born charges, which are the average values of diagonalelements of atomic Born charge tensors (Eq. 11), pro-vide a means of understanding the origin of peaks in thesurface conductivity and surface loss function. These areshown schematically in Fig. 14 for the four Si(001) sur-faces considered in this work. Numerical values for thelargest Born charges at these surfaces are given in TableIV and Born charge tensors for atoms in the p(2x2) and(2x1)H surfaces are given in Table V. Born charges arezero by symmetry in bulk Si and can only be nonzeroclose to the surface. The tensors are highly anisotropicfor near-surface atoms and elements governing chargetransfer parallel to the surface are around 10 times largerthan those governing charge transfer perpendicular to thesurface.

Fig. 14 and Table IV show that only Si atoms in dimersand first and second layers of any of the surfaces stud-ied have Born charge magnitudes greater than 0.1e. Atthe (2x1)-H surface, H atoms have the largest charges(-0.44e) while the Si dimer has relatively small charges(0.27e) , compared with buckled, clean surface dimers. Ineach of the four surfaces studied, the outermost Si atomin the valley between dimer rows has a positive Borncharge ranging from 0.24e at the clean (2x1) surface to0.29e in the c(4x2) surface. Buckled dimer clean surfaceshave large positive charges on the lower atoms in thedimer and larger negative charges on the upper atoms.There are positive charges on Si backbond atoms andnegative charges on atoms immediately below Si dimers.In the (2x1) clean surface there are two types of backbondatom, only the charge for the backbond atoms attachedto the dimer up Si atom is given. Subdimer charges inthe p(2x2) and c(4x2) surfaces are as large as those in

13

the buckled dimers.Born charge tensors in Table V for atoms with large

Born charges at the surface are highly anisotropic. For allfour surfaces studied, the largest elements of the Si dimertensors are the components corresponding to polarizationinduced perpendicular to dimers by atom motion in thatdirection, which is the source of the strong polarizationin the s modes described in Sec. VB.

FIG. 14. Born charges at Si(001)-(2x1)H and clean Si(001)surfaces. Sphere radii indicate the magnitude of the Borncharge, red is a positive charge and blue negative.

TABLE IV. Born charges for atoms at Si(001) surfaces.

Atom (2x1)-H (2x1) c(4x2) p(2x2)H -0.44Si dimer up 0.27 -0.75 -0.61 -0.59Si dimer down 0.27 0.45 0.55 0.59Si backbond 0.03 0.47 0.17 0.15Si subdimer -0.04 -0.22 -0.58 -0.60Si valley 0.26 0.24 0.29 0.27

VI. DISCUSSION

A. RA spectra

RA spectra for clean Si(001) surfaces calculated us-ing first principles LDA-DFT methods have been re-ported previously by Del Sole and coworkers7,10,11 and

TABLE V. Born charge tensors for atoms at the Si(001)-(2x1)H and Si(001)-p(2x2) surfaces. (a) H atom, (b) Si dimeratom and (c) Si valley atom at the Si(001)-(2x1)H surface.(d,e) Outer, inner Si atoms in a tilted Si dimer at the Si(001)-p(2x2) surface, (f) atom immediately below the Si dimer inthe Si(001)-p(2x2) surface (see Fig. 14). Born charges fortensors (a-f) are -0.41, 0.28, 0.26, -0.59, 0.54 and -0.60, inunits of charge, e.

(a)

0

@

−0.45 0.00 0.000.00 −0.64 0.000.03 0.00 −0.13

1

A (b)

0

@

−0.06 0.00 −0.330.00 0.75 0.00−0.05 0.00 0.15

1

A

(c)

0

@

0.75 0.00 0.000.00 0.02 0.000.00 0.00 0.01

1

A (d)

0

@

−1.29 0.00 −0.550.00 −0.55 0.00−0.09 0.00 0.08

1

A

(e)

0

@

−0.07 0.00 −0.530.00 1.69 0.000.00 0.00 −0.01

1

A (f)

0

@

−1.07 0.81 0.000.21 −0.81 0.000.00 0.00 0.07

1

A

by Schmidt et al.8 When an LDA-DFT approach is used,the band gap problem is overcome by rigidly shifting theconduction bands upward by several tenths of an eV. Thisapproach is illustrated in Fig. 4 in this work. Unshiftedspectra obtained using LDA or hybrid DFT Hamiltoni-ans are similar except for a rigid energy shift. RA spectraobtained after conduction bands have been shifted arerather different from those which have not been shiftedand may be significantly distorted. For example, pre-vious RA spectrum calculations7,8,10,11 do not clearlyshow the broad peak around 3 eV in Shioda and Vander Weide’s data5, which is found in the unshifted LDAand hybrid DFT calculations. This peak is significantlyweakened by shifting LDA conduction bands in the RAcalculation (Fig. 4). Furthermore, the shifted LDA peakaround 1.5 eV, which agrees well with the experimentalpeak position, is significantly more than the conductionband shift above the unshifted peak position, which oc-curs below 1 eV. It might therefore be more appropriateto shift the RA spectrum, rather than conduction bands,when LDA-DFT methods are used.

The c(4x2) RA spectrum in Fig. 5 is in excellent agree-ment with experiment, except for the overall difference inthe amplitudes of spectral features and underestimationof the energy of the dip at 1.5 eV. The difference in peakamplitude may be caused by mixed c(4x2) domains, asnoted above. Previous RA spectrum calculations7,8,10,11

show the peaks at 1.5 and 3.5 eV to be of similar ampli-tude and therefore they also overestimate the strength ofthe low energy peak, compared to Shioda and Van derWeide’s data5. The dip in the RA spectrum at 1.5 eVis caused by transitions between surface states (Figs. 2and 3). Correction of the bulk band gap by including asingle weight of Hartree-Fock exchange in the Hamilto-nian may not correct both bulk and surface state energiessimultaneously. In the GW approximation the extent ofscreening of Hartree-Fock exchange is determined by a

14

frequency and wave vector-dependent dielectric function.The extent of screening (and therefore correct weight ofHartree-Fock exchange) may be different for bulk andsurface states. Furthermore, the hybrid DFT methodused here does not include electron-hole attraction effectswhich would be included in a BSE Hamiltonian, whichaffect surface state peak positions and line shapes.

The structure sensitivity of RA is illustrated by com-paring the RA spectra of the (2x1), p(2x2) and c(4x2)surfaces. As noted previously7, the sign of the RA sig-nal below 1 eV depends strongly on dimer tilt and dimerarrangement. The symmetric (2x1) dimer surface has apositive RA signal at around 0.7 eV. This peak is re-placed by one with the opposite sign above 1 eV whenthe dimers are allowed to relax to their tilted configura-tion. These changes are induced by the change in surfacestate dispersion shown in Fig. 2.

B. HREELS

We have presented calculations of cross-sections fordipole scattering by surface modes with small Q|| usingphonon calculations performed at Q|| = 0. It is reason-able to assume that the dielectric functions used in calcu-lating HREEL spectra (Eq. 13) vary slowly with Q|| andthat they do not change significantly at values of Q|| used

in experiment. For example, Takagi et al.20 used Q|| =

0.06 A−1 and Eremtchenko et al.21 used Q|| = 0.04 A−1

for near-specular scattering, which is much closer to Q||

= 0 than the distance to the nearest surface Brillouinzone boundary (∼0.4A−1 for the p(2x2) surface).

Eqs. 8 and 9 (and Eq. 22 which is used to calcu-late HREEL spectra) are derived by considering electricfields propagating into and out of a semi-infinite dielec-tric with a thin surface layer. The factors 1/(ǫb +1)2 andǫ2b/(ǫb + 1)2 which appear in the small Q||d expansion of

Eq. 8 have been interpreted by Evans and Mills18, Ho et

al.19 and many subsequent authors, in terms of screeningof dipole moments associated with vibrations of atoms insurface layers by the underlying dielectric or metal. Thescattering potential in the first Born approximation ap-proach of Evans and Mills18, ∆Q||

(z), a distance, z, abovea crystal surface at wavevector, Q|| is,

∆Q||(z) = 4πenoe

−Q||zǫb

1 + ǫb

(p⊥ − i

ǫb

Q||.p||). (23)

no is the surface unit cell area and p⊥ and p|| areFourier amplitudes which appear in lattice sums of dipolemoments oriented normal and parallel to the surface, re-spectively. ∆Q||

(z) enters the scattering cross-section asits modulus squared and so the contribution from p|| is

reduced by a factor 1/ǫ2b compared to p⊥ and was ne-glected by Evans and Mills18. The factors containing ǫb inthe scattering potential are consistent with the screenedpotential created in vacuum when an electric dipole ori-ented parallel or perpendicular to the surface is placed

just outside an ideal dielectric18. Hence the currently ac-cepted picture of the dipole mechanism for charge scat-tering by surface vibrations is that screening of phonon-induced polarisation parallel to the surface results in neg-ligible dipole scattering by vibrations which have atomicmotion parallel to the surface and only vibrations whichhave atomic motion perpendicular to the surface are ob-served in HREEL spectra. This has come to be known asthe surface dipole selection rule in HREELS17 and it de-pends on screening of charges induced by atomic motionsassociated with surface phonons.

Fig. 14 shows that Born charges at clean or H coveredSi(001) surfaces are confined to the outermost atomic lay-ers. Components of Born charge tensors (Table V) arelarge (∼e) for atom displacements parallel to the surface.They are smaller, by up to an order of magnitude, foratom displacements perpendicular to the surface. Whenan atom is displaced parallel to the surface a large chargetransfer may occur, e.g. between dimer atoms, especiallyat the clean surface where occupied and vacant surfacestates are localized on the ’up’ and ’down’ atoms in thedimer, respectively. Table V shows that when an atom isdisplaced perpendicular to the surface, the zz componentof the Born charge tensor is relatively small, especiallyfor the clean surfaces studied. Presumably this is becausean atom, which is displaced perpendicular to the surface,moves (almost) as a neutral atom, rather than becauseof compensation by charge displacement on neighboringatoms in the opposite sense. The relatively large xx andyy and smaller zz components of dielectric functions andsurface conductivities are the opposite of what would beexpected, were the currently accepted picture and thesurface dipole selection rule to apply. It should be notedthat Born charge tensors express the response of the elec-tronic charge to atomic displacements rather than to anexternal electric field and so there is actually no require-ment for these charges to be screened in the way thatthe field of an external charge is screened by a dielectricsurface. Surface atom Born charge tensors represent anintrisic response property of the surface to surface atomdisplacements.

An alternative interpretation of the factors 1/(ǫb + 1)2

and ǫ2b/(ǫb+1)2 can be given in terms of parallel and per-pendicular components of the electric field strength in thesurface region where Born charges are large. A dielectricsurface treated quantum mechanically can be expectedto have near ideal dielectric behavior, provided that theperturbing charge is in the vacuum region and is at leastseveral interatomic spacings away from the surface. Inthis case the electronic charge responds to the field ofa point charge rather than an atomic displacement. Asnoted in Eqs. 15 to 17, the electric field component par-allel to the surface is reduced by a factor 2/(ǫb +1), com-pared to the field of the bare charge in vacuum, bothinside and outside the dielectric surface. On the otherhand, the field component perpendicular to the surfaceis enhanced by a factor 2ǫb/(ǫb + 1) outside the layerof screening charge and reduced by a factor 2/(ǫb + 1)

15

inside. Since field-field (or equivalently charge density-density) correlation functions appear in the scatteringcross-section, these factors are squared in the scatteringcross-section17. This interpretation also accounts for thefact that vibrational modes of adsorbates on metals withatomic motions parallel to the surface are not observed innear-specular HREEL spectra19, since the parallel com-ponent of the electric field is completely screened at aperfect metal surface.

The charge which accumulates at a surface in responseto the field of an external charge is expected to reside onthe outermost atomic layer(s). Some atoms with largeBorn charges will therefore lie outside that charge layerand others within it. The actual field at the surface,which is the field driving the response of the surfacephonons, requires a microscopic calculation, but this isbeyond the scope of the current work. However, goodagreement between experimental and calculated HREELspectra for H and D covered surfaces, where there is rel-atively little uncertainty in surface structure and surfacecontamination is not a major experimental problem, sug-gests that Eq. 22 can be used to calculate HREEL spec-tra of clean and adsorbate covered semiconductor sur-faces. Eq. 22 requires only Born charges in phonon coor-dinates and phonon frequencies, both of which can eas-ily be calculated for semiconductors and insulators usingvarious electronic structure codes. Since both the paral-lel and perpendicular components of polarization inducedby surface phonons can be observed via HREELS, it isnot sufficient to assign spectra by searching for modeswith atomic motion predominantly perpendicular to thesurface. Instead, Eq. 22 can be used to predict and as-sign spectra.

Experimental data20,21 for the clean Si(001) surface arenot in agreement and there are significant differences be-tween calculated (2x1) and c(4x2) HREEL spectra andbetween calculated and experimental spectra. Differ-ences between experimental spectra may arise because ofdifferent primary beam energies (6.2 versus 2.5 eV), dif-ferences in surface preparation leading to different dimerarrangements, domain sizes, etc. or even differences inlevels of surface contamination. Measurement of HREELspectra of a single domain Si(001)-c(4x2) surface, pre-pared using appropriate techniques5 is needed. Compar-ison to our calculations, which predict a strong loss fea-ture due to the dimer swing mode at 382 cm−1, wouldprovide a further test of this method for calculating andassigning HREEL spectra and the validity of the surfacedipole selection rule at clean and adsorbate covered semi-conductor surfaces.

ACKNOWLEDGMENTS

This work was supported by Science Foundation Ire-land under grant number RFP/11/PHY/3047. Com-puter time was provided by the Trinity Centre for HighPerformance Computing which is supported by the Irish

Higher Education Authority and Science Foundation Ire-land. The author wishes to acknowledge helpful discus-sions with John McGilp and Conor Hogan.

1P. Weightman, D. Martin, R. Cole, and T. Farrell, Rep. Prog.Phys. 68, 1251 (2005)

2C. H. Patterson, (to be published)3T. Yasuda, L. Mantese, U. Rossow, and D. E. Aspnes, Phys.Rev. Lett. 74, 3431 (1995)

4L. Kipp, D. K. Biegelsen, J. E. Northrup, L.-E. Swartz, andR. D. Bringans, Phys. Rev. Lett. 76, 2810 (1996)

5R. Shioda and J. vanderWeide, Phys. Rev. B 57, R6823 (1998)6S. G. Jaloviar, J.-L. Lin, F. Liu, V. Zielasek, L. McCaughan, andM. G. Lagally, Phys. Rev. Lett. 82, 791 (1999)

7M. Palummo, G. Onida, R. DelSole, and B. S. Mendoza, Phys.Rev. B 60, 2522 (1999)

8W. G. Schmidt, F. Bechstedt, and J. Bernholc, Phys. Rev. B63, 045322 (2001)

9Y. Borensztein and N. Witkowski, J. Phys. Condens. Matter 16,S4301 (2004)

10L. Caramella, C. Hogan, G. Onida, and R. DelSole, Phys. Rev.B 79, 155447 (2009)

11M. Palummo, N. Witkowski, O. Pluchery, R. DelSole, andY. Borensztein, Phys. Rev. B 79, 035327 (2009)

12K. Gaal-Nagy, A. Incze, G. Onida, Y. Borensztein, N. Witkowski,O. Pluchery, F. Fuchs, F. Bechstedt, and R. DelSole, Phys. Rev.B 79, 045312 (2009)

13J. E. Northrup, Phys. Rev. B 47, 10032 (1993)14M. K. M. Weinelt, T. Fauster, and M. Rohlfing, Phys. Rev. Lett.

92, 126801 (2004)15A. D. Becke, J. Chem. Phys. 98, 5648 (1993)16P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch,

J. Phys. Chem. 98, 11623 (1994)17H. Ibach and D. L. Mills, Electron energy loss and surface vibra-

tions, Vol. New York (Academic Press, 1982)18E. Evans and D. L. Mills, Phys. Rev. B 5, 4126 (1972)19W. Ho, R. F. Willis, and E. W. Plummer, Phys. Rev. Lett. 72,

1463 (1978)20N. Takagi, S. Shimonaka, T. Aruga, and M. Nishijima, Phys.

Rev. B 60, 10919 (1999)21M. Eremtchenko, F. S. Tautz, R. Ottking, and J. A. Schaefer,

Surf. Sci. 600, 3446 (2006)22H. H. Farrell, F. Stucki, J. Anderson, D. J. Frankel, G. J.

Lapeyre, and M. Levinson, Phys. Rev. B 30, 721 (1984)23M. Matsumoto, K. Fukutani, and T. Okano, Phys. Rev. Lett.

90, 106103 (2003)24F. S. Tautz and J. A. Schaeffer, J. Appl. Phys. 84, 6636 (1998)25J. A. Schaefer, F. Stuckl, J. A. Anderson, G. J. lapeyre, and

W. Gopel, Surf. Sci. 140, 207 (1984)26R. Butz, E. M. Oellig, H. Ibach, and H. Wagner, Surf. Sci. 147,

343 (1984)27J. Eggeling, G. R. Bell, and T. S. Jones, J. Phys. Chem. B 103,

9683 (1999)28D. C. Allan and E. J. Mele, Phys. Rev. B 31, 5565 (1985)29D. C. Allan and E. J. Mele, Phys. Rev. Lett. 53, 826 (1984)30O. L. Alerhand and E. J. Mele, Phys. Rev. B 35, 5533 (1986)31J. Fritsch and P. Pavone, Surf. Sci. 344, 159 (1995)32J. Fritsch and U. Schroder, Phys. Rep. 309, 209 (1999)33H. M. Tutuncu, S. J. Jenkins, and G. P. Srivastava, Phys. Rev.

B 56, 4656 (1997)34R. M. Tromp, R. J. Hamers, and J. E. Demuth, Phys. Rev. Lett.

55, 1303 (1985)35R. J. Hamers, R. M. Tromp, and J. E. Demuth, Phys. Rev. B

34, 5343 (1986)36K. Hata, S. Yoshida, and H. Shigekawa, Phys. Rev. Lett. 89,

286104 (2002)37M.Ono, A. Kamoshida, N. Matsuura, E. Ishikawa, T. Eguchi,

and Y. Hasegawa, Phys. Rev. B 67, 201306 (2003)38K. Sagisaka, D. Fujita, and G. Kido, Phys. Rev. Lett. 91, 146103

(2003)

16

39K. Sagisaka, D. Fujita, G. Kido, and N. Koguchi, Surf. Sci 566,767 (2004)

40L. Perdigao, D. Deresmes, B. Grandidier, M. Dubois, C. Delerue,G. Allan, and D. Stievenard, Phys. Rev. Lett. 92, 216101 (2004)

41S.Yoshida, T. Kimura, O. Takeuchi, K. Hata, H. Oigawa,T. Nagamura, H. Sakama, and H. Shigekawa, Phys. Rev. B 70,235411 (2004)

42Y. J. Li, H. Nomura, N. Ozaki, Y. Naitoh, M. Kageshima, Y. Sug-awara, C. Hobbs, and L. Kantorovich, Phys. Rev. Lett. 96,106104 (2006)

43J. Appelbaum, G. Baraff, and D. Hamann, Phys. Rev. B 14,588 (1976)

44D. J. Chadi, Phys. Rev. Lett. 43, 43 (1979)45J. Ihm, M. Cohen, and D. Chadi, Phys. Rev. B 21, 4592 (1980)46M. Yin and M. Cohen, Phys. Rev. B 24, 2303 (1981)47E. Artacho and F. Yndurain, Phys. Rev. B 42, 11310 (1990)48Z. Zhu, N. Shima, and M. Tsukada, Phys. Rev. B 40, 868 (1989)49N. Roberts and R. Needs, Surf. Sci. 236, 112 (1990)50I. P. Batra, Phys. Rev. B 41, 5048 (1990)51S. Tang, A. Freeman, and B. Delley, Phys. Rev. B 45, 1776

(1992)52J. Dabrowski and M. Scheffler, Appl. Surf. Sci. 56, 15 (1992)53P. Kruger and J. Pollmann, Phys. Rev. B 38, 10578 (1988)54A. Ramstad, G. Brocks, and P. J. Kelly, Phys. Rev. B 51, 14504

(1994)55K. Seino, W. G. Schmidt, and F. Bechstedt, Phys. Rev. Lett.

93, 036101 (2004)56S. B. Healy, C. Filippi, P. Kratzer, E. Penev, and M. Scheffler,

Phys. Rev. Lett. 87, 016105 (2001)57J. D. E. McIntyre and D. E. Aspnes, Surf. Sci. 24, 417 (1971)58M. K. Kelly, S. Zoller, and M. Cardona, Surf. Sci. 285, 282

(1993)

59A. A. Lucas and M. Sunjic, Phys. Rev. Lett. 26, 229 (1971)60D. L. Mills, Surf. Sci. 49, 59 (1975)61P. Lambin, J. P. Vigneron, and A. A. Lucas, Phys. Rev. B 32,

8203 (1985)62B. N. J. Persson, Sol. St. Commun. 24, 573 (1977)63P. Senet, P. Lambin, J. P. Vigneron, I. Derycke, and A. A. Lucas,

Surf. Sci. 226, 307 (1990)64P. Lorrain and D. Corson, Electromagnetic fields and waves, Vol.

San Francisco (Freeman, 1970)65J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique,

Rep. Prog. Phys. 70, 1 (2007)66R. Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-

Wilson, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. Bush,P. D’Arco, and M. Llunell, (2009), Crystal09 User’s Manual,University of Torino, Torino, 2009.

67P. A. M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930)68U. von Barth and L. Hedin, J. Phys. C: Solid State Phys. 5, 1629

(1972)69F. Pascale, C. Zicovich-Wilson, F. Lopez, B. Civalleri, R. Or-

lando, and R. Dovesi, J. Comput. Chem. 25, 888 (2004)70C. Zicovich-Wilson, F. Pascale, C. Roetti, V. Saunders, R. Or-

lando, and R. Dovesi, J. Comput. Chem. 25, 1873 (2004)71S. Dall’Olio, R. Dovesi, and R. Resta, Phys. Rev. B 56, 10105

(1997)72H. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976)73C. H. Patterson, Mol. Phy. 108, 3181 (2010)74J. E. Jaffe and A. C. Hess, Phys. Rev. B 48, 7903 (1993)75F. Pascale, M. Catti, A. Damin, R. Orlando, V. Saunders, and

R. Dovesi, J. Phys. Chem. B 109, 18522 (2005)76F. Pascale, S. Tosoni, C. Zicovich-Wilson, P. Ugliengo, R. Or-

lando, and R. Dovesi, Chem. Phys. Lett. 396, 308 (2004)