Topological aspects of surface states in semiconductors are studied by an adiabatic deformation which connects a realistic system and a decoupled covalent-bond model. Two topological invariants are focused. One is a quantized Berry phase, and the other is

orbitalsinaunitcell.Wesupposetwoatomsinaunit

cellanddenoteorbitalsasα=1,¯1,2,¯2,...,N,N

¯,whereNisanumberoforbitalsinaatomandα¯onaatommeansanorbitaloppositetoanorbitalαontheotheratom(SeeFig.1).Then,HPBC(k)isa4N×4N

matrix.

Fig.1.Modelofadiamondstructure.Spheresstandforatoms,andlinesstandforsp3hybridizedorbitals.Ashadestandsfor(111)surface,and anstandsforbasislatticevectorsrelativeto(111)surface.

Next,wesupposethata1anda2spanaplaneparalleltoasurfaceofasemiconductorasashadedplaneshowninFig. 1.2-dimensionalwavenumbervectorisgivenbyk =2

n=1knbn.Then,thebulkHamiltonianiswrittenas

HPBC= c k ,n HPBC(k )

n,n′ck ,n′,(2)

k n,n′

wherethelabelnisde nedasn=(i3,β)andHPBC(k )isa4NL×4NLmatrix,andi3∈[1,L].Lisanum-beroflayers.Toconsiderthesurface,weintroducetheopenboundarycondition(OBC)fortheonedimensionalsysteminthea3direction.ThematrixHOBC(k )isgivenbytruncatingHPBC(k ).Forexample,anatu-ralwaytotruncationistoprohibitallthematrixele-mentsacrossL.Wede neaboundary-hoppingmatrixHB(k )=HPBC(k ) HOBC(k )as

Hk

HPBC(k ) n,n′[i3,i′3] LB( )n,n′=0others.(3)

HB(k )is4NL×4NLmatrix.Thismatrixrepresents

allhoppingelementsacrosstheboundaryofthesystem.Tode netheBerryphase,weintroduceatwistangleθinahoppingtermacrosstheboundaryasc k ,nck ,n′→

eiθc k ,nck ,n′.Indetail,thetwistangleθisintroducedinaselectedelementnn′ofHB(k )andwedenoteitasHB(θ,k ).Then,theBerryphasecanbede nedas

γnn′(k )=

2π

gs| θ|gs dθ,(4)0

where|gs(θ) isthehalf- lledgroundstateofHPBC(θ,k )=HOBC(k )+HB(θ,k ).Especially,we

denotemainelementsoftheBerryphaseasγασ=γ(Lασ)(1¯ασ).TheBerryphaseisquantizedbecausetheone-dimensionalsystematk de nedbyHPBC(k )hasainversionsymmetryattheboundary:(i3,α) (L+1 i3,α¯).WhenwedenotetheinversionasUandthecom-plexconjugateasK,theanti-unitaryoperatorΘ=KUiscommutablewiththeHamiltonian.Then,theBerryphaseturnsouttobequantizedinthesamewayasinRef.9

Figure2(a)showsthebandstructureofH26PBCofGeandreproducetheindirectgap.Afterintroducingthe(111)surface,thebanddiagramofHOBChasedgestatesinthebandgap,whicharedoubly-degeneratedperspin,asshowninFig.2(b).TheBerryphaseofGe(111)surfaceisactuallyquantizedanddoesnotdependonk .Itturnsouttobeγ1↑=γ1↓=πonthestrongesthoppinginHBandzeroontheothers.Wenotethatthenumberofcandidatesofγnn′is48fortheparametersweused.

(a)

4

2

)

Ve( yg 0

renE-2

-4

L

Γ

X

U,K

Γ

(b)

Fig.2.(a)AbandstructureofbulkGe.(b)Energyspectrumof

Ge(111)surface.ThewhiteareadenotesbandgapofbulkGeandlinesdoedgestatesinGe(111).ThenotationsofpointsinrelativesurfaceBrillouinzoneareaccordingtoIvanovet.al.27Thenumberofthedegeneracyoftheedgestatesisfourincludingspindegreesoffreedom.

ThequantizedBerryphaseandthenumberofedgestatesnearetopologicalinvariantinthefollowingadia-batictransformation.We rstmodifythespin-orbitcou-plingtermadiabaticallytozeropreservingtheenergygapopen,thatis,withoutchangingthetopologicalin-variants.AtthisstagetheHamiltonianisspindecoupledasH=H↑⊕H↓.Nowweignorethespinindexbelowforsimpli cation.Next,wecanmodifythehoppingtermsbetweendi erentorbitalsandobtainaHamiltonianasthesumoftwobandHamiltoniansH=H(1)⊕···⊕H(N),whereH(α)involveshoppingtermsbetweenαandα¯only.Forthisprocess,itisalsopossibletoholdthegapopen.AftertheFouriertransformationofthea3