键结轨道理论在量子半导体之应用与计算课件

TheApplicationandCalculationofBondOrbitalModelonQuantumSemiconductor

鍵結軌道理論在量子半導體之應用與計算TheApplicationandCalculatio1IntroductionIntroduction2WhyisthechoosingtheBOM?ahybridorlinkbetweenthek.pandthetight-bindingmethodscombiningthevirtuesofthetwoaboveapproaches--thecomputationaleffortiscomparabletothek.p

method--avoidingthetediousfittingprocedurelikethe

tight-bindingmethod--itisadequateforultra-thinsuperlattice--theboundaryconditionbetweenmaterialsistreatedin

thestraight-forwardmanner--itsflexibilitytoaccommodateotherwiseawkward

geometriesWhyisthechoosingtheBOM?a3Theimprovementofthebondorbitalmodel(BOM):the(hkl)-orientedBOMHamiltoniantheBOMHamiltonianwiththesecond-neighborinteractiontheBOMintheantibondingorbitalframeworktheBOMwithmicroscopicinterfaceperturbation(MBOM)thek.pformalismfromtheBOMTheimprovementofthebondor4BondOrbitalModelBondOrbitalModel5Whatisthebondorbitalmodel?atight-binding-likeframeworkwiththes-andp-likebasisorbitaltheinteractionparametersdirectlyrelatedtotheLuttingerparametersWhatisthebondorbitalmodel6Zinc-blendeLatticeStructure:Zinc-blendeLatticeStructure:7TheBOMmatrixelements:where：TheinteractionparametersEsandEp:on-siteparameters

Ess,Esx,Exx,Exy,andEzz:thenearest-neighbor

interactionparametersTheBOMmatrixelements:where：8TheBOMmatrix:wherewithH(k)=TheBOMmatrix:wherewithH(k)=9TakingTaylor-expansionontheBOMmatrix:

(uptothesecondorder)whereandH(k)=---TakingTaylor-expansiononthe10RelationsbetweenBOMparametersandLuttingerparametersVBMCVBM/3RelationsbetweenBOMparamete11BulkBandstructure:

(001)-orientationBulkBandstructure:

(001)-orie12SuperlatticeBandstructure:

(001)-orientationSuperlatticeBandstructure:

(013Theorthogonaltransformationmatrix:wheretheanglesandarethepolarandazimuthalanglesofthenewgrowthaxisrelativetotheprimarycrystallographicaxes.

Theorthogonaltransformation14BulkInAsBandstructure:

(111),(110),(112),(113),and(115)-orientationBulkInAsBandstructure:

(111)15InAs/GaSbSuperlatticeBandstructure:

(111),(110),(112),(113),and(115)-orientationInAs/GaSbSuperlatticeBandstr16Thesecond-neighborbondorbital(SBO)model:WhereandH(k)=Thesecond-neighborbondorbit17BulkBandstructure:

WiththeSecondNearestNeighborInteraction:BulkBandstructure:

WiththeS18BulkBandstructureintheAntibondingOrbitalModel:BulkBandstructureintheAnti19BondOrbitalModelwithMicroscopicEffectsBondOrbitalModelwithMicros20Forthecommonatom(CA)heterostructureeg:(AlGa)As/GaAs,InAs/GaAsForthenocommonatom(NCA)heterostructureeg:InAs/GaSb,(InGa)/As/InP--InAs/GaSbwithIn-SbandGa-Asheterobondsattheinterfaces--(InGa)As/InPwith(InGa)-PandIn-AsheterobondsattheinterfacesForthecommonatom(CA)heter21The(001)InAs/GaSbsuperlattice：theplanesofatomsarestackedinthegrowthdirectionasfollows．．．GaSbGaSbInAsInAs．．．．fortheoneinterface;and．．．InAsInAsGaSbGaSb．．．．forthenextinterface.The(001)InAs/GaSbsuperlatti22Theextractingofmicroscopicinformation:thes-andp-likebondorbitalsexpandedintermsofthetetrahedral(anti)bondingorbitals

andinsteadofscalarpotentialbypotentialoperator~thisistheso-calledmodifiedbondorbitalmodel(MBOM)~=(+++),=(+--),=(-+-),=(--+),(R)+),Theextractingofmicroscopic23ThepotentialtermoftheMBOM:apotentialmatrixform,butnotascalarpotentialVV4X4(Rz)=V+0000000000ThepotentialtermoftheMBOM24InAs/GaSbSuperlatticeBandstructure:

(calculatedwiththeBOMandMBOM)InAs/GaSbSuperlatticeBandstr25OrientationDependenceofInterfaceInversionAsymmetryEffectonInGaAs/InPQuantumWellsOrientationDependenceofInte26Inversionasymmetryeffect:themicroscopiccrystalstructure:Dresselhauseffectthemacroscopicconfiningpotential:Rashbaeffecttheinversionasymmetrybetweentwointerfaces:NCAheterostructures--thezero-fieldspinsplitting--in-planeanisotropyInversionasymmetryeffect:the27The73-Å-wide(25monolayers)(001)InGaAs/InPQW:Aandtheplanesofatomsarestackedinthegrowthdirectionasfollows:M+1

CDCDCDA

BABAB

M

forthe(InGa)P-likeinterface;and

N+1

ABABABC

DCDCD

N

fortheInAs-likeinterface,whereA=(InGa),B=As,C=In,andD=P.TheMth(orNth)monolayerislocatedattheleft(orright)interface,whereN=M+25.The73-Å-wide(25monolayers)28WhereRzisthezcomponentoflatticesiter,i.e.,R=R//+RzŽ,andalsotheU(fortheconductionband)andtheV(forthevalenceband)denotethedifferenceofpotentialenergybetweentheheterobondspeciesandthehostmaterialattheinterfaces.00000000000000000000000000WhereRzisthezcomponentof29(001)InGaAs/InPQuantumWellBandstructure:

(calculatedwiththeBOMandMBOM)(001)InGaAs/InPQuantumWell30SpinSplittingoftheLowestConductionSubband:

((001)InGaAs/InPQuantumWell)SpinSplittingoftheLowestC31Whenthein-planewavevectormovesaroundthecircle(=0

2),themixingelementsinEq.(4.2)shouldbestrictlywrittenasforthe(3,5)and(4,6)matrixelementsandforthe(5,3)and(6,4)matrixelements.Therefore,themixingstrengthdependsontheazimuthalangleMoreover,theandtermsequalto–1fororand1foror.Whenthein-planewavevector32The71-Å-wide(21monolayers)(111)InGaAs/InPQW:Thesameorderofatomicplanesasthe(001)QWAandThe71-Å-wide(21monolayers)33theheterobondsinthe[111]growthdirection:theheterobondsaretheremainingthreebondsother

thanthebondalongthe[111]direction:00000000000000000000000000000000000000000000000000000000000000theheterobondsinthe[111]g34(111)InGaAs/InPQuantumWellBandstructure:

(calculatedwiththeBOMandMBOM)(111)InGaAs/InPQuantumWell35SpinSplittingoftheLowestConductionSubband:

((111)InGaAs/InPQuantumWell)SpinSplittingoftheLowestC36The73-Å-wide(35monolayers)(110)InGaAs/InPQW:=+++

=-++-

=-+

and

=-acrossperfect(110)interfaces,planesofatomsarearrangedintheorderof:

M+1

DCDCBABA

CDCDA

BAB

M

fortheleftinterfaceand

N

ABABCDCD

BABADCDC

N+1

fortherightinterface,whereN=M+35The73-Å-wide(35monolayers)37wheretheuppersignisusedfortheMthandNthmonolayer,andthelowersignisusedforthe(M+1)thand(N+1)thmonolayer.0000000000000000000000wheretheuppersignisusedf38(110)InGaAs/InPQuantumWellBandstructure:

(calculatedwiththeBOMandMBOM)(110)InGaAs/InPQuantumWell39SpinSplittingoftheLowestConductionSubband:((110)InGaAs/InPQuantumWell)SpinSplittingoftheLowestC40SymmetrypointgroupofQWs.MicroscopicBOMBulkTdOhCAQW(001)D2dD4hNCAQW(001)C2vD4hNCAQW(111)C3vD3dNCAQW(110)C1horC1D2hSymmetrypointgroupofQWs.Mi41Dresselhaus-likeSpinSplittingDresselhaus-likeSpinSplittin42Dresselhauseffect:Thedegeneracybandsofthezinc-blendsbulkareliftedexceptforthewavevectoralongthe<001>and<111>directions,andthisistheso-calledDresselhauseffect.Dresselhauseffect:Thedeg43SubbandStructureof(110)InAs/GaSbSuperlattice:

(calculatedwiththeBOMandMBOM)SubbandStructureof(110)InA44MBOMBandstructureofInAs/GaSbSuperlattice

(grownonthe(001),(111),(113),and(115)-orientation)MBOMBandstructureofInAs/GaS45MicroscopicInterfaceEffecton(Anti)crossingBehaviorandSemiconductor-semimetalTransitioninInAs/GaSbSuperlatticesMicroscopicInterfaceEffecto46ThisMBOMmodelisbasedontheframeworkofthebondorbitalmodel(BOM)andcombinestheconceptoftheheuristicHbfmodeltoincludethemicroscopicinterfaceeffect.TheMBOMprovidesthedirectinsightintothemicroscopicsymmetryofthecrystalchemicalbondsinthevicinityoftheheterostructureinterfaces.Moreover,theMBOMcaneasilycalculatevariousgrowthdirectionsofheterostructurestoexploretheinfluenceofinterfaceperturbation.Inthischapter,byapplyingtheproposedMBOM,wewillcalculateanddiscussthe(anti)crossingbehaviorandthesemiconductortosemimetaltransitiononInAs/GaSbSLsgrownonthe(001)-,(111)-,and(110)-orientedsubstrates.TheeffectofinterfaceperturbationonInAs/GaSbwillbestudiedindetail.ThisMBOMmodelisbasedonth47(Anti)crossingBehaviorofInAs/GaSbSuperlattice(Anti)crossingBehaviorofInA48(001)SemimetalPhenomenon:

(calculatedwiththeBOMandMBOM)(001)SemimetalPhenomenon:

(c49(111)SemimetalPhenomenon:

(calculatedwiththeBOMandMBOM)(111)SemimetalPhenomenon:

(c50(110)SemimetalPhenomenon:

(calculatedwiththeBOMandMBOM)(110)SemimetalPhenomenon:

(c51k.pFiniteDifferenceMethodk.pFiniteDifferenceMethod52theBOMeigenfunctionsmustbeBlochfunctions,whichcanbeexpressedaswherethenotation

isusedforan-like(=s,x,y,z)bondorbitallocatedatafcclatticesiteR,kisthewavevector,andNisthetotalnumberoffcclatticepoints.

theBOMmatrixelementswiththebond-orbitalbasis(withoutspin-orbitcoupling)aregivenby(ink-space)WhereistherelativepositionvectorofthelatticesiteRtotheoriginand(seechapter2)istheinteractionparameter

takingtheTaylor-expansionontheBOMHamiltonianandomittingtermshigherthanthesecondorderink,thegeneralk‧pformalismiseasilyobtained,whosematrixelementscanbewrittenas[11]theBOMeigenfunctionsmustbe53thekinetictermoftheusualk˙pHamiltonian[inthebasis

)canbewrittenasCT0T00S0000000RRP+QBP-QP-Q-C-BP+Qwherethesuperscript*meansHermitianconjugate,

P=Ev

–[(2Exx+Ezz)/3]a2k2,

Q=–(Exx

–Ezz),a2(k2-)/12,

R=Ec

–Essa2k2

S=–Esxa

(kx+),

T=Esx/,

B=Exya2(–),

C=[(Exx

–Ezz)(–)/4–Exy],

and

Ec=Es+12Ess,Ev=Ep+8Exx+4Ezz.thekinetictermoftheusual54thetime-independentequationcanbeexpressedasafunctionofkz,thatis

]F=EFWiththereplacementofkzby,thisequationcanbeexpressedas=

and=theSchrödingerequationcanbewritteninthelayer-orbitalbasisaswhereistheinteractionbetweenandlayersF=EFThek.pfinitedifferencemethodthetime-independentequation55OptimumStepLengthintheKPFDMethodOptimumStepLengthintheKPF56the-dependenttermsofthek˙pHamiltoniancanbewrittenasandwhereisthespacingofmonolayersalongthegrowthdirectionthereplacementofbyandthentreatedbythefinite-differencecalculation,wehave

andwhereisthepseudo-layer,thesteplengthhisthespacingbetweentwoadjacentpseudo-layers,andFisthecorrespondingstatefunction.Thereasonoftheoptimumsteplength==the-dependenttermsofth57theSchrödingerequationsolvedbytheKPFDmethodcanbewrittenas

whereistheinteractionbetweenandlayers;theintergernis1forthe(001)and(111)samples,2forthe(110)and(113)samples,3forthe(112)and(115)samples,…,etc.Thatistosay,theon-siteand12nearest-neighborbondorbitalsbelongrespectivelyto(2n+1)layers,whichareeasilyclassifiedaccordingtothelongitudinalcomponentofthebond-orbitalpositionvector.Thesteplengthbetweentheon-sitelayerandthenearest-,second-,orthird-neighborinteractionlayeris1ML,2ML,or3MLspacinginthelongitudinaldirection,respectively.theSchrödingerequationsolv58Multi-stepLengthinthe(110)KPFDMethod:Multi-stepLengthinthe(110)59Multi-stepLengthinthe(112)KPFDMethod:Multi-stepLengthinthe(112)60Multi-stepLengthinthe(113)KPFDMethod:Multi-stepLengthinthe(113)61Multi-stepLengthinthe(115)KPFDMethod:Multi-stepLengthinthe(115)62InAsBulkBandstructure

(calculatedwiththek˙pandSBOmethod)InAsBulkBandstructure

(calcu63InAsBulkBandstructure

(calculatedwiththeSBOandKPSFDmethod)InAsBulkBandstructure

(calcu64AnisotropicOpticalMatrixElementsinQuantumWellswithVariousSubstrateOrientationsAnisotropicOpticalMatrixEle65The(11N)44LuttingerHamiltonianattheBrillouin-zonecenter(k1=k2=0)Hk.p(k1=k2=0)=(Ep+8Exx+4Ezz)-+++

whereaisthelatticeconstant,istheanglebetweenthezandX3axes,whichisequaltoThe(11N)44LuttingerHamilt66theopticaltransitionmatrixelementbetweentheconductionandthevalencebandscanbewrittenaswhereisthemomentumoperatorandêistheunitpolarizationvector.thein-planeopticalanisotropycanbecalculatedaswhereandarethesquaredmatrixelementsforthepolarizationparallelandperpendicularto,respectively.theopticaltransitionmatrix67AnisotropicOpticalMatrixElements(inthe(11N)-orientationAnisotropicOpticalMatrixEle68TheApplicationandCalculationofBondOrbitalModelonQuantumSemiconductor

鍵結軌道理論在量子半導體之應用與計算TheApplicationandCalculatio69IntroductionIntroduction70WhyisthechoosingtheBOM?ahybridorlinkbetweenthek.pandthetight-bindingmethodscombiningthevirtuesofthetwoaboveapproaches--thecomputationaleffortiscomparabletothek.p

method--avoidingthetediousfittingprocedurelikethe

tight-bindingmethod--itisadequateforultra-thinsuperlattice--theboundaryconditionbetweenmaterialsistreatedin

thestraight-forwardmanner--itsflexibilitytoaccommodateotherwiseawkward

geometriesWhyisthechoosingtheBOM?a71Theimprovementofthebondorbitalmodel(BOM):the(hkl)-orientedBOMHamiltoniantheBOMHamiltonianwiththesecond-neighborinteractiontheBOMintheantibondingorbitalframeworktheBOMwithmicroscopicinterfaceperturbation(MBOM)thek.pformalismfromtheBOMTheimprovementofthebondor72BondOrbitalModelBondOrbitalModel73Whatisthebondorbitalmodel?atight-binding-likeframeworkwiththes-andp-likebasisorbitaltheinteractionparametersdirectlyrelatedtotheLuttingerparametersWhatisthebondorbitalmodel74Zinc-blendeLatticeStructure:Zinc-blendeLatticeStructure:75TheBOMmatrixelements:where：TheinteractionparametersEsandEp:on-siteparameters

Ess,Esx,Exx,Exy,andEzz:thenearest-neighbor

interactionparametersTheBOMmatrixelements:where：76TheBOMmatrix:wherewithH(k)=TheBOMmatrix:wherewithH(k)=77TakingTaylor-expansionontheBOMmatrix:

(uptothesecondorder)whereandH(k)=---TakingTaylor-expansiononthe78RelationsbetweenBOMparametersandLuttingerparametersVBMCVBM/3RelationsbetweenBOMparamete79BulkBandstructure:

(001)-orientationBulkBandstructure:

(001)-orie80SuperlatticeBandstructure:

(001)-orientationSuperlatticeBandstructure:

(081Theorthogonaltransformationmatrix:wheretheanglesandarethepolarandazimuthalanglesofthenewgrowthaxisrelativetotheprimarycrystallographicaxes.

Theorthogonaltransformation82BulkInAsBandstructure:

(111),(110),(112),(113),and(115)-orientationBulkInAsBandstructure:

(111)83InAs/GaSbSuperlatticeBandstructure:

(111),(110),(112),(113),and(115)-orientationInAs/GaSbSuperlatticeBandstr84Thesecond-neighborbondorbital(SBO)model:WhereandH(k)=Thesecond-neighborbondorbit85BulkBandstructure:

WiththeSecondNearestNeighborInteraction:BulkBandstructure:

WiththeS86BulkBandstructureintheAntibondingOrbitalModel:BulkBandstructureintheAnti87BondOrbitalModelwithMicroscopicEffectsBondOrbitalModelwithMicros88Forthecommonatom(CA)heterostructureeg:(AlGa)As/GaAs,InAs/GaAsForthenocommonatom(NCA)heterostructureeg:InAs/GaSb,(InGa)/As/InP--InAs/GaSbwithIn-SbandGa-Asheterobondsattheinterfaces--(InGa)As/InPwith(InGa)-PandIn-AsheterobondsattheinterfacesForthecommonatom(CA)heter89The(001)InAs/GaSbsuperlattice：theplanesofatomsarestackedinthegrowthdirectionasfollows．．．GaSbGaSbInAsInAs．．．．fortheoneinterface;and．．．InAsInAsGaSbGaSb．．．．forthenextinterface.The(001)InAs/GaSbsuperlatti90Theextractingofmicroscopicinformation:thes-andp-likebondorbitalsexpandedintermsofthetetrahedral(anti)bondingorbitals

andinsteadofscalarpotentialbypotentialoperator~thisistheso-calledmodifiedbondorbitalmodel(MBOM)~=(+++),=(+--),=(-+-),=(--+),(R)+),Theextractingofmicroscopic91ThepotentialtermoftheMBOM:apotentialmatrixform,butnotascalarpotentialVV4X4(Rz)=V+0000000000ThepotentialtermoftheMBOM92InAs/GaSbSuperlatticeBandstructure:

(calculatedwiththeBOMandMBOM)InAs/GaSbSuperlatticeBandstr93OrientationDependenceofInterfaceInversionAsymmetryEffectonInGaAs/InPQuantumWellsOrientationDependenceofInte94Inversionasymmetryeffect:themicroscopiccrystalstructure:Dresselhauseffectthemacroscopicconfiningpotential:Rashbaeffecttheinversionasymmetrybetweentwointerfaces:NCAheterostructures--thezero-fieldspinsplitting--in-planeanisotropyInversionasymmetryeffect:the95The73-Å-wide(25monolayers)(001)InGaAs/InPQW:Aandtheplanesofatomsarestackedinthegrowthdirectionasfollows:M+1

CDCDCDA

BABAB

M

forthe(InGa)P-likeinterface;and

N+1

ABABABC

DCDCD

N

fortheInAs-likeinterface,whereA=(InGa),B=As,C=In,andD=P.TheMth(orNth)monolayerislocatedattheleft(orright)interface,whereN=M+25.The73-Å-wide(25monolayers)96WhereRzisthezcomponentoflatticesiter,i.e.,R=R//+RzŽ,andalsotheU(fortheconductionband)andtheV(forthevalenceband)denotethedifferenceofpotentialenergybetweentheheterobondspeciesandthehostmaterialattheinterfaces.00000000000000000000000000WhereRzisthezcomponentof97(001)InGaAs/InPQuantumWellBandstructure:

(calculatedwiththeBOMandMBOM)(001)InGaAs/InPQuantumWell98SpinSplittingoftheLowestConductionSubband:

((001)InGaAs/InPQuantumWell)SpinSplittingoftheLowestC99Whenthein-planewavevectormovesaroundthecircle(=0

2),themixingelementsinEq.(4.2)shouldbestrictlywrittenasforthe(3,5)and(4,6)matrixelementsandforthe(5,3)and(6,4)matrixelements.Therefore,themixingstrengthdependsontheazimuthalangleMoreover,theandtermsequalto–1fororand1foror.Whenthein-planewavevector100The71-Å-wide(21monolayers)(111)InGaAs/InPQW:Thesameorderofatomicplanesasthe(001)QWAandThe71-Å-wide(21monolayers)101theheterobondsinthe[111]growthdirection:theheterobondsaretheremainingthreebondsother

thanthebondalongthe[111]direction:00000000000000000000000000000000000000000000000000000000000000theheterobondsinthe[111]g102(111)InGaAs/InPQuantumWellBandstructure:

(calculatedwiththeBOMandMBOM)(111)InGaAs/InPQuantumWell103SpinSplittingoftheLowestConductionSubband:

((111)InGaAs/InPQuantumWell)SpinSplittingoftheLowestC104The73-Å-wide(35monolayers)(110)InGaAs/InPQW:=+++

=-++-

=-+

and

=-acrossperfect(110)interfaces,planesofatomsarearrangedintheorderof:

M+1

DCDCBABA

CDCDA

BAB

M

fortheleftinterfaceand

N

ABABCDCD

BABADCDC

N+1

fortherightinterface,whereN=M+35The73-Å-wide(35monolayers)105wheretheuppersignisusedfortheMthandNthmonolayer,andthelowersignisusedforthe(M+1)thand(N+1)thmonolayer.0000000000000000000000wheretheuppersignisusedf106(110)InGaAs/InPQuantumWellBandstructure:

(calculatedwiththeBOMandMBOM)(110)InGaAs/InPQuantumWell107SpinSplittingoftheLowestConductionSubband:((110)InGaAs/InPQuantumWell)SpinSplittingoftheLowestC108SymmetrypointgroupofQWs.MicroscopicBOMBulkTdOhCAQW(001)D2dD4hNCAQW(001)C2vD4hNCAQW(111)C3vD3dNCAQW(110)C1horC1D2hSymmetrypointgroupofQWs.Mi109Dresselhaus-likeSpinSplittingDresselhaus-likeSpinSplittin110Dresselhauseffect:Thedegeneracybandsofthezinc-blendsbulkareliftedexceptforthewavevectoralongthe<001>and<111>directions,andthisistheso-calledDresselhauseffect.Dresselhauseffect:Thedeg111SubbandStructureof(110)InAs/GaSbSuperlattice:

(calculatedwiththeBOMandMBOM)SubbandStructureof(110)InA112MBOMBandstructureofInAs/GaSbSuperlattice

(grownonthe(001),(111),(113),and(115)-orientation)MBOMBandstructureofInAs/GaS113MicroscopicInterfaceEffecton(Anti)crossingBehaviorandSemiconductor-semimetalTransitioninInAs/GaSbSuperlatticesMicroscopicInterfaceEffecto114ThisMBOMmodelisbasedontheframeworkofthebondorbitalmodel(BOM)andcombinestheconceptoftheheuristicHbfmodeltoincludethemicroscopicinterfaceeffect.TheMBOMprovidesthedirectinsightintothemicroscopicsymmetryofthecrystalchemicalbondsinthevicinityoftheheterostructureinterfaces.Moreover,theMBOMcaneasilycalculatevariousgrowthdirectionsofheterostructurestoexploretheinfluenceofinterfaceperturbation.Inthischapter,byapplyingtheproposedMBOM,wewillcalculateanddiscussthe(anti)crossingbehaviorandthesemiconductortosemimetaltransitiononInAs/GaSbSLsgrownonthe(001)-,(111)-,and(110)-orientedsubstrates.TheeffectofinterfaceperturbationonInAs/GaSbwillbestudiedindetail.ThisMBOMmodelisbasedonth115(Anti)crossingBehaviorofInAs/GaSbSuperlattice(Anti)crossingBehaviorofInA116(001)SemimetalPhenomenon:

(calculatedwiththeBOMandMBOM)(001)SemimetalPhenomenon:

(c117(111)SemimetalPhenomenon:

(calculatedwiththeBOMandMBOM)(111)SemimetalPhenomenon:

(c118(110)SemimetalPhenomenon:

(calculatedwiththeBOMandMBOM)(110)SemimetalPhenomenon:

(c119k.pFiniteDifferenceMethodk.pFiniteDifferenceMethod120theBOMeigenfunctionsmustbeBlochfunctions,whichcanbeexpressedaswherethenotation

isusedforan-like(=s,x,y,z)bondorbitallocatedatafcclatticesiteR,kisthewavevector,andNisthetotalnumberoffcclatticepoints.

theBOMmatrixelementswiththebond-orbitalbasis(withoutspin-orbitcoupling)aregivenby(ink-space)WhereistherelativepositionvectorofthelatticesiteRtotheoriginand(seechapter2)istheinteractionparameter

takingtheTaylor-expansionontheBOMHamiltonianandomittingtermshigherthanthesecondorderink,thegeneralk‧pformalismiseasilyobtained,whosematrixelementscanbewrittenas[11]theBOMeigenfunctionsmustbe121thekinetictermoftheusualk˙pHamiltonian[inthebasis

)canbewrittenasCT0T00S0000000RRP+QBP-QP-Q-C-BP+Qwherethesuperscript*meansHermitianconjugate,

P=Ev

–[(2Exx+Ezz)/3]a2k2,

Q=–(Exx

–Ezz),a2(k2-)/12,

R=Ec

–Essa2k2

S=–Esxa

(kx+),

T=Esx/,

B=Exya2(–),

C=[(Exx

–Ezz)(–)/4–Exy],

and

Ec=Es+12Ess,Ev=Ep+8Exx+4Ezz.thekinetictermoftheusual122thetime-independentequationcanbeexpressedasafunctionofkz,thatis

]F=EFWiththereplacementofkzby,thisequationcanbeexpressedas=

and=theSchrödingerequationc