LectureJan.16
Boundaryconditionswithanon-zeroperiodicpotential,U
-Duetothecrystalstructure,itisgenerallynotconvenienttoconsideracubewith
periodicboundaryconditions,aswedidforthefreeelectroncase–aninteger
numberofprimitivecellsofthedirectlatticewouldnotgenerallyfitintothecube.
Instead,itismathematicallyconvenienttoimposethegeneralizedperiodic
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boundaryconditions: ψ ( r + N i ai ) = ψ ( r ), i = 1,2,3wherethe ai areprimitivevectors
oftheBravaislatticeandtheNiare3largenumbers(aroundthecuberootof
Avagadro’snumber).
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-WritingtheeigenfunctionsasinEq.(1),whataretheallowedvaluesof k ?Since
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unk! ( r ) isperiodictheboundaryconditionsreduceto
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e ik ⋅a i N i = 1. !
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Expanding k inprimitivevectorsofthereciprocallattice, bi , €
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(6)
k = f1b1 + f 2b2 + f 3b3 ,
i2πf i N i
= 1.Thusfi=mi/Niwithmiintegers,so
theseconditionsbecome e
! € 3 mi !
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k = ∑ bi .Thisisasimplegeneralizationofthefreeelectroncase.Recallthatwe
N
i=1
!i
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restrict k toaprimitivecellofthereciprocallattice(generallythefirstBrillouin
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zone).Thiscanbechosentobetheparallelepipedwithsides bi soweseethatthe
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allowedwave-vectors k aredenseintheRLunitcell.Thenumberofwave-vectors
€intheunitcellisN1N2N3whichwecallN.Nisthenumberofprimitivecellsinthe
directlattice.WecanchooseaprimitivecelllabeledasinEq.(6)with0<fi<1.
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Alternatively,wecancalculatethevolumeofthetinyparallelepipedwithsides
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bi /N i .Thisis
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!
! !
b1 ⎛ b2
b3 ⎞ 1 !
⋅⎜
×
⎟ = b1 × b2 × b3 ,(1/N)timesthevolumeoftheprimitivecellofthe
N1 ⎝ N 2 N 3 ⎠ N
RL.Therefore,thenumberofallowedwave-vectorsinaRLprimitivecellisN,the
numberofBravaislatticepointsinthecrystalwiththeseboundaryconditions.The
volumeofareciprocallatticeunitcellcanbeproventobe(2π)3/vwherevisthe
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volumeofadirectlatticeunitcell: v = a1 × ( a2 × a3 ) .SeeHomework2.Withthese
boundaryconditions,thenumberofprimitiveunitcellsinthecrystalisN,sothe
totalvolumeofthecrystalisV=vN.Thusthevolumeink-spaceofthetiny
parallelepipedaroundeachallowedk-pointcanbewritten(2π)3/V,thesameresult
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weobtainedforfreeelectrons.
FouriertransformedSchroedingerequation
ItwillbeconvenienttowritetheSchroedinger-likeEq.(5)obeyedbytheperiodic
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function unk! ( r ) inFouriertransformedform–thatisink-space.Wewillusethisfor
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studyingthelimitofsmallbutnon-zeroU.Thatis,doingperturbationtheoryinU.
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1
!
r
Todothis,westartbyobservingthatsinceU(
)isperiodic,itcanbewritten
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iK ⋅ r
U( r ) = ∑U K! e wherethesumisoverRLvectors.TheFouriertransformofa
!
K
generalfunctionwouldinvolveasum(orintegral)overallwave-vectors,but
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periodicityrestrictsthewave-vectorstoRLpoints.Inverting
1
! !! !
U K! = ∫ d re −iK ⋅r U( r )
v cell
Thisfollowssince
! ! !
1
(7)
d 3 r e i( K − K ' )⋅r = δ K! ,K! ' .
∫
v cell
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! !
Thisfollowsfromthefactthat e iK ⋅r areorthogonalfunctions.Inthecaseofasimple
cubiclattice,writing
! ! 2π
K − K' =
n ,n ,n a x y z €
TheintegralinEq.(7)factorizesinto3integralsoftheform:
1 a
1 i(2π / a )n x x x =a
dx e i(2π / a )n x x =
e
∫
x =0 .
a 0
2πi
Thisintegralgiveszerofor n x ≠ 0 and1fornx=0.Thisargumentcanbegeneralized
toanyBravaislattice.TogeneralizethistoanarbitraryBLnotethat,writing
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r = f1a1 + f 2 a2 + f 3 a3 ,wemaywrite
1
1
1
1
3
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∫ d r = ∫ df1 ∫ df 2 ∫ df 3 v cell
0
0
0
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and,writingtheRLvectorsas K = n1b1 + n 2b2 + n 3b3 ,theintegralofEq.(7)becomes:
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1
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)
1
∫ df e
2πin x f1
1
1
∫ df e
2
2πin y f 2
∫ df e
3
2πin z f 3
= δ n x ,0δ n y ,0δ n z ,0 .
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SinceU( r )isreal, U − K! = U K! * .Ifthecrystalhasinversionsymmetry,sothat,fora
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suitablechoiceoforigin, U(−r ) = U( r ) ,then
U K! = U − K! = U K! * .
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€ItwillbeconvenienttoshiftU(
r )byaconstant,wecanmakeitsaverageoveraunit
€
!
r )justshiftsallenergybandsbyaconstantand
cell, U !0 = 0 .(ThisredefinitionofU(
€
0
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0
0
isofnoimportance.)
Theperiodicfactorsintheeigenfunctionshaveasimilarexpansion,whichwewrite:
€ !
! !
!
−iK ⋅ r
!
!
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k liesinthefirstBrillouinzone.Thislabelingofthe
uk ( r ) = ∑ c k − K e
.Here €
!
K
Fouriercoefficient,c,isconvenientbecausethenthefullwave-functioniswritten:
! ! !
!
ψ k! ( r ) = ∑ c k! − K! e i( k − K )⋅r .Weseethatthereisacoefficientcforeachpointinallofk€
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€
K
space(subjecttotheperiodicboundaryconditions).Eq.(5)becomes:
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2
⎡ !2 "
⎤
⎡ !2 " " 2
⎤
" "
" "
" "
" "
2
0 = ⎢−
ik + ∇ + ∑U K" 'e iK '⋅r − ε k" ⎥∑ c k" − K" e −iK ⋅r = ⎢
k − K + ∑U K" 'e iK '⋅r − ε k" ⎥∑ c k" − K" e −iK ⋅r
"
"
⎢⎣ 2m
⎥⎦ K"
⎥⎦ K"
⎣⎢ 2m
K
K
Thenwerewrite:
! !
! !
! !
! !
U K! 'c k! − K! − K! 'e −iK ⋅r = ∑U K! ' − K! c k! − K! 'e −iK ⋅r ∑! U K! 'e iK '⋅r ∑! c k! − K! e −iK ⋅r = ∑
! !
! !
K'
K
K'K
K'K
!
! !
K by K + K ' ,thenreplacingthe
byfirstreplacingthedummysummationvariable
!
! !
!
K
K
'
−
K
K
otherdummysummationvariable
’by
.Thisgivesasumover
ofterms
! !
−iK ⋅ r
withr-dependence e
.Eachofthesetermsmustseparatelyvanishsincethe
!
equationistrueforall r andtheseareorthogonalfunctions,asimpliedbyEq.(7).
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Thuswegetak-spaceSchroedinger-likeequationforthediscreteFouriermodesof
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theperiodicfunctionu:
⎡ ! 2 " "€2
⎤
(7)
( k − K ) −€ε k" ⎥c k" − K" + ∑U K" ' − K" c k" − K" ' = 0 ⎢
"
⎣ 2m
⎦
K'
!
K
WemustsolvethisequationforeachRLvector
togetthesetofeigenfunctions
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andeigenvaluesforaspecifiedvalueof k inthefirstBrillouinzone.The!
eigenfunctionsarenowlabeledby c k! − K! .Eacheigenfunctionhasafixed k .There
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k ,correspondingtothe unk! with
willbeaninfinitenumberofsolutionsforeach
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c k! − K! non-zeroforevery K in
n=0,1,2,3,…Eachsolutionwillgenerallyhaveallofthe
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theBZ.
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