So far, we have assumed that the periodic system under consideration is infinite. Real crystals,ofcourse,haveboundaries.Wecantaketheseboundariesintoaccountbyoneof twocommon choicesof boundaryconditions.First, wecanenforce ψ(x) = ψ(L) = 0 ateach surface,whichis realisticfor electronenergies comparable tothe atomicbound stateenergies.ThisconditionformallyviolatestheassumptionweusedtoproveBloch’s theorem,however, which is that the system is invariant under a translation T´R, for any latticevector ´R.If thesystemis very large,however,this isnot reallyaproblem.Deep insidethecrystal, thesystemis invariantunder translationsbyalargenumberoflattice vectors,andthereforetheBlochwavefunctionswillbeverygoodapproximationsofthe realeigenstates.
Alternatively, we canuse periodic boundary conditions, ψ(0) = ψ(L), ∂ψ (0)/∂x =
∂ψ(L)/∂x. This is known as the Born–von Karman boundary condition. While it is unphysical,ithastheadvantagethatitformallysatisfiestheassumptionsofBloch’stheo- rem,namelythatthesystemisinvariantunderanytranslation T´R.Thisboundarycondition allows travelingwave solutions–a wavecan exitone side and enteronthe otherside.
This simulates the case of waves traveling in the same direction forever in an infinite medium.
The boundaryconditionsforce aquantizationconditiononthe possiblevalues ofthe plane-wavevector ´k,inadditiontotheconstraintderivedabovewhichrestrictedthevalues of ´k totheBrillouinzone. InthecaseoftheBorn–vonKarmanboundarycondition,the phase factor ei´kã´xmust beequalat ´x = 0and ´x = ´R,where |´R| = L.Thisconditionis equivalentto
ei´kã ´R= ei´kã0= 1, (1.7.1)
whichimpliesthat ´k ã ´R = 2πN,where N isaninteger.Thevector ´R mustbeaBravais latticevector,whichimplies
´k ã ´R = ´k ã (N1´a1+ N2´a2+ N3´a3)
= 2πN. (1.7.2)
Thisconditionissatisfiedif ´k hastheform
´k = νN11´q1+ ν2
N2´q2+ ν2
N3´q3, (1.7.3)
wherethe ´qiarethereciprocallatticeprimitivevectorsdefinedinSection1.4and ν1, ν2, and ν3areintegers,inwhichcase
´k ã ´R = (ν1+ ν2+ ν3)2π, (1.7.4) since ´aiã´qj = 2πδij.
Thecondition(1.7.3)impliesthat ´k canonlyhavediscretevalues.Thisisjustthesame typeofquantizationthatoccursduetotheboundaryconditionsofanyfinitesystem.As illustratedinFigure1.20,forperiodicboundaryconditionsinaboxwithsize L,awave canonlyhavediscretewavelengths λn = L/ν,where ν isaninteger,whichimplies kn = 2π/λn= 2πν/L.
E
0 k
(a) (b)
L
(L) = (0) (0)
a 2L
±Fig.1.20 (a)Boundaryconditionsofafinitesystemcausequantizationofthepossiblewavelengths.(b)Forthisreason,an energybandactuallyconsistsofN discretestates,where N isthenumberofcellsinthelattice.
37 1.7 BoundaryConditionsinaFiniteCrystal
Since |ν1| ≤ N1/2, |ν2| ≤ N2/2,and |ν3| ≤ N3/2for ´k-vectorsintheBrillouinzone, thereareonly N = N1N2N3 discretestatesfor ´k in theBrillouin zone.Thismeansthat ifthereare N primitivecells,thereare N statesinaband.Thisisthesamestatementwe madeinSection1.1,namely N cellstatesleadto N bandstates.Alongonedimensionin thedirection ofprimitive vector ´ai, thesestatesareseparatedby dk = 2π/(Ni|´ai|).For large N,thisspacingissosmallthatwecantypicallytreat ´k asacontinuousvariableand ignoreitsdiscretesteps.
Whatifwehadchosentheboundaryconditionofimpenetrableboundaries,insteadof theBorn–vonKarmanperiodicboundaryconditions?Inthatcase,wecouldhaveeitheran oddorevennumberofhalf-wavelengthswithinthesizeofthecrystal,namely
e2i´kã´R= ei´kã0= 1. (1.7.5)
Thiswouldseemtoleadtotheconclusionthatthereare2N statesinabandinonedimen- sion,or8N statesinathree-dimensionalcrystal.Whichisit?Arethere N or8N states?We canresolvethisbyrememberingthatinthecaseofimpenetrableboundaries,theallowed statesare standingwavesinsteadoftravelingwaves,in whichcasewecountonlyposi- tive k states.Thismeanswerestrict thecountingof statesto onlythe firstoctantofthe three-dimensionalspace,whichisone-eighthofthespacethatincludesnegativenumbers.
Therefore,westillhaveexactly N statesfor N cells.
Asmentionedabove,thespacingbetweenthestatevectorsalongonedimensionis ±k = 2π/(Ni|´ai|).Inthreedimensions,insteadoftalkingaboutthespacingbetweenthestates, wecantalkaboutthevolumeperstatein k-space.Onewaytocalculatethis istonotice that(1.7.3)definesanewlatticeinreciprocalspacethathaslatticevectors ´b1/N1, ´b2/N2, and ´b3/N3.AccordingtotheresultofExercise1.4.4,thevolumepercellofthislatticeis
±3k = (2π)3
|N1a´1ã (N2a´2ì N3a´3)|
= (2π)3 N1N2N3Vcell
= (2π)3
V , (1.7.6)
where V isthetotalvolumeofthecrystal.
Exercise1.7.1 Supposewehavearingwithsixidenticalatoms.Thisisaperiodicsytemin onedimension,sotheBlochtheoremapplies.AccordingtotheLCAOapproxima- tion,discussedinSection1.1.2,wewritethewavefunctionasalinearcombination oftheunperturbed atomic waveorbitals.Thisallows usto writeamatrixfor the eigenstatesasfollows:
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝
E0 U 0 0 0 U
U E0 U 0 0 0
0 U E0 U 0 0
0 0 U E0 U 0
0 0 0 U E0 U
U 0 0 0 U E0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝ c1
c2
c3 c4
c5
c6
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
= E
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝ c1
c2
c3 c4
c5
c6
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠ ,
where |ψ ² = c1|ψ1²+ c2|ψ2²+ c3|ψ3²+ c4|ψ4²+ c5|ψ5²+ c6|ψ6². For con- venience,weassumethecoupling term U isreal.Inone dimension,theformulas for the boundary conditions take a simple form: We have b = 2π/a, and k = νb/N = 2πν/Na.Bloch’stheorem tellsusthatthephase factor fromone cellto thenextwillbe e−ika,whichis ei2πν/N.Show thatthesolution cn+1 = e2πνi/6cn
is an eigenvectorof the abovematrix, for ν = 1, ... ,6.(Itis easyto dothis in Mathematica.)
Notethatiftwoeigenvaluesaredegenerate,thentheeigenstatescorrespondingto theseeigenvaluesarenotunique,andcanbewrittenasanylinearsuperpositionof thetwostateswiththesameeigenvalue.Intheabovesolution,therearetwosetsof degenerateeigenstates correspondingto k travelingin oppositedirections.(Since, inthereduced zone,we wouldtake ν from −3to3instead offrom1to6.)One couldthereforeinsteadequallywellwritethecoefficientsas2cos ka = eika+ e−ika, 2isin ka = eika− e−ika.WhicheigenvectorsdoesMathematicagive,ifyouuseits eigenvectorsolver?
Supposethatinsteadofaring,wehaveachainofsixatomswiththeendsuncon- nected.Howshouldyoualtertheabovematrixtodescribethissystem?Foraspecific choiceof E0 and U,namely E0 = 1 and U = 0.1,solve forthe eigenvalues of boththeringandthelinear chain.Howmuchdotheeigenvalues differinthetwo cases?