AsdiscussedinSection1.1,solidstatephysicsdoesnotonlydealwithperiodicstructures.
Nevertheless,thetheoryofperiodicstructuresisextremelyimportantbecausemanysolids dohaveperiodicity. Solids thathave periodicarrays ofatoms arecalled crystals. Most metalsandmostsemiconductorsarecrystals.
Crystalsarecommon innaturebecauseanordered structurehaslowerentropy thana disorderedstructure,and lower entropystatesare favoredatlowtemperatures. Whether or not a system forms an ordered crystal at room temperature depends on the ratio of the thermal energy kBT to the binding energy of two atoms. If kBT issmall compared to the binding energy, then the system is essentially in a zero-temperature state, even if itis quite hot compared to room temperature. We will return to discuss solid phase transitionsinSection5.4.
Bravais lattices.Inordertofillallofspacewithaperiodicstructure,wetakeafinite volume of space, which we call the primitive cell, or unit cell, and make copies of
19 1.4 BravaisLatticesandReciprocalSpace
itadjacent to each other by translatingit without rotation through integer multiples of threevectors, ´a1, ´a2,and ´a3.Thesevectors,knownasthe primitive vectors,mustbelin- earlyindependent,butneednotbeorthogonal.Someexamplesoflatticesgeneratedfrom primitive vectors areshown in Figure1.13. The set of all locations of the unit cells is givenby
´R = N1´a1+ N2´a2+ N3´a3, (1.4.1) where N1, N2,and N3 arethreeintegers.Thissetofallthevectors ´R makesupthe Bravais lattice ofthecrystal.Thesevectorspointtoasetof points whichdefinetheoriginofeach primitivecell.
Theprimitivecellthatiscopiedthroughoutspacedoesnotneedtobecubicorrectan- gular;itcanbeany shapethatwill fillallspacewhencopiedperiodically –itcanbeas complicatedastherepeatedelementsofanEscherprint.Themostnaturalchoice,however, isaparallelepipedwiththreeedgesequaltotheprimitivevectors.
Acrystalcanhavemorethanoneatomperprimitivecell.Withineachprimitivecell,we canspecifya basis,whichisasetofvectorsgivingthelocationoftheatomsrelativeto theoriginofeachcell.Figure1.14showstwoexamplesoflatticeswithabasis.Table1.1 givesthestandardprimitivevectorsandbasisvectorsofsomeofthemorecommontypes ofcrystals.
Theterm“Bravaislattice”istypicallyusedforjustthesetofpointsgeneratedbytrans- lationsofasinglepointthroughmultiplesoftheprimitive vectors.Inthisbook,wewill usethemoregeneralterm lattice torefertothesetofallpointsgeneratedbytheBravais latticevectorsplusthebasisvectorswithineachunitcell.
Exercise1.4.1 UseaprogramlikeMathematicatocreatediagramsanalogoustoFigures 1.13and1.14showingthelocationoftheatomsforthelastfourcrystalstructures fromTable1.1.(InMathematica,itissimpletocreateasetofspheresofradius r centeredatpoints {x1, y1, z1}, {x2, y2, z2}, ... usingthecommand
Show[Graphics3D[Sphere[{{x1, y1, z1}, {x2, y2, z2}, ...}], r]]
Tryusingdifferentviewpointpositionsintheplotting.Howmanynearestneighbors doeseachatomhave?
Exercise1.4.2 Provethatinthewurtzitestructure,eachatomisequidistantfromitsfour nearestneighbors.
The reciprocal lattice.Aswe sawinSection1.2,inaone-dimensionalperiodicsys- temliketheKronig–Penneymodel,thewavenumbers k = ±π/a havespecialproperties because the set ofcells withspacing a form a Bragg reflector, whichperfectly reflects waveswithwavelength2a.Inamulti-dimensionalsystem,therearemanypossibleways toformasetofperiodicreflectors.AsshowninFigure1.15,everysetofatomsthatform aplanearepartofaperiodicreflector.Therefore,todeterminethethree-dimensionalwave vectorsthathavethesameroleasthepoints k =±π/a inaone-dimensionalsystem,we needtofindalltheperiodic,parallelsetsofplanesinthelattice.
Table1.1 Commoncrystalstructures
Structure Standard primitive vectors and basis
Simplecubic(sc) aˆx, aˆy, aˆz
Body-centeredcubic(bcc) aˆx, aˆy,12a(ˆx +ˆy +ˆz)
or sclatticewiththebasis(0,12a(ˆx +ˆy +ˆz)) Face-centeredcubic(fcc) 12a(ˆy +ˆz),12a(ˆz +ˆx),12a(ˆx +ˆy)
or sclatticewiththebasis
(0,12a(ˆx +ˆy),12a(ˆy +ˆz),12a(ˆz +ˆx)) Diamond fcclatticewiththebasis(0,14a(ˆx +ˆy +ˆz)) Simplehexagonal(sh) aˆx,(12aˆx +√23aˆy), cˆz
Hexagonalclose-packed(hcp) shlatticewiththebasisá
0,12aˆx +2√13aˆy + 12cˆzạ , where c =º
83a
Graphite √23aˆx + 12aˆy, −√23aˆx + 12aˆy, cˆz withthebasis
á0,12cˆz,2√13aˆx +12aˆy,−2√13aˆx + 12aˆy + 12cˆzạ
Sodiumchloride fcclatticewiththebasis(0,12a(ˆx +ˆy +ˆz));
thetwobasissiteshavedifferentatoms Cesiumchloride sclatticewiththebasis(0,12a(ˆx +ˆy +ˆz));
thetwobasissiteshavedifferentatoms Zincblende Diamondlatticebutthetwobasissiteshave
differentatoms
Wurtzite shlatticewiththebasisá
0,12aˆx +2√13aˆy + 12cˆz, 38cˆz,12aˆx + 2√13aˆy +78cˆzạ
,where c =º 83a Perovskite sclatticewiththebasis(0,a2(ˆx +ˆy +ˆz),
a2(ˆx +ˆy),a2(ˆy +ˆz),a2(ˆx +ˆz));thelast threebasissiteshaveidenticalatoms Fluorite a2(ˆy +ˆz),a2(ˆx +ˆz),a2(ˆx +ˆy)
withthebasis(0,a4(ˆx +ˆy +ˆz), −a4(ˆx +ˆy +ˆz);
thelasttwositesareidentical Cuprite sclatticewiththebasis(0,a2(ˆx +ˆy +ˆz),
a4(ˆx +ˆy +ˆz),a4(3ˆx + 3ˆy +ˆz),a4(3ˆx +ˆy + 3ˆz), a4(ˆx + 3ˆy + 3ˆz));thefirsttwositesare identicalandthelastfoursitesareidentical
Thisproblemisequivalenttofindingalltheplanewavesthathavetheperiodicityofthe lattice.ThistakesusdirectlytothetheoryofFouriertransforms–theFouriertransformby definitiongivesusalltheperiodicwavesthatmakeupagivenfunction(seeAppendixB).
Wewritethereal-spacelatticeasasetofpointsgivenbyDirac δ-functions, f (´r) =ằ
´R
ằ
i
δ(´r − ´R − ´bi), (1.4.2)
21 1.4 BravaisLatticesandReciprocalSpace
a2 z
y
x a1
a2
a3 a1= (xˆ + yˆ)
a2 a2= (xˆ + zˆ)
a2 a3= (yˆ + zˆ)
z y x
a1 a2
a3 a1= axˆ
a2
a2= (xˆ + √3yˆ) a3= czˆ (a)
(b)
(c)
2a z
y x
a1
a2 a3
a1= axˆ a2= ayˆ
a3= (xˆ + yˆ + zˆ)
±Fig.1.13 (a)Primitivevectorsofabody-centeredcubic(bcc)lattice.Aparallelepipedthathasthesethreevectorsforsideswill fillallofspacewithbody-centeredcubeswhentranslatedbyintegermultiplesoftheprimitivevectors.(b)Primitive vectorsforaface-centeredcubic(fcc)lattice.(c)Primitivevectorsforasimplehexagonallattice.
2a
(a)
(b) z
y x
b1 b1= (xˆ + yˆ + zˆ)
±Fig.1.14 (a)Diamondlattice.Theprimitivevectorsarethesameasaface-centeredcubic(fcc)lattice(lightspheres),butin addition,nexttoeachatominthefcclatticethereisasecondatomashortdistanceaway(darkspheres),makinga two-atombasis.Thisallowseachatomtohavefournearestneighborsatequaldistances.(b)Close-packedhexagonal (hcp)lattice.Theprimitivevectorsarethesameasasimplehexagonallattice,butthereisanadditionalatominthe basis,givingtwohexagonalplanesshiftedrelativetoeachother.
where ´R aretheBravaislatticevectorsthatfillallofspaceand ´bi arethebasisvectorsfor thepositionsoftheatomswithineachunitcell.TheFouriertransformisthengivenby
F(´k) =à ∞
−∞d3rf(´r)ei´kã´r
=ằ
´R
ei´kã ´R
ẳằ
i
ei´kã´bi
ẵ
. (1.4.3)
23 1.4 BravaisLatticesandReciprocalSpace
±Fig.1.15 ThreeofthemanydifferentsetsofparallelplanesthatcanformBraggreflectorsinacrystal.
Thefactorintheparenthesesisknownasthe structure factor,andisthesameforevery primitivecellinthelattice.
The largenumberof differentoscillatingtermsinthe sumover ´R willcancel tozero unless ´k hasaparticularvalue ´Q suchthat
ei´Qã´R= 1, (1.4.4)
or
´Q ã ´R = 2πN, (1.4.5)
forall ´R,where N issomeinteger.Thiscanbesatisfiedfor
´Q = ν1´q1+ ν2´q2+ ν3´q3, (1.4.6) where ν1, ν2,and ν3areintegers,and
´q1 = 2π(´a2×´a3)
´a1ã (´a2ì´a3)
´q2 = 2π(´a3×´a1)
´a1ã (´a2ì´a3)
´q3 = 2π(´a1×´a2)
´a1ã (´a2ì´a3). (1.4.7) Theterminthenumeratorofthesevectors, ´ai×´aj,givesavectorperpendiculartoboth ´ai
and ´aj,whilethedenominatorisanormalizationfactorequaltothevolumeoftheprimitive cellparallelepiped.Thesechoicesofthe ´bivectorsensurethecondition
´aiã ´qj = 2πδij, (1.4.8)
whichimplies
´Q ã ´R = (ν1N1)´a1ã´q1+ (ν2N2)´a2ã´q2+ (ν3N3)´a3ã´q3 = 2πN, (1.4.9) where N isanintegersincethe νiand Niareintegers.Thissatisfiesthecondition(1.4.5).
The relation (1.4.6) impliesthat theFourier transform of the latticeis nonzerofor a specificset of ´Q vectors.Ateach ofthesevalues of ´Q,theFouriertransform hasapeak withheightproportionaltothenumberofBravaislatticesites ´R inthecrystal.Thesepeaks correspondtoplanewavesoftheform ei ´Qã´rwiththesameperiodicityasthelatticeinthe direction of ´Q. Inother words, each of the vectors ´Q pointsin the direction normal to aset ofparallelplanesand hasa magnitudeequalto n(2π/aá), where aá is thedistance betweenadjacent planesand n is an integer.Thisset of vectors ´Q defines alattice just asthereal-space ´R vectorsdo,andiscalledthe reciprocal latticeofthecrystal;itisthe three-dimensional Fourier transform of the original lattice. The spacethat containsthe reciprocallattice,whichhasdimensionsofinversedistance,iscalled reciprocal space,or
“k-space.”
The structure factor that appears in (1.4.3) is an overallmultiplicative factor for the heightofthereciprocallatticepeaks.Whenthereisperiodicityinsideaunitcell,thestruc- turefactorcancausesomepeakstohavezeroamplitude.Thishelpsustounderstandwhat wouldhappenifwechosethe“wrong”cellsize.Forexample,considerthecaseofalat- tice withspacing a betweenplanesinthe x-direction. Inthis case,thereciprocal lattice vectorshave x-component Qx = 2πn/a,forallintegers n.Whatifwehadtreatedthisas alatticewithspacing2a withatwo-atombasis?Thereciprocallatticeinthiscasewould havepointshalfasfar apart,thatis, Q = 2πn/2a = πn/a.But inthiscase,wehavea structurefactorgivenby
C(´Q) =ằ
i
ei ´Qã´bi. (1.4.10)
Forthecaseofthetwo-atombasis(0, aˆx),thesumisequalto
ằ
i
ei´Qã´bi= eiQxã0+ eiQxa. (1.4.11) When Qx= π/a,wethenhave
ằ
´rb
ei ´Qã´rb= ei(π/a)ã0+ e(π/a)ãa= 1 + (−1) = 0, (1.4.12) whilefor Qx= 2π/a,
ằ
´rb
ei ´Qã´rb= ei(2π/a)ã0+ ei(2π/a)ãa= 1 + 1 = 2. (1.4.13) The structurefactor removesthe spuriousnew reciprocallattice vectorsthat camefrom doublingthesizeoftheunitcell,andincreasestheheightoftheremainingFourierpeaks tothesamevaluewewouldhavehadifwehadusedaone-atombasis.
Exercise1.4.3 Prove that you get the same reciprocal lattice peaks from a bcc crystal, whetheryouviewitasasingleBravaislatticeorasasimplecubicBravaislattice withatwo-sitebasisandtheaccompanyingstructurefactor.(SeeTable1.1.)
Hint:Noticethat
ằ
n
f
²n 2
³
=ằ
n
f (n) +ằ
n
f
² n +1
2
³ .
25 1.4 BravaisLatticesandReciprocalSpace
Exercise1.4.4 (a)ShowthatthevolumeofaBravaislatticeprimitivecellis
Vcell=|a´1ã ( ´a2ìa´3)|. (1.4.14) (b)Provethatthereciprocallatticeprimitivevectorssatisfytherelation
´
q1ã ( ´q2ìq´3) = (2π)3
´a1ã ( ´a2ìa´3). (1.4.15) Hint:Write ´q1intermsofthe ´aiandusetheorthogonalityrelation(1.4.8).
With part (a) this proves that the volume of the reciprocal lattice cell is (2π)3/Vcell.
(c) Show that the reciprocal lattice of the reciprocal lattice is the original real-spacelattice,thatis
2π q´2×q´3
´q1ã ( ´q2ì ´q3) =a´1, etc. (1.4.16) Usefulvectorrelations:
´A ì (´B ì ´C) = ´B(´A ã ´C) − ´C(´A ã ´B) (´C ì ´A) ì (´A ì ´B) = (´C ã (´A ì ´B)) ã ´A
(´A ì ´B) ã (´C ì ´D) = (´A ã ´C)(´B ã ´D) − (´A ã ´D)(´B ã ´C). (1.4.17) (ThisexercisewasoriginallysuggestedinAshcroftandMermin(1976).)
The Brillouin zone.Inaone-dimensionalsystem,theBrillouinzonewastherangeof ksfrom −π/a to π/a.Inamulti-dimensionalsystem,theequivalentzonehasboundaries definedby ±π/aá,where aá isthe distancebetweenanyset ofparallelplanes,givenby aá = 2π/|´Q|. In other words,the wave vectorof an electron wavein the solid cannot exceedhalfthemagnitudeof ´Q inanydirection,becausethatwavevectorcorrespondsto awavelengthof2a thatwillbeperfectlyreflectedbyaBraggreflector.
ThisleadstothefollowingrecipefordefiningtheBrillouinzoneinamulti-dimensional periodicsystem:
1. Constructthereciprocallatticeusingtheformulas(1.4.7).
2. Startingfromtheorigin,drawlinestothenearestneighbors,asshowninFigure1.16(b).
3. Bisect theselineswith planes,asshowninFigure1.16(c).
TheseplanesformtheboundariesoftheBrillouinzone.AswewillseeinSection1.9.3, energygapsopenupattheboundariesofthisBrillouinzonejustasintheone-dimensional case.
Thezoneformedinthiswayiscalledthe Wigner–Seitz cell.TheWigner–Seitzmethod canbe used to form a primitive cell for any lattice, which can then be used to fill all spacebytranslationsofthelattice vectors.So, forexample, itcouldbeused insteadof theparallelepipedwechoseaboveforaBravaislattice.TheWigner–Seitzcellhasspecial importanceforthereciprocallattice,however,becausetheWigner–Seitzcellcorresponds totheBrillouinzone.
(b)
(c) (a)
±Fig.1.16 (a)Twoexamplesofreciprocallatticesintwodimensions.(b)Linesfromtheorigintothenearestneighborsinthe reciprocallattice.(c)TheselinesarebisectedbyplanestoformtheWigner–Seitzcell.
Exercise1.4.5 Showthatthereciprocallatticeofasimplehexagonallattice(seeTable1.1) isalsoasimplehexagonal,withlatticeconstants2π/c and4π/√
3a,rotatedthrough 30◦aboutthe c-axiswithrespecttothereal-spacelattice.
Exercise1.4.6 (a)A graphene lattice,or“honeycomb”lattice,isthesameasthegraphite lattice(seeTable1.1)butconsistsofonlyatwo-dimensionalsheetwithlatticevec- tors ´a1and ´a2andatwo-atombasisincludingonlythegraphitebasisvectorsinthe z = 0plane.Showthatthereciprocallatticevectorsofthislatticeare
´q1 = 4π
√3a
ẳ1 2ˆx +
√3 2 ˆy
ẵ
, ´q2= 4π
√3a
ẳ
−1 2ˆx +
√3 2 ˆy
ẵ
. (1.4.18)
(Hint:Althoughthisisatwo-dimensionallattice,itiseasiesttoassumethereisstill alatticevector ´a3= cˆz andusethistocalculatethereciprocalvectorsintheplane.) Makeadrawingofboththereal-spaceandreciprocal-spacelattices,and drawthe Brillouinzoneonthereciprocalspacelattice.
(b)Show thatthestructure factorforthetwo-atom basismultipliesthepeak in reciprocal space at ´Q = 0 by 2, the peak at ´Q = ´q1 by the factor e−iπ/3, and the peak at ´Q = 2´q1 bythe factor eiπ/3. Label the reciprocal lattice vectorson yourdrawingbytheirpeakheightandshowthatthereciprocallatticehasthesame symmetryasthereal-spacehoneycomblattice,andisnotasimplehexagonallattice.
27 1.5 X-rayScattering
(c) Show that the reciprocal lattice of graphene can be viewed as a simple hexagonallatticewithprimitivevectors
´qá1= 4π a
ẳ √3 2 ˆx + 1
2ˆy
ẵ
, ´qá2 = 4π a
ẳ √3 2 ˆx − 1
2ˆy
ẵ
, (1.4.19) andbasis(0, ´q1,2´q1).
(d)Showthatthereciprocallatticeofthisreciprocallatticeisasimplehexagonal latticeinrealspacewithprimitivevectors
´aá1 = a
√3
ẳ1 2ˆx +
√3 2 ˆy
ẵ
, ´aá2 = a
√3
ẳ
−1 2ˆx +
√3 2 ˆy
ẵ
. (1.4.20) (e)Last,showthat thestructurefactorofthereciprocallatticein(c)eliminates oneofeverythreereal-spacelatticepointsfromthelatticeof(d),leavingtheoriginal honeycomblattice.