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1、SpringSchoolFirst-principlesComputationalMaterialResearchIntroductoryLevel(2006)LectureNotesDAY1CrystalStructureandBandTheory講師:梁贊全國立中正大學物理系OutlineCrystalStructure1.CrystalStructure2.ReciprocalLatticeBandTheory1.BlochTheorem2.BandStructure3.DensityofStateT.C.Leung,Na
2、tionalChungChengUniversityP.1Chapter1CrystalStructureAnidealcrystalisconstructedbytheinfiniterepetitionofidenticalstructureunitsinspace.P.2CrystalStructure=Lattice+BasisABravaislatticeconsistsofallpointswithpositionvectorsRKKKKRn=+ana+na112233KKKa,,aa:primitivevector
3、s123Abasisofatomsisattachedtoeverylatticepoint,witheverybasisidenticalincomposition.GKKKτ=xa++xaxai112233where0,≤≤xx,x1123P.3SquarelatticeSquarelatticeconventioncellSquarelatticeFacecentersquarelatticeP.4ExampleofBravaislattice1.SimplecubiclatticeKaa=iˆ1Kaa=ˆj2Kˆaa=k
4、32.BodycentercubiclatticeKaKKKai1=()−+j+k2KaKKKai2=−()j+k2KaKKKai3=+()j−k2conventioncellsimplecubicP.53.Facecenteredcubiclattice(fcc)KaKKaj1=+()k2KaKKai2=+()k2KaKKai3=+()jconventioncell2simplecubic4.HexagonallatticeK⎛⎞13GKaa=+⎜⎟ij1⎜⎟⎝⎠22K⎛⎞13GKaa=−⎜⎟ij2⎜⎟⎝⎠22KKac=k3P
5、.6PrimitivecellWigner-SeitzprimitivecellDrawinglinesconnectingthepointtoallothersinthelattice,bisectingeachlinewithaplane,andtakingthesmallestpolyhedroncontainingthepointboundedbytheseplanesExampleofWigner-SeitzprimitivecellP.7bccWigner-Seitzprimitivecellofbcclattice
6、fccWigner-SeitzprimitivecelloffcclatticeP.8P.9Someexamplesofcrystalstructuresandlatticeswithbasis1.DiamondstructurefcclatticeKDiamond,Si,Geτ=01K1KKKτ=()ij++k242.ZincblendestructurefcclatticeKZnS,GaAsτ=01SiC,ZnSK1KKKτ=()ij++k24P.103.SodiumchloridestructurefcclatticeKτ
7、=01KaKτ=i22NaCl,LiH,MnO,KCl,KBr4.CesiumchloridestructuresclatticeKτ=01KaKKKτ=()ij++k2CsCl,AgMg,AlNi,CuPd2P.115.Hexagonalclosed-packedstructure(hcp)hexlatticeKτ1=0ca≅1.633KK21K1Kτ=++aaa2123332hcp:ABABABAB……fcc:ABCABCABC….fccalong(111)directionP.12ReciprocallatticeKKKK
8、KKKKRn=+ana+naGh=++b12kblb3112233KKKKKKa12,,aa3:primitivevectorsb12,,bb3:reciprocallatticevectorsKK⎧0i≠jab⋅=2πδwhereδ=⎨ijijij⎩1i=jT
