1、固體物理固體物理Chapter 7Energy BandsChapter 7Energy BandsThe problems of the free electron Fermi gas model: The distinction between metals, semiconductors, and insulators Positive value of Hall coefficient The relation of conduction electrons in metals to the valence electrons of free atoms Many transport

2、propertiesTo understand the difference between insulators and conductors, we must extend the free electron model to take account of the periodic lattice of the solid. The possibility of a band gap is the most important new property that emerges.MetalSemimetalSemiconductorInsulatorEnergyEnergy bandVa

3、lence bandConduction bandEnergy gap (band gaps or forbidden band)Nearly free electron modelOn the free electron Fermi gas model, we have22222mmpHThe wavefunctions are)exp()(rkrkwhere, for the periodic boundary conditions ;4 ;2 ; 0 , ,LLkkkzyxThe energy 22222zyxkkkkmThe momentum kpConsider the period

4、ic potentials in the lattice as a perturbation, we have the nearly free electron model.)(22rUmpHwhere .)()(rUTrU絕熱近似：假設晶體中原子實是固定不動，則晶體中的周期勢場是不隨時間變化的。單電子近似：忽略價電子間的相互作用，則價電子的運動是相互獨立的，其波函數可以用單電子在周期勢場中的波函數來描述。For an electronInside the Brillouin zone, by the perturbation theory we have)()()()()()(),()2()

5、1()0()1()0(kEkEkEkErrrkkkwith .2)( and )exp()(22)0()0(mkkErk irkThen we have. )()(, 0)()()0()0(2)0()0()2()0()0()1(kkkkkkkkEErUkErUkEAt the Brillouin zone boundary, we have. and ,Gkkkk)0()0(kkEEThen we have the eigenfunction)0()0(kkBAwith . ,)0()0()0()0()0()0()0()0(kkkkEHEHThe Schrdinger equation is

6、)()()0()0()0()0()0(kkkkBAEBAUH. 0)()( i.e.)0()0()0()0(kkUEEBUEEA0)(0)()0()0()0()0()0()0(BEEAUBUAEEkkkkMultiply and from left side respectively, with)0(k)0(k0)0()0()0()0(kkkkUUSet )0()0(kkkkUU0)(0)()0(*)0(BEEAUBUAEEkkkkthen*)0()0(kkkkkkUUUWe haveThe homogenous linear equations have a solution only if

7、 the determinant of the coefficients of the unknown A, B vanishes.0 )0(*)0(EEUUEEkkkkTherefore kkUEE)0(The periodic potential of the lattice causes the energy gap at the Brillouin zone boundary.The low energy portions of the band structure in (b) are similar to free electrons in (a), but with an ene

8、rgy gap at the Brillouin zone boundary.Due to the Bragg reflection, at the Brillouin zone boundary, the wavefunctions of the electrons are not traveling waves, but are standing waves made up of equal parts of waves traveling to the right and to the left.)./sin(2)/exp()/exp()();/cos(2)/exp()/exp()(ax

9、iaxiBaxiBaxaxiAaxiAIn a linear lattice with the lattice constant a, the standing waves are:Bragg reflection of electron waves in crystals is the cause of energy gaps.The Bragg reflection is a characteristic feature of wave propagation in crystals. The two standing wave (+) and () pile up electrons a

10、t different region. Therefore these two wave have different values of the potential energy. This is the origin of the energy gap.Forwhich concentrates the electrons on the positive ion centers where the potential energy is lowest.Forwhich concentrates the electrons away from the positive ion cores w

11、here the potential energy is highest.),/(cos)()( ),/cos(2)(22axax),/(sin)()( ),/sin(2)(22axaxiOrigin of the energy gapMagnitude of the energy gapWhen we calculate the average value of the potential energy over these three charge distributions, we find that the potential energy of (+) is lower than t

12、hat of the traveling wave whereas the potential energy of () is higher than that of the traveling wave. The width of the energy gap Eg is the potential difference of (+) and ().Suppose the potential energy of an electron in the crystal is)./2cos()(axUxUThen we haveUaxaxaxUdxxUxUEag )/(sin)/(cos)/2co

13、s( 2 )()()()()()(022The gap is equal to the Fourier component of the crystal potential.Bloch functionsThe Bloch theorem:The eigenfunctions of the wave equation for a periodic potential are the product of a plane wave times a function with the periodicity of the crystal lattice. )exp(rk i)(ruk)exp()(

14、)(rk irurkkwhere ).()(ruTrukkThe solutions of the Schrdinger equation for a periodic potential must be of a special form:The one-electron wavefunction is called a Bloch function and can be decomposed into a sum of traveling waves. Bloch functions can be assembled into localized wave packets to repre

15、sent electrons that propagate freely through the potential field of the ion cores.)exp()()(rk irurkkThe proof of the Bloch theorem in one dimensionSuppose N identical lattice points of the lattice constant a. the potential energy is periodic in a, U(x+sa) = U(x).The eigenfunctions of the wave equati

16、on for a periodic potential are k. Due to the lattice translation symmetry, we have)()(xCaxkkwhere C is a constant.Consider the periodic boundary condition, )()()(xxCNaxkkNkThen )/2exp( i.e. , 1NsiCCNTherefore we have )exp()()/2exp()()(ikxxuNasxixuxkkkwith k = 2s/Na.The proof of the Bloch theoremDef

17、ine a translation operation T ,)()(arfrfTwhere ( = 1, 2, 3) is the primitive translation vector.aOne can prove:0TTTTand0HTHTTherefore T and H have same eigenfunctions.Suppose (r) is the eigenfunction of T and H.)()()()(rrTrErHwhere E and are the eigenvalues of H and T respectively.Due to the periodi

18、c boundary condition:)()(raNrThen)()()()(rrrTaNrNNThus)/2exp( i.e. , 1NhiNSet 332211bNhbNhbNhkThen)exp(ak i)exp()( )()(exp )()( )()(332211321321332211321321llllllllRk irralalalk irrTTTalalalrRrDefine a function:)exp()()(rk irrukThen)( )exp()( )(exp)exp()( )(exp)()(rurk irRrk iRk irRrk iRrRruklllllkT

19、herefore)exp()()(rk irurkThe central equationConsider a linear lattice with the lattice constant a. The potential energy U(x) is invariant under a crystal lattice translation.)()(xUaxUWe can expand U(x) as a Fourier series in the reciprocal lattice vectors G.GiGxGeUxU)(In one-electron approximation,

20、 the Schrdinger equation is)()(2)()(222xxeUmpxxUmpGiGxGThe wavefunction (x) can be expresses as the sum over all values of the plane wave permitted in crystal,kikxkkikxeCekCx)()(where k = 2n/L due to the periodic boundary condition.Substitute the wavefunction into the Schrdinger equation: ikxkkeCkmx

21、dxdmxmp2222222)(2)(2GkxGkikGkikxkGiGxGeCUeCeUxxU)()()(kikxkGkxGkikGkikxkeCeCUeCkm)(2220222kikxGGkGkeCUCmki.e. Therefore 0GGkGkkCUCwith the notationmkk222the central equationGxGkiGkkeCx)()(k(x) is the wave packet which is a linear combination of the plane waves with the wavecectors k+G.In principle,

22、there are an infinite number of Ck to be determined. However in practice a small number of Ck will suffice.The central equation is the Schrdinger equation expressed in the reciprocal space. Here we have a set of algebraic equations instead of the differential equation.Restatement of the Bloch theore

23、m The wavefunction is given as )( )()(xueeCeeCxkikxGiGxGkikxGxGkiGkkwhere we define .)(GiGxGkkeCxu)( )()2()(xueCeCeCTxukGiGxGkGsGxiGkGTxiGGkkSuppose T is a crystal lattice vector, TG = 2s.Discussions about the Bloch theorem(1) The Bloch function does not have the same periodicity as the lattice, i.e

24、.)(rk)exp()( )(Rk irRrkkAs proved before,Generally k is not a reciprocal lattice vector G, 1)exp(Rk iTherefore )( )(rRrkkHowever )( )(rRrkkHence)()()()(22rrRrRrkk)( )(rRrkk(2) The Bloch function has the same periodicity as the reciprocal lattice:)(rk)()(rrkGkGrGkiGkkeCr)()()()(GrGGkiGGkGkeCrSet G” =

25、 G G)( )()()()(reCeCeCrkGrGkiGkGGrGkiGkGrGGkiGGkGkkGkEE)()(rErHkkkFrom the Schrdinger equation, one hasand)()(rErHGkGkGk)()( rrkGk)()( )()()( rErErErHrHkGkGkGkkkkGkTherefore (3) If the lattice potential vanishes, U(x) = 0.The central equation reduces to (k )Ck = 0, so that all CkG are zero except Ck

26、, and uk(r) is constant. Thus we haverk iker)(4) The crystal momentum of an electronThe quantity k enters in the conservation laws that govern collision processes in crystal.The crystal momentum of an electronkIf an electron absorbs in a collision a phonon of wavevector , the selection rule is kqGkq

27、kSolution of the central equationFor the potential energy ,)(GrGiGeUrUthe central equation represents a set of simultaneous linear equations.0GGkGkkCUCThese equations are consistent if the determinant of the coefficients vanishes.The determinant in principle is infinite in extent, but in practice a

28、small number of Ck will suffice.The values of the coefficients UG for actual crystal potentials tend to decrease rapidly with increasing magnitude of G.Suppose that the potential energy U(x) contains only a single Fourier component Ug = Ug = U, where g denotes the shortest G.Take five successive equ

29、ations of the central equation. Then the determinant of the coefficients is given by0 0 0 0 0 0 0 0 0 0 0 0 0 22gkgkkgkgkUUUUUUUUFor a given k, the solution of the determinant gives a set of energy eigenvalues nk, which lie on different bands.Kronig-Penney modelAssume the periodic potential is the s

30、quare-well periodic potential.As shown in the figure, in one period the square-well potential is0for ,)(0for , 0)(0 xbUxUaxxUThe periodicity of U(x) is a + b.The wave equation is)()()()(2222xxxUxdxdmiKxiKxBeAex)(In the region 0 x a, U(x) = 0, the eigenfunction is a linear combination of plane waves

31、with the energy mK222In the region b x K and Qb 1, the equation reduces to kaKaKaKaPcoscossinNote: K is not the wavevector k of the Bloch function.Kronig-Penney model in reciprocal spaceWe use the Kronig-Penney model of a periodic delta function potential:0/10cos2)()(GGasGxUsaxAaxUWhere A is a const

32、ant and a the lattice spacing. The sum of s is over all atoms in a unit length, which means over 1/a atoms.Thus AGsaAaGxsaxdxAaGxxUdxUasasssaG cos cos)( cos)( /10/10)1(0We have the central equation:0)(/2nankkkCAChere G = 2n/a.Define GknankkfCf/2Then kkkkfmkmAfAC222/2/2kankfmankmAC222/2/2)/2(/2Sum bo

33、th side over nnknankmankmAfC1222/2/2)/2(/2)cos(cos4sin /2)/2(2/21222KakaKaKaamankmAnThen we havewhere we write 22/2mK Then the final result iskaKaKaKamAacoscossin222Other approximationsEmpty lattice approximationActual band structures are usually exhibited as plots of energy versus wavevector in the

34、 first Brillouin zone. When wavevectors happen to be given outside the first zone, they are carried back into the first zone by subtracting a suitable reciprocal lattice vector.GkkConsider an empty lattice, U(r) = 0. The energies are approximated by the free electron energies . However the plane wav

35、e is modulated by the lattice.mkk2/22The electron energy in a empty lattice is)()()(2 )(2),(222222zzyyxxzyxGkGkGkmGkmkkkwith k in the first Brillouin zone and G allowed to run over appropriate reciprocal lattice points.16,17,18,1912,13,14,158,9,10,114,5,6,7321/a-/a0kx222222222222222)2(2 )2(2 110 ,10

36、1 1,10 011, 916,17,18,1)2()2( )2(2 101 0,11 01,1 10,1 512,13,14,1)2()2( )2(2 110 0,11 101, 110, 8,9,10,11)2( )2( 100 001, 0,10 010, 4,5,6,7)2( )2( 001 100, 2,3 0 000 1)00( )000( 2 band /ak/a/a/ak/a/a/ak/a/ak/a/ak/akkGa/xxxxxxxe.g. the low-lying free electron bans of a simple cubic lattice along 100

37、direction.Approximate solution near a zone boundarySupposemkUUG222For a wavevector exactly at the Brillouin zone boundary,2/ i.e. ,)2/(22GkGkso that at the zone boundary the kinetic energy of the two component waves k = G/2 are equal, i.e. G/2 = G/2.Suppose CG/2 are important coefficients at the zon

38、e boundary and neglect all others.Then we have only two equations with k = G/2 :0)(0)(2/2/2/2/GGGGUCCUCCHere mG2/)2/(22For a nontrivial solution0 UUwhenceUGmU2222The radio of the C:12/2/UCCGGThen the wavefunction at the zone boundary is)2/exp()2/exp()(iGxiGxxHere one solution gives the wavefunction

39、at the bottom of the energy gap, and the other gives the wavefunction at the top of the energy gap.Solutions near the zone boundary G/2With the same two component approximation, the wavefunction is:xGkiGkikxkeCeCx)()(From the central equation we have0)(0)(kGkGkGkkkUCCUCCThe determinant equals zero:0 GkkUUThe energy2/12222UkGkkGkmKUUKGmUmKKGmK2214/2 244/22222222/1222222If we expand the energy in terms of a quantity in the region:2/GkKUmKG2/2UGm2222)(At the Brillouin zone boundaryThereforeUmKK212)()(22Number of orbitals in a bandEach primit