Dept of Phys M.C. Chang Energy bands (Nearly-free electron model) NFE model is good for Na, K, Al… etc, in which the lattice potential is only a small perturbation to the electron sea. • Electron diffraction and energy gap • Bloch theorem • The central equation • Empty-lattice approximation • Tight-binding model (see Chap 9 ) History of band theory (Ref: chap 4 of 半導體的故事 , by 李雅明 ) • 1928 – Bloch theory Both are students of • 1929 – Peierls Heisenberg • nearly-free electron model, diffraction and energy gap • effective mass • filled band does not conduct, hole • umklapp process • 1930 – Kroenig-Penny model • 1931 – Wilson explains metal/semiconductor/insulator Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Bloch recalled, The main problem was to explain how the electrons could sneak by all the ions in a metal so as to avoid a mean free path of the order of atomic distances . Such a distance was much too short to explain the observed resistances, which even demanded that the mean free path become longer and longer with decreasing temperature. By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation. This was so simple that I didn't think it could be much of a discovery, but when I showed it to Heisenberg he said right away: "That's it!" dd d d dik⋅ r d ψ k()r= e ur k () Hoddeson L – Out of the Crystal Maze, p.107 Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Nearly-free electron model Free electron plane wave d d d 2π ψ d (re )=ik⋅ r , kn = ... etc k x x L • Consider 1-dim case, when we turn on a lattice potential with period a, the electron wave will be Bragg reflected when k=±π/a, which forms two different types of standing wave . (Peierls, 1930) π d ixaπ/− ixa π / 2 x ˆ G ψ + =c() e + e = cos , k⋅ G = a a 2 ixaπ/− ixa π / 2 π x ψ − =ce() − e = i sin . a a • Density distribution of the two standing waves Energy gap Bloch theorem The central eq. Empty lattice approx. TBM • These 2 standing waves have different electrostatic energies. This is the origin of the energy gap. If potential V(x)=Vcos(2 πx/a), then Note: Kittel use a potential energy U (=eV) Eg =∫ dx()ρ− − ρ + Vx( ) 0 a 2 2 =∫ dxe( − )|()ψ− ()| x − | ψ + ()| xVx () 0 2ea 2 π xxx π π =−V∫ dxcos sin2 −= cos 2 eV a0 a aa • Lattice effect on free electron’s energy dispersion ψ − ψ + Electron’s group velocity is zero near the boundary of the 1st BZ (because of the standing wave ). π π Q: where are the energy gaps when V(x)=V 1 cos(2 x/a)+V2 cos(4 x/a)? Energy gap Bloch theorem The central eq. Empty lattice approx. TBM A solvable model in 1-dim: The Kroenig-Penny model (1930) (not a bad model for superlattice) Delta functions a • Electron energy dispersion calculated from the Schrödinger eq. Energy bands (Kittel, p.168) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Bloch theorem (1928) The electron states in a periodic potential is of the form dd d d ψ d()r= eik⋅ r ur d () k k , where uk(r+R )= uk(r) is a cell-periodic function . • A simple proof for 1-dim: |Ψ(x)|2 is the same in each unit cell. Ψ(x+a )=C Ψ(x) Consider periodic BC, ψ(xNa+ ) = CN ψ () x = ψ () x s →=Cexp 2π is , = 0,1,2 ,() N − 1 N → ψ(xa+ ) = C ψ () x s Eigenstate of = eika ψ ( x ), k ≡ 2π translation op Na ikx write ψ k(x )= eu k ( x ) then uk( x +a ) = u k ( x ) Similar proof applies to higher dimension. Energy gap Bloch theorem The central eq. Empty lattice approx. TBM important • Schrödinger eq for ψ: Lattice potential p2 d Hψd=ε d ψ d , H = + U (r ) k k k 2m Energy dd d d d ik⋅ r d eigenstate ψ k()()r = e u k r uk(x) depends on the form of the periodic lattice potential . 10 23 times less effort than • Schrödinger eq for u: the original Schrodinger eq. d d d d Hku( ) k= ε k u k within one unit cell d dd d d −ikr ⋅ i k ⋅ r ← where H( k ) ≡ e He Effective Hamiltonian for u k(r) 2 ∇ d 2 d = +k + U() r 2m i Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Allowed values of k are determined by the B.C. important d d d Periodic B.C. d d ψk(rNa+i i ) = ψ k ( ri ), =1,2,3 (3-dim case) d d ik⋅ Ni a i →e =1, ∀ i N d d 3 N2 →kNa ⋅ii = 2π m ii ,m∈ Z , ∀ i (see chap 2) d d d d m1 m 2 m3 N1 → k= b1+ b 2+ b 3 N1 N 2 N 3 d d d d d d db b b 1 ddd b⋅( b × b ) 3 1 2 3 1 2d 3 = ∆=⋅k × = bbb1 ⋅×( 2 3 ) 3 N N1 N 2 N 3 N ∆ k N= NN1 2 N 3 d d d V Vol of 1 st BZ i aa⋅( × a ) == v , Vol of a k-point 1 2 3 N d d d (2π ) 3 i b⋅( b × b ) = , 1 2 3 v (2π ) 3 i ∆3k = , as in the free-electron case. V Therefore, there are N k-points in a BZ (a unit cell in reciprocal lattice), where N = total number of primitive unit cells in the crystal. Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Counting in r-space and k-space r-space k-space a crystal with PBC: G R N points here N points here • N unit cells • Infinite reciprocal lattice points (N lattice points x 1 atom/point) • N k-points in 1 st BZ • If each atom contributes q • N k-points in an energy band conduction electrons, then Nq electrons in total. Q: what if there are p atoms per lattice point? Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Difference between conductor and insulator (Wilson, 1931 ) • There are N k-points in an energy band, each k-point can be occupied by two electrons (spin up and down). ∴ each energy band has 2N “seats” for electrons. conductor E • If a solid has odd number of valence electron per primitive cell, then the energy band is half-filled (conductor). ε For example, all alkali metals are conductors. F k • If evena solid number has even of electronsnumber of per valence primitive electron cell, th peren primitivethere are cell, 2 possibilities:then the energy (a) noband energy can eitheroverlap be or insulator (b) energy or metal overlap.. E.g., alkali-earth elements can be conductor or insulator. Insulator (later) conductor E E k k Filled band partially-filled bands Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Band theory of solids energy Insulator metal semimetal semiconductor (n-type) (p-type) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Some mathematics before moving on • Orthogonal relation iG(− G ') x 1D ∫ dx e= a δG, G ' cell d d d 3iG (− G ') ⋅ r d d See A+M, App. D 3D ∫ dre= v δG, G ' cell d d d d Vδ for discrete k d k, k ' 3ik (− k ') ⋅ r d d note: ∫ d r e = 3 all ()2π δ ()k− k ' for continuous k • Fourier decomposition If f( r) has lattice translation symmetry, that is, f( r)=f( r+R) for any lattice vector R, then it can be expanded as, d d d = iG⋅ r d fr( ) ∑d ef G , where G is the reciprocal lattice vector. G (see Chap 2) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM How do we determine uk(r) from lattice potential U(r)? d d 2 Schrödinger equation ()p+ k d d d +Urur()d () = ε d ur d () 2m k k k Keypoint: go to k-space to avoid derivatives dd dd d d d d Fourier transform: UrR(+= ) Ur (), urRk ( += ) ur k () 1. the lattice potential d d d Ur( ) = Ued iG⋅ r ∑d G G=2 πn/a G d d d d 2. the wave function d= d −iG ⋅ r k=2 πn/L urk()∑d CGe k () G unkown d d 2k 2 Schröd. eq. in k-space ε0dd − ε d+ ddd = ε 0 ≡ ()kG− k CG k( )∑d UG'− G C k (G ') 0, k G' 2m aka. the central eq. Kittel λ uses k Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Matrix form of the central eq . (in 1D) G=ng (g ≡2π/a) 0 εkgk+2− ε U g U 234 g U g U g Ck (− 2 g ) 0 U−gε kgk + − ε UUU g2 g 3 g Ck (− g ) UUε0 − ε UU C (0) = 0 −2g − gkkg2 g k 0 for a UUU−3g − 2 g −− gkgkε− ε U − g Ck ( g ) particular k 0 U−432g U − g U − g U −− gkgkε 2 − ε Ck (2 g ) • For a given k, there are many eigen-values εnk , with eigen-vectors Cnk . ε • The eigenvalues n(k) determine the energy bands . • The eigenvectors { Cnk (G), ∀G} determine the Bloch states : −iGx uxnk()= ∑ CGe nk () G Energy gap Bloch theorem The central eq.