Phys 769 Selected Topics in Condensed Matter Physics Summer 010 Lecture 4: Basic elements of band theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 Introduction Most matter, in particular most insulating matter, is not crystalline! (Even quite a lot of metals are glassy, e.g. PdCuSi). But the theory of electronic behavior of crystalline solids is much better developed than the theory of glasses or liquids. The theory of solids is an 80-year-old subject, but is still capable of surprising (e.g., topological insulators). Model N identical atoms, or N i (commensurate) of species i ionic cores [nuclei+ valence electrons] in a regular crystalline array (e.g., solid Xe, NaCl, Si,...). The open-shell ( conduction ) electrons move in this array; for the moment neglect interactions, so electron wavefunctions shall satisfy the single-particle time-independent Schrödinger equation (TISE). The only many-body effect is Fermi statistics. 3 Bravais lattices We specify (in 3D) three minimal (non-coplanar) unit vectors R 1,R,R 3 which will take us from any point in the crystal to an identical point. These vectors are not, in general, uniquely defined (e.g., (in D) a triangular lattice) but this does not matter. A Unit cell of the lattice is defined as a parallelpiped bounded by R 1,R,R 3 : the volume of the unit cell is given by v 0 = R 1 R R 3 (or in D, R 1 R ). In 3D, there are 14 different Bravais lattices (luckily, we need not know all of them!). The simplest case: simple cubic. If there is more than one atom per unit cell, we have a basis : choose one atom as a reference and specify the position of the other(s) relative to it. This can happen even for a single chemical species, e.g., diamond. 1
4 Reciprocal lattice vectors (RLVs): We can always find three (minimal primitive ) noncoplanar vectors K 1,K,K 3, such that exp {ik i R j } = 1, i, j = 1,, 3. (1) This implies that if we start from any point in the lattice, and go to any equivalent point (not just the nearest one!) by a vector a, e ik a 1. In the simplest case, for a simple cubic lattice of side a, K i = (π/a)r i (i = 1,, 3): in the most general case (e.g., in a honeycomb lattice in D), the RLVs may not be uniquely defined. Again, this does not matter. 5 The first Brillouin zone (FBZ): We will want to consider arbitrary values of the wavevector k ( k-space ). Partition by orthogonally bisecting primitive RLVs: the contained volume is the FBZ. (Ex.: The FBZ of a simple cubic lattice is the region k i π/a, i = 1,, 3, i.e., a simple cube). The number of sites within the FBZ is equal to the number of lattice cells. Solutions of the TISE (single-particle energy eigenstates) in a periodic lattice: m ψ(r) + U(r)ψ(r) = Eψ(r), U(r + R) = U(r) () where R = n 1 R 1 + n R + n 3 R 3 with n 1, n, n 3 positive or negative integers. 6 Bloch (Floquet) theorem: Solutions are of the form ( Bloch waves ): There are two possible conventions: ψ k (r) = e ik r u k (r), u k (r + R) = u k (r). (3) (a) allow k to vary over all of k-space, then no extra suffix is needed on u k (r) (extended zone scheme) (b) restrict k FBZ, then we need a band index n on u k (r): u k (r) u nk (r) (reduced zone scheme)
Choice (a) is useful to illustrate the Fermi-liquid-like approach to band theory (start with plane-wave states, adiabatically turn on the potential U(r): U(r) has large Fourier components only at RLVs K for any given k, mixes in some amplitude of k + K, i.e.: ψ k e ik r ψ k = K c (k) K ei(k+k) r u k (r)e ik r ; u k (r) K c (k) K eik r. (4) However, in most contexts, the reduced zone scheme is more intuitive: we will use this scheme from now on. Thus, the general form of Bloch waves is: ψ kn (r) = u kn (r)e ik r, k FBZ u kn (r + R) = u kn (r) R (5) 7 Simple examples of Bloch wave formation 7.1 Weak-coupling model (1D): Figure 1: Picture of energies E 0 (k) in the free-electron limit within the extended zone scheme (left) and reduced zone scheme (right). The Fourier transform of U(r) couples ks differing by RLVs K. But for weak coupling the effect is only substantial if they are nearly degenerate, i.e. E 0 (k + K) E 0 (k). This happens only for k near K/, i.e., near the edge of the zone. (in the RZS, the states are at the same k). For such points, in the (weak coupling) limit, we can neglect the effect of U(K) for higher RLVs K. Thus, if the relevant Fourier component U(K), (K = K 0 ) is denoted U 0, we have for the original (free-electron) Hamiltonian for this pair of states ( ) ( ) E0 (k) 0 Ĥ(k) = k 0 0 E 0 (k) m 0 (k K 0 ) 3 (6)
and the matrix of the lattice potential in this approximation is ( ) 0 U0 Û(k) =. (7) U 0 0 Thus, the total Hamiltonian is (with the definition ǫ k K 0 q/m): ( ) ǫk U 0 Ĥ(k) = + const.. (8) U 0 ǫ k The eigenvalues are E ± (k) = ±(ǫ k + U 0) 1/ (9) Note that at the edge of the FBZ (k = K 0 /) the explicit form of the Bloch waves is (assuming U 0 < 0): ψ K0 /,+(x) = 1 {e } ik0x/ + e ik 0x/ ( ) K0 cos x ψ K0 /, (x) = i {e } ik0x/ e ik 0x/ ( ) K0 sin x (10) (11) (1) In 1D, one always has a gap at the edge of the FBZ (and higher BZ edges) unless the relevant Fourier component U(K) happens to vanish (pathology). bands are always separated in D or 3D, still get gaps on the edge of the FBZ, but bands may overlap (unusual situation). 7. Tight-binding model In the approximation of well-separated ions with deep potentials, the natural zeroth approximation for the single-electron energy eigenstates would on first sight be the atomic states localized on individual ions (if for simplicity we assume a lattice without basis): ψ0(r) i = ψ at (r R 0 i ), R 0 i = positionof ioni. (13) 4
However, since all the N sites are degenerate and we expect some degree of tunneling between them, the true energy eigenstates should be linear combinations of the ψ0 i(r). According to Bloch s theorem, the appropriate boundary conditions are of the form ψ kn (r) = e ik r u kn (r), u kn (r + R) = u kn (r). (14) This is satisfied by a function of the form ψ kn = 1 e ik R0 i ψat (r R 0 i ) N (15) i with u kn = e ik (r R0 i ) ψ (n) at (r R0 i ) (16) The label n would then normally correspond to different possible atomic states (s, p, etc.). What about the energy E k corresponding to such a tight-binding state? In principle one ought to do a detailed calculation taking into account, inter alia, the possible overlap of functions on neighboring ions, but in practice it is often sufficient to model all such effects by a simpler (n-dependent) nearest-neighbor hopping matrix element t: i.e., the effective Hamiltonian is (for a single value of n): Ĥ = E (n) at a j a j t n j ij=n.n. ( ) a i a j + H.c. We see that E (n) at enters only as a constant, so that while it affects the differences in energies between bands, it does not affect the shape of ǫ n (k) within the band n. From now on, consider a single band and drop the first term and subscript n on t and u: Ĥ = t ij=n.n. (17) a i a j + H.c. (18) Substituting the form (Eq. (16)) for ψ kn, (i.e. ψ kn 1 N i eik R0 i a i vac, where a i creates the atomic state ψ at (r R 0 i )), we find1 E(k) = t i cos (k R i ) R i = primitive lattice vectors (19) E.g. for a simple cubic lattice of side a, E(k) = t (cos(k x a) + cos(k y a) + cos(k z a)). The occurrence of different bands is a quite generic consequence of the lattice periodicity. The number of states (i.e. values of k) is the number of states in the FBZ, which is just the number of cells in the original lattice. Moreover, the typical (though not universal) 1 assumes lattice is inversion symmetric 5
situation is that bands do not overlap. Suppose now that we have Z c open-shell electrons per unit cell. Remembering that two spin states are available for each k-value, we see that if Z c is even, it is likely (though not mandatory) that an integral number of bands is filled and the rest are empty insulator. If, on the other hand, Z c is odd, then it is mandatory that at least one band is only partially filled metal. This textbook result is not the whole truth (if it were, all liquids and glasses with open shell atoms would be metallic!). [Semiconductors, semimetals] From now on, concentrate on crystals where one (or possibly more) band is only partially filled (metals). 8 Fermi surface In a metal with a nontrivial band structure, the concept of a Fermi surface is much less trivial than in the Sommerfeld model. Since electrons are assumed non-interacting, the lowest NZ c states of the lowest band will be filled, up to some energy ǫ F = µ(0). As in the Sommerfeld model, µ(t) is not appreciably a function of T. The Fermi surface is defined by the locus of points such that E(k) = ǫ F. (0) It must have the symmetry of the crystal, but can be of a shape quite different from a sphere. 9 Velocity of Bloch wave states The Bloch waves are not eigenstates of the (true) momentum p (the quantity k, which has many of the properties of p, is often called the quasimomentum ). However, a packet of Bloch waves will have a well-defined velocity given by v k = 1 E(k) k (1) Note that v k 0 not only at k = 0, but at the BZ edges. It is easy to show that k responds to an external force like a momentum, i.e. dk dt = F () 6
Thus, if F is independent of k (e.g., electric field) and there are no collisions, k cycles through the band v oscillates oscillations in real space ( Bloch oscillations ). (In real life, these oscillations are suppressed by impurity or phonon scattering). 10 Density of states Just as in the Sommerfeld model, all the near-equilibrium properties of a metal at temperatures below melting are determined by states within k B T of the Fermi surface. The most important quantity is the single-electron density of states (DOS) (of both spins). Since in k-space the DOS is /(π) 3, we have for the DOS/unit energy: dn dǫ = = (π) 3 (π) 3 F.S. ds k E(k) 1 (note k E(k)is S!), (3) F.S. ds v F (ˆn). (4) 11 Cyclotron resonance The force exerted on an electron by a d.c. magnetic field is ev E. Thus, the equation of motion of a Bloch electron is dk dt = e ( ) E B (5) k For electrons near the minimum of a spherical parabolic band (E(k) = k /m ) this gives dk dt = e m (k B) ω c = eb/m. (6) In the more general case, k z will be conserved (B ẑ) and the motion in the plane perpendicular to B is along a contour of constant energy. This contour may be open or closed. If the contour is closed, the period is given by dk T dt = dk /dt = eb = dk dk eb de i.e. ω c π/t = πe B dk de/dk (7) = A eb E (8) ( ) A 1 (9) E In D, with B perpendicular to the layer, this gives a unique result since E must be taken as the Fermi energy: in 3D, electrons in extremal orbits dominate. 7
1 de Haas van Alphen oscillations These are periodic oscillations in 1/B (not B!) of various equilibrium properties (e.g., magnetization) or transport properties (e.g., electrical resistivity: the Shubnikov de Haas effect). ω c = πeb and the correspondence-limit formula ω c = E/ n: E A (30) ( ) n = A + const. (31) πeb n = A (1/B) (3) πe so the period in 1/B area of the Fermi surface (in D)/extremal area (in 3D). 8