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74 CHAPTER 5. BANDSTRUCTURE OF REAL MATERIALS Bloch’s theorem again. The eigenstates are of the form given by Bloch’s theorem ψ nk (r) = e ik·r u nk (r) (5.3) where u(r) is periodic on the lattice. Notice that if we make the substitution k → k + G, (5.3) continues to hold. This tells us that k can always be chosen inside the first Brillouin zone for convenience, although it is occasionally useful to plot the bands in an extended or repeated zone scheme as in Fig. 4.2. Crystal momentum. The quantity h¯k is the crystal momentum, and enters conservation laws for scattering processes. For example, if an electron absorbs the momentum of a phonon of wavevector q, the final state will have a Bloch wavevector k ′ = k + q + G, where G is whatever reciprocal lattice vector necessary to keep k ′ inside the Brillouin zone. Physical momentum can always be transferred to the lattice in arbitrary units of h¯G. Notice that depending on the energy conservation, processes can thus lead to transitions between bands. Counting states. We saw that the spacing between k-points in 1D was 2π/L, where L is the linear dimension of the crystal. This generalises to 3 dimensions: the volume associated with each k is (∆k) 3 = (2π)3 V (5.4) with V the volume of the crystal. Within each primitive unit cell or Brillouin zone of the reciprocal lattice the number of k states allowed by the periodic boundary conditions is equal to the number of unit cells in the crystal. In practice N is so big that the bands are continuous functions of k and we only need to remember density of states to count. Since electrons are fermions, each k-point can now be occupied by two electrons (double degeneracy for spin). So if we have a system which contains one electron per unit cell (e.g. a lattice of hydrogen atoms), half the states will be filled in the first Brillouin zone. From this, we obtain the even number rule’: Even number rule. Allowing for spin, two electrons per real space unit cell fill a Brillouin zone’s worth of k states. Periodic boundary conditions and volume per k-point A formal proof of the number of allowed k-points uses Bloch’s theorem, and follows from the imposition of periodic boundary conditions: ψ(r + N i a i ) = ψ(r) (5.5)
5.1. BANDS AND BRILLOUIN ZONES 75 where N i are integers, with the number of primitive unit cells in the crystal being N = N 1 N 2 N 3 , and a i primitive lattice vectors. Applying Bloch’s theorem, we have immediately that so that the general form for the allowed Bloch wavevectors is e iNik·ai = 1, (5.6) k = 3∑ i m i N i b i , for m i integral. (5.7) with b i primitive reciprocal lattice vectors. Thus the volume of allowed k-space per allowed k-point is just (∆k) 3 = b 1 N 1 · b2 N 2 ∧ b 3 N 3 = 1 N b 1 · b 2 ∧ b 3 . (5.8) Since b 1 · b 2 ∧ b 3 = (2π) 3 N/V is the volume of the unit cell of the reciprocal lattice (V is the volume of the crystal), (5.8) shows that the number of allowed wavevectors in the primitive unit cell is equal to the number of lattice sites in the crystal. We may thus rewrite (5.8) (∆k) 3 = (2π)3 V (5.9) Metals and insulators in band theory The last point is critical to the distinction that band theory makes between a metal and an insulator. A (non-magnetic) system with an even number of electrons per unit cell may be an insulator. In all other cases, the fermi energy must lie in a band and the material will be predicted 2 to be a metal. Metallicity may also be the case even if the two-electron rule holds, if different bands overlap in energy so that the counting is satified by two or more partially filled bands. Notation The bandstructure E n (k) defines a function in three-dimensions which is difficult to visualise. Conventionally, what is plotted are cuts through this function along particular directions in k-space. Also, a shorthand is used for directions in k-space and points on the zone boundary, which you will often see in band structures. • Γ = (0, 0, 0) is the zone centre. • X is the point on the zone boundary in the (100) direction; Y in the (010) direction; Z in the (001) direction. Except if these directions are equivalent by symmetry (e.g. cubic) they are all called ”X”. • L is the zone boundary point in the (111) direction. • K in the (110) direction. 2 Band theory may fail in the case of strongly correlated systems where the Coulomb repulsion between electrons is larger than the bandwidth, producing a Mott insulator
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