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126 CHAPTER 9. ELECTRONIC INSTABILITIES describe them – parameters which are usually estimated, or derived from experiment.
9.1. CHARGE DENSITY WAVES 127 9.1 Charge Density Waves The crystal structures of solids are much more complex than one might have expected. Even if you take the elements, rather few form simple close-packed structures. For example Ga metal has a complicated structure with 5 nearest neighbours, Se crystallises in a structure that can be thought of as an array of spiral chains with three atoms per unit cell, As, Sb and Bi have puckered sheets where each atom has three near neighbours. Of course, all of this reflects the production of chemical bonds inside the solid, and a complicated balance of forces. But the fundamental principle of bonding is that by placing the chemical potential in a gap, the occupied states are lowered in energy (and the unoccupied states go up in energy). Getting the chemical potential to lie in a gap involves making sure that the Brillouin zone boundary lies ”in the right place”, i.e. at a momentum that will contain exactly the correct number of states to account for all of the electrons in the solid. 9.1.1 The Peierls transition As a concrete example we consider a one-dimensional chain of atoms, with lattice constant a, and an electron density chosen such that the Fermi wave-vector k F falls somewhere in the middle of the band. It is a metal. Notice that we could turn this metal into an insulator by applying an external potential with a periodicity of 2π/Q where Q = 2k F : following the earlier lectures, we know that a periodic potential V o cos(Qx) produces Bragg scattering at a wavevector Q/2 (hence a new Brillouin zone boundary). If Q/2 = k F then there is a gap induced on the fermi surface. Rather than applying an external potential, we could get the same effect by making a periodic lattice distortion (PLD) with the same periodicity: namely move the n th atom in the chain to a new position R n = na + u o cos(Qna) . (9.2) We assume that the amplitude of the PLD is small, u o ≪ a. [We have already met this phenomenon in the diatomic chain, studied earlier, but where we considered the case of a halffilled band — in that case k F = π/2a and Q = π/a, and the periodicity of the distorted lattice is twice that of the undistorted one.] If the atoms have a PLD with period 2π/Q, they will produce a new potential seen by the electrons with the same period. It is also evident that the amplitude of the Fourier component V Q ∝ u o is linearly proportional to the displacement (for small displacements). We may write V Q = g Q u o , where the coefficient g Q is the electron-phonon coupling constant. Now remember that the energy gap on the zone boundary is |V Q |. That means that an energy level at a momentum just below k F is lowered by an energy proportional to the atomic displacement |u o |, (and the unoccupied one just above k F is raised in energy by the same amount). So overall there is an energy lowering as a result of the PLD. The magnitude of this can be computed (see the problem on sheet 4) by adding up all the energy changes of all occupied states: the answer can be written as E electronic = A(u o /a) 2 ln |u o /a| (9.3)
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