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128 CHAPTER 9. ELECTRONIC INSTABILITIES Figure 9.1: The Peierls transition. The upper figure shows the familiar one-dimensional chain with lattice constant a and the corresponding lowest electronic band, plotted for momenta between 0 and π/a. In the lower figure (b) a periodic lattice modulation is introduced, with u(r) of the form of (9.2). The period is cunningly chosen to be exactly 2π/2k F , so that a band gap of amplitude 2g Q u 0 is introduced exactly at the chemical potential. in the limit u o /a ≪ 1, and A is a constant (depending on g Q ). Note the logarithm — this varies faster than quadratically (just). It is negative - the energy goes down with the distortion. By an extension of the standard band structure result, it should be clear that there is an electronic charge modulation accompanying the periodic lattice distortion - this is usually called a charge density wave (CDW). (9.3) is just the electronic contribution to the energy from those states very close to the fermi surface. But as we have argued before, it is sensible to model the other interactions between atoms just as springs, in which case we should add an elastic energy that is of the form E elastic = K(u o /a) 2 (9.4)
9.1. CHARGE DENSITY WAVES 129 0.3 0.2 x 2 log (x) 0.1 0 −0.1 −0.2 −1.5 −1 −0.5 0 0.5 1 1.5 x Figure 9.2: Sketch of (9.5) showing the double minimum of energy Adding the two terms together gives a potential of the form E(x) = Ax 2 ln |x| + Bx 2 (9.5) which always has a minimum at non-zero displacement. The system lowers its energy by distorting to produce a PLD and accompanying CDW, with a period that is determined by the fermi wave-vector, viz. 2π/2k F . Such a spontaneous lattice distortion is a broken symmetry phase transition (see Fig. 9.2), that goes by the name of its discoverer, Peierls. It tells us that a one-dimensional metal is always unstable to the formation of a CDW, even if the electronphonon coupling is weak. 1 Materials that are strongly anisotropic in their electronic structure are thus prone to a spontaneous lattice distortion and accompanying charge density wave. (The logarithmic singularity does not appear in dimensions greater than one — although CDW’s indeed happen in higher dimension, they don’t necessarily occur in weak coupling). Commonly there will be a phase transition on lowering the temperature that corresponds to the onset of order — one can monitor this by the appearance of new Bragg peaks in the crystal structure, seen by electron, neutron, or X-ray scattering (see Fig. 9.3). More subtly, the onset of a CDW can be seen in the phonon spectrum. Notice that by calculating the energy change as a result of a small lattice displacement, we have in the coefficient of the quadratic term in the energy as a function of displacement, the phonon stiffness for a mode of the wavevector 2k F . Consequently, the onset of a CDW is when the stiffness becomes zero (and negative below the transition), so there is no restoring force associated with the displacement. Then the phonon spectrum ω(q) (even in the high temperature undistorted phase) will be expected to show a sharp minimum in the vicinity of q = 2k F , as seen in Fig. 9.4. 1 There are of course other periodicities produced by beating of the spatial frequencies Q with 2π/a. These need not concern us if the amplitude is small, because they will generally occur at momenta different from k F , so the gap will lower and raise the energy of pairs of states that are either both unoccupied or both occupied, cancelling in the total energy.
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