Skip to content

Peierls Transition

  • In absence of e-e interaction and e-p interaction, the electron states are filled up to Fermi level and the ground state corresponds to the situation shown in Fig. 1(a)
  • In presence of e-p interaction, it is favorable to introduce a periodic lattice distortion with a period of \(\lambda\) related to the Fermi wave vector \(k_F\) by, \(\lambda = \frac{\pi}{k_F}\)
  • This distortion opens up a gap at the Fermi level, which leads to lowering of the electronic energy.

Peierls_Transition.png

  • In 1D, the single-particle gap \(\Delta\) is proportional to the amplitude of the periodic lattice distortion u, and the decrease of the electronic energy is small is for small displacements proportional to \(u^2 ln u\).
  • For a small distortion, the total energy of the coupled electron-phonon system is smaller than that of the undistorted metal.
  • The modification of the dispersion relation also leads to a position-dependent electron density, which is a function of position x with period, given by equation \(\lambda = \frac{\pi}{k_F}\).
  • At finite temperatures, normal electrons excited across the single-particle gap screen the e-p interaction. This in turn leads to the reduction of the gap and of the lattice distortion, and eventually a Second-order transition at the so-called Peierls temperature \(T_p\).
  • The material is a metal above transition temperature while it is a semiconductor below \(T_p\) with a temperature-dependent gap \(\Delta(T)\).
  • For small distortions, the energy saving is linear in the distortion, while the elastic energy cost is quadratic in the distortion. Therefore, at zero temperature, when we can ignore entropic effects, the undistorted state is never stable.

Extending Peierls picture to 2D and 3D

The standard practice is to calculate the susceptibility (\(\chi(\vec{q},\omega)\)) for a given electronic configuration and use the zero energy value of the Lindhard response function \(\chi_0(\vec{q}) \equiv \chi_0(\vec{q},\omega=0)\) to determine whether the electron response can drive a Peierls phase transition―there should be a peak in the imaginary part of the response function \(Im[\chi_0(\vec{q})]\) at the Fermi surface nesting (FSN) vector \(q_{CDW}\) as well as in the real part \(Re[\chi_0(\vec{q})]\), because the real part defines the stability of the system.

Lindhard_function.png

There is a logarithmic divergence in \(Re[\chi_0(\vec{q})]\) for the 1D system, but no peaks in the 2D and 3D electron gases.