Peierls instability in a nearly-free-electron model, including nonlinear screening

We determine the conditions under which a Peierls instability (PI) will occur at $T=0\mathrm{}$°K for a nearly-free-electron model that includes electron-electron interaction. We include nonlinearities in the screening, which are found to be very important, and we do not limit the calculation to $q=2\mathrm{}{k}_F$. It is found that a PI requires $\gamma >1\mathrm{}$, where $\gamma$ is a parameter characterizing the electron-phonon interaction. In a special case, $\gamma =\frac{{\omega }_{\mathrm{pl}}^2}{{\omega }_q^2}$, where ${\omega }_{\mathrm{pl}}$ is the (bare) plasma frequency of the ions, and ${\omega }_q$ is the bare phonon frequency. We find the renormalized phonon frequency generally does not have to vanish to have an instability. Furthermore, we find that when $1<\mathrm{}\gamma \approx 1$ the instability is likely to occur at $q<\mathrm{}2\mathrm{}{k}_F$ rather than at $q=2\mathrm{}{k}_F$, thus putting a gap below the Fermi level. We find that the size of the instability gap is probably limited by anharmonicities, rather than being limited by minimization of the total energy calculated in the harmonic approximation. It is significant that the most important contributions to the total energy for the screened PI are not present for the unscreened PI. Thus the gap is not expected to have a BCS-type of temperature dependence, contrary to the case of the unscreened PI.