Electron-phonon interaction, Kohn anomalies, and the Peierls transition in semiconductor quantum wires

We calculate the magnitude of phonon softening due to the electron–acoustic-phonon interaction in a continuum one-dimensional model as appropriate for semiconductor quantum wires. We compare the relevant quantum-wire parameters with those of a typical Peierls-unstable one-dimensional compound ${\mathrm{K}}_2$Pt(CN${)}_4$${\mathrm{Br}}_{0.3}$⋅${3\mathrm{H}}_2$O. We obtain quantitative results for the phonon softening, and discuss the possibility of a Peierls instability in quantum-wire structures. We discuss quantitative effects of electron-electron interaction on Kohn anomalies in doped wires. We obtain conditions for a continuous phonon softening as the temperature reaches the mean-field transition temperature. We conclude that for realistic system parameters it is unlikely for quantum wires to exhibit Peierls instability even though strong renormalization of phonon modes may be observable under suitable experimental conditions.