Vibrational excitations of a one-dimensional electron-phonon system in strong coupling

Abstract

We consider an electron interacting with optical phonons through the deformation potential in one dimension. We calculate the energy spectrum of such a system to one order in $\frac{1}{{\ensuremath{\alpha}}^{2}}$ beyond the strong-coupling limit. We calculate the self-energy, effective mass and modified phonon spectrum to this order. The modified phonon spectrum is determined by the solution of a homogeneous linear integral equation. We are able to solve this integral equation in closed form for the odd-parity phonon modes. The result is a frequency spectrum ${\ensuremath{\Omega}}_{n}={\ensuremath{\omega}}_{0}{[1\ensuremath{-}\frac{4}{n(n+3)}]}^{\frac{1}{2}}$, where $n=1,3,5,\dots{}$ and ${\ensuremath{\omega}}_{0}$ is the unperturbed optical-phonon frequency. For $n=1$, ${\ensuremath{\Omega}}_{1}=0$ and this mode is a translation. For even-parity modes, the phonon frequency spectrum is determined numerically.