Mobility of Slow Electrons in Polar Crystals

Abstract

An intermediate-coupling calculation of polaron-phonon scattering has been carried out by expressing the scattering amplitude directly in terms of matrix elements between exact eigenstates of the Hamiltonian corresponding to the initial and final states of the scattered particle. The basic approximation made is in the use of the rather accurate polaron wave functions of Lee, Low, and Pines for these initial and final states. The result for the mobility, which is only valid in the low-temperature region ($T\ensuremath{\ll}\ensuremath{\theta}$) is $\ensuremath{\mu}=\frac{1}{2\ensuremath{\alpha}\ensuremath{\omega}}\left(\frac{e}{m}\right)\frac{1}{{[1+(\frac{\ensuremath{\alpha}}{6})]}^{3}}f(\ensuremath{\alpha})\mathrm{exp}(\frac{\ensuremath{\omega}}{\ensuremath{\kappa}T}),$ where $\ensuremath{\alpha}$ is the coupling constant for the electron-phonon interaction and $f(\ensuremath{\alpha})$ is a slowly varying quantity of order $\frac{5}{4}$ for $3<\ensuremath{\alpha}<6$.