Perturbation Theoretic Calculation of Polaron Mobility

The low-temperature drift mobility of the polaron is calculated in perturbation theory with the aid of the Kubo formula. The result is $\mu ={\mu }_0\left(\frac{1-\alpha }{6}\right)$, where ${\mu }_0$ is the weak coupling mobility ${\mu }_0=\left(\frac{e}{2\alpha \omega m}\right)\exp \left(\frac{\hslash \omega }{\mathrm{kT}}\right)$. A comparison is made with the perturbation expansion of various intermediate coupling mobility theories. The expansion of Osaka, $\mu ={\mu }_0(1-0.173\mathrm{}\alpha +\cdots )$, agrees most closely with the exact perturbation expansion. It is concluded that the Osaka formula is probably the best in the intermediate coupling range $\alpha <6\mathrm{}$. It is explicitly shown to lowest nontrivial order in $\alpha$ that various quasiparticle concepts are valid, viz., that $\mu =\frac{e\tau }{m^{*}}$, and that the electron density is a momentum integral over $f\left(E\right(\mathbf{p}\left)\right)$.