Transverse Conductivity of a Degenerate System of Landau Electrons and Optical Phonons

The transverse conductivity of a system of independent electrons weakly interacting with a gas of optical phonons in an external magnetic field is obtained for the case $\omega ={\omega }_0$, where $\omega =\frac{\mathrm{}\left|e\right|H}{m^{*}}$ ($\hslash =c=1\mathrm{}$) and where ${\omega }_0$ is the optical-phonon frequency, in the limit $\beta \omega \gg 1$. For the densities considered ($n_e={10}^{12}-{10}^{15}$ ${\mathrm{cm}}^{-2\mathrm{}}$) the effects of the electron-electron interaction are negligible. The only mechanism used to remove the logarithmic infinity predicted, as Gurevich and Firsov have pointed out, by the usual Titeica expression is the electron-optical-phonon interaction itself (i.e., "collision broadening"). The value of ${{\sigma }^{\mathrm{xx}}|}_{\omega ={\omega }_0}$ is found to be ${{\sigma }^{\mathrm{xx}}|}_{\omega ={\omega }_0}=\frac{3}{4}{\{1+(\frac{2}{\surd \pi }\left){\left(\beta \omega \right)}^{\frac{1}{2}}F\right(\beta \omega {\alpha }^{\frac{2}{3}}\left){\sigma }^{\mathrm{xx}}\right|}_{\omega \ne {\omega }_0},$ where ${{\sigma }^{\mathrm{xx}}|}_{\omega \ne {\omega }_0}=\left(\frac{4}{3}\right)n_e\alpha \beta e^2{e}^{-\beta \omega }{\left(m^{*}\right)}^{-1\mathrm{}},$ $\alpha$ being the Fröhlich coupling constant and $F$ of order unity for $\beta \omega \gg 1$ and $\alpha \ll 1$.