Determination of the Electron-Scattering Cross Section in Degenerate Semiconductors from Low-Field Magnetoresistance Measurements

The usefulness of low-field, low-temperature ($\frac{\omega \hslash }{{\epsilon }_F}\ll 1,\frac{k_{B}T}{{\epsilon }_F}\ll 1$) magnetoresistance measurements for the determination of the electron-scattering cross section in semiconductors in the "metallic" regime ($n_e\gtrsim {10}^{16}$ electrons/${\mathrm{cm}}^3$ for InSb) is pointed out, and specific application of the method is made for the case of $n$-type InSb. In this treatment the impurity electrons are assumed to be nearly free, scattering weakly from the assumed randomly distributed spherically symmetric impurity centers. In this case we may compute the low-field transport coefficients from the solution of the transport equations obtained by one of us (L. D.). In this, a knowledge of the effective mass $m^{*}$, the effective $g$ value $g^{*}$, and the scattering matrix $t_{\mathrm{k}{\mathrm{k}}^{\prime }}\left({\epsilon }_{k_{\sigma }}\right)$ for an electron of spin $\sigma$ scattering from a single impurity center in the absence of a magnetic field (but including the spin energy) is required. If we neglect the momentum dependence of $t_{\mathrm{kk}}\left({\epsilon }_{{\mathrm{k}}_{\sigma }}\right)$, we may express the conductivity tensor completely in terms of the first three coefficients $r_i(k_F,{\epsilon }_F)(i=0,1,2\mathrm{})\mathrm{}$ and their derivatives with respect to $k_F$ of an expansion of the differential cross section in Legendre polynomials. A determination of these parameters is made using the zero-field mobility and the transverse magnetoresistance measurements of Isaacson, and the adequacy of a screened Coulomb model (pseudo-) potential is discussed. It is expected that additional measurements of electrical and thermal magnetoresistance and magnetothermopower (especially recommended) at several electron densities greater than ${10}^{16}$ electrons/${\mathrm{cm}}^3$ would greatly improve the accuracy with which the parameters may be determined.