Simple Derivation of the Hall Anisotropy Factors for Cubic and Octahedral Constant-Energy Surfaces

An elementary method is described for the derivation of the weak-field Hall anisotropy factor ($r$ in $R_0=\frac{r}{\mathrm{ne}}$) for the cases of cubic and octahedral energy surfaces. The magnetoconductivity approach is used (i.e., an electric field which is fixed in direction and magnitude, and a current which rotates when the magnetic field is turned on). The method simply computes the longitudinal current due to the electric field and the transverse current due to the magnetic force. The latter results entirely from carriers which drift across an edge of the energy surface, thereby changing the direction of their velocity. Then the Hall angle, Hall field, and finally the Hall coefficient are easily determined, and the results agree with the results of the much more complicated approach (the kinetic method of Shockley, McClure, and Chambers) previously used to derive the Hall anisotropy factor for these two cases.