Magnetoresistance ofn-Type Germanium in the Phonon-Assisted Hopping Conduction Range at High Magnetic Fields

The transverse magnetoresistance $\frac{\rho }{{\rho }_0}$ of phonon-assisted hopping conduction in $n$-type germanium samples having phosphorus concentrations $N_d$ between 5×${10}^{15}$ and 2×${10}^{16}$ ${\mathrm{cm}}^{-3\mathrm{}}$ has been measured at 4.2°K as a function of the strength and orientation of the magnetic induction B in a (110) plane up to much higher values of $B$ (78 kG) than used in previous work (30 kG). It is found that $\frac{\rho }{{\rho }_0}$ is an increasing function of $\frac{B^2}{N_d}$ up to the highest values of $B$ and depends on the direction of B, the anisotropy being more complicated at higher $B$. For $\mathbf{B}\parallel \mathrm{}\left[001\right]\mathrm{}$, $\frac{\rho }{{\rho }_0}$ increases most rapidly (almost exponentially) with $B$. Following Sladek and Keyes these effects are explained qualitiatively in terms of the influence of the magnetic field on the donor wave functions. A simple extension of the magnetoresistance theory of Mikoshiba is made and compared with our anisotropy curves. A reasonably good fit requires that the difference in phase between wave functions on adjacent donors have a much smaller effect than expected of following Mikoshiba's method of choosing a certain parameter $\epsilon$. A more reasonable choice of $\epsilon$, which also includes the influence of the decrease in size of the donor wave functions due to the magnetic field, greatly reduces the calculated phase effect. A final choice of values for $\epsilon$ and a parameter relating the spacing to the concentration of impurities yields a calculated anisotropy curve which reproduces the main features of the experimental curve in remarkable detail.