Hop-Conduction Magnetoresistance inp-Type Germanium

Measurements of the resistivity $\rho$ of $p$-type, Ga-doped germanium samples having room-temperature carrier concentrations between 2.2×${10}^{15}$ and 1.4×${10}^{16}$ ${\mathrm{cm}}^{-3\mathrm{}}$ have been made at liquid-helium temperatures and for comparison at 77°K employing magnetic inductions $B$ up to 25 kG. At liquid-helium temperatures $\frac{\rho }{{\rho }_0}\approx \exp \left[\gamma B^2\right]$ with $\gamma \sim {N_A}^{-0.7\mathrm{}}$, where $N_A$ is the acceptor concentration, $\frac{\rho }{{\rho }_0}$ is almost independent of temperature and, in a transverse magnetic field, $\frac{\rho }{{\rho }_0}$ exhibits a different type of anisotropy than is characteristic of valence-band conduction. The dependence of $\frac{\rho }{{\rho }_0}$ on the strength and orientation of the magnetic field is explained qualitatively in terms of the influence of the magnetic field on the acceptor wave functions and thereby on the phonon-assisted hopping of holes between acceptor sites, which is the process responsible for conduction at low temperatures. The ratio of the transverse to the longitudinal magneto-resistance at 4.2 and at 3.5°K is found to be in qualitative agreement with that predicted by theory for hop-conduction magnetoresistance in weak fields for a semiconductor with a simple spherical band at $k=0\mathrm{}$.