High-field magnetoresistance of hopping transport in the disordered impurity system of transmutation-doped Ge

Abstract

The magnetoresistance behavior for impurity conduction of a series of transmutation-doped Ge samples (majority impurity Ga, compensation $K=0.4\mathrm{}$) with Ga impurity density $N_A$ ranging from 3 ± ${10}^{15}$ ${\mathrm{cm}}^{-3\mathrm{}}$ to 5 ± ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ up to fields of 110 kOe is presented. The resistivity approximates the form $\rho ={\rho }_0\exp \left(\frac{{\epsilon }_3}{\mathrm{kT}}\right)$. At low densities ($N_A\lesssim 2\times {10}^{15}$ ${\mathrm{cm}}^{-3\mathrm{}}$) the pre-exponent obeys $\ln {\rho }_0\mathrm{}\left(H\right)=\ln {\rho }_0\mathrm{}(H=0\mathrm{})+C{H}^{2}N_A^{}{}_{}{}^{-1\mathrm{}}$ as Milkoshiba's theory predicts, and ${\epsilon }_3$ is constant. At moderate densities, both ${\epsilon }_3$ and $p_0$ increase with field, with the effect on ${\rho }_0$ being larger than expected from the simple theory, but with a weaker functional dependence on $H$ and $N_A$. The ratios of transverse to longitudinal effects are consistent with percolation theory. Contrary to the results of Gadzhiev and Shlimak, we see a smooth decrease of ${\rho }_0\mathrm{}\left(H\right)\mathrm{}$ with $N_A$ throughout the moderate-density region. At high densities, a metal-to-semiconductor transition is induced which appears to be an Anderson transition. The data are discussed in terms of the impurity polarization model of Mott and Davis, the Anderson delocalization model Shklovskii and Shlimak, and the menyelectron hopping model of Knotek and Pollak. The data indicate that while the former two models may contribute at the higher densities, the latter model more reasonably accounts for the lower-density behavior.