Analytical description of hopping electrical conductivity of compensated semiconductors and calculations on the example of $p$-Ge : Ga

Analytical expressions are proposed for the prefactor $\sigma_{03}$ and the thermal activation energy of direct current electrical $\varepsilon_3$-conductivity $\sigma_h=\sigma_{03}\exp(-\varepsilon_3/k_\mathrm{B}T)$ of compensated $n$- and $p$-type semiconductors for hydrogen-like impurities. The obtained formulas are applicable to describe the hopping migration of both holes via acceptors and electrons via donors. For certainty, we considered $p$-type crystalline semiconductors in the range of doping levels corresponding to the insulator side of the insulator–metal (Mott) concentration phase transition. For simplicity, it was assumed that the majority and compensating impurities form a single simple nonstoichiometric cubic lattice in the crystal matrix. The calculation of $\sigma_{03}$ and $\varepsilon_3$ values is based on the preliminary determination of the characteristic temperature $T_3$, in the region of which phonon-assisted tunnel hopping of holes via nearest neighbor acceptors is observed. The shift of the top of the $v$-band deep into the band gap due to the formation of a quasi-continuous band of allowed energy values for $v$-band holes from the excited states of neutral acceptors is taken into account. The distribution of the density of hole states in the acceptor band was assumed to be Gaussian. The influence of the configurational entropy and thermal entropy of holes in the acceptor band on the values of $\sigma_{03}$ and $\varepsilon_3$ was also taken into account. The values of $\sigma_{03}$ and $\varepsilon_3$ calculated from the obtained formulas for moderately compensated $p$-Ge : Ga are in quantitative agreement with the known experimental data on the entire insulator side of the Mott transition.