Maximum hopping direct current conductivity via hydrogen-like impurities in semiconductors

A quasi-classical model for calculating DC (direct current) electrical conductivity in crystalline semiconductors with hydrogen-like impurities is developed at the transition from band conduction to impurity hopping conduction with decreasing temperature. This transition from the minimum band conductivity to the maximum hopping conductivity via impurities has the form of a characteristic “kink” in the temperature dependence of the electrical resistivity. The idea of the calculation is to preliminarily determine the transition temperature $T_j$ using the standard approach within the framework of the two-band model. The shift of the top of the $v$-band (the bottom of the $c$-band) into the depth of the band gap due to the formation of a quasi-continuous band of allowed energy values from the excited states of acceptors (donors) is taken into account. This leads to a decrease in the value of a thermal ionization energy of the majority shallow impurities due to a decrease in the maximum localization radius of a hole on an acceptor (an electron on a donor) with increasing impurity concentration. The values of the observed maximum hopping conductivity and drift hopping mobility corresponding to the temperature $T_j$ are calculated. The numerical calculation within the framework of the proposed model is consistent with the known experimental data on the electrical conductivity and Hall coefficient of moderately compensated $p$-Ge crystals doped by neutron transmutation and non-intentionally compensated metallurgically doped $n$-Ge, as well as $n$- and $p$-Si crystals on the insulator side of the Mott insulator-metal concentration phase transition.