Transition temperature from band to hopping direct current conduction in crystalline semiconductors with hydrogen-like impurities: Heat versus Coulomb attraction

Abstract

For nondegenerate bulk semiconductors, we have used the virial theorem to derive an expression for the temperature T_j of the transition from the regime of “free” motion of electrons in the c-band (or holes in the v-band) to their hopping motion between donors (or acceptors). Distribution of impurities over the crystal was assumed to be of the Poisson type, while distribution of their energy levels was assumed to be of the Gaussian type. Our conception of the virial theorem implementation is that the transition from the band-like conduction to hopping conduction occurs when the average kinetic energy of an electron in the c-band (hole in the v-band) is equal to the half of the absolute value of the average energy of the Coulomb interaction of an electron (hole) with the nearest neighbor ionized donor (acceptor). Calculations of T_j according to our model agree with experimental data for crystals of Ge, Si, diamond, etc. up to the concentrations of a hydrogen-like impurity, at which the phase insulator-metal transition (Mott transition) occurs. Under the temperature T_h ≈ T_j /3, when the nearest neighbor hopping conduction via impurity atoms dominates, we obtained expressions for the electrostatic field screening length Λ_h in the Debye-Hückel approximation, taking into account a nonzero width of the impurity energy band. It is shown that the measurements of quasistatic capacitance of the semiconductor in a metal-insulator-semiconductor structure in the regime of the flat bands at the temperature Th allow to determine the concentration of doping impurity or its compensation ratio by knowing Λ_h.