Role of electrostatic fluctuations in doped semiconductors upon the transition from band to hopping conduction (by the example of $p$-Ge:Ga)

The electrostatic model of ionization equilibrium between hydrogen-like acceptors and $v$-band holes in crystalline covalent $p$-type semiconductors is developed. The range of applicability of the model is the entire insulator side of the insulator–metal (Mott) phase transition. The density of the spatial distribution of acceptor- and donor-impurity atoms and holes over a crystal was assumed to be Poissonian and the fluctuations of their electrostatic potential energy, to be Gaussian. The model takes into account the effect of a decrease in the energy of affinity of an ionized acceptor to a $v$-band hole due to Debye–Hückel ion screening by both free $v$-band holes and localized holes hopping over charge states (0) and (–1) of acceptors in the acceptor band. All donors are in charge state (+1) and are not directly involved in the screening, but ensure the total electroneutrality of a sample. In the quasiclassical approximation, analytical expressions for the root-mean-square fluctuation of the $v$-band hole energy $W_p$ and effective acceptor bandwidth $W_a$ are obtained. In calculating $W_a$, only fluctuations caused by the Coulomb interaction between two nearest point charges (impurity ions and holes) are taken into account. It is shown that $W_p$ is lower than $W_a$, since electrostatic fluctuations do not manifest themselves on scales smaller than the average de Broglie wavelength of a free hole. The delocalization threshold for $v$-band holes is determined as the sum of the diffusive-percolation threshold and exchange energy of holes. The concentration of free $v$-band holes is calculated at the temperature $T_j$ of the transition from dc band conductivity to conductivity implemented via hopping over acceptor states, which is determined from the virial theorem. The dependence of the differential energy of the thermal ionization of acceptors at the temperature 3$T_{j}$/2 on their concentration $N$ and degree of compensation $K$ (the ratio between the donor and acceptor concentrations) is determined. Good quantitative agreement between the results of the calculation and data on the series of neutron transmutation doped $p$-Ge samples is obtained up to the Mott transition without using any fitting parameters.