Ionization equilibrium at the transition from valence-band to acceptor-band migration of holes in boron-doped diamond

Abstract

A quasi-classical model of ionization equilibrium in the p-type diamond between hydrogen-like acceptors (boron atoms which substitute carbon atoms in the crystal lattice) and holes in the valence band (v-band) is proposed. The model is applicable on the insulator side of the insulatormetal concentration phase transition (Mott transition) in p-Dia:B crystals. The densities of the spatial distributions of impurity atoms (acceptors and donors) and of holes in the crystal are considered to be Poissonian, and the fluctuations of their electrostatic potential energy are considered to be Gaussian. The model accounts for the decrease in thermal ionization energy of boron atoms with increasing concentration, as well as for electrostatic fluctuations due to the Coulomb interaction limited to two nearest point charges (impurity ions and holes). The mobility edge of holes in the v-band is assumed to be equal to the sum of the threshold energy for diffusion percolation and the exchange energy of the holes. On the basis of the virial theorem, the temperature T j is determined, in the vicinity of which the dc band-like conductivity of holes in the v-band is approximately equal to the hopping conductivity of holes via the boron atoms. For compensation ratio (hydrogen-like donor to acceptor concentration ratio) K % 0.15 and temperature T j , the concentration of "free" holes in the v-band and their jumping (turbulent) drift mobility are calculated. Dependence of the differential energy of thermal ionization of boron atoms (at the temperature 3T j /2) as a function of their concentration N is calculated. The estimates of the extrapolated into the temperature region close to T j hopping drift mobility of holes hopping from the boron atoms in the charge states (0) to the boron atoms in the charge states (À1) are given. Calculations based on the model show good agreement with electrical conductivity and Hall effect measurements for p-type diamond with boron atom concentrations in the range from 3 Â 10 17 to 3 Â 10 20 cm À3 , i.e., up to the Mott transition. The model uses no fitting parameters. Published by AIP Publishing.

Drift-diffusion model of hole migration in diamond crystals via states of valence and acceptor bands

“…In a p-type semiconductor, the density of the stationary hopping current of holes via hydrogen-like acceptors along the x axis is [13,26]:…”

“…If the thermal energy k T B (e.g., at T T 3 2; ) . Taking into account the excited states of electrically neutral acceptors, the quantity a b in equation (14) should be replaced by [13,41]:…”

“…The root-mean-square fluctuation of the electrostatic potential energy U p of a v-band hole W p is smaller than W a due to smoothing of the potential (with characteristic amplitude W a ) by a hole within the scale of its de Broglie wavelength [13,30]:…”

“…] which is the average distance between nearest neighbor acceptors, donors, and vband holes responsible for the screening and for the maintenance of the electrical neutrality and ionization equilibrium [13,30]. At the distances r d s  the total electrostatic potential s F of an acceptor in the charge state 1 -( ) with a screening cloud of charges is determined by the solution of the linearized Poisson equation [19,53]: ) .…”

“…Let us consider the algorithm for solving the system of equations (1), (8), (13) and (24). Taking into account the excited states of electrically neutral boron atoms the electrical neutrality condition (1) takes the form (see, e.g., [19,21,37]):…”