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

N A Poklonski,O N Poklonskaya,A I Kovalev,A G Zabrodskii,S A Vyrko

doi:10.1063/1.4954281

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 insulator–metal 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 Tj 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 Tj, 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 3Tj/2) as a function of their concentration N is calculated. The estimates of the extrapolated into the temperature region close to Tj hopping drift mobility of holes hopping from the boron atoms in the charge states (0) to the boron atoms in the charge states (−1) 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 × 1017 to 3 × 1020 cm−3, i.e., up to the Mott transition. The model uses no fitting parameters.

Highlights

Boron-doped p-type diamonds (p-Dia:B) are attractive for use as electrodes in electrochemistry,1 as well as in power semiconductor electronics.2 Experimental studies of p-Dia:B superconductivity at liquid helium temperatures are a topic of current interest
The estimates of the extrapolated into the temperature region close to Tj 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
We developed a quasi-classical theory for calculating the concentration of holes in the v-band, as well as their drift mobility in boron-doped p-type diamond crystals in the temperature range when the dc band-like conductivity of v-band holes rp is roughly equal to the hopping conductivity via boron atoms rh

Summary

INTRODUCTION

Boron-doped p-type diamonds (p-Dia:B) are attractive for use as electrodes in electrochemistry, as well as in power semiconductor electronics. Experimental studies of p-Dia:B superconductivity at liquid helium temperatures are a topic of current interest (see, e.g., Ref. 3). The purpose of this work is to develop a quasi-classical model of ionization equilibrium and of hole migration in boron-doped p-type diamond crystals in the temperature range of the transition from band-like to hopping dc electrical conduction. The jump of hole includes a thermally activated transition of hole from an electrically neutral acceptor to the mobility edge Emb, a flight with thermal velocity in crystal matrix to the ionized acceptor and capture on it. Where e is the elementary charge, pmb is the concentration of holes with energies greater than the mobility edge Emb that unrestrictedly migrate via the v-band states within the entire crystal sample, lp is the drift mobility of holes, % is the strength of an external electric field along the x axis, Dp is the diffusion coefficient, x is the coordinate, and rp 1⁄4. According to Refs. 27 and 28, the probability density function of potential energy Up fluctuations of a hole in the v-band of a bulk crystal is normal (Gaussian)

Wp exp

Wa exp

PiðrÞ dr

RH r

CONCLUSION