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

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 bas...

Ionization equilibrium at the transition from valence-band to acceptor-band migration of holes in boron-doped diamond N 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. 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 (see, e.g., Ref. 3). The effect of compensation of acceptors by donors in semiconductor oxide ceramic of p-type on its transition from a superconducting to an insulating state is discussed in Ref. 4. Studies of electrical properties of diamond become especially relevant due to the development of effective technologies for the synthesis of perfect crystalline films and massive (bulk) diamond samples, [5][6][7] and to the broader application of such materials.
Currently, there is not enough information about the ionization equilibrium and the migration of holes in the temperature range where band-like and hopping dc conductivities in p-Dia:B crystals are comparable. This information is important, for example, in designing diamond based bolometers (see, e.g., Ref. 8), when it is necessary to predict the temperature of the transition from band-like to hopping conduction to block the migration of holes between acceptors. Generally speaking, this line of research of electrical properties of diamond crystals is not developed in the desirable limits. Indeed, in Ref. 9, the model of transition of p-Dia:B from the insulating to the metallic state (Mott transition 10 ) under the increase in boron concentration is proposed in the presence of electrical conduction of v-band holes. The critical concentration of boron atoms (N M % 4 Â 10 20 cm À3 ) at which the Mott transition occurs is found experimentally in Ref. 11. The calculation of dc hopping conductivity of p-type diamond by means of hole hopping via the (0) and (À1) charge states of boron atoms is performed in Ref. 12. However, no quantitative comparison of calculations of the concentration and the mobility of holes with experimental data specifically at the temperature of mass transition from "free" migration of holes in the v-band to hopping migration of holes via acceptors has been performed to date.
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 temperature dependence of the logarithm of the electrical resistivity q on the reciprocal temperature 1/T for crystalline semiconductors doped and compensated by a) Electronic mail: poklonski@bsu.by hydrogen-like impurity atoms is schematically shown in Fig.  1(a). Note that at the temperature T j the values of resistivities q b and q h are approximately equal. Band-like conduction (the BC regime) is realized at temperatures T > T j , where an average hole in the v-band mostly moves "freely" through the crystal matrix between successive acts of scattering on phonons and impurity ions. Note that usually the energy of thermal ionization of the major (doping) impurity is measured in the temperature range close to T 1 % 3T j /2. In the hopping conduction regime (the HC regime) at T < T j , the hole is localized near a single acceptor most of the time. Holes tunnel through the crystal lattice between acceptors in the charge states (0) and (À1) due to the "assistance" of phonons, so the electrical conductivity r ¼ 1/q decreases exponentially with 1/T as temperature decreases in the range close to T 3 % T j /3. It should be noted that in the vicinity of the temperature T j the average lifetime of a hole at an acceptor is approximately equal to the "flight" or jump time of hole from an acceptor in the charge state (0) to an acceptor in the charge state (À1). The jump of hole includes a thermally activated transition of hole from an electrically neutral acceptor to the mobility edge E mb , a flight with thermal velocity in crystal matrix to the ionized acceptor and capture on it. This migration regime of the holes can be tentatively called "relay," 13 "turbulent," 14 or "jumping" 15 (the JC regime). In the JC regime, mass capture of holes from the v-band by acceptors in the charge state (À1) occurs. This results in increased mobility of the remaining holes in the v-band due to the decrease in the concentration of scattering ions. 16 A stationary current density of "free" v-band holes J p along the x axis is given by (see, e.g., Refs. [17][18][19] J p ¼ ep mb l p % À eD p dp mb dx ¼ r p % À eD p dp mb dx ; where e is the elementary charge, p mb is the concentration of holes with energies greater than the mobility edge E mb that unrestrictedly migrate via the v-band states within the entire crystal sample, l p is the drift mobility of holes, % is the strength of an external electric field along the x axis, D p is the diffusion coefficient, x is the coordinate, and r p ¼ ep mb l p ¼ 1=q b is the electrical conductivity of the holes in the v-band. The ratio D p =l p ¼ n p k B T=e is higher than the classical value k B T=e, where k B is the Boltzmann constant, and T is the absolute temperature; parameter n p ! 1 increases with increasing hole concentration. 20,21 A stationary hopping current density J h of holes migrating via hydrogen-like acceptors in charge states (0) and (À1) is given by 12,22,23 23 It is clear that Eqs. (1) and (2) are applicable in the driftdiffusion approximation. It means that the strength of an external electric field %ðxÞ, as well as the concentrations p mb ðxÞ and N 0 ðxÞ ¼ N À N À1 ðxÞ significantly change along the x axis only for distances much greater than the mean free path of the hole and the mean hop distance of a hole between the acceptors.

II. STATISTICS OF HOLES AND CHARGE STATES OF HYDROGEN-LIKE ACCEPTORS IN p-TYPE DIAMOND CRYSTALS
We assume that the impurity atoms are randomly distributed over the sites of the diamond crystal lattice. Let p be the average over the crystal volume V concentration of holes in the v-band, N ¼ N 0 þ N À1 be the sum of the average concentrations of acceptors (boron atoms) in the charge states (0) and (À1), and KN be the concentration of donors, all in the charge state (þ1), where 0 < K < 1 is the compensation ratio of acceptors by donors. Then, the electrical neutrality condition has the form (see, e.g., Refs. 18 and 19) where p and N À1 are temperature dependent. The average over the crystal volume V concentration of all holes in the v-band 9,24,25 where g p is the energy density of states of holes, f p ¼ f1 þ exp½ðE p À E F Þ=k B Tg À1 is the Fermi-Dirac function, E p ¼ E kin þ U p is the total energy of the hole (the sum of kinetic E kin and potential U p energies), E F is the Fermi level (E F < 0 in the band gap), and k B T is the thermal energy. The top of the v-band (E p ¼ 0) of an undoped crystal is set as the reference point for E F [ Fig. 1(b)]. Holes in the v-band with an energy E p above the mobility edge E mb (E p > E mb ) are delocalized in a crystal. 26 According to Refs. 27 and 28, the probability density function of potential energy U p fluctuations of a hole in the v-band of a bulk crystal is normal (Gaussian) where W p is the root-mean-square fluctuation of the hole energy [see Fig. 1 Taking into account Eq. (5) in the quasi-classical approximation [27][28][29] for the energy density of states of the holes in the v-band, we find 24,25 where m p is the density-of-states effective mass of holes in the v-band of undoped diamond, and h ¼ h=2p is the reduced Planck's constant. Taking into account three subbands of the diamond valence band (v-band) [heavy (h) and light (l) holes, as well as holes in the subband split-off due to the spin-orbit interaction (so)], the hole effective mass m p is equal 30 where m h ¼ 0:78m 0 is the heavy hole effective mass, m l ¼ 0:14m 0 is the light hole effective mass, m so ¼ 0:394m 0 is the hole effective mass in the spin-orbit split-off subband of the v-band, and m 0 is the electron mass in vacuum.
The average concentration of mobile v-band holes (characterizes the dc electrical conductivity r p and the stationary Hall effect) is determined as where V is the volume of the diamond sample and E mb is the mobility edge of holes [see Fig. 1 The average concentration of the acceptors in charge state (À1), taking into account deviations of their energy levels E a from the average value E a , can be written as 31,32 where f À1 ¼ 1 À f 0 is the probability that an arbitrary acceptor with energy level E a > 0 higher than the top of the v-band is ionized, G a is the probability density function for the distribution of energy levels E a relative to E a in the diamond band gap [ Fig. 1(c)].
Excluding the excited states of all the acceptors for f À1 , we find 18,19 where b a ¼ 4 þ 2 expðÀe so =k B TÞ is the degeneracy factor of level E a considering all three hole subbands in the diamond v-band, e so ¼ 6 meV is the value of so-subband split-off (due to spin-orbital interaction) from the degenerate heavy and light hole subbands. If the thermal energy k B T [e.g., at T ¼ T 1 ; Fig. 1(a)] is substantially larger than e so, then b a % 6.
Taking into account only the excited states of electrically neutral acceptors, the quantity b a in Eq. (9) should be replaced by 18,32 where l m ! 1 is the largest number of possible excited states of the average acceptor in the charge state (0). The number l m is determined by the equality of the hole orbit radius l 2 m a H in the excited state l m to half the average distance between impurity atoms d im /2. This quantity is taken to be continuous, i.e., l 2 The Bohr orbit radius a H for a hole is determined by the thermal ionization energy I a ¼ e 2 =8pea H of a solitary acceptor in the ground (non-excited) state l ¼ 1 in the lightly doped diamond crystal. Here, e ¼ e r e 0 is the static permittivity, e r ¼ 5.7 is the relative permittivity of the diamond due to v-band electrons, 33 and e 0 is the electric constant. To this end, the quantity l m can be viewed as l m ¼ ðd im =2a H Þ 1=2 , where a H ¼ e 2 /8peI a . In Eq. (10), the summation over l taking into account l m ! 1 can be approximately replaced by integration 34 where l m % 0:526ð8pe r e 0 I a =e 2 Þ 1=2 ½ð1 þ KÞN À1=6 . In numerical calculations of the ionization equilibrium in p-Dia:B crystals at T j in Eqs. (8) and (9) instead of b a , the quantity b am determined by Eq. (11) will be used. Note that the condition l m ¼ 1 for I a ¼ 370 meV and compensation ratio K % 0.05 corresponds to the boron concentration N % 5 Â 10 20 cm À3 that is close to the experimental Mott transition concentration N M % 4 Â 10 20 cm À3 after Ref. 11. According to Refs. 35 and 36, we assume a Gaussian probability density function G a for the acceptor energy level distribution E a over the crystal (cf. representation of G a in Ref. 37) where W a is the effective width of the acceptor band [see Fig. 1(c)]. Let us find according to Refs. 31 and 32, and Fig. 1(b) the position of the "center" of the acceptor band E a (relative to the top of the v-band in the undoped crystal E p ¼ 0). The quantity E a is the energy needed for dissociation of the bound state of hole on a statistically average acceptor due to the thermal fluctuations. Specifically, E a is equal to the energy of ionized acceptor with screening "cloud" of charges E À1 ¼ I À1 þ E cor plus the energy of hole passed into the v-band on its mobility edge E mb and minus the energy of neutral acceptor where I a ¼ I À1 À I 0 is the energy level (thermal ionization potential) of a solitary acceptor (for E cor ¼ E mb ¼ E p ¼ 0), I a ¼ 370 meV for a boron atom as a substitutional impurity in the diamond crystal lattice, 38 E cor < 0 is the energy of the correlative reduction of thermal ionization energy of the acceptor in the charge state (0) due to the screening of the acceptor in the charge state (À1) by a "cloud" of charges (with the total charge þe), E mb ¼ E per þ E exc < 0 is the mobility edge of holes in the v-band, 39 E per is the percolation threshold for the diffusion of holes in the v-band, and E exc < 0 is the shift of the top of the v-band into the band gap due to the exchange interaction of holes of the v-band.
Note that when the tail of the density of states of holes in the v-band can be neglected (W p ( W a ), the mobility edge coincides with the top of the v-band: E mb ¼ E p ¼ 0 (see, e.g., Refs. 9, 31, and 32). If the energy of the hole E p is less than the mobility edge E mb , then the hole can only move within a limited volume in the crystalline sample.
Let us specify in Eq. (13), the energy quantities According to Refs. 9, 31, 39, and 40, the total electrostatic correlation energy E cor of the system "acceptor impurity ion þ cloud of screening point charges" is equal to 41 where K s is the screening radius of the Coulomb field of the impurity ion in the Debye-H€ uckel approximation, d s % 0:554½N þ KN þ p À1=3 is the average distance between nearest acceptors, donors, and v-band holes (point particles randomly distributed over the crystal; see, e.g., Refs. 42 and 43). When deriving Eq. (14) in accordance with numerical estimates, 9 it was assumed that an average acceptor in the charge state (À1) has the time to gather around it a cloud of screening charges before capturing a hole from the v-band, i.e., prior to electrical neutralization of the ionized acceptor. 31 The screening radius of the impurity ion Coulomb field, taking into account holes in the v-band, is defined by the expression 12,21,22,44 where p ¼ N À1 À KN is the total concentration of holes in the v-band and N h ¼ N 0 N À1 =N f 0 f À1 N is the effective concentration of holes that hop between acceptors in charge states (0) and (À1); n p ! 1 and n h ! 1 are dimensionless parameters Note that for W p ( k B T and W a ( k B T, according to Eq. (16), the parameters are n p ¼ n h ¼ 1. Analytical approximations and numerical calculations of n p and n h for W p ) k B T and W a ) k B T are given in Refs. 12, 23, and 44-46. Assuming 24,28,47 that a critical part of the volume of a three-dimensional semiconductor sample that is available for the diffusive motion of holes in the v-band and that corresponds to the energy level of their diffusion percolation E per is we obtain 48 E per % À0:955W p < 0: As the concentration of v-band holes increases, their exchange interaction becomes significant. 39,40,49 Consequently, the energy position of the top of the v-band (E p ¼ 0 for an undoped crystal) shifts deeper into the band gap, decreasing the energy level of diffusion percolation E per [see Eq. (17)] by where E exc (U p ) is the exchange energy for a v-band hole, G p ðU p Þ is given by Eq.
In the limit W p ! 0, when G p ðU p Þ takes the form of the Dirac delta function dðU p Þ, Eq. (18) (17) and (18) give the mobility edge of v-band holes [see Eq. (7)] To calculate using Eq. (20), the dependence of E mb on the concentration of impurity atoms and v-band holes, and on the temperature we need to establish in Eqs. (17) and (18) the dependence of W a and W p on the same quantities. Following Refs. 9 and 53, we assume that an insulator-metal concentration phase transition in the limit of zero temperature (T ! 0) takes place when the Fermi level E F crosses the mobility edge E mb of v-band holes (which is equal to the sum of the percolation threshold E per and the exchange energy E exc ), i.e., when E F ¼ E mb .

III. ELECTROSTATIC FLUCTUATIONS OF ENERGY OF A HOLE IN THE ACCEPTOR AND VALENCE BANDS
The concentration of all point charged particles N ch that satisfies the condition of electrical neutrality (3) is Taking into account only Coulomb interaction between the two nearest "point" charges in the crystal, for the effective width of the acceptor band W a , which is equal to the root-mean-square fluctuation of the energy of the ionized acceptor, we find 31,32,54 where P 3 i¼1 P i ðrÞ dr ¼ 4pr 2 N ch expðÀ4pr 3 N ch =3Þ dr is the Poisson probability 55 that the charged particle nearest to a given charged point particle (charged acceptor (À1), charged donor (þ1) or v-band hole) is located at a distance from r to r þ dr, and that there are no other charged particles in the volume 4pr 3 /3 of a ball centered on the given charge.
The average distance between the nearest point charged particles in the crystal matrix, taking into account the probability density P i ðrÞ from Eq. (21) takes the form 42,43 where N ch ¼ 2N À1 .
Let us now find an expression for the mean fluctuation W p of the v-band hole potential energy. Note that Eq. (21) for the root-mean-square fluctuation W a of the potential energy of an acceptor in the charge state (À1) is written for "frozen" point particles with charge 6e randomly distributed (after Poisson) over the volume of a crystal. 56 As noted in Ref. 28, an average hole in the v-band with energy above E mb does not notice potential energy fluctuations on a spatial scale smaller than or comparable to its mean wavelength k p (after de Broglie). However, no explicit analytical expression for the root-mean-square fluctuation of the electrostatic potential energy of v-band hole (or c-band electron) is given in Ref. 28 (see also Refs. 47 and 57).
Let us randomly select a region in the crystal with volume k 3 p % 2=p. Here, the factor 2 takes into account that, according to the Pauli principle, two v-band holes with opposite spins can simultaneously occupy the same volume % k 3 p . The corresponding wavelength of an average hole 58,59 is k p % ðp=2Þ À1=3 . There are q ¼ k 3 p =d 3 ch charged particles located within this volume. Each particle is localized in a region with linear dimensions d ch % 0:554N À1=3 ch , where N ch ¼ p þ KN þ N À1 . We assume that the potential energy U s of each sth charged particle has the same distribution law, thus the average potential energy of the considered v-band hole U p ¼ ð1=qÞ P s U s . Using the relation for variance of the arithmetic mean of mutually independent identically distributed random variables [60][61][62] where the corresponding variances are VarðU p Þ ¼ W 2 p and VarðU s Þ ¼ W 2 a , we get 63 where W a is determined by Eq. (21). Since for a p-type semiconductor p < N ch , it follows from Eq. (24) that W p < W a . We note that, according to Eqs. (17), (20), and (24), as well as condition (3), the energy value E mb of the mobility edge of v-band holes depends on their concentration p, which does not agree with the arguments presented in Ref. 64.
Let us rewrite the condition of electrical neutrality (3) for weakly doped with boron (N ( N M ) p-type diamond crystals taking into account Eqs. (4)- (20). For an "ideal" diamond of p-type, when W a ( k B T, W p ( k B T, and E a ¼ I a ¼ 370 meV, taking into account excited states of the electrically neutral boron atoms from Eq. (3), we yield (see, e.g., Refs. 18, 19, and 25) where p a ¼ ðp v =b ai Þ expðÀI a =k B TÞ; p v ¼ 2ð2pm p k B TÞ 3=2 = ð2p hÞ 3 ; the degeneracy factor b ai is determined by Eq. (11) for E a substituted by I a .
For the low temperatures, when KN ) p þ p a , from Eq. (25) we get standard expression relating the concentration p of delocalized holes in the v-band to the thermal ionization energy I a of a solitary boron atom Note that the condition KN ) p þ p a combined with Eq. (26) gives the restriction on hole concentration p ( K(1 À K)N.
For the high temperatures, when p a ) KN þ p, from Eq. (25), we get p ¼ (1 À K)N and p a ) N.
In short, we suggest the following algorithm of the numerical calculations using Eqs.  (20). The computed value E F allows to refine the values of W a , W p , N À1 , and p. (iv) The solution of the electrical neutrality equation (3) and refinement of the values of W a , W p , N À1 , and p are iteratively performed until their relative error becomes less than the desired one e goal . The relative error e(X) of the value of a calculated quantity X ¼ fE F ; W a ; W p ; N À1 ; pg is calculated as: eðXÞ ¼ 2jX k ÀX kÀ1 j=jX k þ X kÀ1 j, where X k is the value of a quantity X at the kth step of the iterative procedure.
Let us test this algorithm and the developed quasiclassical theory on the experimental results of Ref. 65. In this work using photoelectric spectroscopy at T ¼ 300 K, a crystalline film of p-Dia:B was studied. The film was grown by CVD method and then was annealed in microwave hydrogen plasma at temperature 800 8C during 5 min. According to Ref. 65, the Fermi level E F at the boron concentration N ¼ (2-3) Â 10 20 cm À3 is located in the band gap at 90-120 meV above the top of the v-band of the undoped diamond. Calculation using Eqs. (3)-(24) at temperature T ¼ 300 K gives E F % 100 meV for N ¼ 2.5 Â 10 20 cm À3 at K % 0.28. Such a compensation ratio can be attributed to the compensation of boron atoms by hydrogen, or, more specific, due to the formation of electrically neutral associates of boron with hydrogen.

IV. CALCULATION OF THE IONIZATION EQUILIBRIUM PARAMETERS IN p-Dia:B UNDER TRANSITION FROM BAND-LIKE TO HOPPING CONDUCTION
In Refs. 63 and 66, on the basis of the virial theorem, the characteristic temperature T j of the transition from the hopping migration of holes via boron atoms (the HC regime) to the band-like migration of holes via states of the v-band (the BC regime) is determined. According to these works, the value of T j temperature, for which r p % r h [see Fig. 1(a)], is found from the equation where N À1 ðT j Þ and pðT j Þ ¼ N À1 ðT j Þ À KN depends on T j and the compensation ratio K, according to the electroneutrality condition (3).
In the limit of low concentration of holes in the v-band pðT j Þ ( N À1 ðT j Þ % KN from Eq. (27), we yield where KN is the concentration of hydrogen-like donors which compensate the boron atoms. An algorithm of calculation of the values W a , W p , N À1 , and p at temperature T j according to Eq. (27) is analogous to that given in the end of the Section III. At the beginning, a seed value of T j using Eq. (28) is calculated. The described iterative procedure for calculation of the values W a , W p , N À1 , and p at the seed temperature is performed. Using Eq. (27) the value of T j is refined. The iterative procedure for determination of W a , W p , N À1 , and p and refinement of T j are performed until the relative error of T j becomes less than e goal . Fig. 2 shows the results of calculations of temperature T j ðN; KÞ, determined from Eq. (27), applied to boron-doped p-type diamonds for compensation ratios K ¼ 0.15 and K ¼ 0.5 as a function of boron atom concentration N. The values of T j obtained from experimental data [67][68][69][70][71][72][73][74][75] are also presented. We should note that the compensation ratio of the diamond samples with boron concentration N % 3 Â 10 17 cm À3 studied in Refs. 67-75 can reach K % 0.5, and for N % 3 Â 10 20 cm À3 , it can reach K % 0.05. The value of K affects the position of the Fermi level E F relative to the edges of the allowed energies. 76 It leads to the change in the relation between concentrations of ionized and electrically neutral impurities. 18,19 Note that in Ref. 77, the temperature dependence of electric conduction of n-type homoepitaxial diamond films doped with donor atoms of phosphorus (thermal ionization energy of a solitary atom %0.6 eV) was measured. The transition from conduction-band to donor-band migration of electrons for donor concentration N % 4 Â 10 19 cm À3 occurs at the temperature T j % 400 K. The calculation using Eq. (28) for the compensation ratio K % 0.15 gives T j % 390 K. Fig. 3 shows the concentration p mb of delocalized v-band holes [with energy E p higher than the mobility edge E mb ðT j Þ ¼ E per þ E exc according to Eq. (20)] as a function of the dopant concentration N at K ¼ 0.15, calculated using Eq. (7). Fig. 3 also shows a comparison of these calculations with measurements of the dc electrical conductivity r ¼ r p þ r h ¼ 1=q 3 is the limit of the total thermal ionization of acceptors p mb ¼ (1 À K)N. and of the stationary Hall effect in homoepitaxial diamond films and p-type diamond crystals. [71][72][73][74] For the analysis of the experimental data, we used the relation that takes into account the contribution of both band-like (r p ) and hopping (r h ) conductivities to the experimental value of the Hall coefficient 78,79 where R p ¼ 1=ep H is the Hall coefficient due to v-band holes (when the Hall factor equals to unity in a sufficiently strong magnetic field), R p r p is the Hall mobility of v-band holes, R h is the Hall coefficient due to holes hopping between acceptors (R h ( R p ), and R h r h is the hopping Hall mobility of holes in the acceptor band. Then, according to Eq. (29), the experimental value of hole concentration p H (T j ) determined from Hall effect measurements and electric conductivities at temperature T ¼ T j ðN; KÞ, for which r p % r h , takes the form 18 We should note that, according to calculations using Eqs. (4) and (7), in the range of boron concentrations indicated in Fig. 3, the concentration of mobile v-band holes p mb hardly differs from the total concentration of holes p (deviation ðp À p mb Þ=p < 4%).
The calculation of p mb ¼ p using Eq. (26) for K ¼ 0.15 is represented in Fig. 3 by the curve 1 0 . The calculation of p % p mb ¼ ð1 À KÞN for K ¼ 0.15 is shown in Fig. 3 by the curve 3.
By analogy with Eq. (26) the differential energy of thermal ionization E 1 of boron atoms in the vicinity of the temperature T 1 ¼ 3T j =2, when p mb ( Kð1 À KÞN, is determined as (see, e.g., Refs. 31 and 32) where the concentration p mb of delocalized holes in the vband (with energy E p higher than the mobility edge E mb within the volume of crystalline diamond sample) is given by Eq. (7). The values of E 1 calculated using Eq. (31) for temperature T 1 ¼ 3T j /2 are compared in Fig. 4 to the experimental data of Refs. 72, 73, and 80. Experimental values E 1 were determined in the vicinity of the temperature T 1 ¼ 3T j /2 using linear approximation of the concentration of holes p H [obtained from the temperature dependences of the Hall coefficient R H according to Eq. (29)] plotted in the Arrhenius scale: lnðp H T À3=2 Þ against 1/T. For the temperature T % T j ðN; KÞ, the band-like electrical conductivity r p equals to the hopping conductivity r h . Then, according to Eqs. (29) and (30), the experimentally determined Hall mobility l H ðT j Þ of v-band holes is related with the theoretically determined drift mobility l p ðT j Þ ¼ r p =ep mb of v-band holes in the following way: 63 where R H ¼ R p is the Hall coefficient of v-band holes, rðT j Þ ¼ r p þ r h ¼ 2r p is the total conductivity of v-band holes and of holes hopping between boron atoms, p H ðT j Þ % p mb ðT j Þ is the Hall concentration of v-band holes; l p ðT j Þ is the drift mobility of v-band holes at temperature T j . We assume that in the vicinity of the temperature T j , the average time of "free" motion t j of a hole in the crystal matrix with thermal velocity v t ¼ ð3k B T j =m pr Þ 1=2 , where m pr is the conductivity effective mass, is approximately equal to the average time of hole localization on an acceptor. Thus, for turbulent transfer of holes (JC regime), the time of flight of a v-band hole on the mobility edge E mb between acceptors is roughly equal to the time of "settled life" of hole at an average acceptor (see also Ref. 81). Therefore, at T % T j , the drift mobility l p ðT j Þ of v-band holes is approximately equal to the jumping drift mobility l j of holes via acceptors. Using the Drude model (see, e.g., Refs. 17, 79, and 82), we can estimate the value of l j as follows: l j ¼ et j =m pr , where t j ¼ L j =v t is the average time of free "flight" in the form of a jump of hole with the speed v t between two acceptors in the charge states (0) and (À1), and L j is the projection of the hole jump distance between acceptors on the direction of the strength of the constant external electric field in the crystal. Taking into account the distribution 42,61 of angles between a half-line with fixed direction and a half-line for which all the directions are equally probable, we get L j ¼ S a /2, where S a ¼ 1:28½ð1 þ KÞN À1=3 is the average distance between the closest acceptors in the crystal matrix (determined in Ref. 43 by the Voronoi-Dirichlet polyhedra method). 83 Finally, for the drift mobility of v-band holes l p ðT j Þ at T % T j ðN; KÞ in the JC regime, we find where m pr is the conductivity effective mass of v-band holes in a diamond crystal 19,25,30,79  We should note that we also calculated T j and l j for the experimental data 84 on a ZnO crystal, containing processrelated donors with the concentration N % 6.6 Â 10 18 cm À3 , compensation ratio K % 0.2, and thermal ionization energy I d % 18.3 meV. According to the experiment: l H (T j ) ¼ 73 cm 2 / (V s) at T j ¼ 95 K. The calculation using Eq. (27) gives the temperature T j % 270 K, and the calculation using Eq. (33) gives the jumping (turbulent, relay) drift mobility of electrons via donors l j /2 % 53 cm 2 /(V s). In the calculations, the following parameters of the ZnO crystal were used: 30  Finally, let us estimate the value of drift hopping mobility of holes M h between boron atoms in diamond at the temperature T ¼ T j (N, K). From the equality of the electrical conductivities r p and r h ¼ eN h M h at T ¼ T j , the expression for the hopping drift mobility of holes via acceptors follows: Whence taking into account that r p ðT j Þ ¼ ep mb ðT j Þl p ðT j Þ % ep mb l j and N h ðT j Þ ¼ N 0 ðT j ÞN À1 ðT j Þ=N, we get where all the values in Eq. (34) are determined at T j found from Eq. (27). In Eq. (34) at T j the turbulent (jumping, relay) hole mobility l j is found using Eq. (33), the concentration of delocalized v-band holes p mb is found using Eq. (7), and the concentration of holes hopping between acceptors N h ¼ N 0 N À1 =N is found using Eq. (8) and taking into account Eq. (3). If p ( N À1 then from Eq. (3) it follows that N h ¼ ð1 À KÞKN.
The hopping drift mobility of holes calculated using Eq. (34) is shown in Fig. 6 as a function of a dopant (boron atoms) concentration. One can see that the quantity M h ðT j Þ on the insulator side of the Mott transition is small even at T ¼ T j and is decreased exponentially at the increase in the average distance % ½ð1 þ KÞN À1=3 between boron atoms. According to Refs. 12 and 44, in the temperature range close to T 3 % T j /3, the dc hopping conductivity dominates (r h > r p ). The effective concentration of hopping holes N h ¼ ð1 À KÞKN does not depend on temperature, 85 and the mobility M h ðT 3 Þ is proportional to expðÀe 3 =k B T 3 Þ, where e 3 is the thermal activation energy for migration of holes via acceptors (boron atoms) in the charge states (0) and (À1). Note that at temperature T 3 [see Fig. 1(a)], the hopping mobility is smaller than at temperature T j . In fact, the ratio of hopping mobilities of holes  confirmed by theoretical estimates 77 of hopping drift mobility of electrons via donor atoms of phosphorus in the diamond of n-type.
From the comparison of Figs. 5 and 6, it is seen that at the temperature T ¼ T j ðN; KÞ the calculated jumping (turbulent, relay) drift mobility of v-band holes l j ¼ l p ðT j Þ is comparable in value with the calculated hopping drift mobility via boron atoms M h ðT j Þ for the boron concentration N > 10 20 cm À3 and the compensation ratio K ¼ 0.15.
The above estimates may explain the fact that the Hall mobility R h r h of holes hopping between acceptors is not registered yet up to the acceptor concentration corresponding to the Mott transition. For example, even for such perfect semiconductor crystals as neutron transmutation doped p-Ge:Ga with compensation ratio K % 0.35 and the concentration of hopping holes via gallium atoms N h ¼ ð1 À KÞKN % 0:23N, mobility M h ðT 3 Þ cannot be determined from measurements of Hall effect (see Refs. [86][87][88].

VI. CONCLUSION
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 r p is roughly equal to the hopping conductivity via boron atoms r h . The disorder in crystals is caused by a random spatial distribution of impurity ions and mobile v-band holes. Comparison with the experiment and numerical calculations (without the use of any adjustable parameters) were carried out for the example of p-type diamond crystals doped with boron atoms with concentration N % 3 Â 10 17 -3 Â 10 20 cm À3 with an approximately constant ratio of their compensation by donors K % 0.15. We assumed that boron atoms can be in the charge states (0) and (À1), and donors that compensate them can be only in the charge state (þ1) on the background of an electroneutral diamond matrix. Within the frame of the proposed model, a theoretical description of the dependence of the differential energy of thermal ionization of boron atoms as a function of their concentration N in the vicinity of the temperature T 1 ¼ 3T j /2 is given. The calculation of the concentration of mobile v-band holes p mb and their jumping (turbulent, relay) drift mobility l j were performed for the temperature T j . At this temperature, determined on the basis of the virial theorem, the mass capture of holes from the vband by negatively charged boron atoms ends, and r p % r h . The comparison of the calculation results for p mb and l j with experimental data for the temperature T j shows quantitative agreement for concentrations of boron atoms up to the concentration corresponding to the insulator-metal concentration phase transition (Mott transition), from the insulator side. Estimates of hopping mobility of holes via boron atoms extrapolated from the low temperature region (close to T 3 % T j /3) into the high temperature region (close to T j ) are carried out.