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

Ionization equilibrium and dc electrical conductivity of crystalline diamond are considered, for the temperature T j in the vicinity of which valence band (v-band) conductivity is approximately equal to hopping conductivity via acceptors. For the first time, we find explicitly (in the form of definite integrals) the fundamental ratio of diffusion coefficient to drift mobility for both v-band holes and holes hopping via hydrogen-like acceptors for the temperature T j . The known ratios follow from the obtained ones as particular cases. The densities of the spatial distributions of acceptors and hydrogen-like donors as well as of holes are considered to be Poissonian and the fluctuations of electrostatic potential energy are considered to be Gaussian. The dependence of exchange energy of v-band holes on temperature is taken into account. The thermal activation energy of hopping conduction as a function of the concentration of boron atoms (as acceptors) is calculated for temperature T 3 ≈ T j / 2 . Without the use of any adjustable parameters, the results of calculations quantitatively agree with data obtained from the measurements of hopping conductivity of diamond with boron concentration from 3 × 10 17 to 3 × 10 20 cm−3, i.e. on the insulating side of the Mott phase transition.


Introduction
The ionization equilibrium, e.g. between v-band (valence band) holes and acceptor impurity atoms in p-type semiconductor crystals, and the coexistence of v-band and hopping via acceptors migration of holes are considered separately (see, e.g., reviews [1][2][3]). However, at such a temperature T j , when the dc v-band electrical conduction is approximately equal to the dc hopping electrical conduction via hydrogen-like acceptors it is necessary to consider jointly the ionization equilibrium and the drift-diffusion migration of holes. It is important for the wide bandgap semiconductors, like boron-doped crystalline diamond, for which T j is in the region from liquid nitrogen to room temperatures and higher. For the narrow bandgap semiconductors, like boron-doped crystalline silicon, temperature T j is usually in the region from liquid helium to liquid hydrogen temperatures [4]. Such a difference in temperatures T j between boron-doped diamond and boron-doped silicon is connected with that the thermal energy of ionization by electrically neutral boron atoms in diamond is eight times greater than the one in silicon. Let us note that the wide bandgap semiconductors (for which the energy of electron affinity is less than the band gap) are important for applications in high-temperature electronics and power optics [5]. In particular, heavily boron-doped diamonds are attractive for studies of superconductivity at liquid helium temperature and higher (see, e.g., [6,7]) and application as electrodes for detection of neurochemicals in the human brain at room temperature [8].
The purpose of this article is to calculate theoretically (in the framework of the drift-diffusion model) the ratio of the diffusion coefficient to the drift mobility for both v-band holes and acceptor band holes in borondoped diamond crystals of p-type (p-Dia:B). In contrast to other works we consider the temperature region where the contributions of free v-band holes and holes hopping via acceptor band are comparable.
Let us consider a bulk homogeneous diamond crystal that contains v-band holes with the average concentration p, a majority impurity-hydrogen-like acceptors (boron atoms) in the charge states 0 ( ) and 1 -( ) with the total average concentration N, and a minority impurity-hydrogen-like donors, all in the charge state 1 + ( ), with the average concentration KN, where K 0 1 < < is the compensation ratio of acceptors by donors.
The electrical neutrality condition at ionization equilibrium for a random distribution of impurity atoms and holes over the crystal volume is is the average concentration of ionized acceptors, i.e. acceptors in the charge state 1 -( ), N 0 is the average concentration of electrically neutral acceptors.
Holes in the v-band with a total energy E p above the threshold of drift-diffusion migration E 0 v d < (see figure 1) are delocalized over the entire volume of the crystal (between scattering or recombination events) holes in the v-band. These completely free v-band holes characterize the dc electrical conductivity p s and the stationary Hall effect. We will denote their concentration as p mob . On the other hand, holes in the v-band with energy E E v p d < can only move within restricted regions of a crystal and are probably responsible for the increase of the macroscopic permittivity in doped semiconductors (see, e.g., [1,[9][10][11]). Let p loc be the concentration of partially free v-band holes. The total concentration of v-band holes is p p p mob loc = + . We will show that p p 1 loc  on the insulating side of the concentration transition insulator-metal (Mott transition) for K 0.5  in the limit of zero temperature. In boron-doped moderately compensated (K 0.1 < ) diamond of p-type the critical concentration of boron atoms corresponding to the Mott transition is N N 4 M = »1 0 20 cm −3 [12]. Impurity atoms are assumed to be localized at sites of diamond crystal lattice. However, because holes can transfer from acceptors into the v-band and back, as well as hop between acceptors, the charge states of immobile acceptors 0 ( ) and 1 -( ) migrate within crystal [13]. Conversely, all donors permanently remain in the charge state 1 + ( ), so their charge states do not migrate. At temperatures T T j < the electrical conductivity is mainly determined by holes hopping from acceptors in the charge state 0 ( ) to acceptors in the charge state 1 -( ): the HC regime in figure 1. Turbulent (jumping, relay) migration of holes is realized in the vicinity of the temperature T j , when the v-band p s and the hopping h s via acceptors electrical conductivities are approximately equal to each other (JC regime). At temperatures T T j > the electrical conductivity is mainly determined by v-band holes, that move 'freely' in the crystal matrix between events of scattering on phonons and impurity atoms (BC regime).
The expression for the density of the stationary (direct) electric current of mobile ions and conduction electrons in solids taking into account drift and diffusion components of the current was first written by Wagner [14]; see also [15]. The expression for the density of stationary hopping current of electrons via hydrogen-like donors in the charge states 0 ( ) and 1 + ( ) in the drift-diffusion approximation was first obtained in [16,17]. In a crystalline semiconductor of p-type the density J p of a direct (stationary) current of v-band holes in the direction of the coordinate x axis has the form [14,18,19]: where e is the elementary charge, p is the concentration of v-band holes, p m is the hole drift mobility, is the value of the external electric field strength directed along x axis; j is the electric potential, x is the coordinate, D p is the diffusion coefficient, ep is the dc conductivity of v-band holes. Figure 1. The logarithm of the dc electrical conductivity σ as a function of the reciprocal temperature T 1 for a p-type crystalline semiconductor in the regime of hole migration in the v-band ( p s ) at the total ionization of acceptors for T T i » and in the regime of hole hopping via nearest neighbor acceptors ( h s ) at T T 2 j 3 » . In the region T T 3 2 j 1 » the energy of thermal ionization of acceptors is determined. Here BC is the conduction of holes in the valence band, HC is the conduction of holes in the acceptor band, JC is the jumping migration of holes-the mixture of the v-band and hopping conductions. Inset: E p is the total energy of v-band hole; E v =0 is the top of the v-band of undoped crystal; E 0 F < is the Fermi level (relative to E v = 0); W p is the root-mean-square fluctuation of the v-band hole potential energy; E v d is the shift of the valence band top into the band gap; E 0 a > is the energy difference between E v =0 and the center of the acceptor band with effective width W ; a E n is the single-electron energy in a crystal.
Note that in the framework of the linear stationary theory of electron and hole transport in semiconductors it is assumed that in the expressions for the current densities the quantities that appear in front of the differentiation by coordinate operator do not depend on the coordinate [18][19][20]. We will use this approximation in the following.
From equation for all x we get [19][20][21]: is the electrochemical potential of holes in the point with the coordinate x; E F is the Fermi level for holes (that is determined from the electroneutrality condition (1)).
From equations (1)-(3) follows the expression for the static electrical conductivity of v-band holes e D p e D p E d d d d , that is used for analyzing the conditions and quantitatively describing the insulator-metal transition in semiconductors for given doping and compensation levels [22].
For an 'ideal' nondegenerate gas of holes, where the root-mean-square fluctuation of the potential energy of a v-band hole W p is much smaller than the thermal energy k T B , the shift of the v-band top E 0 v d = , and the  (3) we get (see, e.g., [24]): (4) and (5) shows that for degenerate gas of v-band holes the D p p m ratio become greater than the one for nondegenerate gas of v-band holes. This is due to the fact that D l v

Comparison of equations
is the mean free path [25] of hole in a crystal, F t is the quasimomentum relaxation time is the Fermi velocity [19,20]. Thus, for degenerate gas, D E 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]: is the effective concentration of holes hopping via acceptors in the charge states 0 ( ) and 1 ; -( ) N N 0 is the fraction of electrically neutral acceptors that belong to the infinite cluster of acceptors 'connecting' the electrodes to the crystal under the conditions of stationary hopping electrical conduction, N N 1 -is the fraction of ionized acceptors that belong to the infinite cluster of acceptors providing hopping dc migration of holes between electrodes, M h is the drift mobility of holes hopping via acceptors, is the value of the external electric field in the crystal, D h is the diffusion coefficient of holes hopping along x axis, eN M h h h s = is the dc hopping electrical conductivity via states of the acceptor band. It follows from equation (6) that at J 0 h = , but 0 E ¹ , the drift and diffusion components of the hopping current density in the x-direction compensate each other. Taking into account that in crystal the external electric field strength  [31][32][33][34] used particular forms of the expressions for N hp and N hn that are only suitable for limiting values of the compensation ratio (K 1  or K 1 1 - ) of the majority impurity by the minority impurity. In this paper we will show how to lift this limitation for the example of p-Dia:B within the framework of the linear drift-diffusion model of hopping migration of holes via boron atoms.

Statistics of holes in valence and acceptor bands
In the thermodynamic equilibrium state for the concentration of v-band holes p averaged over the volume V of a bulk crystalline diamond sample according to [30,[35][36][37] we obtain where g p is the energy density of states of v-band holes (taking into account their spin that is equal to the electron is the total energy of hole, the sum of its kinetic energy E kin and the potential energy U p of its interaction with impurity atoms and other holes, E F is the Fermi level (chemical potential for holes), k T B is the thermal energy. Values of E p , E F , E a , E v d , and U p are given relative to the top of the v-band (E v =0) of the undoped crystal (figure 1).
According to [38,39] the probability density function of potential energy U p fluctuations of a hole in the v-band of a bulk crystal is Gaussian where W p is the root-mean-square fluctuation of the potential energy of the hole, see equation (18).
In the quasi-classical approximation [38,39], taking into account the probability density function p  of potential energy U p fluctuations given by equation (9), for the energy density of states of holes in the v-band we find where m p is the density-of-states effective mass of a hole in the v-band of undoped diamond, E U E p p k i n -= is the kinetic energy of the hole, and Note that in the 'ideal' diamond crystal, i.e. without fluctuations of the potential energy of v-band holes, when U is the Dirac delta function, equation (10) takes the standard form [18][19][20]: is the kinetic energy of motion of a free v-band hole with the module of quasiwave vector k k = | | . Taking into account three subbands of the diamond valence 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 [40]: The average concentration of the acceptors in the charge state (−1), taking into account deviations of their energy levels E a from the average value E 0 a > , can be written as [30]: where a  is the probability density function for the distribution of energy levels in the diamond band gap, is the probability that an arbitrary acceptor with energy level E a is ionized, f N N; Excluding the excited states of all the acceptors, for f 1 -we find [19,21]: )is the degeneracy factor of energy level E a considering all three hole subbands in the diamond v-band, 6 so e = meV is the value of hole so-subband split-off from the degenerate heavy and light hole subbands. If the thermal energy k T B (e.g., at T T 3 2; j 1 » figure 1) is substantially larger than so e then 6 a b » . The reference point for the acceptor energy level E 0 a > and the Fermi level E 0 F < in equation (14) is chosen at E v =0. For calculation of f f 1 1 0 = --by equation (13) in the exponent of the function f 1 -given by equation (14) the quantity E a should be replaced by E E a a -( )and E F should be replaced ) . Taking into account the excited states of electrically neutral acceptors, the quantity a b in equation (14) should be replaced by [13,41]: is the largest number of possible excited states of the average acceptor in the charge state 0 is the average distance between the nearest impurity atoms considered as point particles randomly (Poissonianly) distributed over a crystalline matrix [30,42], a e I 8 meV for boron atoms in diamond [43]. In numerical calculations of the ionization equilibrium in p-Dia:B crystals at T j in equations (13) and (14) the quantity am b determined by equation (15) will be used instead of a b .
According to [9,44], we assume a Gaussian probability density function a  for the fluctuations of acceptor energy levels E a relative to the average value E a over the crystal where W a is the effective width of the acceptor band ( figure 1).
Taking into account only Coulomb interaction between the two nearest point charges in the crystal, we find the following expression for the effective width of the acceptor band W a , which is equal to the root-mean-square fluctuation of the electrostatic energy of the ionized acceptor [35,45]: 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]: where p is the concentration of v-band holes; W a is determined by equation (17). From equations (17) and (18) it is seen that the quantities W a and W p depend on the concentration of v-band holes p, the concentration of ionized acceptors N 1 -, and the compensation ratio K. From formulas (3) and (7) for v-band holes and holes in acceptor band we obtain  Equations (20) and (21) are supported by experiments [47] measuring the ratio of the diffusion coefficient to the drift mobility of holes ( 1 p  x and 1 h  x ) in amorphous hydrogenated silicon at temperature T j , where the turbulent (jumping or relay) regime of hole transport probably occurs [13,48].
For the region of high temperature (T T i » ), at which practically all acceptors are ionized, the concentration In particular, at T T i » the fluctuations of the potential energy of holes are (20) taking into account equations (8)- (11) we find (see also [49] Two particular cases follow from equation (22): equation (4) for an ideal nondegenerate gas of v-band holes of p-type diamond, when y 0 F < and y 1 F  | | , and equation (5) for an ideal degenerate gas of v-band holes, when y 1 F  . (Let us note that in [50] it is unreasonably stated that the relation D k T e p p B m = is also valid for an ideal degenerate gas of holes.) For the region of low temperature (T T j < ), when p is found from the condition K 2 1 erf 2 a g = -( ), which is followed from equation (13).
Note that D M h h value according to equations (21) and (23) increases with the amplitude of electrostatic potential energy fluctuations W a . This means that the hopping mobility M h of holes decreases more rapidly comparing to their diffusion coefficient D h . This is explained by the fact that the actual trajectory of a hole diffusing via acceptors, on average, passes through the lower barriers as compared to the barriers that are produced by an external electric field and are responsible for the drift mobility M h of this hole. The parameter h x enters the expression for the coefficient of the differential thermo-emf for hopping migration of holes via acceptors [51]. To calculate p x and h x using equations (20) and (21) we need to establish the dependence of the energy of the acceptor band center E a on the acceptor concentration N, the compensation ratio K, and the temperature T. According to [13,30] and figure 1 the center of the acceptor band E 0 a > (relative to the top of the v-band in the ideal crystal E v =0) is given by the following expression where I a is the energy level of a solitary acceptor in the ideal crystal, E 0 v d < is the shift of the top of the v-band into the band gap due to the doping of crystal by impurity atoms, E 0 cor < is the energy of the correlative reduction of the affinity of an acceptor in the charge state 1 -( ) to a v-band hole due to the screening of the Coulomb field of the ionized acceptor by a cloud of charges surrounding it (with total charge e + ), E 0 per < is the percolation threshold for v-band holes, i.e. the minimum energy needed for v-band holes to migrate within the entire crystalline sample [13,52], E 0 exc < is the shift of the top of the v-band into the band gap due to the exchange interaction of holes in the v-band.
Let us consider three cases for energy quantities that compose E v d in equation (24) ] 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] where s L is the screening radius of the Coulomb field of the impurity ion (and generally of a point charge) in an electrically neutral crystal.
Calculation [55][56][57] The quantity E cor can be treated as the local energy of a hole at the top of the v-band (with zero kinetic energy) with a cloud of charges screening it, i.e. as the energy of a plasma hole polaron [45,52]. Note that in the Debye-Hückel approximation [53,54] the charge density of charges screening an ion is proportional to the total electrostatic potential of the ion (acceptor in the charge state 1 -( ), i.e. having charge e 0 -< ) with the screening cloud of mobile charges (v-band holes and holes in the acceptor band); see linearized Poisson equation (25). According to [46,57]   . Taking into account equations (3) and (7), the radius of static screening of the Coulomb field of the impurity ion by v-band holes and holes of the acceptor band migrating over the crystal is determined by the following expression (see also [26,28,29,46]):  (1) and (6) is the effective concentration of holes hopping via acceptors in the vicinity of the screening ion; p x and h x are the dimensionless parameters defined by equations (20) and (21).
Let us point out that the screening length s L is defined for the state of the thermodynamic equilibrium, i.e. at zero current densities J J 0 p h = = . According to equation (28) at low temperatures (T T j < , when ) the quantity s L is determined solely by holes hopping via acceptors N ) . The problem of the physical meaning of N h was formulated already in [58,59] and solved in [16,17], where it is shown that N h is the concentration of holes hopping via hydrogen-like acceptors, while (b) Calculation of the percolation threshold E per for v-band holes. Following [1,36,39] we assume that the critical part of the volume of a bulk semiconductor sample that is unavailable for the diffusive motion of v-band holes is equal to 0.17. Thus, taking into account equation (9) the threshold energy E per , above which the percolation of v-band holes sets on, is determined from the relation where W p is given by equation (18).
(c) Calculation of the exchange energy E exc for v-band holes. As the concentration of v-band holes increases, their exchange interaction becomes significant [52,57,60]. As a result, the energy of the top of the v-band (E v = 0 for an undoped crystal) shifts deeper into the band gap, decreasing the energy level E per for hole percolation, equation (29), by the value of the exchange energy E exc for a single v-band hole. That is equivalent to the top of the v-band approaching the center of the acceptor band E a and is taken into account in equation (24).
The energy of the exchange interaction of v-band holes E 0 exc < results from the symmetry of the wave function of holes relative to their transposition. In the effective mass approximation the Bloch wave functions of c-band electrons (and v-band holes) are replaced with the plane waves. In the absence of fluctuations of the potential energy of holes (i.e. when the density of states of v-band holes g pi in the ideal diamond crystal is determined by equation (11)) the energy of a hole with the absolute value of quasi-wave vector k k = | | and the effective mass m p is equal to k m E 2 . For these conditions the single-particle exchange energy E 0 exc < is given by [60]:  1), the quantity E exc based on equation (31) can be tentatively averaged over the fluctuations of the potential energy U p of holes [52]: is the exchange energy of a v-band hole with kinetic energy E E U ; kin F p » the probability density function p  of potential energy U p fluctuations relative to E v =0 is given by equation (9).
It should be pointed out that equation (27) for E cor was derived at E 0 per = and E 0; exc = equation (29) for E per -at E 0 cor = and E 0 exc = , and equation (30) for E exc -at E 0 cor = and E 0 per = . We based the estimate of the ratio of concentration of v-band holes p loc that have the energy E E v p d < and may migrate only in restricted regions of a bulk crystal sample to the total concentration of holes in the v-band on equation (8) and found

Algorithm of numerical calculations
Let us consider the algorithm for solving the system of equations (1), (8), (13) and ( The algorithm for solving of the system of equations (1), (8)-(30) numerically consists of four steps (see also [13,30]): (i) Initial parameters of a diamond doped with boron atoms are set: r e , a b , I a , m p , the temperature T, the compensation ratio K, the range of variation of the doping impurity concentration (boron) N, and the relative error of calculations 10 goal 5 e = -. (ii) Using the initial estimate, from equation (33), the concentration of v-band holes p in an ideal p-Dia:B is calculated. Then, according to equations (17) and (18), where N p KN 2 ch = + ( ) , seed values of the root-meansquare fluctuations W a and W p are evaluated.
(iii) The nonlinear electrical neutrality equation (1) is solved for the unknown Fermi level E F using the values of W a and W p obtained in the previous step. The quantities p and N 1 -in equation (1) are determined using equations (8) and (13) taking into account equations (9), (10), (14)- (30). The computed value of E F allows to refine the values of W a , W p , E a , N 1 -, and p. (iv) The solution of the electrical neutrality equation (1) and refinement of the values of W a , W p , E a , N 1 -, and p are iteratively performed until their relative error becomes less than a cutoff value goal e . The relative error X e( ) for the value of a calculated quantity X where X k is the value of a quantity X at the kth step of the iterative procedure.
Let us test this algorithm on the experimental results of [63], where a homoepitaxial crystalline film of p-Dia:B treated in hydrogen plasma in order to obtain a negative electron affinity was studied using photoelectric spectroscopy at T = 300K. According to [63]  In [4,13,30] the characteristic temperature T j of the transition from the HC regime to the BC regime of hole migration is determined on the basis of the virial theorem. According to these references, the value of temperature T j , where p h s s = (figure 1), is found from the equation where N T j 1 -( )and p T j ( ) are determined according to the electroneutrality condition (1). Note that in [30] for calculation of T j by equation (34) the concept of the nearest neighbors (ionized acceptors and mobile v-band holes) was introduced at their Poisson distribution over a crystal. For instance, the v-band hole and the acceptor in the charge state (−1) are the nearest neighbors when this acceptor is the nearest to this hole and this hole is the nearest to this ionized acceptor.
In the limit of a low concentration of v-band holes, p T N T KN The algorithm for calculating W a , W p , E a , N 1 -, and p at temperature T j using equation (34) is similar to that described in section 3 and consists of four steps: (i) The seed value T j using equation (35) is initially set.
(ii) Using the described iterative procedure the values W a , W p , E a , N 1 -, and p at the seed temperature T j are calculated until their relative error becomes less than goal e . (iii) Using equation (34) the value of T j is refined. (iv) Steps (ii) and (iii) are performed until the relative error of T j becomes less than goal e . Figure 2 shows the temperature T j calculated using equation (34), the Fermi level E 0 F < obtained using the electrical neutrality condition (1) (18), other things being equal, the root-mean-square fluctuation W a of the acceptor energy levels E a relative to the average value E a is larger than the root-mean-square fluctuation of the v-band hole potential energy W p .  x = (for acceptor band holes) calculated using equations (20) and (21) at the temperature T T j = obtained from equation (34) on the concentration N of the doping impurity (boron atoms) in a p-type diamond for the compensation ratios: K = 0.03 (curves 1) and 0.3 (curves 2).