DC hopping photoconductivity via three-charge-state point defects in partially disordered semiconductors

The stationary (DC) hopping photoconductivity caused by the migration of electrons via intrinsic point t-defects of the same type with three charge states (−1, 0, and +1 in units of elementary charge) is theoretically studied. It is assumed that t-defects are randomly (Poissonian) distributed over a crystal and hops of single electrons occur only via t-defects in the charge states (−1), (0) and (0), (+1). Under the influence of intercenter illumination nonequilibrium charge states (−1) and (+1) of defects are generated due to photostimulated electron transitions between pairs of defects in the charge states (0). During the recombination of nonequilibrium charge states (−1) and (+1) of defects, pairs of defects in the charge states (0) are formed. It is assumed that illumination does not heat the crystal, i.e. does not increase the coefficient of thermal ionization of t-defects. The dependence of the ratio of photoconductivity to dark hopping electrical conductivity on the ratio of photoionization coefficient (γ) of neutral t-defects to coefficient of ‘capture’ (α) of an electron from a negatively charged to a positively charged t-defect is calculated. The calculations of hopping photoconductivity were carried out for the partially disordered silicon crystal with total concentration of t-defects of 3·1019 cm−3, compensated by shallow hydrogen-like donors. The ratios of donor concentration to t-defect concentration (compensation ratios) are 0.25, 0.5, and 0.75. It is taken into account that an electron localization radius on t-defect in the charge state (−1) is greater that on t-defect in the charge state (0). The calculated value of the dark hopping electrical conductivity is consistent with the known experimental data. A negative DC photoconduction at γ > α is predicted, due to a decrease in the concentration of electrons hopping via states (−1), (0) and (0), (+1).


Introduction
In the experimental study of germanium and silicon crystals moderately doped with hydrogen-like impurities in the temperature range of liquid helium, Dobrego and Ryvkin [1,2] introduced the concept of hopping photoconductivity as a change in the magnitude of DC (stationary) hopping electrical conductivity under the influence of optical radiation.When the intrinsic absorption band of crystalline semiconductors is excited by light, 'free' electrons in the c-band and holes in the v-band are created as a result of interband transitions.Nonequilibrium electrons and holes are captured by donors and acceptors that are present in the semiconductor, which provide hopping migration of electrons and holes: the value of hopping electrical conductivity changes-hopping photoconductivity is observed.Somewhat later, the study of hopping photoconductivity under interband excitation of germanium crystals was carried out by Davis [3].
Note that even in early works on hopping conductivity [4] in lightly doped semiconductors the connection between conductivity and the degree of compensation of a majority (doping) impurity by a minority (compensating) impurity was mentioned.This dependence was considered as empirical.Later, this relationship was supported theoretically both for the ordinary (dark) hopping electrical conductivity [5] and for hopping photoconductivity [6].
A method for excitation of hopping photoconductivity, when there are several impurity energy levels in the band gap of the semiconductor and hopping migration of electrons (or holes) can be conducted via one of them (shallow, hydrogen-like) was proposed and experimentally implemented in [7].With compensation corresponding to the absence of electrons (or holes) at the shallow energy level hopping conduction will not be observed.However, when the semiconductor is illuminated with a photon energy equal to the energy difference between the impurity levels, it is possible that carriers are transferred from the deeper level to the shallow one and hopping conduction occurs.In this case, hopping photoconductivity arises due to the excitation of interimpurity transitions of electrons by light.
Deep impurities in the crystal matrix tend to form precipitates as their concentration increases, and only irradiation-induced defects can be introduced in significant concentrations sufficient to realize hopping electrical conductivity at room temperature.To introduce three-charge-state ^-defects into covalent semiconductors, they are irradiated with ionizing radiation (gamma rays, fast electrons, neutrons, etc.).In contrast to 'metallurgical' doping with impurities, the concentration of intrinsic radiation defects can be largely controlled by annealing [8].In some crystals, e.g., in Si, GaAs, SiC, GaN, diamond, the Fermi level is pinned in the lower half of the band gap as the radiation fluence increases, i.e. with the accumulation of intrinsic irradiation-induced defects (atomic vacancies, interstitial atoms, and their associates) [9].
Previously, the photoconductivity of amorphous hydrogenated silicon (a-Si:H) films containing t-defects was considered due to nonequilibrium c-band electrons and v-band holes [10,11], but the possibility of hopping migration of electrons via t-defects in these works was not taken into account.
A model of hopping photoconductivity in amorphous semiconductors when they are excited by longwavelength infrared radiation was proposed in [12].According to this model, stimulation of hopping photoconductivity by long-wavelength radiation implies an initial transition of the electron to a neighboring energetically higher localized state in the vicinity of the Fermi level.Subsequently, in most cases, a reverse transition occurs without affecting the electrical conductivity, however, some spatially separated localized states are associated with others, allowing the electron and hole to separate over a long distance.This is the only case where hopping photoconductivity is observed.
Hopping stationary photoconductivity via three-charge-state impurity atoms in the three-dimensional crystal was considered in [6], but the value of electrical conductivity was not calculated.In [13], for the first time, a Peltier element with hopping migration of electrons via intrinsic point defects randomly (Poissonian) distributed over the crystal matrix was proposed.However, in [13] hopping photoconductivity via three-chargestate point defects (defects of t-type) was not considered and was not calculated.
The dependence of hopping electrical conductivity on temperature in disordered semiconductors with point structural defects was studied earlier [14][15][16][17][18].For example, in [14], hopping photoconductivity of a-Si:H films was measured before and after their irradiation with protons, and the data were interpreted according to the Mott model of hopping electrical conductivity [19].In [15], the electrical conductivity of a-Si doped with donor phosphorus atoms with the concentration of 5.5-10 18 cm" 3 was measured.The temperature dependence of hopping electrical conductivity in amorphous carbon films was also measured [20,21 ].To explain the dependences obtained, it was assumed in [20] that small polarons migrate in the variable range hopping (VRH) regime.(VRH regime is the regime of electron hopping via defects, when the hops with the lowest activation energy are realized primarily and the hops with the shortest hopping length are realized secondarily.)The interpretation of the experimental data in [21] on photo-and electrical conductivity was based on the parameterization of the Gaussian electronic density of states and the postulation of the temperature and energy dependence of the hopping mobility with fitting parameters without its theoretical justification.Moreover, in [14][15][16][17][18][19][20][21], hopping electrical conductivity via three-charge-state point defects could not be described without using fitting parameters (see also [22,23]).
The purpose of this work is to calculate the change in the DC hopping photoconductivity of the semiconductor with point ^-defects in the charge states (-1,0 and+1) under the change in the illumination intensity, which stimulates electron transitions between t-defects in the charge states (0).

Hopping photoconductivity model
Let us consider a three-dimensional partially disordered crystalline semiconductor sample positioned between metal ohmic electrodes (M).The sample dimensions are much larger than the average distance between defects.The crystal contains point two-level defects in three charge states (^-defects) with their concentration sufficient to pin the Fermi level £ F in the energy band gap and to realize hopping conductivity via t-defects.Defects of ttype in the charge states (-1) and (0) form a 11) -band with an average energy level E Y and the ones in the charge states (0) and (+1) form a 12) -band in the band gap (with an average energy level £ 2 )> located closer to v-band than 11) -band (figure 1).We assume that the thermal ionization energies of hydrogen-like donors |d) and acceptors | a) are much less than E Y and E g -E 2 , respectively.Note that typical shallow hydrogen-like impurity atoms contribute energy levels of about 40 meV deep in the energy gap of crystalline silicon.They can be introduced up to the Mott concentration (see, e.g., [24]).
Let us consider semiconductor under conditions of only hopping electron migration via immobile irradiation-induced defects (of t-type) in the charge states (-1) and (0), as well as in the charge states (0) and (+1).The total concentration of ^-defects is N t = N u -i+ N ty0 + N ty+ 1 = N_ i + N o + N + 1 in the charge states (-l,0and+l).Weassume that the shallow hydrogen-like donors |d) and acceptors |a) are completely ionized and their concentrations in the charge states (+1) and (-1) are N d < N t andN a < N*, respectively.
The electrical neutrality condition for the partially disordered semiconductor with t-type defects, taking into account totally ionized hydrogen-like donors and acceptors with concentrations N d = KJSf t and N a = K a N^ has the form: where 0 < K d < 1 and 0 < K a < 1 are the compensation ratios of t-type irradiation-induced defects by hydrogen-like donors |d) and acceptors |a), respectively.Note that for K d = 0.5 and K a <C K d the Fermi level £jj c) practically coincides with the top energy level E Y (figure 1).If the Fermi level in metallic contacts coincides with the energy level E ly then these contacts are ohmic, and in this case hopping conduction via ^-defects in charge states (-1) and (0) dominates.
The average concentrations of ionized and neutral ^-defects can be written as [25]: where/z is the probability that the defect is in the one of three possible charge states Z = -1,0, +1 (in units of the elementary charge e).
If we neglect the excited states of ^-defects, then the inverse distribution functions \/f z of defects in 11) -and 12) -bands over charge states are [26,27]: where E$ c) is the Fermi level (counted from the c-band electron mobility edge [28,29]); E^c ) < 0 for the Fermi level in the band gap; E x > 0 is the average energy of the electron detachment from the negatively charged fdefect and its transition to the c-band mobility edge E$ = 0; E 2 > 0 is the average energy of electron detachment from f-defect in the charge state (0) and its transition to the c-band mobility edge E$ = 0; Ai = Si -Ei and A 2 = S 2 -E 2 are the differences between the f-defect energy level and the average f-defect energy level in | l)-band and |2)-band, respectively, k B is the Boltzmann constant; Tis the absolute temperature; g\ =go/g-i ^ 1>#2 = go/g+i ^ 1> where g z is the number of quantum states off-defect in the charge state Z We assume that the energy levels £i = Ai + £iand£ 2 = A 2 + £2 off-defects in the band gap of semiconductor have a normal (Gaussian) distribution [25]: where Wi and W 2 are the effective widths of 11) -band and 12) -bands, respectively (figure 1).
With the total concentration of charged defects and impurity ions N c h = N_ i + N+1 + K d N trandomly (Poissonian) distributed over the crystal, we have equal rms fluctuations of the electrostatic energy, i.e. the widths of 11) -and 12) -bands are [30]:

\a-i 2k B T )\ I \a 0 2k B T )\
where the Coulomb interaction of each charged defect is considered only with its nearest charged defect (ion); e is the elementary charge; £r£ 0 is the static dielectric permittivity; e r is the relative dielectric permittivity of partially disordered semiconductor, £ 0 is the electric constant.
In equation ( 5), taking into account equation ( 1), the concentration of ionized f-defects and shallow hydrogen-like impurity ions is N ch = 2(N+1 + K d N t ) = 2(N_ 1 + KjSff).Further we consider the compensation off-defects only by shallow hydrogen-like donors (0.25 < K d < 0.75; K a = 0).Thus, in the darkN+i < KJSfâ nd the concentration of impurity ions is The hopping frequencies r_i )0 = r_i )0 (r, £_i, 0 ) and r o ,+i = r o ,+i(r, £o,+i) of electrons via f-defects in the charge states (-1), (0) and (0), (+1) are given by the relations (see, e.g., [23,31,32]): where r is the electron hopping length via f-defects in the charge states (-1), (0) and (0), (+1); £_ 1, 0 is the difference between the ionization energy off-defect in the charge state (-1) and the electron affinity energy of another f-defect in the charge (0), between which the electron hops in 11) -band; £o,+i is the difference between the ionization energy off-defect in the charge state (0) and the electron affinity energy of another f-defect in the charge (+1), between which the electron hops in |2)-band; v\ At is the characteristic frequency of crystal matrix phonons; a_i oc (miEi)~1^2 and a 0 oc (m 2 E 2 )~1^2 are the radii of localization of an electron on a f-defect in the charge state(-l)in |l)-bandandin the charge state(0)in |2)-band;mi « m 2 are the effective masses of an electron on a f-defect with average energy levels Ei and E 2 .In numerical calculations, we assumed a_ 1 = d im and UQ = a-i(Ei/E 2 ) 1 / 2 , where d xm = 2[47r(l + K d + K a )N f /3]~1/ 3 is the average diameter of a spherical region per one point defect (f-defect, donor or acceptor) in semiconductor; a_ 1 > OQ since E 2 > E\.
Note that the value a_ 1 = d im is a reasonable estimate (see, e.g., [33]), suitable for describing electron hopping via the excited state of a pair off-defects in the charge states (-1) and (0).Our model is most applicable for the nearest neighbor hopping regime rather than for the variable range hopping regime, since the quantities S-1 )0 and e o ,+i do not depend on the electron hopping length between f-defects.
(7) 6 6 From equation (7), taking into account equation (6), we obtain interval (r, r + dr); P Ot+l (r)dr = 47rr 2 (N 0 + N+i)exp [-47rr 3 (N 0 + N+i)/3]dr is the probability density function of mutually nearest t-defects in the charge states (0) and (+1) [30]; the Gaussian distribution of difference £_i )0 between the ionization energy of t-defect in the charge state (-1) and the electron affinity energy of another t-defect in the charge (0), between which the electron hops in 11) -band, is (cf.equation ( 4)): the Gaussian distribution of difference £ 0 ,+i between the ionization energy of t-defect in the charge state (0) and the electron affinity energy of another t-defect in the charge (+1), between which the electron hops in 12) -band, is The normalization conditions for the Poissonian probability density functions P_ 1)0 (r) and P 0 ,+i( r ) are satisfied: Note that in the case of dependence of the hopping frequency F_ i )0 (or r o ,+i) on the variable hopping length r, according to [37] and equations ( 7) and ( 8), it is necessary to average their product (f 2 r_ 1>0 ) (or {T^YQ^I)) in contrast to averaging separately (f 2 ) and (r_ 1>0 ) (or separately (f 2 ) and (r 0>+1 )), as proposed in [38].

•/•/-oo
Dimensionless parameters ^_1)0 ^ 1 and £ 0 ,+i ^ 1 characterize the difference in the extent to which the diffusion coefficient and the mobility of electrons hopping in 11) -and 12) -bands are affected by fluctuations of the electrostatic potential energy in the crystal.Note that when obtaining equations (10) it was assumed that W oc N^3 given by equation ( 5) and, hence, the (Gaussian) distributions G Y and G 2 given by equations ( 4), or more exactly, the electronic densities of states in 11) -and 12) -bands, do not depend on the position of the Fermi level E^c ) (see, e.g., [39,40]).In our case, this is accomplished at relatively high temperatures and moderate degrees of compensation of ^-defects by shallow hydrogen-like donors (e.g., for 0.25 < K d < 0.75).

Calculation results and discussion
As an example, let us consider a partially disordered silicon crystal.For irradiation-induced ^-defects in silicon we assume [42]: E x = 400 meV, E 2 = E x + A, = 700 meV, where A, = 300 meV is the difference between centers of 11) -and 12) -bands.The ratio of hopping photoconductivity to dark conductivity cr h (7)/cr h according to equations ( 11)-( 14) was calculated numerically as a function of the decimal logarithm of the ratio 7/a of coefficient 7 of the illumination stimulated transition of an electron between two ^-defects in the charge states (0) to coefficient a of electron transition from t-defect in the charge state (-1) to t-defect in the charge state (+1).
The value of the dark hopping electrical conductivity cr h (in the absence of illumination, i.e. for 7=0) was calculated according to equation (11) for the following parameter values: the characteristic frequency of crystal matrix phonons t\ at « 10 THz [43]; the total concentration of ^-defects N t = 3-10 19 cm" 3 was chosen sufficiently high to realize hopping conductivity via ^-defects and to pin the Fermi level in the energy band gap [9,44^46]; the compensation ratios off-defects by donors K d = 0.25,0.5,0.75 and acceptors K a = 0; the absolute temperature T = 300 K.The radius of electron localization on t-defect in the charge state (-1) is a_i = d imy where d im = 3.7 3.48,3.3lnm for K d = 0.25,0.5,0.75, respectively, The Fermi level E^ in the absence of illumination (7 = 0) is found from the electrical neutrality condition (1), where concentrations of ionized t-defects N +i andN_i are calculated from equation ( 2), taking into account equations ( 3) and ( 4).The Fermi level £^c ) coincides with the energy level £1 (half rilled with electrons in the absence of illumination) for K d = 0.5 andK a = 0 (see figure 1).
Figure 2 shows the results of calculations according to equations (10) ratios of diffusion coefficients to drift mobilities^-i,o = eD_i )O /M_i )O fcBT(bluelme),£o,+i = eD o , + i/M o , + i/:BT(red line), and ^_i, 0 = £ 0 ,+i = 1 for W < fc B T(dashed line) as functions of the concentration of t-defects N t for K d = 0.5 and K a = 0 at T = 300 K.It can be seen that the parameters £_ 1)0 and £ 0 ,+i characterizing the ratio of the hopping diffusion coefficient to the drift hopping mobility increase with the concentration of t-defects N t .Calculations in figure 2 find support in experiments [47] performed on a-Si:H films under conditions of photoconductivity by c-band electrons and vband holes, in the sense that the parameters £_i,o ^ 1 and £ 0 ,+i ^ U for details of our calculations, see [48].
Figure 3 shows the results of calculations according to equations (12), taking into account equations ( 2) and (3), of the ratios of concentrations of ^-defects in the different charge states: N_i(7) (blue line), N 0 (7) (black line), and N +1 (7) (red line), to total concentration N t of ^-defects depending on the ratio of photoionization coefficient 7 of neutral ^-defects, proportional to the illumination intensity, to coefficient of'capture' a of electrons from 11)-to |2)-band.It can be seen that with an increase in 7/a, the concentrations N_i( 7) andN+i( 7) of ^-defects in the charge states (-1) and (+1) increase, and their concentration N 0 (7) in the charge states (0) decreases, due to -2 -1 0 1 logio(y/a) Figure 3. Dependences of the concentrations off-defects in the different charge states (in units of N t ): N_i(j) (blue line), Nod) =N t -N_i(i) -N +l (i) (black line), and N+1(7) (red line) calculated by equations (12), taking into account equations ( 2) and (3), on the decimal logarithm of the ratio 7/a of illumination intensity 7 to recombination rate a of (-1) and (+1) charge states, for s r = 11.47;Nt = 3-10 19 cnr 3 ;lC d = 0.5andiC a = 0atT= 300 K. electron transitions from neutral ^-defects of 12) -band to neutral ^-defects of 11) -band.These results can be useful for measuring the concentrations of ^-defects by the electron spin resonance method.
Note that such a nonmonotonic character of the change in photoconductivity was described in [41] for hopping photoconductivity via shallow hydrogen-like acceptors and donors.
The above equations are also applicable for the compensation ratios K d = 0 and K a = 0.5, i.e. when the Fermi level E^c ) coincides with the energy level E 2 .In this case, the photoconductivity cr h (7) will be mainly determined by the change in the concentration of holes N 0)+ 1 (7) hopping via ^-defects in the charge states (+1) and (0) in 12)-band.
In this paper the photoconductivity is considered for the case of ohmic contact electrodes to a semiconductor sample, i.e. when K& = 0.5 and K a = 0 (or K& = 0 and K a = 0.5).In this case, the Fermi level Eî n the metallic electrode coincides with the t-defect energy level Ei (or E 2 ) in semiconductor.IfK& = K a , an energy barrier A f /2 appears on the cathode for transitions of electrons from cathode to semiconductor, and the value of hopping conductivity will be limited by this barrier.
Note that in the calculation of hopping photoconductivity it was assumed that the sample is under isothermal conditions (at T = 300 K).However, it should be noted that under photogeneration of electrons from 12) -band to 11) -band [appearance of ^-defects in the charge states (-1)] and electron vacancies in 12) -band [appearance off-defects in the charge states (+1)], for K d = 0.5 and K a = 0, the charge states (+1) recombine at the cathode and heat is released (figure 1).For small photocurrents the thermoelectricity effects in such systems can be neglected (see also [13]).

Conclusion
A new model is proposed for the DC hopping photoconductivity in the partially disordered semiconductor with ^-defects [three-charge-state (-1,0, +1) point defects with the concentration N t ].Calculations for the case of compensation of ^-defects by shallow hydrogen-like donors and acceptors with concentrations of N^ and N a and compensation ratios K d = N^/N t = 0.5, K a = N a /N t = 0 show that the ratio of the hopping diffusion coefficient to drift mobility (parameters £_i,o and £o,+i) increases with the concentration off-defects.The photoconductivity was calculated for highly defective silicon crystal with the compensation ratios K d = 0.25,0.5,0.75 and K a = 0.It is shown that positive hopping photoconductivity increases with the intensity of intercenter illumination, when the photoionization coefficient 7 of neutral ^-defects is less than the coefficient a of electron 'capture' from a negatively to a positively charged t-defect.The photoconductivity increases due to electron transitions between neutral t-defects (from the bottom energy level E 2 = 700 meV to the top energy level £1 = 400 meV), which generate charged states (-1) and (+1).In this case, the concentration of neutral states is comparable with the concentration of (-1) states and illumination increases the concentration of (+1) states.Negative photoconductivity is predicted, when 7 becomes greater than a, due to a decrease in the concentration of neutral ^-defects (up to total their extinction, when there are no places to hop).The value of the dark hopping electrical conductivity «0.32 Ohm~1-cm~1 calculated for N,« M0 19 cm~3,^ = 0.5, Ka = 0, and the temperature T = 300 K is close in magnitude with the known experimental data.When calculating the diffusion coefficient of hopping electrons, we averaged over the energy and hopping length the product of the variable hopping length and the hopping frequency.The proposed model of positive and negative DC hopping photoconductivity via ^-defects can be applied for the development of light-absorbing device structures.

Figure 1 .
Figure 1.Single-electron energy E n as function of x coordinate in semiconductor with two ohmic electrodes (M) in equilibrium: E^ = 0 and E^ are the drift mobility edges for c-band electrons and v-band holes, E^c ) < 0 is the Fermi level, counted from the electron mobility edge (E^ = 0), E£> -E^ = E g is the width of the energy band gap of semiconductor, A t = E 2 -E l is the width of energy gap between 11) -and 12) -bands of point three-charge-state f-defects, W x and W 2 are the widths of 11) -and 12) -bands.Arrows indicate hops of single electrons e~ via |l)-band as well as generation [gen: 2(0) -> (-1) + (+1)] and recombination [rec: (-1) + (+1) -> 2(0)] electron transitions between |1)-and |2)-bands; |d) and |a) are the states of shallow hydrogen-like donors and acceptors in the charge states (+1) and (-1), respectively, K d « 0.5 and K a <C K d are the compensation ratios off-defects by donors and acceptors (the position of the Fermi level E^c ) < 0 practically coincides with the energy level E{).[Number of hops of single electrons from the charge states (0) to the charge states (+1) of f-defects in 12) -band is small and they are not shown.]