The Theory of Giant Splash of Photoresponse in Semiconductors at Low-Level Illumination with Increasing Concentration of Deep Recombination Impurity

Recombination of excess (nonequilibrium) electrons and holes in semiconductors through impurity recombination centers (traps) known as trap-assisted (Shockley-Read-Hall) recombination is in many cases the dominant process. In this chapter, we develop the general theory of trap-assisted recombination and study in detail two key characteristics: (1) dependences of excess charge carriers’ lifetime and photoelectric gain on concentration N of recombination centers and (2) effectiveness of band-to-band photoexcitation of charge carriers and photo-emf in semiconductors at low-level illumination considered outside quasi-neutrality approximation. We have done systematic mathematical and detailed physical analysis of considered characteristics. Giant splash of photoresponse in semiconductors with increasing re‐ combination center concentration N is caused mainly by the growth of charge carri‐ ers’ lifetime in orders of magnitude. Also, this factor is sufficient to provide an increase, in order of magnitude and more, in efficiency of charge carriers’ photoexcita‐ tion with increasing N. Results of strict analytical calculations (i.e., outside commonly used local approximation of quasi-neutrality) show, that, photoinduced local space charge affects substantially on giant splash of semiconductor photoelectric response with increasing concentration of recombination centers. The theory of giant splash of photoresponse in semiconductors at low-level illumination with increasing concentration of recombination centers could develop further through generalization of boundary conditions on semiconductor surfaces and current contact electrodes, accounting for nonuniformity of charge carriers’ photoexcitation along the line of current flow and fluctuation processes. The study of nonstationary (frequency domain and transient) characteristics is of particular interest. It follows from physical essence of considered effects that similar effects can occur in other mediums with recombination of dissociative or ion-ion type, for example, in multicomponent plasma as well. Also, ignoring approximation of quasi-neutrality, we study, at low-level illumination, the effectiveness of band-to-band photoexcitation of charge carriers and photo-emf V ph in semiconductors with dominant trap-assisted recombination. Analytical expressions for photo-emf V ph and mean, with respect to light propagation length, concentrations of photoelectrons <Δ n > and photoholes <Δ p > are given. It is shown that target values of V ph , <Δ n > , and <Δ p > can be improved radically by increasing concentration of recombina‐ tion centers; moreover, approximation of quasi-neutrality can lead to errors of several orders of magnitude.

We have done systematic mathematical and detailed physical analysis of considered characteristics. Giant splash of photoresponse in semiconductors with increasing recombination center concentration N is caused mainly by the growth of charge carriers' lifetime in orders of magnitude. Also, this factor is sufficient to provide an increase, in order of magnitude and more, in efficiency of charge carriers' photoexcitation with increasing N. Results of strict analytical calculations (i.e., outside commonly used local approximation of quasi-neutrality) show, that, photoinduced local space charge affects substantially on giant splash of semiconductor photoelectric response with increasing concentration of recombination centers.
The theory of giant splash of photoresponse in semiconductors at low-level illumination with increasing concentration of recombination centers could develop further through generalization of boundary conditions on semiconductor surfaces and current contact electrodes, accounting for nonuniformity of charge carriers' photoexcitation along the line of current flow and fluctuation processes. The study of nonstationary (frequency domain and transient) characteristics is of particular interest.

Introduction
Recombination of excess (nonequilibrium) electrons and holes in semiconductors through impurity recombination centers (traps) referred to in this chapter as trap-assisted (Shockley-Read-Hall) recombination is in many cases the dominant process [1][2][3][4][5][6][7][8][9]. Hall [10] and Shockley and Read [11] have proposed the theory of trap-assisted recombination as early as 1952. Further, the theory of trap-assisted recombination has been developed in many aspects and details [1][2][3][4][5][12][13][14]. At the same time, due attention was not paid to study dependences of lifetimes of excess electrons τ n (N ) and holes τ p (N ) on concentration of traps N . In some cases, traps are produced intentionally by doping semiconductor (e.g., by bombarding with highenergy ions [15,16]) to reduce time of transient processes. It seems that lack of attention is caused by traditional understanding that the larger the concentration of traps N , the greater the capture rate of excess charge carriers on impurity level traps and, therefore, the shorter the lifetimes of excess charge carriers.
That reasonable understanding is incompletely adequate to reality. As shown below, lifetimes of excess electrons and holes (see our definition of τ n and τ p in Section 2) may grow strongly (in order of magnitude and more) with increase of concentration N .
In this chapter, we generalize the theory and give systematic mathematical and detailed physical analysis of dependences τ n (N ) and τ p (N ) on concentration of recombination centers. semiconductor parameters and temperature. We give detailed physical interpretation of obtained results. In particular, it is shown that physical mechanisms responsible for strong non-monotonic dependences τ n (N ) and τ p (N ) differ from each other.
It is reasonable to expect that the growth of lifetimes τ n (N ) and τ p (N ) with increasing N will lead to the growth of photoresponse of semiconductor sample (including photoelectric gain G). However, specificity of dependences τ n (N ) and τ p (N ) does not determine the type of dependence G(N ) in total. As it follows from [18,19], G increases with increasing charge carriers' lifetime, if ambipolar mobility [2,13] μ a = 0 or there is no recombination at current contact electrodes (x = 0 and x = W ; see insert in Figure 1a). In reality, recombination at current contact electrodes is always happening to some extent [5][6][7][8][9]. Therefore, in normal conditions (μ a ≠ 0), increase in lifetimes, after reaching some values, does not lead to an increase in photocurrent I ph [5,9,18,19]. Saturation of I ph becomes apparent in the case of high-rate recombination at the contact electrodes (sweep-out effect [5,9,18,19]) when where Δn(x) = n(x) − n e and Δp(x) = p(x) − p e are deviations of electron n and hole p concentrations from their equilibrium values n e and p e , respectively. In trap-assisted recombination, function μ(N ) can vanish at the same (up to small corrections) concentration N , at which dependences τ n (N ) and τ p (N ) reach points of maximal extremum τ n and τ p (Figure 1b and 1c). Therefore G and, hence, I ph grow to the extent of increase in τ n (N ) and τ p (N ). These are physical grounds of giant splash of photoelectric gain with increasing N (Figure 1a).
It was first reported in [20] that vanishing μ(N ) in points of maximal extremum of dependences τ n (N ) and τ p (N ) allows avoiding highly undesirable effect -saturation of G in intrinsic photoconductors, when applied bias voltage V increases [5,9,18,19,21,22]. As is known [19], this disadvantage is the most evident in photoconductors with sweep-out effect on contact electrodes, i.e., when relations (1) are fulfilled. Result presented in Ref. [20] was obtained in approximation of quasi-neutrality [1-9, 13, 18, 19, 21, 22], which was usually used at moderate electric fields, i.e., when we neglect in Poisson equation by term Δρ ≡ (ε × ε 0 ) × divΔE, which determines the density of photoinduced space charge Δρ. In our case, ΔE ≡ E(x) − E 0 is the variation of electric field caused by deviation of concentrations of free charge carriers and their traps from equilibrium values by reason of band-to-band absorption of radiation, E(x) and E 0 are the electric field intensity in the presence and absence of illumination, ε is the relative dielectric permittivity of semiconductor, and ε 0 is the vacuum permittivity. However, even at moderate electric fields (≈ 1÷10 V/cm), approximation of quasineutrality is not always acceptable [23].
Below, in case of single recombination level, we consider in detail the impact of photoinduced space charge Δρ on value Ĝ of photoelectric gain G in semiconductors with sweep-out effect on contact electrodes at the point of maximal extremum of function G(N ) (Figure 1a). Considering semiconductor as base material for making intrinsic photoconductors with threshold electro-optical performance, we assume that photocarriers are excited by photons of lowintensity optical radiation with wavelength range responding to fundamental absorption band of semiconductor. Figure 2a shows that we cannot use approximation of quasi-neutrality, when voltage V across the sample becomes larger than some particular value.
Also, ignoring approximation of quasi-neutrality, we study, at low-level illumination, the effectiveness of band-to-band photoexcitation of charge carriers and photo-emf V ph in semiconductors with dominant trap-assisted recombination. Analytical expressions for photo-emf V ph and mean, with respect to light propagation length, concentrations of photoelectrons < Δn > and photoholes < Δ p > are given. It is shown that target values of V ph , < Δn > , and < Δ p > can be improved radically by increasing concentration of recombination centers; moreover, approximation of quasi-neutrality can lead to errors of several orders of magnitude.
Analyzing above-mentioned problems, we do not use conventional (Shockley-Read) expression-based form [1-5, 9, 11-14] of generation-recombination rate. This form does not allow to express explicitly dependences τ n and τ p and, even more, I ph and V ph on N . And therefore, because of the need for solving complex transcendental equations, conventional (Shockley-Read) expression-based form leads to serious mathematical difficulties in study (especially analytical) dependences τ n, p (N ) and, even more, I ph (N ) and V ph (N ).
These difficulties are dramatized by the fact that under certain conditions, τ n, p , I ph and V ph are dependent very strongly on concentration N in a particular interval of N (Figure 1а, 1b, 7). Perhaps, it was the main reason for longtime absence of complete theoretical analysis of lifetime dependences τ n, p (N ), while detailed analysis of lifetime dependences on concentrations of free charge carriers was made in pioneering paper by Shockley and Read [11] concerning the theory of recombination through impurity level trap. In present chapter, we use the method of calculation assuming that N is expressed in terms of the ratio of the number of filled recombination level states to the number of empty. This allows to avoid transcendental equations, in other words, to avoid the need to solve inverse problem. As a result, the described above approach simplifies greatly the mathematical analysis and physical interpretation of calculations of desired parameters.

Model and basic relations
Consider nondegenerated semiconductor doped by shallow fully ionized single type impurity (for definition donors) with concentration N D . Recombination of excess charge carriers occurs in said semiconductor through the energy level of acceptor impurity atoms with concentration N , which can be in two charge states (assume in neutral and singly negatively charged). Concentration of recombination impurity atoms in neutral state corresponds to concentration of acceptor atoms N 0 , which are simultaneously centers of electron capture and centers of thermal emission of holes. Concentration of recombination impurity atoms in negatively charged state corresponds to concentration of atoms N − = N − N 0 , which are capture centers of holes and, at the same time, centers of thermal emission of electrons. Described above is recombination through single-level trap [10][11][12][13] (Figure 5b), which is often dominant [1-5, 9, 14] and called Shockley-Read-Hall recombination.
impurity atoms with concentration N , which can be in two charge states (assume in neutral and singly negatively charged).
Concentration of recombination impurity atoms in neutral state corresponds to concentration of acceptor atoms 0 N , which are simultaneously centers of electron capture and centers of thermal emission of holes. Concentration of recombination impurity atoms in negatively charged state corresponds to concentration of atoms 0 N N N    , which are capture centers of holes and, at the same time, centers of thermal emission of electrons. Described above is recombination through single-level trap [10][11][12][13] (Figure 5b), which is often dominant [1-5, 9, 14] and called Shockley-Read-Hall recombination.   [3]. Schematic view of photoconductor on insert in Figure 1a The Let either band-to-band excitation (Figure 5b) or injection on the contacts produce excess electrons and holes. Then, in stationary case, equation determines the charge state of recombination impurity atoms.
Recombination-generation rates of electrons R n and holes R p due to capture of charge carriers by acceptor impurity traps and their thermal emission from recombination level states into permitted conduction or valence bands are equal to Here, w n and w p are electron and hole capture probabilities, respectively, at appropriate recombination level state, δ = N − e / N 0 e (superscript indicates equilibrium values of concentration of recombination impurity atoms N in relevant charge states).
For small deviation of charge carriers' and their capture centers' concentrations from equilibrium values, we can linearize relations (2) and (3) with respect to proper deviations. Then taking into account Poisson equation ΔE is change in electric field caused by deviation of charge carriers' and capture centers' concentrations from equilibrium values, and q is absolute electron charge value and θ = w p / w n . First terms in (5) and (6) mean recombination rates of excess electrons and holes (and therefore, symbols τ n and τ p mean their lifetimes) in quasi-neutrality with respect to electric field ΔE, i.e., at sufficiently small values divΔE [1-3, 5, 9-14, 18, 24, 25]. We will use the same terminology for τ n and τ p in the case of failure to comply quasi-neutrality (see below); therefore values τ n and τ p will not depend on value divΔE in the present study.
High-performance photoconductors operate with extremely low-level illumination. Therefore, linear for g approximation, usually used in the theory of high-performance photodetectors [5-7, 9, 21, 22, 26], is correct in calculation of photoelectric gain G, where g is density of charge carriers' photoexcitation rate.
In view of the above provision, we can write expressions for the density of photocurrent components as follows: where μ n and μ p are electron and hole mobility and D n and D p are electron and hole diffusion constants. The density of electron ΔI n and hole ΔI p components of photocurrent ph n p must satisfy continuity equations: that allows to neglect by the dependence of μ n and μ p on electric field, where W is distance between current contact electrodes (see insert in Figure 1a).
where n µ and p µ are electron and hole mobility and n D and p D are electron and hole diffusion constants.
and also Let limit voltage be applied to sample by value that allows to neglect by the dependence of n µ and p µ on electric field, where W is distance between current contact electrodes (see insert in Figure 1a). (see Figure 1a) on bias voltage across the sample V (distance between current contacts where Ĝ max is maximal value Ĝ for given W (see Figure 2a); (d), value V op on W , where V op is optimal voltage, at which Ĝ = Ĝ max (see Figure 2c). Voltage V in V; length W in cm. Physical parameters of semiconductors and temperature are the same as in Figure 1. Solid curves GaAs, dashed curves Si

Lifetime of excess charge carriers
Using distribution function of electrons over acceptor level states [12,27] for nondegenerated semiconductor at thermodynamic equilibrium as follows: n t and p t are equilibrium concentrations of electrons and holes when Fermi level energy coincides with recombination level energy E t , and n i is intrinsic charge carriers' concentration.
When derived (20), we have adopted that spin degeneracy factor of acceptor state is equal to 1/2 [2,12,14,27] Expressions (20), (24), and (25) determine dependences τ n (N ) and τ p (N ) in parametric form. Figure 1b shows that, as usual, dependences τ n (N ) and τ p (N ) fall with increased N , but in some interval of concentration N , dependences can rise up sharply. Further, we give analytical solution of extremum problem for dependences τ n (N ) and τ p (N ) at θ ≥ 1, because hole is captured on attractive center and electron -on neutral.

Mathematical analysis of hole lifetime
The analysis of equation which determines extremum points of dependence τ p (N ), shows that well-defined nonmonotonic behavior of this function occurs at 2 1 3 2 To determine minimum point, let us set out equation (26) into the form where absolute value of function 3 2

29)
is much less than unity at It means that the first root of equation (26) δ 1 p can be found by the method of successive iterations using Λ 1 p (δ 1 p ) as small parameter. Zeroth-order approximation (30) leads to the formula for concentration of recombination centers N = N 1 p , where dependence τ p (N ) reaches its minimum τ ̮ p ( Figure 1b): It follows from this formula that ratio (Figure 3a). Further, it will be demonstrated that extremum, like the maximum of dependence τ p (N ) as well as τ n (N ), can occur only at values N closer to N D . Therefore, in expression for upper limit of value N 1 p , small correction has been remained, which is primal.
To determine maximum point of dependence τ p (N ), let us set out equation (26) into the form where absolute value of function is much less than unity at It means that the second root of equation (26) δ 2 p can be found by the method of successive iterations using Λ 2 p (δ 2 p ) as small parameter. It follows from relations (20) and (34) that concentration of recombination centers N = N 2 p , where dependence τ p (N ) reaches its maximum τ p (Figure 1b), is determined in first-order approximation for small parameter Λ 2 p (δ 2 p ) by expression ( ) , where dependence ) (N p τ reaches its maximum p τˆ (Figure 1b), is determined in first-order approximation for small parameter ) ( 2 2 p p δ Λ by expression It follows from (22), (25), (27), (30), and (34) that Relation (37) shows that function K p (N D ) is non-monotonic and can vary by several orders of magnitude ( Figure 3b). Value Optoelectronics -Materials and Devices 312 increases with decreasing recombination level energy E t (Figure 4a).
The dependence of τ p on temperature T is determined by the location of recombination level in forbidden gap of semiconductor ( Figure 4b). Value τ p decreases with lowering temperature where k is Boltzmann constant and N v is effective density of states in valance band.

Mathematical analysis of electron lifetime
The analysis of equation which determines extremum points of dependence τ n (N ), shows that well-defined nonmonotonic behavior of this function occurs at To determine minimum point let us set out equation (40) into the form where absolute value of function is much less than unity at ( ) It means that the first root of equation (40) δ 1n can be found by the method of successive iterations using Λ 1n (δ 1n ) as small parameter. Zeroth-order approximation (44) leads to the formula for concentration of recombination centers N = N 1n where dependence τ n (N ) reaches its minimum τ ̮ n : ( ) It follows, from this formula, that ratio N 1n / N D decreases from ½ when N D < < 2 × θ × p t to We can transform equation (40) to form (32) where Λ 2 p (δ) will be replaced by function: At value δ = δ 2n , where dependence τ n (N ) has its maximum τ n , absolute value | Λ 2n (δ) | < < 1.
Therefore, in zeroth-order approximation for small parameter Λ 2n (δ 2n ), value δ 2n = δ 2 p (0) and concentration N = N 1n , where τ n (N ) = τ n equals to N D (as for holes). And where κ = 4 at θ = 1 and κ = 1 at θ > > 1. It follows from relation (47) that function K n (N D ), in contrast to K p (N D ), grows monotonically with increased N D and this growth can be many orders of magnitude ( Figure 4b).
Optoelectronics -Materials and Devices 314 increases, as for τ p , with the decrease of recombination level energy ( Figure 4a) and, in contrast to τ p , always falls with temperature rise (Figure 4b).

Physical interpretation
Let's explain physical mechanisms of the above regularities.
At value has its maximum n τˆ, absolute value 1 ) ( 2 << Λ δ n . Therefore, in zeroth-order approximation for small parameter , grows monotonically with increased D N and this growth can be many orders of magnitude (Figure 4b). Value increases, as for p τˆ, with the decrease of recombination level energy ( Figure 4a) and, in contrast to p τˆ, always falls with temperature rise (Figure 4b).

Physical Interpretation
Let's explain physical mechanisms of the above regularities.

Hole lifetime
Reciprocal hole lifetime consists of three partial components.
First component corresponds to the change of capture rate of holes p e × w p × ΔN − = Δ p / τ p2 caused only by deviation of concentration of hole trapping centers from its equilibrium value (capture of equilibrium holes p e at nonequilibrium trapping centers ΔN − ).
Third component corresponds to the change of thermal emission rate of holes from impurity level states into valence band 2 × p t × w p × ΔN 0 = Δp / τ p3 caused by deviation of concentration of hole generation centers from its equilibrium value (thermal emission of holes from nonequilibrium centers ΔN 0 ).
Lifetime τ p1 can be interpreted as capture time of excess holes by equilibrium traps, lifetime τ p2 can be interpreted as relaxation time of excess holes due to capture of equilibrium holes by nonequilibrium traps, and lifetime τ p3 can be interpreted as time of thermal emission of holes from nonequilibrium centers.
If conditions (27) are fulfilled and N < N D , then recombination centers are almost completely filled with electrons (δ ≡ N − e / N 0 e > 1). For this reason, even if θ ≡ w p / w n = 1, capture time of hole τ p1 is much shorter than capture time of electron τ n1 for the relevant equilibrium trapping centers. In other words, equilibrium traps capture holes much more intensively than electrons. Therefore, the generation of excess free charge carriers initiates the formation of additional nonequilibrium centers of thermal generation of holes and, simultaneously, reducing concentration of trapping centers of electrons (ΔN 0 = − ΔN − > 0). This change of charge state of recombination impurity atoms results in negative values of components 1 / τ p2 and 1 / τ p3 in expression (36); moreover | τ p3 | < < | τ p2 | , because δ > > 1 at N < N D . This means that hole lifetime τ p exceeds capture time of holes τ p1 at equilibrium traps due to dominating thermal emission of holes from relevant nonequilibrium centers. As long as N < N D , concentration Optoelectronics -Materials and Devices decreasing | τ p3 | than decreasing τ p1 . As a result, starting with concentration N = N 1 p , the rate of hole thermal emission from nonequilibrium centers and the capture rate of nonequilibrium holes at equilibrium traps become closer to each other. For this reason, τ p starts to grow ( Figure   1b).
When N becomes larger than N D , the concentration N − e of equilibrium hole capture centers practically stabilizes, while concentration N 0 e of equilibrium electron capture centers grows with the increase of recombination centers' concentration (N − e ≅ N D ,N 0 . This means that the ratio τ p1 / τ n1 increases with increasing N . For this reason, concentration of nonequilibrium hole thermal emission centers decreases, and concentration of hole traps N − increases. As a result, τ p (N ) decreases with increased N ( Figure   1b). When N prevails N D 2 / 2p t , the concentration N − e of equilibrium hole capture centers grows again with increased N due to thermal emission of electrons from valence band to impurity level (N − e ≅ 2N × p t ). However, concentration of equilibrium capture centers of electrons grows much faster (N 0 e ≅ N ). Therefore, the decrease of τ p (N ) continues. As it is seen from (36)-(39), τ p becomes less than τ p1 , when product δ × θ becomes less than unity.
As shown above, minimum point N = N 1 p of dependence τ p (N ) shifts toward N D with growth N D (Figure 3a). The main reason is that equilibrium electrons are being captured at centers of hole thermal emission and decreased concentration ΔN 0 of these centers. The higher the concentration of equilibrium electrons n e , the more ΔN 0 decreases. Concentration n e grows with increased N D . When N ascends, then n e descends and N − e increases that causes increased ΔN 0 at N < N D . In other words, decreased ΔN 0 with increased N D is compensated by increased ΔN 0 with increased N . This is the reason why the greater the N D , the closer the N 1 p to N D .
For the same reasons, non-monotonic dependence τ p on N cancels out, as shown above ( Figure   3b), at N D > 2 × (θ × p t ) 2 / n t (increased ΔN 0 with increasing N is not able to compensate decreas- Non-monotonic character of dependence τ p on N does not occur and at low concentrations N D {see inequities (27), expressions (22), and Figure 3b}, when equilibrium electron population at recombination level is determined mostly by electron-hole transitions between that level and free bands. In this case, values δ cannot provide prevailing growth of hole thermal emission rate from nonequilibrium centers over the growth of capture rate of nonequilibrium holes at equilibrium hole traps with increasing N .
Maximal value of ratio K p ≡ τ p / τ ̮ p is achieved at N D ≅ 2 × θ × p t and equals to approximately Note that with increasing energy E t of recombination level, non-monotonicity of dependence τ p (N ) fades out ( Figure 3b) and then cancels out absolutely. This is caused by the increase in concentration n e of equilibrium electrons and decrease in value δ and fall of the probability of hole thermal emission from recombination level into valence band with increasing energy of recombination level referred to the top of valence band. For the same reason, value τ p decreases with increasing E t (Figure 4a).
The character of dependence τ p on temperature (Figure 4b) is determined by the following dependences on temperature: Values p t (T ) and n e (T ) increase always with temperature T rise. Increased p t means increasing probability of thermal emission of hole from recombination center into valence band. Therefore, the above-mentioned process facilitates increasing τ p with T rise. At the same time, growth n e (T ) facilitates decreasing τ p with T rise due to decreasing concentration of nonequilibrium centers ΔN 0 of hole thermal emission.
Value δ 2 p decreases with T rise at E t ≤ E g / 3 due to approaching N 2 p closer and closer to N D (see expression (36)). Value δ 2 p decreases also at E t ≥ E g / 2 up to temperature at which nonmonotonic dependences τ p and τ n on N cancel out. Falling δ 2 p decreases ΔN 0 that facilitates decreasing τ p with T rise. When E t ≥ E g / 2, then p t (T ) increases faster and δ 2 p (T ) falls and n e (T ) grows. As a result, τ p increases with temperature rise (Figure 4b). If E t ≤ E g / 3, then increased p t with temperature rise cannot compensate decreased δ 2 p (T ) and growth n e (T ). As a result, τ p decreases with temperature rise (Figure 4b). If E g / 3 < E t < E g / 2, then at T < T , dependence τ p (T ) will be increasing, and at T > T dependence τ p (T ) will be falling for the same reasons that in previous cases (see expression (39) and insert in Figure 4b).

Electron lifetime
By analogy with hole lifetime, reciprocal electron lifetime consists of three partial components: First component corresponds to the change of electron capture rate n e × w n × ΔN 0 = Δn / τ n2 caused solely by deviation of concentration of electron capture centers from equilibrium value (capture of equilibrium electrons n e on nonequilibrium capture centers ΔN 0 ).
Value δ > > 1 as long as N < N D , and hence ratio τ p1 / τ n1 < < 1. Therefore, the occurrence of excess  Component 1 / τ n2 falls with growth N due to the decrease in concentration n e of equilibrium electrons, so τ n ≈ τ n2 and increases with growth N (Figure 1b).
At values N greater than N D , the ratio τ p1 / τ n1 increases with increasing N . This, again, leads to decreasing concentration of nonequilibrium capture centers of electrons with increasing N . Value n e continues to fall as well. As a result, partial component 1 / τ n1 becomes critical component defining reciprocal electron lifetime 1 / τ n , and therefore, τ n (N ) falls with increasing N (Figure 1b).
As shown above, in contrast to dependence τ p (N ), ratio N 1n / N D decreases (Figure 3a) and ratio K n ≡ τ n / τ ̮ n always increases monotonically with increasing N D (Figure 3b). Such regularities are caused by increased n e with increasing N D . Because of this, capture rate of equilibrium electrons at nonequilibrium traps becomes greater than capture rate of nonequilibrium electrons at equilibrium traps at lower concentrations ΔN 0 , i.e., at lower values of ratio N / N D . In contrast to the situation with holes, here, decreasing ΔN 0 with increased N D is compensated by increasing n e .
Similar to the behavior of hole lifetime, non-monotonic dependence τ n (N ) fades gradually and then cancels out (Figure 3b) with decreasing N D or increasing E t . First regularity is caused by decreased n e and δ = 2n e / n t with decreasing N D . Second regularity is caused by decreasing δ and, hence, ΔN 0 , with increasing energy E t of recombination level. In this case, however, due to growth n e , non-monotonicity of dependence τ n (N ) cancels out at larger values E t than in the case of holes.

Solitary Illuminated Sample
In this section, we will consider opportunities for improving photoexcitation of charge carriers and photo-emf V ph by increasing concentration N of recombination centers.

Preliminaries: Basic relations
We will call the sample as solitary, if it is not in external electric field and external electrical circuit is open.

Figures 6 and 7 show calculated dependences < Δn > (N ) and V ph (N ).
We have not used in study quasi-neutrality approximation [2-9, 13, 18, 21, 22, 28-31] because it can lead to unacceptable errors in calculation of dependences < Δn > (N ), < Δ p > (N ) (Figures  8 and 9), and V ph (N ) (Figures 10 and 11) due to the fundamental contribution of photoexcited space charge into photoelectric effects in semiconductors. In other words, even in solitary sample, photoexcited electron-hole plasma in semiconductor may not always be quasi-neutral.
Let's consider a homogeneous semiconductor sample ( Figure 5) with no voltage applied, i.e., in absence of illumination and intensity of electric field E 0 = 0. The density of photogeneration rate of charge carriers, in view of multiple internal reflections, is determined by the following expression: where R and γ are coefficients of light reflection and absorption, F 0 is density of incident photon flux, and W is sample thickness along incident light direction ( Figure 5a). As is clear from (66) and (81), in discussed conditions, relation between concentrations of excess holes Δ p and electrons Δn is as follows: where . a n where R and γ are coefficients of light reflection and absorption, 0 F is density of incident photon flux, and W is sample thickness along incident light direction (Figure 5a). As is clear from (66) and (81), in discussed conditions, relation between concentrations of excess holes p ∆ and electrons n ∆ is as follows: ) ( x where ξ D D D a n + = . (89)  In quasi-neutrality approximation, parameters ξ, ξ n and ξ p are equal to zero; therefore, in this approximation, the distribution of excess electrons' concentration is determined by equation Error! Not a valid embedded object.  Figure 6 Exact solution of equation (88) is

Effectiveness of charge carriers' photoexcitation
We define the mean value < y > of variable y(x) as 0 1 ( ) .      dependences < Δn > (N ) and < Δp > (N ). We will call found dependences (see Figures 6, 8, and 9) as exact, because, in linear approximation with respect to F 0 , dependences are exact in contrast to quasi-neutrality approximation case.
Denote desired dependences in quasi-neutrality approximation as < Δñ (N ) > and < Δ p(N ) > . In quasi-neutrality approximation, (113) Figure 6 shows that the effectiveness of charge carriers' photoexcitation may grow significantly with increasing N . Up to small corrections, dependences < Δn > (N ) and < Δ p > (N ) reach maximums < Δn> max and < Δp> max at the same concentration N = N = N D as for lifetimes ( Figure   1b) and after that fall very strongly. Figures 8 and 9 illustrate the influence of photoexcited space charge in point N = N on the validity of results. It is clear from Figures 8 and 9 that with thinning W of sample, using quasi-neutrality approximation leads to error up to several orders of magnitude. Let's clarify Figures 6, 8, and 9.
Lengths L and L 1 ≡ 1 / k 1 , up to small corrections, equal to L a ; moreover At W < < L a , we have where function ψ(W , γ) is independent on τ n and τ p . On the other hand, if inequality W < < L 2 is sufficiently strong, and when L p > > W as well, from expressions (104) despite the fact that τ p < < τ n (Figure 1b).
When γ −1 < < W and inequity W < < L n are sufficiently strong, then the total quantity of photoelectrons is proportional to W due to reducing loss on unilluminated surface. Therefore, there is a "plateau" on dependence < Δn> max on W. For the same reason, at γ −1 < < W < < L a , dependence < Δñ > max on W has a "plateau" as well. However, in the last case, "plateau" height is much higher, and its width is much wider than true "plateau" (Figure 8, curve 1). Moreover, shielding length of photoinduced space charge L 2 ≈ L a when impurity level energy equals to E t 2 . Therefore, solution obtained in quasi-neutrality approximation, even when W → ∞, differs from exact solution at least by several times (Figures 8 and 9, curve 2).

Effectiveness of photo-emf excitation
In view of the fact that under considered conditions 0, from expressions (12)- (14) and (87), we obtain that photo-emf i.e. potential ϕ(x) of illuminated surface with respect to unilluminated (Figure 5а) expressed by formula It is clear from expression (121) that at infinite surface recombination rate, i.e., when conditions (97) are fulfilled, illumination produces no photo-emf. In this regard, assume that photoexcited charge carriers are not captured on surfaces and there is no charge on surfaces. In these conditions, photoexcited electric field intensity And densities of electron ΔI n and hole ΔI p photocurrents It is clear from expression (121) that at infinite surface recombination rate, i.e., when conditions (97) are fulfilled, illumination produces no photo-emf. In this regard, assume that photoexcited charge carriers are not captured on surfaces and there is no charge on surfaces. In these conditions, photoexcited electric field intensity 0 ) And densities of electron n I ∆ and hole (125)  Therefore, we may write boundary conditions, in view of relation (87), as 0 0, Find consistently dashed lines -solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as in Figure 6 Therefore, we may write boundary conditions, in view of relation (87), as where (cm) Figure 9. Dependences of ratio < Δp> max / < Δn> max in the point of maximum functions < Δn > (N ) and < Δ p > (N ) on thickness W for GaAs. Curves 1 and 2, recombination level energy E t equals to E t 1 and E t 2 , respectively; solid lines -exact solutions; dashed lines -solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as in Figure 6 Optoelectronics -Materials and Devices If we utilize relation  where   2  2  1  2  1  2  1  2  2  1  2  2  2  2  2  2  2  2  2  2  1  2  2  1  2 In quasi-neutrality approximation, we get We refer to the dependences (141) of photo-emf V ph on N and W as exact. The reason is that, in contrast to the case of quasi-neutrality approximation, said dependences are exact in linear approximation with respect to flux density F 0 .
Figures 7, 10, and 11 show these dependences and solution in quasi-neutrality approximation as well. It is clear in Figures 10 and 11 that, in quasi-neutrality approximation, maximal value V ph (N ) far exceeds the "true" value; moreover, exceedance may be several orders of magnitude.
We can explain results by the fact that very long shielding length (114) of photoexcited space charge L 2 in the point of maximum N = N ≡ N D and nearby will cause diffusion of photoelectrons independently on photoholes (see Section 6.2) at W ∼ 0.1 cm. In other words, electron diffusion constant D n determines the distribution of photoelectron concentration Δn(x). Aside from that, at N ≅ N , due to charge coupled to impurities, inequity D n > > D n a is fulfilled. It means that true effectiveness of photoelectrons' spreading is much higher, than that given by quasineutrality approximation. Therefore, ratio (V ph ) max / (Ṽ ph ) max < < 1, when W > > L 2 ( Figure 11).
where n is arithmetic mean with respect to x concentration of excess electrons (insert in Figure  1а Values of parameters Q, D ξ , μ ξ , and D E are dictated by photoexcitation of space charge Δρ (4).
Analysis shows that inequality ξ < < 1, which is typically fulfilled with large margin, allowing to omit in equation (151)    Curves 1 and 2 -recombination level energy E t equals to E t 1 and E t 2 . V ph , exact solutions; Ṽ ph , solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as in Figure 6 The Relations (18) It can be shown that equation μ n a (N ) = 0, where μ n a is given by expression (74), has a solution when inequalities (27) and (41) are fulfilled. In zeroth-order with respect to small parameters (27) and (41), the root of this equation coincides with maximal extrema of functions τ n (N ) and τ p (N ) and equals to N D (Figure 1b and 1c).
Let's explain dependence μ n a on N shown in Figure 1c.
Product a a n n p p m t m t =´ ( 158) determines drift length and direction in electric field of concentrational perturbation -quasineutral cloud of positive and negative charges [2,31], including bound at deep impurity (here μ n a and μ p a are electron and hole ambipolar mobility). Last mentioned bounding explains dependence μ n a on ratio τ p (N ) / τ n (N ) in trap-assisted recombination. It would appear reasonable that charge carriers, which prevail in quantity, can easily shield photoexcited space charge, i.e., they are adjusted to drift of charge carriers of another type. That is why, in the case of band-to-band recombination (τ n = τ p ), quasi-neutral cloud of positive and negative charges drifts in electric field with the same velocity and in the same direction as minority charge carriers, whereas in intrinsic material, cloud is out of control by electric field at all [2,31] (μ n a = μ p a ≅ μ p >0 at n e > > p e , μ n a = μ p a ≅ − μ n <0 at n e < < p e and μ n a = μ p a =0 at n e = p e ). Similar situation, but not exactly identical, happens in the case of trap-assisted recombination.

Optoelectronics -Materials and Devices
In this case, due to the fact that τ p < τ n (Figure 1c), vanishing μ n a occurs in n-type material (for specified parameters in Figure 1, at n e ≅ 10 × p e in silicon and n e ≅ 10 4 × p e in gallium arsenide).
Positive sign of perturbation charge bound at deep impurities (ΔN − < 0) causes such behavior.
Ratio p e / n e begins to increase significantly, and very sharply, only when N ≅ N D . At the same time, ratio τ p / τ n may not have so many orders of smallness as ratio p e / n e may have. Therefore, μ n a vanishes when N ≅ N D , if deep level, according to conditions (27) and (41)  Value L a is conventional ambipolar diffusion length at maximal extremum of function G(N ) (Figure 1a) calculated in quasi-neutrality approximation, i.e., when parameter ξ is set to zero.
From expression (159), it follows that function Ĝ depends non-monotonically on applied bias voltage V (Figure 2a). This is caused by increased L ef with increasing E 0 = V / W that provides progressive loss of photoexcited charge carriers resulting from increasing diffusive inflow of photocarriers to current contacts' electrodes with follow-up recombination.
Increase in effective ambipolar diffusion constant D ef = D n a + D E (coefficient before second derivative in equation (152)

Summary
Capture rate of excess charge carriers increases with increasing concentration N of deep impurity levels, i.e., recombination centers (traps). However, as shown in this chapter on the example of single-level acceptor, this increase does not lead to unavoidable decrease in lifetime of excess electrons τ n and holes τ p , when nonequilibrium filling of recombination level states is very low. The matter is that lifetimes are determined not only by capture of excess charge carriers at equilibrium traps but also by bound-to-free transitions of electrons and holes from nonequilibrium capture centers due to thermal emission and by capture of equilibrium charge carriers at nonequilibrium traps as well. Therefore, lifetimes of excess charge carriers can be either more or less than the time of their capture at equilibrium traps and can be strongly nonmonotonic functions of recombination center concentration (Figure 1b). In the case of acceptor recombination level, it can happen if recombination level is located in lower half of forbidden gap. In the case of donor recombination level, it must be located in upper half of forbidden gap. It is essential that the ratio of lifetimes in maximum and minimum of functions τ n (N ) and τ p (N ) can be several orders of magnitude (Figures 1b and 3b).
It seems, authors of article [25] have reported first about the availability of minimum and portion of weak growth (up to 24 %) on experimental dependence of excess charge carriers' lifetime on recombination center concentration, which increased because of bombarding sample by high-energy electrons. Many years later, increase in lifetime, presumably, caused by increasing N , but already gained in several times, was observed experimentally [32].
The main reason for giant splash of photoresponse in semiconductors with increasing recombination center concentration N (Figures 1a, 6, and 7) is the growth of charge carriers' lifetime in orders of magnitude.
This reason is also sufficient to provide increase, in order of magnitude and more, in efficiency of charge carriers' photoexcitation with increasing N (Figure 6). However, increase in orders of magnitude in charge carriers' lifetime with increasing N (Figure  1b) is not a good reason for the development of giant splash in photoelectric gain G with increasing N (Figure 1a). As follows from [18,19], G increases with increasing charge carriers' lifetime, if ambipolar mobility μ a (see (74), (158), [2]) is equal to zero, or if there is no recombination on current contact electrodes (x = 0 and x = W ; see inset in Figure 1a).
In reality, recombination on contact electrodes occurs always to more or less extent [5,9]. Therefore, under normal conditions (μ a ≠ 0), increase in lifetimes, beginning from some lifetime values, does not increase in photocurrent density I ph [5,18,19].
Saturation in I ph is most clear in the case of high-rate recombination at contact electrodes (sweep-out effect on contacts [5,18,19]), when there are no photocarriers at contacts, i.e., conditions (1) are fulfilled. At trap-assisted recombination, function μ(N ), under the same conditions (27) and (41), when there are non-monotonic dependences τ n (N ) and τ p (N ), vanishes at the same, up to small, correction value N ≅ N , at which functions τ n (N ) and τ p (N ) reach their maximal extrema τ n and τ p (Figure 2b and 2c). Therefore, I ph and, consequently, G increase to the extent of increasing τ n and τ p . These are physical grounds of giant splash in photoelectric gain G with increasing N (Figure 2a). Above mentioned results of strict analytical calculations (i.e., outside commonly used local approximation of quasi-neutrality) show that photoinduced local space charge affects substantially on giant splash of semiconductor photoelectric response with increasing concentration of recombination centers.
Strict solutions of problems concerning the quantity of photoexcited electrons N ph and holes P ph and Dember photo-emf V ph may be fundamentally different from solutions obtained in approximation of quasi-neutrality Ñ ph , P ph , and Ṽ ph , respectively.
It may be that P ph / N ph > > 1 even if hole lifetime τ p is much less than electron lifetime τ n (Figure  2b). At the same time, in approximation of quasi-neutrality, P ph / Ñ ph = τ p / τ n < < 1 (Figure 2b).
At point N = N , at which functions N ph (N ), P ph (N ), and V ph (N ) reach maximum values, and for thin samples (with thickness along light propagation W < 0.1 cm), in surroundings of point N = N , solutions obtained in quasi-neutrality approximation may differ from solution outside quasi-neutrality in several orders of magnitude (Figures 8-11). Moreover, even W → ∞, neglecting by photoinduced space charge is not always possible, i.e., it is not always possible to solve problem in quasi-neutrality approximation. The reason is that when recombination level is deep enough, then shielding length of photoexcited space charge may be of the order of ambipolar diffusion length of charge carriers.
At sweep-out effect on contact electrodes, splash of G(N ) with increasing N depends nonmonotonically on applied voltage V across the sample (Figure 2a). That non-monotonic behavior is not related to heating of charge carriers or lattice and charge carriers injecting contacts. The reason is the increase in effective ambipolar diffusion constant D (coefficient before second derivative in equation (81) determining the distribution of photocarriers) with increasing V , leading to huge loss of photocarriers due to faster diffusion to contacts and subsequent recombination. In turn, increase in D ~ V 2 is caused by photoinduced local space charge. What is important is that, at optimum voltage V op (Figure 2d), value G can have several orders of magnitude ( Figure 2c) at high concentrations of recombination centers.
As shown in [33], when recombination impurity N has three charged states (two-level recombination center), then, again, strong increase in τ n (N ) and τ p (N ) with increasing N may occur; moreover dependences τ n (N ) and τ p (N ) may have two charged states and two minimums and maximums. Opposite to single-level recombination center, in the case of two-level recombination center, maximum G(N ), as shown in [34], can be reached at lower concentrations N and have greater peak value. Photoelectric gain G, to the left of maximum value G(N ), is larger in the case of two-level recombination center, than in single level. The reason is the low ambipolar mobility of charge carriers in the case of two-level recombination center.
As shown in [35], significant growth of charge carriers' lifetimes with increase in concentration of recombination impurity in certain range could be stimulated by uncontrolled (background) doping by deep impurities. Even two maximums can occur.