Open access peer-reviewed chapter

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

Written By

Viacheslav A. Kholodnov and Mikhail S. Nikitin

Submitted: October 5th, 2014Reviewed: June 10th, 2015Published: October 7th, 2015

DOI: 10.5772/61028

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Abstract

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.

Keywords

  • Impurity recombination center (trap)
  • Trap-assisted (Shockley-Read-Hall) recombination
  • Excess (nonequilibrium) charge carriers
  • Band-to-band photogeneration of excess charge carriers
  • Electron lifetime
  • Hole lifetime
  • Photoresponse
  • Photoelectric gain
  • Photo-emf
  • Small deviation from equilibrium state in semiconductor
  • Photoinduced space charge
  • Approximation of quasi-neutrality
  • Giant splash of photoresponse

1. 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 [1-9]. 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 [1-5, 12-14]. At the same time, due attention was not paid to study dependences of lifetimes of excess electrons τn(N)and holes τp(N)on concentration of traps N. In some cases, traps are produced intentionally by doping semiconductor (e.g., by bombarding with high-energy 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 τnand τpin 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 τn(N)and τp(N)on concentration of recombination centers.

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2. Preliminaries

Let’s consider the case of small deviation of free charge carriers’ concentrations from equilibrium values. This situation occurs often in semiconductors used for registration of low-level signals, for example, optical signals. It will be shown that both τn(N)and τp(N), under certain conditions, will grow very strongly with increasing Nin a particular interval of Nvalues. Completely different physical mechanism causes this increase in lifetimes. It differs from mechanisms available for many years [4], as well as proposed in [17]. We analyze extreme points (corresponding formulas are derived) of dependences τn(N)and τp(N)as functions of 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 τn(N)and τp(N)differ from each other.

It is reasonable to expect that the growth of lifetimes τn(N)and τp(N)with increasing Nwill lead to the growth of photoresponse of semiconductor sample (including photoelectric gain G). However, specificity of dependences τn(N)and τp(N)does not determine the type of dependence G(N)in total. As it follows from [18, 19], Gincreases with increasing charge carriers’ lifetime, if ambipolar mobility [2, 13] μa=0or there is no recombination at current contact electrodes (x=0and x=W; see insert in Figure 1a). In reality, recombination at current contact electrodes is always happening to some extent [5-9]. Therefore, in normal conditions (μa0), increase in lifetimes, after reaching some values, does not lead to an increase in photocurrent Iph[5, 9, 18, 19]. Saturation of Iphbecomes apparent in the case of high-rate recombination at the contact electrodes (sweep-out effect [5, 9, 18, 19]) when

Δn(0)=Δp(0)=Δn(W)=Δp(W)=0,E1

where Δn(x)=n(x)neand Δp(x)=p(x)peare deviations of electron nand hole pconcentrations from their equilibrium values neand pe, respectively. In trap-assisted recombination, function μ(N)can vanish at the same (up to small corrections) concentration N, at which dependences τn(N)and τp(N)reach points of maximal extremum τ^nand τ^p(Figure 1b and 1c). Therefore Gand, hence, Iphgrow to the extent of increase in τn(N)and τp(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 τn(N)and τp(N)allows avoiding highly undesirable effect – saturation of Gin intrinsic photoconductors, when applied bias voltage Vincreases [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×ε0)×divΔE, which determines the density of photoinduced space charge Δρ.

In our case, ΔEE(x)E0is 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 E0are the electric field intensity in the presence and absence of illumination, εis the relative dielectric permittivity of semiconductor, and ε0is the vacuum permittivity. However, even at moderate electric fields (≈ 1÷10 V/cm), approximation of quasi-neutrality is not always acceptable [23].

Below, in case of single recombination level, we consider in detail the impact of photoinduced space charge Δρon value G^of photoelectric gain Gin 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 low-intensity 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 Vacross 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 Vphin semiconductors with dominant trap-assisted recombination. Analytical expressions for photo-emf Vphand mean, with respect to light propagation length, concentrations of photoelectrons <Δn>and photoholes <Δp>are given. It is shown that target values of Vph, <Δn>, and <Δp>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 τnand τpand, even more, Iphand Vphon 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 τn,p(N)and, even more, Iph(N)and Vph(N).

These difficulties are dramatized by the fact that under certain conditions, τn,p, Iphand Vphare dependent very strongly on concentration Nin 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 τn,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 Nis 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.

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3. Model and basic relations

Consider nondegenerated semiconductor doped by shallow fully ionized single type impurity (for definition donors) with concentration ND. 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 N0, 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=NN0, 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 [10-13] (Figure 5b), which is often dominant [1-5, 9, 14] and called Shockley-Read-Hall recombination.

Figure 1.

Dependences on concentration of single-level recombination centersN(сm – 3):(a)G− photoelectric gain;(b)τ− lifetime of electrons (curve 1) and holes (curve 2) (s);(c)μ− ambipolar mobility of charge carriers (in units of hole mobility). Adopted: room temperature,W=101cm,θwp/wn=102,wn=108cm3/s,ni/nt=104,ND=1015cm-3,E0=10V/cm. Solid curves,GaAs, and dashed curves,Si. Physical parameters of semiconductors are obtained from monograph [3]. Schematic view of photoconductor on insert inFigure 1a

Let either band-to-band excitation (Figure 5b) or injection on the contacts produce excess electrons and holes. Then, in stationary case, equation

Rn=RpE2

determines the charge state of recombination impurity atoms.

Recombination-generation rates of electrons Rnand holes Rpdue 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

Rn=(n×N0δ1×ne×N)×wn,Rp=(p×Nδ×pe×N0)×wp.E3

Here, wnand wpare electron and hole capture probabilities, respectively, at appropriate recombination level state, δ=Ne/N0e(superscript indicates equilibrium values of concentration of recombination impurity atoms Nin relevant charge states).

For small deviation of charge carriers’ and their capture centers’ concentrations ΔN0=N0N0e=ΔN=NeNfrom equilibrium values, we can linearize relations (2) and (3) with respect to proper deviations. Then taking into account Poisson equation

Δρε4π×ΔEx=q×[ΔpΔnΔN],E4

we obtain

Rn=Δnτn+an×divΔE,E5
Rp=Δpτpap×divΔE,E6
Δp=τpτn×Δn+(an+ap)×τp×dΔEdx,E7

where

1τn=wn×N×δ×θ1+δ×N+(1+δ)×(1+δ1)×(ne+pe)δ×θ×N+(1+δ)×(1+δ1)×(ne+δ×θ×pe),E8
1τp=wp×N×δ1+δ×δ×N+(1+δ)2×(ne+pe)δ×N+(1+δ2)×(ne+δ×θ×pe),E9
an=ε4π×q×wn×N×ne×θ×(1+δ)δ×θ×N+(1+δ)×(1+δ1)×(ne+δ×θ×pe),E10
ap=ε4π×q×wp×N×pe×1+δN+(1+δ)×(1+δ1)(ne+δ×θ×pe),E11

ΔEis change in electric field caused by deviation of charge carriers’ and capture centers’ concentrations from equilibrium values, and qis absolute electron charge value and θ=wp/wn. First terms in (5) and (6) mean recombination rates of excess electrons and holes (and therefore, symbols τnand τpmean 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 τnand τpin the case of failure to comply quasi-neutrality (see below); therefore values τnand τpwill not depend on value divΔEin the present study.

High-performance photoconductors operate with extremely low-level illumination. Therefore, linear for gapproximation, usually used in the theory of high-performance photodetectors [5-7, 9, 21, 22, 26], is correct in calculation of photoelectric gain G, where gis density of charge carriers’ photoexcitation rate.

In view of the above provision, we can write expressions for the density of photocurrent components as follows:

ΔIn=q×μn×(E0×Δn+ne×ΔE)+q×Dn×Δnx,E12
ΔIp=q×μp×(E0×Δp+pe×ΔE)q×Dp×Δpx.E13

where μnand μpare electron and hole mobility and Dnand Dpare electron and hole diffusion constants. The density of electron ΔInand hole ΔIpcomponents of photocurrent

Iph=ΔIn+ΔIpE14

must satisfy continuity equations:

ΔInx=q×(Rng),E15
ΔIpx=q×(gRp),E16

and also

Iphx=0.E17

Let limit voltage be applied to sample by value

V=E0×WE18

that allows to neglect by the dependence of μnand μpon electric field, where Wis distance between current contact electrodes (see insert in Figure 1a).

Figure 2.

Dependences:(a), photoelectric gainG=G^in point of maximal extremum of functionG(N)(seeFigure 1a) on bias voltage across the sampleV(distance between current contactsW=101cm);(b), ratioζG^appr/G^exactonVatW=101cm, whereG^apprandG^exactare approximate and exact valuesG^, respectively;(c), valueG^maxonW, whereG^maxis maximal valueG^for givenW(seeFigure 2a);(d), valueVoponW, whereVopis optimal voltage, at whichG^=G^max(seeFigure 2c). VoltageVin V; lengthWin cm. Physical parameters of semiconductors and temperature are the same as inFigure 1. Solid curvesGaAs, dashed curvesSi

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4. Lifetime of excess charge carriers

Using distribution function of electrons over acceptor level states [12, 27], we can write neutrality equation

ne+Ne=pe+NDE19

for nondegenerated semiconductor at thermodynamic equilibrium as follows:

N=nt×1+δ2×δ2×f(δ),E20

where

δ=NeN0e,f(δ)=B+A×δδ2,E21
A=2×NDnt,B=4×ptnt=(2×nint)2,E22

ntand ptare equilibrium concentrations of electrons and holes when Fermi level energy coincides with recombination level energy Et, and niis 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]. In considered conditions

ne=δ2×nt,pe=2δ×pt.E23

From expressions (8), (9), (20), and (23), it follows that

τn=2δ2f(δ)×θ×f(δ)+(1+δ)×(θ×B+δ)[δ×A+(2+δ)×B+δ3]wp×nt,E24
τp=2δf(δ)×B+(A+θ×B)×δ+(θ×B+δ)×δ2[δ×A+(2+δ)×B+δ3]wp×nt.E25

Expressions (20), (24), and (25) determine dependences τn(N)and τp(N)in parametric form. Figure 1b shows that, as usual, dependences τn(N)and τp(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 τn(N)and τp(N)at θ1, because hole is captured on attractive center and electron – on neutral.

4.1. Mathematical analysis of hole lifetime

The analysis of equation

δ×1τp=0,E26

which determines extremum points of dependence τp(N), shows that well-defined non-monotonic behavior of this function occurs at

ξ13θ×B<<1<<1ξ2A24×B,ξ31A<<1.E27

To determine minimum point, let us set out equation (26) into the form

δ2+2×θ×B×δθ×A×B×[1+Λ1p(δ)]=0,E28

where absolute value of function

Λ1p(δ)=B+(A+θ×B)×δθ×A×B×δBA×δ+B×δ3+(δ2+B)[2×B+(A+B)×δ]θ×A×B×δ6××[B+(A+θ×B)×δ+θ×B×δ2+δ3]+B+A×δδ2θ×A×δ5××{A+(2×θ1)×B+4×θ×B×δ+[3+(A+B)×θ]×δ22(θ1)×δ3}E29

is much less than unity at

δ=δ1p(0)θ×B+θ×B×(A+θ×B).E30

It means that the first root of equation (26) δ1pcan be found by the method of successive iterations using Λ1p(δ1p)as small parameter. Zeroth-order approximation (30) leads to the formula for concentration of recombination centers N=N1p, where dependence τp(N)reaches its minimum τp(Figure 1b):

N1p=ND+2×θ×pt2×θ×pt×(ND+2×θ×pt).E31

It follows from this formula that ratio N1p/NDincreases from ½ when ND<<2×θ×ptto (12×θ×pt)1when ND>>2×θ×ptwith increased ND(Figure 3a). Further, it will be demonstrated that extremum, like the maximum of dependence τp(N)as well as τn(N), can occur only at values Ncloser to ND. Therefore, in expression for upper limit of value N1p, small correction has been remained, which is primal.

To determine maximum point of dependence τp(N), let us set out equation (26) into the form

[1Λ2p(δ)]×δ2=A+B,E32

where absolute value of function

Λ2p(δ)=1θ×δ4×[A+(2×θ1)×B+3δ22×(θ1)×δ3]++B+δ2θ×B×δ5×[2×B+(A+B)×δ]×[1+δ×(δ+1)×(θ×B+δ)B+A×δδ2]E33

is much less than unity at

δ=δ2p(0)=A+B.E34

It means that the second root of equation (26) δ2pcan be found by the method of successive iterations using Λ2p(δ2p)as small parameter. It follows from relations (20) and (34) that concentration of recombination centers N=N2p, where dependence τp(N)reaches its maximum τ^p(Figure 1b), is determined in first-order approximation for small parameter Λ2p(δ2p)by expression

N2pND=11A+BA+BA×Λ2p×(A+B)1.E35

Figure 3.

Dependences: point of minimumN1of functionsτp(N)иτn(N)(a)and ratioKbetween maximal to minimal charge carriers’ lifetimes(b)on concentration of shallow donorsNDat different locations of recombination level. Solid curves, holes; dashed curves, electrons. Rationt/nivalues: curve 1, 10-4; curve 2, 10-2; curve 3, 1; curve 4, 102. ConcentrationN1is measured in unitsND, concentrationNDin unitsni. Adopted:θ=102

In particular

N2pND={1nt2×ND×(1+2θ+2×ND×ntθ×pt),forND>>2×pt1(2×niND)2,for2ni<<ND<<2pt.E36

It follows from (22), (25), (27), (30), and (34) that

Kp(ND)τ^pτp={ND8×ni,for8×ni<<ND<<2×ptND32×ni,forND<<2θ×pt<<θ×ND2×θ×ptND×nt,for2×ND×nt<<2×θ×pt<<ND1,for2×θ×pt<<2×ND×nt.E37

Relation (37) shows that function Kp(ND)is non-monotonic and can vary by several orders of magnitude (Figure 3b). Value

τ^p=(1+2×θ×ptND×nt+2×ni2)×(ND×wp)1E38

increases with decreasing recombination level energy Et(Figure 4a).

The dependence of τ^pon temperature Tis determined by the location of recombination level in forbidden gap of semiconductor (Figure 4b). Value τ^pdecreases with lowering temperature if EtEg/2and increases if EtEg/3(value Etis measured from top of valence band and Egis the energy gap of semiconductor). If Eg/3<Et<Eg/2, then dependence τ^phas maximum value at

T=T˜Etk×ln1(2×NvND×Eg2×Et3×EtEg),E39

where kis Boltzmann constant and Nvis effective density of states in valance band.

4.2. Mathematical analysis of electron lifetime

The analysis of equation

δ×1τn=0,E40

which determines extremum points of dependence τn(N), shows that well-defined non-monotonic behavior of this function occurs at

2×ξ3<<1,ξ4=2×BA<<1.E41

To determine minimum point let us set out equation (40) into the form

(θ1)×δ22×θ×(A+B)×δ+θ×A×(A+B)×[1+Λ1n(δ)]=0,E42

where absolute value of function

Λ1n(δ)=(θ1)×Bθ×A×(A+B)2×B+A×δθ×A×(A+B)×δ××[(1+2×θ×Bδ)×(1+A+Bδ2+2×Bδ3)+1δ×(A+B+2×Bδ)×(θ×A+Bδθ+1)]++B+A×δδ2θ×A×(A+B)×δ[2+(θ1)×A+Bδ+6×θ×Bδ+2×Bδ2×(2×θ21δ)]E43

is much less than unity at

δ=δ1n(0)θ×(A+B)θ1×(θ×(A+B)A+θ×B).E44

It means that the first root of equation (40) δ1ncan be found by the method of successive iterations using Λ1n(δ1n)as small parameter. Zeroth-order approximation (44) leads to the formula for concentration of recombination centers N=N1nwhere dependence τn(N)reaches its minimum τn:

N1n=ND+2×θ×ptθ1×(θ×(ND+2×pt)ND+2×θ×pt).E45

It follows, from this formula, that ratio N1n/NDdecreases from ½ when ND<<2×θ×ptto 1/(θ+1)when ND>>2×θ×ptwith increased ND(Figure 3a).

We can transform equation (40) to form (32) where Λ2p(δ)will be replaced by function:

Λ2n(δ)=2×δA+2×BA×θ×(A+B)×δ+2×θ×B+δθ×(A+B)×δ4×(2×A+B+2×Bδ)+(2×θ×B+δ)××A×(A+B+δ2)+2×B×δθ×A×(A+B)×δ32×B+A×δδ2θ×A×(A+B)×δ4×(δ3+3×θ×B×δ22×B)(θ1)××[2×B×[4×B+(4×A+B)×δ+(δ2)×δ2]+2×A×(A+B)×δ2+δ×(Bδ2)×(A+Bδ2)θ×A×(A+B)×δ4].E46

At value δ=δ2n, where dependence τn(N)has its maximum τ^n, absolute value |Λ2n(δ)|<<1. Therefore, in zeroth-order approximation for small parameter Λ2n(δ2n), value δ2n=δ2p(0)and concentration N=N1n, where τn(N)=τ^nequals to ND(as for holes). And

Kn(ND)τ^nτn={ND8×ni,for2×ni<<ND<<2×pt1κ×ND2×nt,forND>>2×ptE47

where κ=4at θ=1and κ=1at θ>>1. It follows from relation (47) that function Kn(ND), in contrast to Kp(ND), grows monotonically with increased NDand this growth can be many orders of magnitude (Figure 4b).

Value

τ^n=ND+2×pt2×ni×(ND×wn)1E48

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).

4.3. Physical interpretation

Let’s explain physical mechanisms of the above regularities.

Figure 4.

Dependences of maximal lifetime of holesτ^p(N)(solid lines) and electronsτ^n(N)(dashed lines) on recombination level energyEt(eV) forGe(1),Si(2), andGaAs(3) atT=300K(a)and on temperatureT(K) forSiat different valuesEt(b): curve 1, 0; curve 2, (-101/152); curve 3, (-3/4). Recombination level energy is reckoned from the middle of forbidden gap. Physical parameters forGe,Si, andGaAsare obtained from monograph [3].Adopted:ND= 1015 сm-3,wn=108cm3 /s,θ=102

4.3.1. Hole lifetime

Reciprocal hole lifetime

τp1=τp11+τp21+τp31E49

consists of three partial components.

First component

τp11=wp×Ne=wp×nt2×B+A×δδ2δE50

corresponds to the change of capture rate of holes Δp×wp×Ne=Δp/τp1caused only by deviation of hole concentration from its equilibrium value (capture of excess holes Δpat equilibrium trapping centers Ne).

Second component

τp21=wp×pt×2δ×(δ×θ1)×(B+A×δδ2)A×δ+B×[1+δ×θ×(1+δ)]+δ3E51

corresponds to the change of capture rate of holes pe×wp×ΔN=Δp/τp2caused only by deviation of concentration of hole trapping centers from its equilibrium value (capture of equilibrium holes peat nonequilibrium trapping centers ΔN).

Third component

τp31=δ×τp21E52

corresponds to the change of thermal emission rate of holes from impurity level states into valence band 2×pt×wp×ΔN0=Δp/τp3caused by deviation of concentration of hole generation centers from its equilibrium value (thermal emission of holes from nonequilibrium centers ΔN0).

Lifetime τp1can be interpreted as capture time of excess holes by equilibrium traps, lifetime τp2can be interpreted as relaxation time of excess holes due to capture of equilibrium holes by nonequilibrium traps, and lifetime τp3can be interpreted as time of thermal emission of holes from nonequilibrium centers.

If conditions (27) are fulfilled and N<ND, then recombination centers are almost completely filled with electrons (δNe/N0e>1). For this reason, even if θwp/wn=1, capture time of hole τp1is much shorter than capture time of electron τn1for 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 (ΔN0=ΔN>0). This change of charge state of recombination impurity atoms results in negative values of components 1/τp2and 1/τp3in expression (36); moreover |τp3|<<|τp2|, because δ>>1at N<ND. This means that hole lifetime τpexceeds capture time of holes τp1at equilibrium traps due to dominating thermal emission of holes from relevant nonequilibrium centers. As long as N<ND, concentration NeNof equilibrium capture centers of holes grows with increased N, but concentration N0e<<Nof capture centers of electrons still remains low. Therefore, concentration ΔN0of nonequilibrium centers of hole thermal emission increases as well. This increase causes faster decreasing |τp3|than decreasing τp1. As a result, starting with concentration N=N1p, 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, τpstarts to grow (Figure 1b).

When Nbecomes larger than ND, the concentration Neof equilibrium hole capture centers practically stabilizes, while concentration N0eof equilibrium electron capture centers grows with the increase of recombination centers’ concentration (NeND,N0eNNDat nt/2<<NND<< ND2/2pt). This means that the ratio τp1/τn1increases with increasing N. For this reason, concentration of nonequilibrium hole thermal emission centers decreases, and concentration of hole traps Nincreases. As a result, τp(N)decreases with increased N(Figure 1b). When Nprevails ND2/2pt, the concentration Neof equilibrium hole capture centers grows again with increased Ndue to thermal emission of electrons from valence band to impurity level (Ne2N×pt). However, concentration of equilibrium capture centers of electrons grows much faster (N0eN). Therefore, the decrease of τp(N)continues. As it is seen from (36)-(39), τpbecomes less than τp1, when product δ×θbecomes less than unity.

As shown above, minimum point N=N1pof dependence τp(N)shifts toward NDwith growth ND(Figure 3a). The main reason is that equilibrium electrons are being captured at centers of hole thermal emission and decreased concentration ΔN0of these centers. The higher the concentration of equilibrium electrons ne, the more ΔN0decreases. Concentration negrows with increased ND. When Nascends, then nedescends and Neincreases that causes increased ΔN0at N<ND. In other words, decreased ΔN0with increased NDis compensated by increased ΔN0with increased N. This is the reason why the greater the ND, the closer the N1pto ND.

For the same reasons, non-monotonic dependence τpon Ncancels out, as shown above (Figure 3b), at ND>2×(θ×pt)2/nt(increased ΔN0with increasing Nis not able to compensate decreasing ΔN0with increasing ND).

Non-monotonic character of dependence τpon Ndoes not occur and at low concentrations ND{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 δcannot 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 Kpτ^p/τpis achieved at ND2×θ×ptand equals to approximately

(Kp)maxθ×ptnt.E53

Note that with increasing energy Etof recombination level, non-monotonicity of dependence τp(N)fades out (Figure 3b) and then cancels out absolutely. This is caused by the increase in concentration neof equilibrium electrons and decrease in value δand 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^pdecreases with increasing Et(Figure 4a).

The character of dependence τ^pon temperature (Figure 4b) is determined by the following dependences on temperature:

δ2p(T)=1ni(T)×2×ND×pt(T)+4×pt2(T),E54
pt(T)=Nv×exp(EtkT),ne(T)=δ2p(T)×ni2(T)2×pt(T).E55

Values pt(T)and ne(T)increase always with temperature Trise. Increased ptmeans increasing probability of thermal emission of hole from recombination center into valence band. Therefore, the above-mentioned process facilitates increasing τ^pwith Trise. At the same time, growth ne(T)facilitates decreasing τ^pwith Trise due to decreasing concentration of nonequilibrium centers ΔN0of hole thermal emission.

Value δ2pdecreases with Trise at EtEg/3due to approaching N2pcloser and closer to ND(see expression (36)). Value δ2pdecreases also at EtEg/2up to temperature at which non-monotonic dependences τpand τnon Ncancel out. Falling δ2pdecreases ΔN0that facilitates decreasing τ^pwith Trise. When EtEg/2, then pt(T)increases faster and δ2p(T)falls and ne(T)grows. As a result, τ^pincreases with temperature rise (Figure 4b). If EtEg/3, then increased ptwith temperature rise cannot compensate decreased δ2p(T)and growth ne(T). As a result, τ^pdecreases with temperature rise (Figure 4b). If Eg/3<Et<Eg/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).

4.3.2. Electron lifetime

By analogy with hole lifetime, reciprocal electron lifetime consists of three partial components:

τn1=τn11+τn21+τn31E56

First component

τn11=wn×N0e=wn×nt2×B+A×δδ2δ2E57

corresponds to the change of electron capture rate Δn×wp×N0e=Δn/τn1caused by deviation of electron concentration from equilibrium value (capture of excess electrons Δnon equilibrium traps N0e).

Second component

τn21=wn×nt2×(δ×θ1)×(B+A×δδ2)θ×(B+A×δδ2)+(1+δ)×(θ×B+δ)E58

corresponds to the change of electron capture rate ne×wn×ΔN0=Δn/τn2caused solely by deviation of concentration of electron capture centers from equilibrium value (capture of equilibrium electrons neon nonequilibrium capture centers ΔN0).

Third component

τn31=(δ×τn2)1E59

corresponds to the change of thermal emission rate of electrons from impurity level into valence band (1/2)×nt×wn×ΔN=Δn/τn3caused by deviation of concentration of electron thermal emission centers from equilibrium value (thermal emission of electrons from nonequilibrium centers ΔN).

Times τn1, τn2and (τn3) have physical sense similar to times τp1, τp2and (τp3), respectively.

Value δ>>1as long as N<ND, and hence ratio τp1/τn1<<1. Therefore, the occurrence of excess free charge carriers leads to the formation of additional (nonequilibrium) capture centers of electrons and, at the same time, decrease in concentration (ΔN0=ΔN>0) of electrons’ generation centers. Partial components 1/τn2and 1/τn3are positive values at such change of charge state of recombination centers; moreover τn2<τn3, because of δ>1at N<ND. It means that, due to preferable capture of equilibrium electrons on nonequilibrium traps, lifetime of electrons τnis shorter, than capture time of electrons on equilibrium traps. With further increased N, the number of equilibrium capture centers of holes grows, but the number of equilibrium capture centers of electrons remains still small. As a result, concentration ΔN0of nonequilibrium capture centers of electrons increases. For this reason, starting from concentration N=N1n, capture rate of equilibrium electrons on nonequilibrium traps becomes higher than capture rate of nonequilibrium electrons on equilibrium traps. In other words, partial component 1/τn2becomes critical component defining reciprocal lifetime of electrons 1/τn. Component 1/τn2falls with growth Ndue to the decrease in concentration neof equilibrium electrons, so τnτn2and increases with growth N(Figure 1b).

At values Ngreater than ND, the ratio τp1/τn1increases with increasing N. This, again, leads to decreasing concentration of nonequilibrium capture centers of electrons with increasing N. Value necontinues to fall as well. As a result, partial component 1/τn1becomes critical component defining reciprocal electron lifetime 1/τn, and therefore, τn(N)falls with increasing N(Figure 1b).

As shown above, in contrast to dependence τp(N), ratio N1n/NDdecreases (Figure 3a) and ratio Knτ^n/τnalways increases monotonically with increasing ND(Figure 3b). Such regularities are caused by increased newith increasing ND. 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 ΔN0, i.e., at lower values of ratio N/ND. In contrast to the situation with holes, here, decreasing ΔN0with increased NDis compensated by increasing ne.

Similar to the behavior of hole lifetime, non-monotonic dependence τn(N)fades gradually and then cancels out (Figure 3b) with decreasing NDor increasing Et. First regularity is caused by decreased neand δ=2ne/ntwith decreasing ND. Second regularity is caused by decreasing δand, hence, ΔN0, with increasing energy Etof recombination level. In this case, however, due to growth ne, non-monotonicity of dependence τn(N)cancels out at larger values Etthan in the case of holes.

Due to decreasing δ2nwith increasing Et, value τ^ndecreases as well (Figure 4a). The type of dependence τ^non temperature (Figure 4b) is determined only by dependence δ2n(T), because in maximum point τn=τn1/21/N0eδ2n. In zeroth approximation, δ2n(T)coincides with δ2p(T), determined by expression (56). Therefore, τ^ndecreases always with temperature rise (Figure 4b).

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5. Relation between concentrations of photoholes and photoelectrons outside approximation of quasi-neutrality

Note first, from formulas (10), (11), (20), and (23) follow

an=ε8π×q×(1+δ)×f(δ)×wp×ntθ×f(δ)+(1+δ)×(θ×B+δ),E60
ap=ε2π×q×(1+δ1)×f(δ)×wp×ptB+(A+θ×B)×δ+(θ×B+δ)×δ2.E61

Differentiating (12) with respect to x, we obtain

ΔEx=1q×μn×ne×ΔInxE0ne×ΔnxDnμnne×2Δnx2.E62

Recall that ΔEE(x)E0is the change of electric field intensity caused by deviation of concentrations of free charge carriers and their capture centers from equilibrium values by reason of band-to-band absorption of optical radiation: E(x)and E0are electric field intensities with and without illumination.

From equation (16) and formula (5), we find

1q×ΔInx=Δnτn+an×ΔExg.E63

Eliminating 1q×ΔInxfrom equations (62) and (63), we obtain that

ΔEx=11ξ˜n×[1μn×ne×(Δnτng)E0ne×ΔnxDnμn×ne×2Δnx2],E64

where

ξ˜n=anμn×ne.E65

Eliminating ΔExfrom equations (7) and (64) and taking into account expressions (23) and (65), we find that relation between concentrations of excess holes Δpand electrons Δnis determined by the following formula:

Δp=τpτn×Δn+χ×(τpτn×Δng×τpμn×τp×E0×ΔnxDn×τp×2Δnx2),E66

where

χ=an+apμn×nt×δ2an.E67

In quasi-neutrality approximation an=ap=0; therefore, in this case,

Δp=τpτn×Δn.E68
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6. Derivation of equation for distribution function of photoexcited charge carriers’ concentration outside quasi-neutrality

From expressions (12)–(14), it follows that

ΔE=Iphq×(μn×Δn+μp×Δp)×E0+q×Dp×Δpxq×Dn×Δnxq×(μn×ne+μp×pe).E69

Plugging this expression for intensity ΔEof photoinduced electric field in (12) and taking into account (17), we obtain

ΔIn=b×ne×Iphb×ne+pene×Δppe×Δnb×ne+pe×q×μn×E0+(ne×Δpx+pe×Δnx)×q×Dnb×ne+pe,E70

where b=μn/μp=Dn/Dp. Formulas (17) and (70) allow writing

1q×ΔInx=ne×Δpxpe×Δnxb×ne+pe×μn×E0+(ne×2Δpx2+pe×2Δnx2)×Dnb×ne+pe.E71

Plugging relation (66) between Δpand Δnin (71), we find that

1q×ΔInx=(μna+ξ1ξ˜n×τpτn×μp)×E0×Δnx++(Dna+ξ1ξ˜n×(τp×μp×μn×E02+τpτn×Dp))×2Δnx2++ξ1ξ˜n(τp×μpμn×Dn2×4Δnx4+τp×μp×E0×gxτp×Dp×2gx2),E72

where

Dna=ne×τp+pe×τn(pe+b×ne)×τn×Dn=δ2×τp+B×τn(B+b×δ2)×τn×Dn,E73
μna=ne×τppe×τn(pe+b×ne)×τn×μn=δ2×τpB×τn(B+b×δ2)×τn×μn.E74

Formulas (73) and (74) are, none other than, well-known (in quasi-neutrality approximation) expressions for ambipolar diffusion constant Dna, and ambipolar mobility μnaof electrons and dimensionless parameter

ξ=an+apμn×ne+μp×pe=2×(an+ap)×δ(δ2×μn+B×μp)×ntE75

is much less than unity (see below). On the other hand, as it follows from equations (63) and (64),

1q×ΔInx=11ξ˜n×(Δnτng)ξ˜n1ξ˜n×(μn×E0×Δnx+Dn×2Δnx2).E76

Equating (72) and (76), we obtain equation

ξ×Dn×Dp×τp×4Δnx4[(1ξ˜n)×Dna+ξ×τp×μp×μn×E02+ξ×τpτn×Dp+ξ˜n×Dn]×2Δnx2++[(1ξ˜n)×μna+ξ×τpτn×μpξ˜n×μn]×E0×Δnx+Δnτn==g+ξ×τp×(μp×E0×gxDp×2gx2).E77

Because

DnDna=bτp/τnb×ne+pe×ne×Dn,μna+μn=b+τp/τnb×ne+pe×ne×μn,E78

then

(DnDna)×ξ˜n+ξ×τpτn×Dp=ξn×Dn+ξp×τpτn×Dp,
ξ×τpτn×μp(μna+μn)×ξ˜n=ξp×τpτn×μpξn×μn,E79

where

ξn=anμnne+μppe<<1,ξp=ξξn=apμnne+μppe<<1.E80

Therefore, we can rewrite equation (77) as follows:

Q×4Δnx4D×2Δnx2+μ×E0×Δnx+Δnτn=g+ξ×τp×(μp×E0×gxDp2gx2),E81

where

D=Dna+DE+Dξ,μ=μna+μξ,E82
DE=ξ×τp×μp×μn×E02,E83
Q=ξ×Dn×Dp×τp,Dξ=ξp×τpτn×Dp+ξn×Dn,μξ=ξp×τpτn×μpξn×μn.E84

Equation (81) is the desired equation. It depicts adequately the continuity of electron, hole, and total currents [see (15)-(17)].

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7. Solitary Illuminated Sample

In this section, we will consider opportunities for improving photoexcitation of charge carriers and photo-emf Vphby increasing concentration Nof recombination centers.

7.1. Preliminaries: Basic relations

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

For simplicity, we will characterize effectiveness of charge carriers’ photoexcitation by light-propagation-length averaged concentrations of photoelectrons <Δn>and photoholes <Δp>(Figure 5).

Figures 6 and 7 show calculated dependences <Δn>(N)and Vph(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), <Δp>(N)(Figures 8 and 9), and Vph(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 E0=0. The density of photogeneration rate of charge carriers, in view of multiple internal reflections, is determined by the following expression:

g(x)=γ×[a×exp(γx)+a+×exp(γx)],E85

in which

a=(1R)×F01R2×exp(2γW˜),a+=a×R×exp(2γW˜),E86

where Rand γare coefficients of light reflection and absorption, F0is 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 Δpand electrons Δnis as follows:

Δp=τpτn×Δn+χ×(τpτn×Δng×τpDn×τp×2Δnx2),E87

and function Δn(x)obeys the equation

Q×4Δnx4D×2Δnx2+Δnτn=g(x)ξ×τp×Dp×2gx2,E88

where

D=Dna+Dξ.E89

Figure 5.

Layout of sample illumination (а) and generation-recombination processes in semiconductor (b).F0, density of incident photon flux;Vph, photo-emf;ϕ(x), electric potential;W˜, thickness of sample along light propagation;EcandEv, energy of conduction band bottom and valence band top

Figure 6.

Dependences of mean concentration of photoelectrons<Δn>inGaAsfor levels with energy (eV):Et=Et1= –0.42 (curve 1) andEt=Et2= 0.24(curve 2) on concentration of recombination impurityN; layout of sample illumination and axisxare shown on insert. Adopted: light absorption coefficientγ=104cm-1, diffusion constants of electronsDn=221cm3/s and holesDp=10.4 cm3/s [2-9];W˜=103cm;F0=1cm-2×s-1;T=300К; concentration of shallow donorsND=1015cm-3;θwp/wn=102,wn=108cm3/s [1-9], wherewpandwnare capture probabilities of hole and electron

Denote: τnand τpare electron (24) and hole (25) lifetimes; Dn,pare their diffusion constants and values Dna, Dξand Qand dimensionless parameters χand ξ<<1are determined by expressions (73), (84), (67), and (75), respectively.

Figure 7.

Dependence of photo-emfVph(arbitrary units) inGaAsfor levels with energyEt1(curve 1) andEt2(curve 2) on concentration of recombination impurityN. Parameters are the same as inFigure 6

Exact solution of equation (88) is

Δn(x)=i=14Ci×exp(ki×x)+Tn×g(x),E90

where

k1,2=L2L44×ξ×Ln2×Lp22×ξ×Ln2×Lp2,k3=k1,k4=k2,E91
Tn=(1ξ×Lp2)γ2Dn×(γ2k12)×(γ2k22),E92
L2=La2+ξn×Ln2+ξp×Lp2,E93
La2=pe×Ln2+b×ne×Lp2p+b×ne=B×τn+δ2×τpB+b×δ2×DnDna×τn=Dpa×τp,E94

Ln=Dn×τnand Lp=Dp×τpare electron and hole diffusion lengths, Dpais quasi-neutral ambipolar hole diffusion constant, and Lais quasi-neutral ambipolar diffusion length of charge carriers.

In quasi-neutrality approximation, parameters ξ, ξnand ξpare equal to zero; therefore, in this approximation, the distribution of excess electrons’ concentration is determined by equation

Dna×2Δnx2Δnτn=g(x).E95

7.2. Effectiveness of charge carriers’ photoexcitation

We define the mean value <y>of variable y(x)as

<y>=1W˜0W˜y(x)dx.E96

Let’s analyze the worst case, when recombination of excess charge carriers on illuminated (x=0)and unilluminated (x=W˜)surfaces of the sample (Figure 5a) is so intensive that

Δn(0)=Δp(0)=Δn(W˜)=Δp(W˜)=0.E97

From (87), (90), and (97), it follows that

Δn(x)=Δn1(x)+Δn2(x)+Tn×g(x),E98
Δn1(x)=k12×(Ln2×k221)Dn×(k22k12)×(γ2K12)×g(0)×sh[k1×(W˜x)]+g(W˜)×sh(k1×x)sh(k1×W˜),E99
Δn2(x)=k22×(Ln2×k121)Dn×(k22k12)×(γ2k22)×g(0)×sh[k2×(W˜x)]+g(W˜)×sh(k2×x)sh(k2×W˜),E100
Δp(x)=τpτn×Δn(x)τpτn×χ×(Ln2×k121)×(Ln2×k221)Dn×{A1+A2k22k12+γ2×g(x)(γ2k12)×(γ2k22)}E101

where

A1=k12γ2k12×g(o)×sh[k1×(W˜x)]+g(W˜)×sh(k1×x)sh(k1×W˜),E102
A2=k22γ2k22×g(o)×sh[k2×(W˜x)]+g(W˜)×sh(k2×x)sh(k2×W˜).E103

Thus, according to definition (96), we find

<Δn>=ηn×<g>,<Δp>=ηp×<g>,E104

where

ηn=Tn+γ×cth(γ×W˜/2)(k22k12)×Dn×i=12(1)i1×ki×α3iγ2ki2×th(ki×W˜2),E105
α1,2=k1,22×Ln21,E106
<g>=(1R)×F0W×1exp(γ×W˜)1R×exp(γ×W˜),E107
ηp=4πε×(an+ap)×τp×ηρ+τpτn×ηn,E108
ηρ=ε4π×γ×α1×α2(μn×nean)×Ln2×{Λ3cth(γ×W˜/2)k22k12×i=12(1)i×kiγ2ki2×th(ki×W˜2)}<Δρ><g>,E109
Λ3=γ(γ2k12)(γ2k22),E110

and Δρis photoexcited space charge density.

Outside quasi-neutrality approximation, expressions (20)–(25), (60), (61), (75), (80), (91)–(94), and (104)–(110) determine, in parametric form (value δ=Ne/N0eis used as parameter), 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 F0, dependences are exact in contrast to quasi-neutrality approximation case.

Denote desired dependences in quasi-neutrality approximation as <Δn˜(N)>and <Δp˜(N)>. In quasi-neutrality approximation,

Δn(x)=Δn˜τn1γ2×La2×{g(x)g(0)×sh[(W˜x)/La]+g(W˜)×sh(x/La)sh(W˜/La)},E111
<Δp>=<Δp˜><Δn˜>×τpτn,E112
<Δn>=<Δn˜>(1R)×F0×τn(1γ2×La2)×W˜×1exp(γW˜)γ×La×[1+exp(γW˜)]×th(W˜/2La)1R×exp(γW˜).E113

Figure 6 shows that the effectiveness of charge carriers’ photoexcitation may grow significantly with increasing N. Up to small corrections, dependences <Δn>(N)and <Δp>(N)reach maximums <Δn>maxand <Δp>maxat the same concentration N=N^=NDas 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 Land L11/k1, up to small corrections, equal to La; moreover

Ln>L1>L21k2.E114

At W˜<<La, we have

<Δn˜>=ψ(W˜,γ)Dna,<Δp˜>=ψ(W˜,γ)Dpa,E115

where function ψ(W˜,γ)is independent on τnand τp. On the other hand, if inequality W˜<<L2is sufficiently strong, and when Lp>>W˜as well, from expressions (104), (105), and (108)–(110), it follows that

<Δn>=ψ(W˜,γ)Dn,<Δp>=ψ(W˜,γ)Dp.E116

This means that diffusion of photoelectrons and photoholes is independent from each other.

Therefore, L2has physical meaning as shielding length of photoinduced space charge.

Length L2<<La,W˜in the vicinity of the point N=N^, i.e. quasi-neutrality is valid. At values N<N^, length La<W˜due to small value Dna. When Npassing through the point N^toward larger values N, then Dnaincreases very strongly (by several orders of magnitude), and after that it is remaining substantially constant. As a result, length La>W˜at values Nhigher N^. This explains the asymmetry of dependence <Δn>(N)about point N=N^, and “plateau,” when N>N^as well (Figure 6).

At point N=N^and about it, length L2is so long that even at W˜∼ 0.1 cm, solution in quasi-neutrality approximation is unacceptable; moreover, with decreasing W˜

<Δn>max<Δn˜>maxDnaDn<<1,E117

and

<Δp>max<Δn>maxDnDp20,E118

despite the fact that τp<<τn(Figure 1b).

When γ1<<W˜and inequity W˜<<Lnare 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>maxon W˜. For the same reason, at γ1<<W˜<<La, dependence <Δn˜>maxon 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 L2Lawhen impurity level energy equals to Et2. 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).

7.3. Effectiveness of photo-emf excitation

In view of the fact that under considered conditions

Iph=0,E119

from expressions (12)–(14) and (87), we obtain that photo-emf

Vph=0W˜ΔE(x)dx=ϕ(0)ϕ(W˜),E120

i.e. potential ϕ(x)of illuminated surface with respect to unilluminated (Figure 5а) expressed by formula

Vph=Dn×ΔnDp×Δpμn×ne+μp×ne=(Ln2Lp2)×Δn+χ×Lp2×(τn×Δg+Ln2×ΔnΔn)(μn×ne+μp×ne)×τn,E121

where

Δn=Δn(0)Δn(W˜),Δp=Δp(0)Δp(W˜),E122
Δg=g(0)g(W˜),Δn=2Δnx2|x=02Δnx2|x=W˜.E123

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

ΔE(0)=ΔE(W˜)=0E124

And densities of electron ΔInand hole ΔIpphotocurrents

ΔIn(0)=ΔIp(0)=ΔIn(W˜)=ΔIp(W˜)=0.E125

Figure 8.

Dependences of<Δn>maxin the point of maximum functions<Δn>(N)and<Δp>(N)on thicknessW˜forGaAs. Curves 1 and 2, recombination level energyEtequals toEt1andEt2, respectively; solid lines - exact solutions; dashed lines - solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as inFigure 6

Figure 9.

Dependences of ratio<Δp>max/<Δn>maxin the point of maximum functions<Δn>(N)and<Δp>(N)on thicknessW˜forGaAs. Curves 1 and 2, recombination level energyEtequals toEt1andEt2, respectively; solid lines - exact solutions; dashed lines - solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as inFigure 6

Therefore, we may write boundary conditions, in view of relation (87), as

Δnx|x=0=Δnx|x=W˜=0,E126
Dn×3Δnx3|x=0=gx|x=0,Dn×3Δnx3|x=W˜=gx|x=W˜.E127

From (85), (86), (90), (126), and (127), it follows that

C1,2=(1)2,1×b1,2×g0×exp(k1,2×W˜)gW˜2×sh(k1,2×W˜),E128
C3,4=C1,2×g0×exp(k1,2×W˜)gW˜g0×exp(k1,2×W˜)gW˜,E129

where

b1,2=k1,2×(k2,12×Ln21)Dn×(k22k12)×(k1,22γ2),E130
g0gx|x=0=γ2×a×[1R×exp(2γW˜)],E131
gW˜gx|x=W˜=(1R)×exp(γW˜)1R×exp(2γW˜)×g0.E132

Find consistently

Δn(0)=Tn×g0+i=12(1)i+1bi×[g0×cth(ki×W˜)gW˜×cosech(ki×W˜)],E133
Δn(W˜)=Tn×gW˜+i=12(1)i+1bi×[g0×cosech(ki×W˜)gW˜×cth(ki×W˜)],E134

where

g0g(0)=γ×a×[1+R×exp(2γW˜)],gW˜g(W˜)=(1+R)×exp(γW˜)1+R×exp(2γW˜)×g0.E135
Δn=Tn×Δg+i=12(1)i+1×bi×th(ki×W˜2)×Σg,E136
Δn=γ2×Tn×Δg+i=12(1)i+1×ki2×bi×th(ki×W˜2)×Σg,E137

where

Δg=g(0)g(W˜)=γ×F0×(1R)×1exp(γW˜)1+R×exp(γW˜),E138
Σg=g0+gW˜=γ2×F0×(1R)×1+exp(γW˜)1+R×exp(γW˜).E139

If we utilize relation

ZLn2Lp2χ×Lp2×(k12×Ln21)×(k22×Ln21)=0,E140

then expressions (121) and (136)–(139) allow us to obtain

Vph=(1R)×γ×F0×(Ln2Lp2)μn×ne+μp×pe×M,E141

where

M=k12×k22×f(γW˜)(k12γ2)×(k22γ2)+γ×k1×k2k22k12×[k1×th(k2×W˜/2)k22γ2k2×th(k1×W˜/2)k12γ2]×f+(γW˜),E142
f,+(γW˜)=1exp(γW˜)1+R×exp(γW˜).E143

In quasi-neutrality approximation, we get

Vph=V˜(1R)×γ×F0×(Ln2Lp2)μn×ne+μp×pe×M˜,E144

where

M˜=γ×La×f+(γW˜)×th(W˜/2La)f(γW˜)(γ×La)21.E145

We refer to the dependences (141) of photo-emf Vphon Nand 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 F0.

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 Vph(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 L2in the point of maximum N=N^NDand nearby will cause diffusion of photoelectrons independently on photoholes (see Section 6.2) at W˜∼ 0.1 cm. In other words, electron diffusion constant Dndetermines the distribution of photoelectron concentration Δn(x). Aside from that, at NN^, due to charge coupled to impurities, inequity Dn>>Dnais fulfilled. It means that true effectiveness of photoelectrons’ spreading is much higher, than that given by quasi-neutrality approximation. Therefore, ratio (Vph)max/(V˜ph)max<<1, when W˜>>L2(Figure 11).

Evidently, with thinning W˜, spreading area of photoexcited charge carriers becomes wider, i.e., values Δn=Δn(0)Δn(W˜)и Δp=Δp(0)Δp(W˜)decrease. Therefore, values (Vph)maxand (V˜ph)maxfall with thinning W˜(Figure 10).

Note that for impurity level energy equal to Et2, the solution obtained in quasi-neutrality approximation, even when W˜, differs from the exact solution more than two times (Figure 10). This is due to the fact that in considered case the shielding length of photoexcited space charge L2=1/k2La, where Lais ambipolar diffusion length of charge carriers.

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8. Photoelectric gain

We will consider uniform spatial distribution of density of charge carriers’ photoexcitation rate gand ignore surface recombination of photoexcited charge carriers.

We will facilitate mathematical description of photoelectric gain (see Figures 1а and 2)

G(N,V)Iphq×W×gE146

through utilizing small dimensionless parameter (75) ξ<<1, which will characterize the degree of deviation of semiconductor from local neutrality under illumination. Here Nis concentration of recombination centers; Iphis photocurrent density (14); qis absolute value of electron charge; Vis bias voltage applied across the sample; Wis distance between current electrodes (see insert in Figure 1a).

Using linearized expressions for electron (12) and hole (13) components of photocurrent Iphand expressions (60) and (61), we may rewrite equations (5)–(7) as follows:

Rn=Δnτnξn×jx,Rp=Δpτp+ξp×jx,Δp=τpτn×Δnξ×τp×jx,E147

where

j=(μn×Δn+μp×Δp)×E0+Dn×ΔnxDp×Δpx,E148

E0=V/W, and dimensionless small parameters ξn<<1and ξp<<1as defined in (80). Relations (147) show that, by dimensionless small parameter ξ, we may really characterize the degree of deviation of semiconductor from local neutrality under illumination. If there is no external load (inset in Figure 1a), then illumination does not change voltage Vacross the sample.

Therefore, expressions (4), (5), (8), and (9) with boundary conditions (1) allow us to write

Iph={(b+τpτn)×Δn+ξ×Lp2W×(Δpxb×Δnx)|x=0x=W}×q×μp×E0,E149

where nis arithmetic mean with respect to xconcentration of excess electrons (insert in Figure 1а)

Δn=1W×0WΔn(x)dx,E150
b=μn/μp.

Actual distribution Δn(x)at g(x)=constis defined by equation

Q×4Δnx4(Dna+DE+Dξ)×2Δnx2+(μna+μξ)×E0×Δnx+Δnτn=g.E151

Values of parameters Q, Dξ, μξ, and DEare 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) terms with parameters Q, Dξand μξ. However, term with parameter DEmust be retained, because, even at moderate electric fields, DEmay exceed Dnadue to square-law dependence DEon E0. For the same reason, we can omit in equation (149) terms including small parameter ξ. Thus, we arrive at relations

(Dna+D)E×2Δnx2μna×E0×ΔnxΔnτn+g=0,E152
Iph=q×(μn+τpτn×μp)×Δn×E0.E153

Figure 10.

Dependences of maximal valueVph(N)inGaAson thicknessW˜. Curves 1 and 2, recombination level energyEtequals toEt1andEt2, respectively; solid curves, exact solutions; dashed curves, solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as inFigure 6

Figure 11.

Dependence of ratiort=V˜ph/Vphin the point of maximal valueVph(N)inGaAson thicknessW˜. Curves 1 and 2 – recombination level energyEtequals toEt1andEt2.Vph, exact solutions;V˜ph, solutions in approximation of quasi-neutrality. Adopted parameters and other symbols are the same as inFigure 6

Equation (152), with boundary condition (1), and relation (153) allow us to obtain the formula for photoelectric gain:

G=(μn×τn+μp×τp)×E0W×{14×LefW×(da2Lef)2+1×sh(W2L1)×sh(W2L2)sh(W2L1+W2L2)},E154

where quasi-neutral ambipolar drift length of charge carriers

da=μna×τn×E0,E155

effective diffusion length of charge carriers

Lef=(Dna+DE)×τn,E156

and effective reciprocal diffusion-drift lengths L1and L2are defined by expressions

1L1,2=±da2Lef2+(da2Lef2)2+1Lef2.E157

Relations (18), (20)–(22), (24), (25), (73)–(75), (83), and (153)–(157) determine, in parametric form dependence G(N,V)(see Figures 1a and 2а). Ratio δ=Ne/N0eis used as parameter in said relations.

It can be shown that equation μna(N)=0, where μnais 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 τn(N)and τp(N)and equals to ND(Figure 1b and 1c).

Let’s explain dependence μnaon Nshown in Figure 1c.

Product

μna×τn=μpa×τpE158

determines drift length and direction in electric field of concentrational perturbation – quasi-neutral cloud of positive and negative charges [2, 31], including bound at deep impurity (here μnaand μpaare electron and hole ambipolar mobility). Last mentioned bounding explains dependence μnaon ratio τp(N)/τn(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 (τn=τp), 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] (μna=μpaμp>0 at ne>>pe, μna=μpaμn<0 at ne<<peand μna=μpa=0 at ne=pe). Similar situation, but not exactly identical, happens in the case of trap-assisted recombination.

In this case, due to the fact that τp<τn(Figure 1c), vanishing μnaoccurs in n-type material (for specified parameters in Figure 1, at ne10×pein silicon and ne104×pein gallium arsenide). Positive sign of perturbation charge bound at deep impurities (ΔN<0) causes such behavior. Ratio pe/nebegins to increase significantly, and very sharply, only when NND. At the same time, ratio τp/τnmay not have so many orders of smallness as ratio pe/nemay have. Therefore, μnavanishes when NND, if deep level, according to conditions (27) and (41), lies in lower half of bandgap. If that level lies in upper half of bandgap, then, again, due to the fact that τp/τnmay not have so many orders of smallness as ratio pe/nemay have, function μna(N)is always positive for actual values N, wherein lifetimes τpand τndecrease always with increasing N(see Section 3). We denote solution based on relations (152) and (153) as approximate. Parameter ξ(N), still remaining small, reaches its global maximum near point N=N^=ND, where function G(N)reaches maximal extremum G^(Figure 1a). Deviation of approximate value G^=G^apprfrom exact value G^=G^exact{calculated with due regard for all terms in relation (149) and equation (151)} is shown in Figure 2b. Exact solution at N=N^is not so difficult to find, because at this point μna=0. It is clear from Figure 2b that agreement G^apprwith G^exactis quite good.

From expression (154), it follows that

G^={12×L^efW×th(W2L^ef)}×(μn×τ^n+μp×τ^p)×E0W,E159

where

L^ef=L^a2+L^E2E160

is effective ambipolar length (156) at N=N^(i.e., at δA+B) and τ^nand τ^pare relevant electron (48) and hole (38) lifetimes. We can write that

τ^n=A+B2×wn×ND,τ^p=2×A+θ×B×A+B2×wp×ND×(A+B),E161
L^a2=(A/θ)+B×A+B(A+B)×Dn+B×Dp×Dn×Dpwn×ND,E162
L^E2=ε8×π×q×2×A+(B+1)×θ×A+Bwp×ND×nt×A+B×μn×μp×E02(A+B)×μn+B×μp.E163

Value L^ais conventional ambipolar diffusion length at maximal extremum of function G(N)(Figure 1a) calculated in quasi-neutrality approximation, i.e., when parameter ξis set to zero.

From expression (159), it follows that function G^depends non-monotonically on applied bias voltage V(Figure 2a). This is caused by increased L^efwith increasing E0=V/Wthat 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 Def=Dna+DE(coefficient before second derivative in equation (152)) causes elongation L^efwith increasing E0. In turn, photoinduced space charge Δρ(4) causes monotonic increased Defwith increasing E0. The analysis of expressions (159)–(163) shows that function G^(V)reaches its maximum value (Figure 2c)

G^max(W){2572×WL^a×G^maxmax,whenW<2×L^aG^maxmax,whenW>2×L^a,E164

when bias voltage is applied across sample V=Vopt(W), where optimal bias voltage (Figure 2d)

Vopt(W){E˜0×W,whenW<2×L^aE˜0×W22×L^a,whenW>2×L^aE165

Threshold value G=G^maxmax(for given physical parameters of semiconductor) and electric field intensity E0=E˜0, at which L^ef=2×L^a, are defined by the following expressions:

G^maxmax=(μn×τ^n+μp×τ^p)×3×E˜025×L^a,E166
E˜=8π×kT×niε×(1+AB)1/4.E167
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8. Summary

Capture rate of excess charge carriers increases with increasing concentration Nof 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 τnand holes τp, 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 non-monotonic 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 τn(N)and τp(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).

At and about point N=N^ND, where charge carriers’ lifetime is maximal, equilibrium concentration of charge carriers becomes small, where NDis shallow dopant concentration. Therefore, increase in Dember photo-emf Vphin several orders of magnitude (Figure 7) is caused by both strongly non-monotonic dependences τn(N)and τp(N)(Figure 1b) and highly non-monotonic dependence of sample dark resistance on concentration N[1-3, 8, 30, 31].

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 Gwith increasing N(Figure 1a). As follows from [18, 19], Gincreases with increasing charge carriers’ lifetime, if ambipolar mobility μa(see (74), (158), [2]) is equal to zero, or if there is no recombination on current contact electrodes (x=0and 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 (μa0), increase in lifetimes, beginning from some lifetime values, does not increase in photocurrent density Iph[5, 18, 19].

Saturation in Iphis 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 τn(N)and τp(N), vanishes at the same, up to small, correction value NN^, at which functions τn(N)and τp(N)reach their maximal extrema τ^nand τ^p(Figure 2b and 2c). Therefore, Iphand, consequently, Gincrease to the extent of increasing τ^nand τ^p. These are physical grounds of giant splash in photoelectric gain Gwith 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 Nphand holes Pphand Dember photo-emf Vphmay be fundamentally different from solutions obtained in approximation of quasi-neutrality N˜ph, P˜ph, and V˜ph, respectively.

It may be that Pph/Nph>>1even if hole lifetime τpis much less than electron lifetime τn(Figure 2b). At the same time, in approximation of quasi-neutrality, P˜ph/N˜ph=τp/τn<<1(Figure 2b). At point N=N^, at which functions Nph(N), Pph(N), and Vph(N)reach maximum values, and for thin samples (with thickness along light propagation W˜<0.1cm), 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 Ndepends non-monotonically on applied voltage Vacross 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~ V2is caused by photoinduced local space charge. What is important is that, at optimum voltage Vop(Figure 2d), value Gcan have several orders of magnitude (Figure 2c) at high concentrations of recombination centers.

As shown in [33], when recombination impurity Nhas three charged states (two-level recombination center), then, again, strong increase in τn(N)and τp(N)with increasing Nmay occur; moreover dependences τn(N)and τp(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 Nand 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.

Above-mentioned regularities occur at arbitrarily low-level photoexcitation and they become the more evident, the wider the semiconductor bandgap.

The theory of giant splash of photoresponse in semiconductors at low-level illumination with increasing concentration of recombination centers could develop further through the 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.

From physical essence of considered effects, it follows that similar effects can occur in other mediums with recombination of dissociative or ion-ion type, for example, in multicomponent plasma [36]. More details about topic are given in [37-46].

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Acknowledgments

Authors are grateful to Prof. Sergey A. Nikitov for valuable support of publication.

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Written By

Viacheslav A. Kholodnov and Mikhail S. Nikitin

Submitted: October 5th, 2014Reviewed: June 10th, 2015Published: October 7th, 2015