J. Phys. Chem. 1994, 98, 12653-12662
12653
Computer Simulation of the Photoluminescence Decay at the GaAs-Electrolyte Junction. 1. The Influence of the Excitation Intensity at the Flat Band Condition 0. Kriiger' and Ch. Jung FB Chemie, Humboldt-Universitat zu Berlin, Walther-Nernst-Institutf i r Physikalische & Theoretische Chemie, Hessische Strasse 1-2, 0-10115 Berlin, Germany
H. Gajewski Institut f i r Angewandte Analysis und Stochastik im Forschungsverbund Berlin e. V., Hausvogteiplatz 5-7, 0-10117 Berlin, Germany Received: October 26, 1993; In Final Form: June 20, 1994@
The influence of excitation intensity, surface recombination velocity, and photocunent on the time-resolved photoluminescence (PL) at the nlGaAs-electrolyte junction under flat band conditions was investigated using computer simulations. The mathematical background of the two-dimensional semiconductor analysis package (TOSCA) will be presented. We will show that the effects may be characterized qualitatively by exponential fit function(s), although in most cases the simulated PL decay is nonexponential. For a free carrier concentration no = 5 x 1017cmP3, under the conditions of low injection density (pi)and with flat bands, both the surface recombination and the photocurrent lead to a remarkable decrease in the PL decay time either for surface recombination velocities SO> lo3 cm/s or for exchange photocurrent densities j@,O > lo-'' A/cm2. Under the conditions chosen in this work, the PL decay time approximated by the monoexponential decay time T decreases to a minimum value in Tfin = 1.64 ns when SO I lo6 cm/s or jph,O L lov8A/cm2. This minimum indicates that the diffusion of the minority carriers toward the surface where they immediately nonradiatively vanish by means of surface recombination and/or charge transfer, respectively, is limited by the thermal velocity. In both cases, diffusion competes effectively against bulk recombination. For injection levels (p,/no)> 0.1, the PL decay time decreases with increasing injection density because the quadratic recombination process dominates. At high injection densities, the surface recombination velocity may not remain constant during the PL decay, but instead varies as a function of time S(t); S may decrease from its maximum value (SO)by up to 1 order of magnitude during the PL decay. The results will be discussed in relation to experimental results published previously by other groups.
(I) Introduction Recently, interesting studies on photoelectrochemical cells (which are important in the context of liquid junction solar cells, waste treatment, material synthesis and processing, and sensors) have focused on the charge transfer at semiconductor-liquid electrolyte junctions.',* In this context, the time-resolved photoluminescence (PL) seems to be a powerful tool fulfilling the following requirements: PL is contactless, destructionless, and under certain circumstances leaves the electrodes unchanged. Although there are many papers about this method, there is up to now no universal analytical solution (including all important influences: high-injection effects, space charge field and band bending, generation attributed to selfabsorption-photon recycling) for the calculation of the PL decay in the nanosecond range, in spite of the different models propo~ed.~-'~ To solve the problems concerning the quantitative interpretation of the experimental results obtained for the PL decay of semiconductor electrodes under different experimental conditions, several approximations, simplifications, and assumptions were made: (1) The intensity of the exciting light was chosen or assumed to justify either the low injection10s12or the high injection limit,8911.15 which is defined in terms of the injected density of minority carriers (pi) and the free carrier concentration (no) as @
Abstract published in Advance ACS Abstracts, November 1, 1994.
0022-365419412098-12653$04.50/0
pi > no, respectively. For the calculation of the luminescence signal, one assumed either a linear dependence on the minority carrier concentration in the low injection limit10~'2*'6 or a quadratic dependence in the high injection limit.338*11In this way it is possible to analytically treat the limiting cases, but not the transition range between both injection limits. (2) The influence of the space charge field on the carrier dynamics was neglected in the time-dependent (diffusion) model by either implicitly assuming the flat band case3-7,9,12,16 or assuming the excess minority carrier density is zero at the inner edge of the depletion region.'O The flat band case was realized experimentally by an externally applied p ~ t e n t i a l ' ~or , ' ~by the high injection limit.8311915 (3) The rate of the bulk recombination r is assumed to be linear in p, generally expressed by r = plt in the continuity equation for the minority carrier c~ncentration.~ (4) The surface recombination velocity S, characterizing the process of nonradiative surface recombination, is usually considered to be independent of the injection density and is incorporated in the model as a constant in the boundary condition of the continuity equation. Sometimes contradictionary assumptions were used: the bands were assumed to be flat in the low injection limit without externally applied p~tential,'~ or at high injection, the recombination kinetics was assumed to be first order.I3 Recently, we have shown the influence both of the excitation intensity at constant externally applied potential18 and of the externally 0 1994 American Chemical Society
12654 J. Phys. Chem., Vol. 98, No. 48, 1994
Kriiger et al.
applied potential in the low injection limitlg on the PL decay at PL decay curves for n-GaAs electrodes. These simulations were n-GaAs electrodes. obtained by a numerical solution of the differential equations including the dynamic terms, for example, for pulse-shaped Corresponding to the law of absorption, the generation of illumination. The aim of this paper is to investigate the excess carriers will be nonhomogeneous, and therefore, it is influence of the excitation intensity and the band bending on difficult to collect the emitted PL from a spatial range of the the PL decay by means of computer simulation. In the first sample in only one of the two limits. part of this paper, we focus on the flat band case. In the second To illustrate the statements made above, some aspects of a part of this paper, the results obtained for the case of a depletion study recently published by Shumaker et al.14 will be considered. layer are described.z4 These authors investigated the PL decay in the nanosecond range The structure of part 1 of the paper is as follows: In section for n-CdSe electrodes in 0.5 M KOH under different conditions. 11,the computer simulation program TOSCA is described. Here The concentration of free electrons was 0.3 x 10I6 ~ m - and ~, the injection density was varied between l O I 4 and loL8~ m - ~ . we present the mathematical background, a short summary of the models used for generation and recombination, including The PL decay was measured near the flat band condition (U = considerations about the dependence of the different lifetimes -0.903 V vs SCE) and under reverse bias (U = -0.203 V vs on the injection density, and the model used for the semiconSCE). By measuring the PL decay at different emission ductor-electrolyte junction. Section I11 describes the analysis wavelengths while keeping the excitation conditions constant, of the simulated PL decay curves. In section IV, the influence they tried to measure the effects of self-absorption-photon of the surface recombination and the photocurrent will be recycling and the inner filter effect. Shumaker et al. fitted their discussed for both the low and high injection cases. In the case results to an analytical solution for the decay law of the oneof high injection, we discuss the time dependence of the surface dimensional diffusion equation by assuming flat bands and linear recombination velocity during one excition pulse. Finally, we recombination processes (low injection limit).g discuss the magnitude of the parameters that characterize the With respect to the following two points, the interpretation surface processes. of their results allowed them to go beyond the limits imposed by the chosen analytical decay law: (11) Computer Simulation Program TOSCA (1) Using a random walk algorithmz0 for the computer simulation of the decay law, they varied the occupation time of (1) Mathematical Background. The PL decay was simusurface trap sites. Shumaker et al. visualized the limiting role lated on the basis of the two-dimensional semiconductor analysis that the number of these sites has on the surface recombination package (TOSCA) which was developed by Gajewski and covelocity and the effect of their saturation. The latter effect worker~.*~The physical background of TOSCA is briefly introduces nonlinear surface recombination. They showed that explained in this section. the PL decay time increases with increasing occupation time if TOSCA is based on the fundamental equation system the ratio of the number of surface trap sites to the number of formulated in 1950 by van Roosbroeck.26 This equation system initially injected carriers is kept constant. This assumption does consists of Poisson’s equation (eq 1) and the continuity equations not seem reasonable to us. for electrons and holes (eq 2). (2) They considered photon recycling by including the inner -V(EE~V@) = e, e = e@ - II N,, - NA-) (1) filter effect in the decay law according to ref 16 and showed the influence of the emission wavelength on the observed PL decay. For longer emission wavelength, they observed longer decay processes which could be modeled excellently by the modified decay law, although their parameters for the diffusion eaP Vjp= e(g - r ) and the absorption coefficient deviated considerably from the at experimental values. The first trials measured at a single electrode under the same The symbols are defined in the list at the end of the paper. The conditions showed a trend toward shorter decay times caused diffusion of carriers is due to the gradient of the corresponding by a change of the electrode surface. In spite of the shortelectrochemical potentials (quasi-Fermi potentials). The current comings of the decay law used in their interpretation, and of density can be calculated according to the time evolution of the electrode due to the very long collection times (between 1 and 11 h) that were necessary for the j , = -enp,VEF, measurement of a PL decay curve (pulse repetition rate 300.4 kHz), the following important conclusions from this work should (3) be noted:21 (1) The observed PL decay time is shortened going from low In the case of nonstationary conditions, the total current is given injection density to high injection density by varying the density by the continuity equation over 4 orders of magnitude. v-j = 0 (2) It was shown experimentally that the PL decay time is shortened by moving from flat band into the depletion range. a4 j =j , j , - ccoV (3) It was demonstrated that the rate of the recombination (4) at processes changes nonlinearly with the excitation density. (4) The inner filter effect may have important consequences The carrier concentration in the energy bands at thermal for the PL decay depending on the observed emission waveequilibrium is calculated by TOSCA according to the Fermilength. Dirac statistics.*’ An alternative way of interpreting the PL results was given TOSCA solves the equation system of van Roosbroeck for for the case of a steady-state illuminated semiconductor electrode any two-dimensional area. The interaction of the semiconductor using the numerical solution of the differential equation^.^^^^^ bulk with different ambient environments is defined by suitable In this paper we present results of a computer simulation of the boundary conditions.28 In this paper, the following three cases
+
+
+
J. Phys. Chem., Vol. 98, No. 48, 1994 12655
Computer Simulation of Photoluminescence Decay are considered: (i) ohmic contact (eq 5 ) , (ii) gate contact (eq 7), and (iii) insulator and symmetry conditions (eq 8).
According to Slotboom, the effective intrinsic density ni, represents the dependence on temperature ni(T) and on dopant concentration niG(no).29
According to the boundary conditions for a gate contact (eq 7), the intrinsic charge in surface states Qs varies with the thickness of oxide layer dox,de and may be described by the following expression
Q, = ~ ( E E o ' @ )
where the nominator represents the total amount of the injected carriers and the denominator represents the approximate volume in which most of the excess carriers are generated. pi is a useful parameter for characterizing the excitation intensity, whereas Sp is more useful in the mathematical considerations following in this subsection. A comparison of the results obtained for computer simulations with samples of different free carrier concentrations (dopant densities) is possible on the basis of the ratios pilno. This ratio is called the injection level. Presently, the self-absorption of the luminescence (photon recycling) is not included in the treatment, although this process may have important implications for the carrier dynamics.14J6s31,32 The rate of radiative recombination r, can be calculated according to eq 12 from carrier densities, provided that the
-I- (Eo,ideEddoxide)(@ - @A -
kj = kj = 0 n
P
r, = Bnp
(7)
where i is the unity vector perpendicular to the corresponding edge of the semiconductor. The second condition in eq 7 implies that no current flows out of the semiconductor. It was modified as described in subsection 3 (eq 24). At the other edges of the semiconductor bulk (without special boundary conditions), the natural boundary condition (eq 8) is applied. In TOSCA, the carrier transport bV@=kj" = k J P
=o
coefficient of radiative recombination B is known. From eq 12, which is used to calculate the rate of radiative recombination in TOSCA, the two limiting values for the radiative lifetime of minority carriers (holes) can be derived. All of the following explanations refer to a n-doped semiconductor. In the case of low injection (Sp E, if E, > Ei
(17) or
nl andpl denote the concentration of electrons in the conduction band and that of holes in the valence band, respectively, if the energy of the recombination center in the band gap is situated at the Fermi level. t n 0 is the lifetime of the electrons when all recombination centers are occupied with holes, and tpo is the lifetime of the holes when all recombination centers are occupied with electrons. Its reciprocal value is the maximum of the rate of Shockley-Read-Hall recombination. Provided that no >> n1, p1, PO, it can be shown that ZSW derived from eq 16 depends on the excitation intensity ( 6 ~ ) : ZSN increases with increasing injection levels up to a limiting value. 11,12bs35 Another process leading to a decrease of excess carrier concentration is the surface recombination. According to Stevenson and Keyes, the rate of surface recombination rs is given by eq 1836and is implemented in TOSCA. n p , - n;
= (n,
+ nl)/Sp + (P, + PJS,
Sp= Nsvpop, Sn= Nsvnon It should be noted that p s and ns are the (time-dependent) nonequilibrium densities of holes and electrons at the surface. The surface recombination velocity S is defined as the velocity of diffusion of the minority carriers to the surface where they nonradiatively recombine. The physical upper limit of S is the thermal velocity of carriers (for GaAs: vn = 4.4 x lo7 c d s and vp = 1.65 x lo7 cm/s3'). In the following discussion, we show that, in the case of steady-state illumination, S depends on the injection density. In the case of a n-doped semiconductor with one electronic surface state situated at Es, the surface recombination velocity is given by eq 20 when the following conditions (eqs 19) are used: (i) thermal equilibrium is established between the space charge layer and the field-free bulk (which is rather improbable during pulse-shaped illumination of nanosecond-pulse duration), (ii) the excess carrier densities are equal in each spatial element (which is rather improbable for the space charge layer because electrons and holes move in opposite directions), and (iii) the capture cross sections for electrons and holes as well as the thermal velocity of electrons and holes do not differ significantly from each other.
(ii) n = n o + 6 n , p = p o + 6 p , (iii) Sp= S, = So
dn=dp (19)
Here, nl and p1 are the surface values corresponding to eq 17. From eqs 19i and 20, it is obvious that the surface recombination velocity depends strongly on the surface concentration of caniers (surface potential, band bending ~ A ~ ~ s c L ) :on ~ ,the ~ ~dopant -~~ and on the injection density ( d ~ ) In . ~the~ flat band condition (no,s = no, p0.s = PO,Ei,s = Ei), the values of nl and PI are negligible compared to no when (compare eq 17)
(B) EF 2Ei - E, if E, < Ei So one can distinguish the case of low injection resulting in S' = Sp= Nsvpop
(21)
from the high injection condition resulting in
This result holds for the steady-state case. The surface recombination velocity decreases with increasing injection density.lZb Therefore, in the case of a time-dependent (high) injection density caused by a short light pulse, one can expect that S also varies during the light pulse. A similar effect has been found experimentally and discussed in a few s t ~ d i e s . ' ~ J ~ ~ ~ " Taking into account that the Stevenson-Keyes expression involves only one kind of surface state, it should be noted that surface states lying closer to the intrinsic level at the surface Ei,s than the Fermi level (EF) are particulary efficient (I& &.SI < IEF - E ~ , s I ) ~ ~ Of course, the superposition of all recombination processes yields the effective lifetime of minority carriers
From the above considerations, it is clear that the individual lifetimes depend on the injection level in quite a different manner. The question arises whether and how the injection level effects the PL decay time. In TOSCA, the number of recombining carriers is calculated separately for each individual recombination process. The PL decay curves were obtained by plotting the integrated rate of radiative recombination versus time (eq 12). (3) Model of the Semiconductor-Electrolyte Junction. The computer simulations of time-resolved PL at the semiconductor-electrolyte junction are based on the following assumptions: At the front side of the semiconductor, there is a 0.3 nm thick oxide layer with the dielectric constant of water (EW = 78). At the back, there is an ohmic contact. The bulk of the semiconductor considered here is characterized geometrically by its cross section of AL = 0.1 x 100 pm2 and a thickness of z = 200 p m (Figure 1). The illumination of the cross section was assumed to be homogeneous in the ydimension and to be Gaussian shaped in the x dimension with the center at xo = 50 pm (see eq 9). The physical constants of the semiconductor corresponding to those of GaAs are listed in Table 1. In the model of the Shockley-Read-Hall recombination, we assumed that tn0 = t p o = 5 ns and that the recombination center is situated at the intrinsic level (Et = Ei because of nl = p1 = ni = 2.25 x lo6 ~ m - ~ )The . same energy position was assumed for the surface state (Es = Ei,s). The space charge layer (depletion layer) in equilibrium in the dark was simulated by a contact potential ( 4 ~at) the gate contact corresponding to a band bending of -600 mV. By varying the potential @A, it is possible to change the band bending. To ensure the flat band condition, we used q 5 = ~ 600 mV for complete compensation of q 5 in ~ each case. Taking these boundary conditions into consideration, the initial spatial
Computer Simulation of Photoluminescence Decay
J. Phys. Chem., Vol. 98, No. 48, 1994 12657 at the surface is reduced because of the relative high surface recombination velocity SO = lo5 c d s . In this paper, we do not distinguish between S,, and S, in our simulations, and so the TOSCA input parameter is denoted SOaccording to eq 19iii. The resulting maximum decreases, flattens, and shifts into the bulk with increasing time. This observation corresponds to the results of V a i t k u ~ ,Boulou ~ and and of Tyagi et aL5 obtained by analytically solving the time-dependent continuity equation for the minority carriers. To add a photocurrent at the semiconductor-electrolyte junction, the TOSCA model of the gate contact (eq 7) was modified by a time-dependent photocurrent density j p h ( f ) . According to Geri~cher:~it may be written
x=100 vm
y = 0.1 vm Y
2.200
um
X
E "0xide"t
semiconductor
Ohmic
gate contact
contact
Figure 1. Geometrical and physical model for a semiconductorelectrolyte junction used in TOSCA. The electrolyte is simulated by an oxide layer (gate contact) with the dielectric constant of water: solid line, equilibrium state with a band bending (e&); dotted line, state with an extemal potential @A.
Presently, an equivalent expression for (cathodic) electron current is not included in TOSCA. (111) Analysis of the PL Decay
The PL decay curves were fitted with (mono- or bi-) exponential decay functions
i= 1
s
4 ~ 1 0 ~ ~
The quality of the curve fit was checked with the residuals (leastsquares values SQ, eq 26)
8
Y
a 2x10'0
WEAf -MINIMUM n
SQ =
i= 1
n
0
&lo3
Zr103
6x103
Ai = ?(ti) - ?,(ti), w = l/(n - P )
= [ml Figure 2. Simulations with TOSCA. Profiles of hole density at different times r during and after pulse-shaped excitation according to eq 10, with, , r = 2 ns and t H = 0.5 ns at the flat band case and at the low injection condition pi/no = 1.2 x with SO = 105 c d s : (1) t = 1.04 ns, (2) t = 2.00 ns, (3) t = 3.12 ns, (4) 1 = 4.24 ns, ( 5 ) r = 6.00 ns, and (6) t = 10.00 ns.
physical parameter
value = 12.85
1
n-i
j = 1, 2,
n
..., -
ref
EG = 1.423 eV ~ ~ = 4 . 2x 11017cm-3 N" = 9.51 x 1018 cm-3 ,un= 8000 cm2Ns ,up = 320 cm2Ns B= cm3/s
37 37 37 37 37 37 46
C, = C, = cm6/s a(670 nm) = 3 x lo4 cm-'
35 47
n, = 2.25 x lo6cm-3 no = 5 x 10'' cmW3
37
E
and with the autocorrelation function C(t) (eq 27, with i and j including only the n points from the fitted range of the curve).50
2
TABLE 1: Physical Parameters of GaAs Used in the TOSCA Simulations dielectric constant band gap state density in conduction band state density in valence band electron mobility hole mobility coefficient of radiative recombination coefficient of Auger recombinatic)n absorption coefficient for exciting photons intrinsic carrier pair density dopant density
(26)
profiles of the carriers as well as of the potential in the dark were given by the (stationary) solution of the van Roosbroeck's equation system (eqs 1 and 2). Subsequently, the nonstationary solutions (time behavior) resulting from the pertubation by pulseshaped excitation (eq 9) were calculated. In Figure 2, the density of holes at different times during and after the excitation pulse is plotted for the case of low injection and flat bands. It is obvious that the density of holes
P denotes the number of fit parameters in the fit function Z(t). The simulated PL decay curve j(t), consisting of 250 points, is considered to be a convolution (eq 28) of the PL decay and the "apparatus function" &t) (excitation pulse).
The parameters of the fit function (eq 25) were calculated by iterative deconvolution. In Figures 3 and 4, typical simulated PL curves and least-squares fits are presented for the flat band condition under low injection with no photocurrent. For low surface recombination velocities (SO < lo4 c d s ) , the quality of a monoexponential fit seems to be satisfactory, although the rising part of the curve cannot be described exactly by one exponential term. Here, the calculated fit function differs systematically from the simulated PL curve (see A-curve). The quality of the monoexponential fit decreases significantly as the surface recombination velocity increases (SO 2 lo4 c d s ; compare Figures 3 and 4i). A quite similar fitting curve of the PL decay is observed forjph,O =- lo-" A/cm2. As shown in
Kriiger et al.
12658 J. Phys. Chem., Vol. 98, No. 48, 1994
1 ns -
-r (4)
Figure 3. Monoexponentialfit (curve 3) of a PL decay curve simulated
with TOSCA (2) assuming a Gaussian-shaped excitation pulse (1) according to eq 10, with tmax= 2 ns and t~ = 0.5 ns. The starting point of the fitting procedure (5% level of P L U ) is indicated by the line (4). The figure includes the residuals A and the autocorrelation function C(t). Simulation conditions: low injection level piho = 1.2 x flat bands, without photocurrent (for the other parameters, see Table 1). Figure 4ii, an additional exponential term improves the fit, but the initial part of the curve is still fit poorly. Application of the fitting procedure only to the descending part of the curve does not significantly improve the quality of the fit (Figure 4iii). Therefore, all the following results refer to fits of the nearly complete decay curve-the starting point of the fit procedure was always at the 5% level of P L m . In all cases, the autocorrelation functions (C(t)curves) show clear oscillations, indicating that the chosen fit functions do not correspond exactly to the decays observed. It is well-known that the initial part of nonexponential PL decay can be due to surface r e c ~ m b i n a t i o n . ~Nevertheless, ~ ~ ~ ~ ~ - ~ ~our results show that it is possible to characterize the PL decay by the parameters Ai and Ti of exponential function(s). In order to quantify the PL decay, we also tested the characterization of the curves by an effective decay time (z) (eq 29) as proposed by Shumaker et a1.14
i= 1
i= 1
Although this effective decay time is not justified by theoretical reasons, it is indeed a useful parameter for comparisons of decay curves. We found that both the monoexponential decay time T and the effective decay time (t) reflect the observed effects in a similar manner (see Figures 6 and 7 below).
j
(4)
(iii)
t
[nr]
Figure 4. Different kinds of exponential fits (curve 3) of a PL decay curve simulated with TOSCA (2) assuming a Gaussian-shapedexcitation pulse (1) according to eq 10, with tmax= 2 ns and t H = 0.5 ns. The different starting points of the fitting procedure are indicated by the lines (4). The figures include the residuals A and the autocorrelation function C(t). Simulation conditions: low injection level piho = 1.2 x flat bands, without photocurrent (for the other parameters, see Table 1). (i) Monoexponential fit (start at the 5% level of PL,,,& (ii) biexponential fit (start at the 5% level of PL,,,=),(iii) biexponential fit (start at the 100% level of PL,,,=).
(IV) Results and Discussion (1) Influence of Surface Recombination Velocity and Photocurrent under Conditions of Low Injection and Flat Bands. There are a number of analytical approaches to the PL decay under the flat band condition in the case of low Solving the one-dimensional continuity equation in a semiinfinite semiconductor slab, the PL intensity I p ~ ( t )after an infinitely short excitation pulse at t = 06,7*9916 is
where S, z, and D denote the surface recombination velocity, the bulk luminescence lifetime, and the diffusion coefficient of the minority carriers, respectively, and erfc(u) is the complementary error function of u. In Figure 5, the PL decay curves obtained with eq 30 (t = 4 ns, D = 8.2 cm%, a = 3 x cm-I) are compared with those of corresponding TOSCA simulations under the flat band
Computer Simulation of Photoluminescence Decay
J. Phys. Chem., Vol. 98, No. 48, 1994 12659
PL-intensity ( a u )
2
0-0
10-12
I(J-11
IO-^ IO-^
2: analytical solution
0
5
15
10
Figure 7. Influence of the photocurrent on the mono- ( r ) and biexponential decay times (Tl, 72) as well as on the effective decay time ((t))obtained with TOSCA for the low injection pilno = 1.2 x flat bands, and without considering a surface recombination (SO = 0). The values shown at j p . 0 = 10-12 A/cmZindicate the case of zero photocurrent jph.0 = 0.
20
t (ns)
Figure 5. PL decay curves simulated using the analytical solution (eq 30 with z = 4 ns, D = 8.2 cm%, a = 3 x cm-I) and TOSCA simulations (for other parameters, see Table 1) for different surface recombination velocities S under the conditions of low injection p i l ~ = 1.2 x flat bands, and without considering a photocurrent. (ns)
T (ns)
51
4
3
2
1 103
10-1
101
103
S (cmls)
105
io7
Figure 6. Influence of the surface recombination velocity S on the monoexponential ( r ) and effective ((t)) decay times, respectively, obtained with the analytical solutions (eq 30 with t = 4 ns, D = 8.2 cm2/s,a = 3 x cm-') with TOSCA for the low injection pi/^ = 1.2 x flat bands, and without considering a photocurrent (for other parameters, see Table 1).
condition with low injection. These curves are shown for different surface recombination velocities. In Figure 6, the corresponding results of exponential fits (curves 1 and 2, effective decay time (r);curves 3 and 4, monoexponential decay time T, are shown. It should be noted that the slight differences may be due to the different mathematical treatment. In TOSCA, the models of the generationhecombination mechanisms are incorporated explicitly (see section II,2), whereas in the analytical solution (1) the recombination processes are assumed to be linear, i.e., characterized by the constants (S and r), (2) the surface recombination completely consumes the minority carriers arriving at the surface (boundary condition of the continuity equation), and (3) the decay curve was subsequently convoluted with the corresponding Gaussian time function (eq 10 with Q = 1). Keeping these facts in mind, we feel that the agreement between the results is good.52 For low values of surface recombination velocity, the
observed monoexponential decay time nearly corresponds to the effective lifetime of holes, which can be calculated from the radiative and Shockley-Read-Hall lifetime (rp,,ff= 4 ns, eq 23). The negligible difference may be due to numerical transformations during the evaluation procedure (convolution, deconvolution, etc). For a penetration depth of the light pulse of l / a = 0.33 p m (which corresponds to an excitation of A,,, = 670 nm47),the exponential fit parameters change drastically in the range of lo3cm/s < SO< lo6 c d s (see also Tyagi et aL5 and Ahrenkiel16). If the surface recombination velocity is SO 2 lo6 c d s , a monoexponential decay time T = 1.64 ns is obtained. With decreasing light penetration depth, the influence of surface recombination on the whole decay process of minority carriers increases. If the penetration depth is l / a = 1.39 x pm, corresponding to an excitation at A,, = 337 nm?7 the monoexponential decay time decreases in the range of lo3 c d s < SO < lo7 c d s (i.e., T also decreases above SO = lo6 cds). Considering the biexponential fits, it was found that, in general, with increasing SO (for SO 2 104cm/s), the amplitude of the rapidly decaying exponential term (short decay time) increases relative to the slower decaying one (long decay time). This trend indicates that the surface recombination mainly influences the initial part of the decay p r o c e s ~ . ~ ~ ~ , ~ ~ , ~ ~ In a similar manner, the photocurrent influences the PL decay at low injection levels as shown in Figure 7. Note that the values shown at j P , O = A/cm2 indicate the case of zero photocurrent (jph,O = 0). For exchange photocurrent densities j p , O Ilo-" A/cm2,the PL decay is not significantly influenced. At exchange photocurrent densities jph,o > lo-" A/cm2, the PL response becomes shorter and a biexponential fit of the curve improves its quality. The exchange photocurrent densities published in the literature for GaAs differ remarkably. Jung and Kolb assumed jph,O = 10-l A/cm2 for mathematical simulations of experimentally observed sub-band-gap photocurrent curves at rutheniummodified electrode^.^^ This value can also be estimated from the band edge shift measurements of Allongue et al.54 Reichman used jph,o = lops Ncm2 for calculations of the photocurrentvoltage behavior.55 The exponential decay time(s) depends on the exchange photocurrent density in a manner similar to the results pictured in Figure 6. This indicates that the effect of both surface recombination and photocurrent on the decay of excess caniers in the low injection case is quite similar. At the surface, the arriving carriers nonradiatively recombine or transfer into the
12660 J. Phys. Chem., Vol. 98, No. 48, 1994
Kriiger et al.
T (ns)
4 3
2 1
1
1
IO4
IO0
IO2
Pl‘nO
Figure 8. Dependence of the monoexponential decay time T on the injection level @,/no) for different surface recombination velocities SO under the conditions of flat bands without considering a photocurrent: (1) so = 0, ( 2 ) so= 104 cm/s, (3) so = 105 cm/s, (4) so = 106 cm/s, and ( 5 ) SO = lo7 c d s .
electrolyte. If one or both of these processes produce a significant decrease in the minority carriers at the surface, the minority carriers will diffuse toward the surface with thermal velocity. Therefore, the diffusion of carriers toward the surface is competitive with bulk recombination as indicated by a decrease in the PL decay. Under the conditions chosen in our computer simulations (Table l), nearly the same monoexponential decay time T = 1.64 ns results for both SO ? lo6 cm/s andjph,O 1 A/cmZ, indicating that the rate of both surface processes is limited by the thermal velocity of holes moving toward the surface. (2) PL Decay under Conditions of High Injection and Flat Bands. As shown in Figure 8, the influence of injection level pJn0 on the PL decay depends on the value of SO. In the case of low injection, the monoexponential decay time T (which has to be considered as a mean decay time) remains constant up to an injection level of @,/no) a 0.1 and then decreases more or less drastically with increasing injection density. The different role which the photocurrent plays in this context was discussed recentlyLs and should not be considered further here. In this section, we will focus on the PL decay in the transition range (0.1 5 (PI/%) I l), as well as in the range of high injection ( ( P h o ) > 1). To determine why T decreases with increasing injection density beginning at p,/no a 0.1, we compared the results of the TOSCA simulations with those obtained by simulation of the PL decay curves applying only the “law of quadratic recombination”. In Figure 9, the results of monoexponential fits are shown together with the corresponding results of TOSCA simulations. The TOSCA calculations were carried out using the parameters given in Table 1 with So = 0 under the flat band condition (we did not include surface recombination in order to compare the results with those obtained using exclusively the “law of quadratic recombination”). Curves 2 and 3 clearly show that the decrease of the monoexponential decay time in the high injection region (PI > no) is due to quadratic recombination and not due to Auger recombination. The difference between the curves calculated with TOSCA (curves 2 and 3) and those calculated applying only the “law of quadratic recombination” (curve 1) can be traced to the different parameters characterizing the excitation density. In the TOSCA simulations (curves 2 and 3), p 1 is a mean value in both time and space. This value contrasts with the homogeneous initial concentration of excess carriers dp(0) presumed in the calculations of curve 1. Hence, the decrease of the PL decay time in the range of @,/no) > 0.1 is due to the increasing influence of quadratic recombination (eq 13).
p i( ~ m - ~ )
Figure 9. Monoexponential decay times T for different injection densities pi: (i) PL decay applying only the law of quadratic recombination with B = 1O-Io cm3/s;(2) TOSCA simulations with C, = C, = cm6/s(SO= 0, flat bands, without photocurrent; for other parameters, see Table 1); and (3) as with ( 2 ) but with C, = C, = 0
(without Auger recombination).
(3) Surface Recombination under Conditions of High Injection. Recently, Kauffman et al. reported upon studies of the n-Gas-electrolyte junction by time-resolved PL at different injection levels under the flat band condition.13 They fit their PL decay curves using the analytical solution for the low injection limit (eq 30), although it seems to us that the injection density approaches the dopant density. The authors reported that the assumption of a time-dependent capture velocity of holes in surface states (corresponding to the surface recombination velocity) according to s(t) = SO x exp{-P(t)/(kT)} improves the quality of the fit of PL decay curves measured at different injection levels (corresponding to pilno 2, 0.5, and 0.15, respectively). The time dependence of s is determined by a time-dependent activation barrier p(t) = p-{ 1 - exp(-t/zc)} with a characteristic time constant zc. The authors justified their approach with the assumption that the capture of carriers in surface states has an activation barrier that depends on the nature of the states. A distribution of the activation barrier is expected which corresponds to the character of different surface states. From these assumptions, they postulated that the surface capture velocity decreases with increasing injection levels as more surface states are occupied. Within this model, the surface recombination velocity depends on the injection level. But in our opinion, it is not possible to explain the result by Kauffman and co-workers that the activation barrier P” decreases with increasing injection level. Inspired by the paper of Kauffman et al. and the statements mentioned above in section 11.2 concerning the surface recombination velocity$3 we reviewed the TOSCA simulations to determine whether the surface recombination velocity is a constant for pulse-shaped excitation of different intensities. By evaluating the TOSCA simulations, the time-dependent value of surface recombination velocity S(t) has been calculated according to S(t) = rs(t)/ps(t)for different injection levels. The results are shown in Figure 10. It is obvious that (A) the surface recombination velocity may change by nearly 1 order of magnitude during one decay measurement (under the conditions we used), (B) the amount of change depends on the injection level, and (C) the surface recombination velocity changes nonexponentially. Thus, one can conclude that rs(t) increases slower than ps(t) as a result of the limitation imposed by the constant value of SO. Physically, this means that ~ ( tbecomes ) limited by the constant capture velocities. This is shown in Figure 11, where the smaller SO causes a relatively stronger decline of S. It seems that fact C and the application of the analytical
Computer Simulation of Photoluminescence Decay 104
I Y)
5x103
obd .s.k W
lo3
'
'
'
lo!& . t
'
15!&
. d 0 O '
[=I
Figure 10. Time-dependent change of the surface recombination velocity S for different injection levels @i/no) under the conditions of flat bands, exchange photocurrent density j p . 0 = lo-" A/cm2,and So = lo4 c d s : (1) pilno = 12, (2) pi/no = 1.2, and (3) pi/% = 0.12.
m
'06
S
i
I
I
I
l
103 0.00
5.00
10.00
15.00
200
t
[=I
Figure 11. Time-dependent change of the surface recombination velocity S under the conditions of high injection pJn0 = 12, flat bands, and an exchange photocurrent density j P , o = lo-" A/cmZ: (1) SO= lo4 c d s and (2) SO= lo6 c d s .
function for PL derived for low injection conditions may explain why the curve fit obtained by Kauffman et al. is rather poor.
(V) Summary and Conclusions The interaction between the different processes-injection (local nonhomogeneous excitation), changes in the potential distribution in the semiconductor, photocurrent, recombination, drift, and diffusion of carriers-makes it very difficult to sbtain a comprehensive analytical solution to the differential equation system describing the generatiordrecombinatioddiffusion of carriers and to extract physical parameters (S, different individual lifetimes z,i) by a fit of experimentally observed PL decay curves. If approximationdsimplificationswere used during the mathematical treatment (e.g., flat band condition, low injection condition), the applicability of the solution is restricted, and one must conscientiously check the validity of approximations used in the experiment. For instance, the term dphP usually used for the recombination rate in the continuity equation3-5,7-9,10a.1 1 ~ 1 2 ~implies 16 that the recombination rate is linear with the excess carrier concentration7,l6and does not seem valid in the high injection case as zp is not a constant. In this context, the possibility of simulating the PL decay curves with the help of TOSCA for a wide range of the different parametersinjection density pi, surface recombination velocity SO,and exchange photocurrent density j,h,o-represents important progress. The simulations with TOSCA offer excellent possibilities for studying the influence of the different individual processes on the behavior of the PL decay. Our results of computer simulations of the PL decay show that it is possible to characterize the PL decay by exponential functions, although the real decay process is nonexponential,
J. Phys. Chem., Vol. 98, No. 48, 1994 12661 particularly for high values of surface recombination velocity (SO=. lo3 c d s ) and of exchange photocurrent density (iph.0 > lo-" A/cm2). In the case of low injection density, the PL decay time approximated by use of a mean monoexponential decay time Tis most sensitive to changes in the surface recombination velocity and exchange photocurrent density in the range of lo3 c d s So 106 c d s , and lo-" A/cm2 j p , O A/cm2, provided that the mean penetration depth of the exciting light is approximately 0.33 pm. For lower values of SOandjph,o,the PL decay is determined by the bulk recombination processes. If either the surface recombination velocity SO 2 lo6 c d s or the exchange photocurrent density Jph,O Ilop8A/cm2, a lower limiting value of the PL decay time (T- = 1.64 ns) could be observed in the low injection limit. Under these circumstances, the velocity of diffusion of the minority carriers toward the surface is limited by the thermal velocity. These facts are important in estimating the range of application of the method of time-resolved PL under low injection density. The decrease of the monoexponential-approximatedPL decay time with increasing injection density in the range of pilno > 0.1 is due to the increasing influence of the quadratic recombination. In the region of the injection level 5 (pi/no) I 10 investigated in this work, the Auger recombination does not play an important role in the whole decay process. As shown above, in the case of high injection, the surface recombination velocity changes nonexponentially and cannot be assumed to be constant. In order to estimate parameters characterizing experimentally investigated systems, further work comparing the results obtained with TOSCA with those measured experimentally will be published e1~ewhere.l~~
Acknowledgment. The authors are very grateful to the reviewers who recommended we discuss ref 14 in more detail.
amplitude of the exponential function geometrical cross section of the semiconductor bulk coefficient of the radiative recombination coefficients of the Auger recombination elementary charge energy of conduction band edge (valence band edge) quasi-Fermi-potentials of electrons (holes) intrinsic Fermi level (at the surface) energy of recombination centers at the surface (in the bulk) rate of generation of carriers current density of electrons (holes) photocurrent density (exchange photocurrent density) Boltzmann's constant density of electrons in the conduction band (in equilibrium) intrinsic density surface density of electrons in the conduction band (in equilibrium) effective state density in the conduction (valence) band density of ionized donators (acceptors) density of recombination centers at the surface (in the bulk) density of holes in the valence band (in equilibrium) injected density of holes surface density of holes in the conduction band (in equilibrium) maximum of the intensity of photoluminescence rate of radiative recombination (Auger recombination) rate of Shockley-Read-Hall recombination (surface recombination)
Phys. Chem., Vol. 98, No. 48, 1994 maximum of surface recombination velocity when all recombination centers are occupied with holes (electrons) maximum of surface recombination velocity (input parameter of TOSCA) time half-width time (time of maximum of intensity) of Gaussian time function temperature; monoexponential decay time biexponential decay times thermal velocity of electrons (holes) direction perpendicular to the surface into the bulk absorption coefficient of the exciting light excess density of electrons (holes) dielectric constant of the semiconductor (of the vacuum) mobility of electrons (holes) charge density capture cross section of the recombination center for electrons (holes) lifetime of holes electrostatic potential potential applied at the ohmic contact contact potential at the gate contact potential difference across the space charge layer
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