J. Pkys. Ckem. 1992, 96, 10371-10379 method of preparation of dimer 3.
Referenem and Notes (1) Liebermann, C. Chem. Eer. 1889, 22, 124, 782. (2) Bentein, H. I.; Quimby, W. C. J. Am. Chem. Soc. 1943, 65, 1845. (3) Cohcn. M. D.:Schmidt, G. M. J. Pure A d . Chem. 1971, 27, 647. (4) Hasegawa. Mi Chem. Rev. 1983,83,507: (5) Quina, F. H.; Whitten, D. G. J. Am. Chem. Soc. 1977, 99, 877. (6) Takagi, K.; Suddaby, B. R.; Vadas, S. L.; Backer, C. A.; Whitten, D. G. J. Am. Chem. Soc. 1986,108,7865. (7) Allara, D. L.; Swalen, J. D. J. Phys. Chem. 1982, 86, 2700. (8) Kimura, F.; Umcmura, J.; Takcnaka, T. Langmuir 1986, 2, 96. (9) Naselli, C.; Swalcn, J. D.; Rabolt, J. F. J. Chem. Phys. 1989,90, 3855.
10371
(10) Umemura, J.; Kamata, T.; Kawai, T.; Takenaka, T. J. Phys. Chem. 1990, 94,62. (11) Kawai, T.; Umcmura, J.; Takenaka, T. Langmuir 1990,6,672. (12) Stroeve, P.; Sapentein, D. D.; Rabolt, J. F. J. Chem. Phys. 1990,92, 6958. (13) Allara, D. L.; Nuzzo, R. G. Langmuir 1985, 1, 45. (14) Allara, D. L.; Nuzzo, R. G. Langmuir 1985, 1, 52. (15) Takagi, K., private communication. (16) Itoh, K.; Hayashi, K.; Hamanaka, Y.; Yamamoto, M.; Araki, T.; Iriyama, K. Lungmuir 1992,8, 140. 117) Meic. Z.: Gusten. H. Soectrochim. Acra 1978. 34A. 101. (18j Wajima,'T.; Yankmot;, M.; Itoh, K., unpublkhcd work. (19) Snyder, R. G. J. Chem. Phys. 1%7,47, 1316. (20) Shimomura, M.; Hashimoto, H.; Kunitake, T. Langmuir 1989,5, 174.
Carrier Relaxation at Semiconductor Interfaces and Essential Features of a Quantltative Model M. L. Shumaker, W. J. Dohrd, and D. H. Waldeck* Department of Ckemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (Received: May 18, 1992; In Final Form: July 22, 1992)
Experimental studies of the time-resolved bandgap emission of CdSe are used to probe the attributes required in a theoretical model for the carrier dynamics of semiconductor electrodes. By variation of excitation conditions and material properties, the validity of approximations in a commonly used model are tested. The experimental results show that the details of bulk recombination, the effects of self-absorption, and the influence of the space charge field are quite important factors for the carrier dynamics. Simulation results show that the detailed kinetics of the interfacial recombination can also have important consequences for the carrier relaxation. Appropriate modeling of these processes is needed before a surface recombination velocity can be extracted from such studies in a quantitative manner.
htroduction
Heterogeneous charge-transfer processes are ubiquitous in nature and crucially important in many technologies. Recombination of minority carriers at the semiconductor/electrolyte interface is an important physical proteas which nceds to be clearly understood before an adequate mechanistic description of heterogeneous charge transfer is possible. Such recombination processes may be either desirable or undesirable, depending upon one's goal. For example, such recombination processes may be the first step in corrosion of electrodes. Also they may limit the operating efficiencies of photovoltaic and photocatalytic devices. In contrast, these processes are desirable when corrosion is used in the fabrication of devices. Also these processes may act as mediators that enhance heterogeneous charge-transfer rates. Although recombination of carriers in the bulk of semiconductor materials has bem studied for quite some time, less effort has been expended on interfacial recombination.lS2 Yet in many cases, chemical reactions are dominated by interfacial properties and processes. The experimental methods used to quantify the minority carrier recombination at interfaces can be divided into two major c a t e gories, steady-state measurements and transient (or AC) mea~urements.~.'Clearly the steady-state measurements do not directly probe the relaxation kinetics, and they only provide 'effective" values (or time-integrated values) of the kinetic parameters. For this reason these methods are less useful as a guide to understanding the dynamics of carrier relaxation; however, it should be reaiized that these steady-state methods are generally quite useful for the characterization of devices which operate on long time males. A variety of direct methods have been used to probe the carrier evolution in electrodes. Transient electrical methods, such as photopotmtial and photocurrent, arc limited by the RC of the electrochemical cell.43 All optical methods provide the opportunity to probe the carrier dynamics on the picosecond 0022-3654/92/2096-1037 1$03.00/0
and subpicosecond time scales. Both timeresolved fluorescence61o and transient grating".'* spectroscopic methods have been used to study carrier recombination. This study experimentally probes some of the limits of a diffusion mode11J0J3-16which has been widely usedb12to analyze the recombination kinetics of photogenerated minority carriers from the observed photoluminescence decay. The emphasis of this work is on the relaxation of minority carriers under low injection conditions (Le. the photogenerated minority carrier density is significantly smaller than the thermal majority carrier population density) and the transition from high injection to low injection. If possible, one d e s k to monitor the carrier dynamics under low injection conditions for two reasons. First the system's response is expected to be linear and analytical models can be evaluated. Second this regime is the same as that obtained under solar irradiation, making such studies directly relevant to solar powered devices and processes. The relaxation kinetics of photogenerated minority carriers in n-CdSe single crystals and single crystal electrodes are used as a test system in order to probe the available models. In these studies an ultrashort light pulse (approximately 10 ps) with suprabandgap energy is used to excite the material, and the bandgap emission is monitored from the front surface as a function of time (see Figure 1 for a sketch of the experimental geometry). The geometry and the use of intense laser excitation is consistent with the requirements of the model13used to describe the carrier dynamia. The limitations of these models are probed by variation of the excitation conditions and of the material properties. A variety of approximations are inherent in the modeling which is typically used, and four of these are studied in this work. First the influence of the space charge field on the carrier dynamics is not treated in the time dependent models, although it has been included in steady-state analyses1' and in numerical solutions of the diffusion e q ~ a t i o n . ~ *Usually J~ this neglect is justified by 0 1992 American Chemical Society
10372 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 considering the excitation to be in the high injection limit so that the bands are flattened. Second the bulk recombination kinetics is assumed to be “quasi” first order for doped materials such as n-CdSe. This approximation breaks down under high excitation conditions and may be incompatible with the first approximation which requires high excitation in order to flatten the bands. Third the effects of self-absorption, sometimes referred to as photon recycling, are considered. Although this effect is wellkn0wn,199J0*20*21 it has not been widely used in modeling the dynamics. Fourth the recombination velocity at the interface is assumed to be a constant, and the detailed kinetics of the interfacial recombination is not considered to impact the observed dynamics. Lastly, other approximations which are not analyzed herein are discussed on a qualitative level. The material used in these studies, n-CdSe, has been studied by a variety of g r o ~ p s . ~ . ~The ~ . ~11-VI ~ J ” materials (CdS, CdSe, ...) have direct bandgaps in the visible region of the spectrum (the bandgap of CdSe is 1.7 eV), making them amenable to these methods. This previous work demonstrates quite clearly the sensitivity of the fluorescence decay and fluorescence quantum efficiency to the interfacial recombination dynamics. The focus of this work concerns one’s ability to quantitatively extract a surface recombination velocity from the measured fluorescence decay. The structure of the paper is given here. In the next section various experimental considerationsare described. Subsequently a more detailed discussion of the carrier diffusion and recombination is presented, and an algorithm for modeling the carrier diffusion is given. In the Results section of the paper the experimental study of the assumptions made in the diffusion model are discussed. Lastly conclusions concerning the requirements of an appropriate model are presented.
Experimental Section The temporal behavior of the bandgap emission was measured using the time-correlated single photon counting method. A detailed description of this particular apparatus has been reported.25 For the experiments described herein the excitation wavelength is 595 nm which is well above the bandgap of CdSe. The optical skin depth for CdSe at this wavelength is 1200 A (the absorption coefficient a! is 8.1 X lo4 cm-1).26 The emission wavelength is 725 nm (unless otherwise indicated) for which the absorption coefficient is 400 times smaller (2.0x 102 cm-I). The emission wavelength is isolated using a bandpass filter (f5 nm) and cutoff filter (Schott 630). The instrumental response function in these experiments typically had a full width at half-maximum (fwhm) of 70 p and a full width at tenth-maximum (fwtm) of 130 ps. The geometry of the experiment can have an important impact on the quantitative determination of the emission decay law. Wilson and Pester13have found an expression for the timeresolved emission decay under the conditions of finely focussed laser pulse excitation. Using their result it is possible to choose an experimental geometry in which the lateral diffusion of the carriers is unimportant for the observed decay law, and only a one-dimens i d treatment of the diffusion equation is required. A schematic diagram of the experimental geometry is given in Figure 1. The angle of incidence of the excitation beam is 60° (as measured from the surface normal), and the beam is focussed with a 1254x1focal length lens to a spot diameter of about 250 pm. The collection lens for the emission is aligned with the surface normal and has a 5-cm focal length. The emission light is first filtered by the bandpass filter and subsequently passes through a cutoff filter before being detected with a microchannelplate photomultiplier tube. The emisson decay curve3 were measured for the n-CdSe single crystal in air or in an electrochemical cell. The specifics of the elacttochemical cell design have been described previously?7 All of the studies reported herein were with the n-CdSe electrode in contact with 0.5 M KOH solution. This solution was chosen in order to compare directly with other studies of electrochemical properties and photocapacitance spactra; however, the electrode
Shumaker et al. is unstable in this solution. A typical fluorescence decay curve in air can take from 1 to 2 h to collect, and the average power incident on the sample during this time is 5 mW or less. Unless stated otherwise the laser repetition rate is 300.4 KHz, and the pulse duration is approximately 10 ps. The form of the fluorescence decay curve in air was not found to depend on the irradiation time of the sample for time scales of a few hours. It may be that photocorrosion of the electrode24.28occurs on time scales very different than those used here or that it has a small effect on the fluorescence decay curve. In the KOH solution the fluorescence decay curve was found to vary with irradiation time under flat band conditions. This effect is discussed later. The electrodes were constructed from single crystals of n-CdSe (purchased from Cleveland Crystals) with the front surface (Cd rich) cleaved perpendicular to the c-axis. Details of the electrode preparation have been reported previ~usly.~~ Briefly, an ohmic contact was made to the back surface (Se rich) of the electrode with In/Ga eutectic and connected to a wire using silver epoxy. These contacts have an ohmic I-V curve with a resistance less than 2 ohms. The front surface of the electrode was mechanically polished with alumina through a series of decreasing grit size down to 0.05 pm. After this mechanical polishing the electrode was chemically etched. Although a variety of etching procedures were explored, the work reported here concerns electrodes which were etched with a 0.25% Br2 methanol solution for 30 s. The bromine concentration was chosen to optimize the reproducibility of the surface properties as determined by the fluorescence emission. Some of the electrodea were also studied using SIMS and ESCA. It was found that these “clean surfaces”contain oxides and carbon re~idue.~~’~ The simulations and the data fitting were performed on the chemistry department computer system (FPS Model 500) and the university’s VAX cluster. Decay curves were fit by iterative convolution of the measured instrumental response function with the guessed functional form for the decay law. The sum of squares was minimized using the Marquardt-Levenburg algorithm.
Theoretical Background The light emission from a semiconductor surface is very sensitive to the surface properties and is commonly used to measure the quality of surfaces for processing. A number of workers have treated the form of the fluorescence decay law’oJs16 so that the surface recombination velocity may be obtained quantitatively. The treatment of Wilson and Pester is discussed here because its results are similar to the other treatments in the simple limit. Second the simulation of this model using a random walk algorithm is discllssed. Third the generalization of the Wilson and Pester result to a case including self-absorption is presented. A Model. Wilson and Pester begin their treatment with the full three-dimensional diffusion equation for the initially created minority carrier (in this case photogenerated holes) concentration, and the loss of minority carriers at the interface is obtained by application of a suitable boundary condition. The generation function g(i;r) describes the spatial and temporal distribution of the photogenerated carriers, and it depends on the time profile of the laser pulse and the spatial profde of the laser, laterally along the surface plane of the crystal ( x and y directions) and normal to the crystal surface (z direction). Figure 1 gives a schematic diagram of the experimental geometry used in these studies. The laser can be well approximated as having a Gaussian profile, and the focussed laser beam has cylindrical symmetry. Normal to the surface, the laser beam is strongly attenuated because the excitation energy is above the bandgap. Wilson and Pester show that in the limit of low numerical aperture the lateral shape of the excitation is immaterial. For the studies reported here the numerical aperture used in the excitation is 0.10, and the influence of the lateral shape on the form of the generation function along the z direction is leap than 0.25%. CMleeqUeatly a omdimdona1 treatment of the charge camers diffusion is possible, and the generation term is treated as being exponential into the bulk of the material. In fact this approximation is even more reasonable than suggested above because the collection optics only image the
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10373
Carrier Relaxation at Semiconductor Interfaces
CB E,
I
IX
I VB
Figure 1. This figure provides a schematic diagram of the experimental arrangement used in the fluorescence studies. The arrow indicates the direction of the excitation laser pulse, and the fluorescence is collected normal to the surface. The optical elements are 12.5-cm focal length lens (Ll), a 5-cm focal length lens (L2), 726-nm interference filter (IF), and cutoff filter(s) (F). The box labeled D represents the detector. The cross hatched region is the CdSe single crystal, and the boxes around it are meant to indicate the electrochemical optical cell. The inset shows the laboratory coordinate system which is defined with respect to the surface normal of the crystal and the plane of propagation of the laser beam.
center portion of the focussed laser spot. The diffusion model which has found widest application in treating the dynamics of charge carrier relaxation is based on a one-dimensional diffusion equation, namely
In this equation p(z,t) is the nonequilibrium probability distribution of minority carriers (e.g., in n-CdSe it is holes) in the z-direction at time t . More generally, one should consider the diffusion of both the minority carriers and the excess majority carriers and their influence on each other. In the limit that the photogenerated minority carrier concentration is small with respect to the equilibrium majority camer concentration, this more general approach reduces to that used in writing eq 1. The bandedge emission studied in this report is created by the radiative recombination of holes in the valence band with electrons in the conduction band or in shallow donor states (see Figure 2).29 Equation 1 models the diffusive motion of holes in the valence band. The recombination of minority carriers in the bulk of the material is assumed to follow an exponential decay law with a bulk relaxation time given by 7. The recombination rate constant is a sum of a radiative rate constant and a nonradiative rate constant (7-l = k, + k,). This assumption treats the nonradiative recombination of holes as being first order in the hole concentration. The radiative transition involves a recombination of an electron and hole with simultaneous emission of a photon and is necessarily i.e., the emission intensity is proportional to n(3,t) lip(?,?). The concentration of electrons n is given by nq 6n where nq is the equilibrium concentration of electrons (about 0.3 X 10l6cm-3 for the n-CdSe used in these studies) and 6n is the nonequilibrium concentration of electrons, i.e., those created by the optical pulse. The concentration of holes is simply 6p since the equilibrium population is close to zero. When the concentration of holes (and hence the nonequilibrium concentration of electrons) is small with respect to the concentration of electrons (n >> 6n), this bimolecular process becomes quasi first order. ?he function g(z,t) describes the spatial and temporal distribution of the generated carriers. The excitation light energy is suprabandgap, and the light field decays exponentially into the bulk of the electrode. As discussed above the lateral shape of the excitation does not effect the observed decay law. The temporal form of the excitation acts to distort the form of the intrinsic decay. In fact, Wilson and Pester show that the laser pulse time profile is convoluted with the intrinsic system response. The solutions quoted herein will be for impulse excitation. The solution presented will be convoluted with the instrumental response function of the apparatus when theory and experiment are compared. The diffusion coefficient, D,is interpreted as that of the minority carriers. It is probably better viewed as an effective diffusion
+
CdSe
I z=o
Figure 2. This figure contains a schematic energy level diagram of the material n-CdSe. Because the material is n-type the Fermi level (Ef)is closer to the conduction band (CB) than the valence band (VB). Defect states localized at the interface are indicated by the solid box. The various recombination processes (bulk nonradiative recombination (knr), radiative recombination (k,) and surface recombination ( S ) )are indicated as well.
coefficient, however. For the more general situation in which both electron and hole diffusion are treated, this diffusion coefficient becomes an ambipolar diffusion coeffi~ient.~~ To the extent that the electron motion is important this diffusion coefficient will differ from that for the holes. The diffusion coefficient will also be modified by the possible importance of radiative trapping, or “photon recycling”,gJ0which is discussed below. Both of these effects will change the value of the diffusion coefficient from that expected for the motion of the minority carriers. This diffusion equation ignores two, possibly important, factors in describing the time evolution of the photogenerated carriers, photon recycling and a space charge field. Equation 1 ignores the possibility of the emission light being reabsorbed by the semiconductor, referred to as photon recycling. The major effect of this process is to change the observed diffusion constant and the observed bulk recombination time for the material. The importance of this effect can be estimated by comparison of the diffusion length of the minority carriers, defined as LD = ds, with the photon absorption length in the material, defined as L(u) = l / a ( u ) . For n-CdSe the photon absorption length for 725-nm light is 45 pm, whereas the diffusion length of minority carriers is on the order of 1 pm.31 However for 700-nm light the photon absorption length is only 0.5 pm, and self-absorption will be an important effect. The effects of self-absorption are discussed more quantitatively below. The space charge field is not included in this model, and the usual justification given is that the high intensity of the laser excitation creates a large number of minority carriers and “flattens” the bands. Clearly when the concentration of photogenerated carriers is much higher than the dopant density of the material, the static charges from the dopant atoms will be well shielded and the space charge field will be effectively counteracted. Interestingly, this assumption is commonly made at the same time that one assumes the radiative decay is quasi first order, which requires that the excitation not create large carrier densities. It is unlikely that these two assumptions can be simultaneously met in real systems. The space charge field however can be eliminated by the application of a bias potential to electrodes in contact with an electrolyte. The dependence of the emission decay on bias potential is discussed later. Equation 1 is solved subject to certain boundary conditions, namely p(z,t)I,=.n = 0
(2)
and (3)
The first boundary condition (eq 2) requires that the equilibrium
10374 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
carrier distribution be preserved deep in the bulk of the material. This boundary condition is consistent with these studies since the sample thickness (1 mm) is much larger than the penetration depth of the light field (about 0.1 pm) and the diffusion length of the minority carriers (about 1 pm). The second boundary condition is a requirement that the net flux of carriers to the interface must be matched by their recombination at the interface and reflects the conservation of carriers. The quantity, us, is the surface recombination velocity which describes the rate of minority carrier capture at the interface. This quantity is the reaction rate for charge transfer at the interface and can be quite high in magnitude (106 cm/s and higher). Time-resolved emission studies of the sort performed here are sensitive to recombination velocities in the range of 10’ to IO6 cm/s. The expression obtained by Wilson and Pester for the form of the decay law under the above limits is given by
where 7 is the bulk recombination time, A is the reduced absorption coefficient at the excitation wavelength (A t aL = a d z ) ,S is the reduced surface recombination velocity (S usdG) and erfc(r) is the complimentary error function of z. In the limit that the surface recombination velocity is 0 this expression reduces to
Equation 5 holds true rigorously in the case when focussing of the laser beam is important as well. In fact at very low surface recombination velocities and at very high surface recombination velocities, the emission decay is independent of S.” Another interesting limit of (4) is the case of long time (t >> T ) for which one obtains
Even at long times the emission decay is not exponential if the surface recombination velocity is finite. It has been shown clearly by a number of workers that the emission decay law is extremely sensitive to the surface quality and the nature of surface attached species.6922*27*32 Interestingly however, the surface recombination velocity is treated as a constant even though it has a more complex form us = N,(u)u (7) in which u is the capture cross section for a carrier by a trapsite at the interface or by a redox species in solution, ( u ) is the thermal velocity of carriers, and N, is the concentration of available trap sites, and it could be effected by complicated kinetics. When the number of photogenerated camm is high compared to the number of trapsites, the trap population can saturate. Consequently the surface recombination velocity will appear to be time dependent. Such an effect has been discussed by a few This phenomenon is explored through the use of simulation studies of the diffusion equation. A Simulation Algorithm. In order to analyze the data fitting routines and to probe the limits of the diffusion equation it is worthwhile to simulate eq 1. The algorithm used for this simulation is adapted from Holloway.’* The diffusion equation is simulated by a random walk of particles which represent the minority carriers. The algorithm used for a single carrier is as follows. (1) Inject the carrier at a position chosen from a probability distribution having the same shape as the generation term used above, Le., exp(-az). Set the
Shumaker et al. jump counter j to zero and the magnitude of the carrier to 1. (2) Obtain new coordinates (X,Y,Z)for the carrier by choosing a random number for each possible direction. The random number determines the sign of the motion in x, y , and I,and the step size is u, where u = 2LM/L2. The coordinate space is normalized such that X = x / L , Y = y / L , and 2 = z / L . ( 3 ) If the carrier is at 2 = 0, and at an unoccupied trap site in X and Y,multiply its magnitude by zero and set the occupation time of the trapsite to its initial value f. (4) Decrease the occupation times of the traps by 1. ( 5 ) Multiply the magnitude of the carrier by exp(-ju2/2). (6)If the carrier magnitude is above a minimum value go back to step 2. The step size u must be chosen to be small compared to unity in order to accurately simulate the population of carriers in time. Typically it was chosen to be 0.01 or smaller. When the trapsites have a zero occupation time (Le., always ready to annihilate a carrier), the number density of trapsites determines the magnitude of the surface recombination velocity. The multiplication of the carrier by exp(-ju2/2) accounts for the bulk recombination of carriers, which is exponential. This algorithm docs not directly simulate that process but incorporates it by the propagation of ”fractional” particles. In the simulations performed for this study a vector of carriers was propagated through the algorithm described above. The first five operations above could be performed in parallel. Operation 6 was W i e d so that the sum of the carrier vector was calculated and used to determine the carrier population at each time step as well as to decide whether the simulation should end. Typically for a carrier vector with 2000 carriers the propagation would be stopped when only one carrier remained. When the occupation time of a trapsite is chosen to be zero, this algorithm simulates eq 1 for a semiinfinitesolid subject to the two boundary conditions described above. When the occupation time of the trapsites is finite, it is possible to simulate and assea the importance of trap filling on the overall carrier evolution. The results of the simulations will be discussed later. Selt-AbsOrption. Because of the large optical density of a solid such as n-CdSe above and near the bandgap, self-absorption of the emitted light could have an important impact on the emission deca~.~JO It is possible to identify two effects of self-absorption which we term photon recycling and the inner filter effect. Photon recycling is the process whereby an electron and hole recombine to create a photon; however, the photon is reabsorbed creating another minority carrier (Le., hole) leading to no net change in minority carrier concentration. This process changes the diffusion constant of the minority carriers since the absorption length of the photon is generally quite different than the diffusion length of the minority carrier. This process also increases the lifetime of the minority carriers. The importance of photon recycling is determined by the inherent emission spectrum of the material, not the observation wavelength. Hence if the emission spectrum overlaps considerably with the absorption spectrum this process will contribute to the population dynamics of the minority carriers. The effect of photon recycling on the observed parameters, diffusion constant, and lifetime, has been addressed by others? In the studies discussed here the lifetime and diffusion constant are considered to be adjustable parameters. The form of the excited-state decay law, as developed by Wilson and Pester,is assumed to remain the same when photon recycling is present, but the value of D and 7 are allowed to change. The inner filter effect modifies the form of the emission spectrum which is observed in the laboratory and is caused by the decrease in absorptivity as one moves through the emission band of the solid away from the absorption band. This effect determines the depth into the solid from which fluorescence can be observed. If the minority carrier population is given by P(z,t) and the absorption coeficient at frequency Y is given by N u ) , then the observed emission spectrum will be given by Z’(u)
a
L - P ( z , t ) exp(-@(v)z) dz
where the prime symbol indicates that photon recycling is incorporated. The modification of the emission intensity by the
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10375
Camer Relaxation at Semiconductor Interfaces absarption coefficient 4 @ (B(u) ) is averaged Over the detection bandwidth) also modifies the observed time dependence. This effect may be included in the Wilson and Pester treatment for the emission intensity with time. By restating Wilson and Pester's eq 17 so that the emission intensity in time is given by JO
where p is the Laplace transform variable to the time, Z is the reduced distance, and B is the reduced absorption coefficient (B = OL). For impulse excitation and a generation function which is a decaying exponential into the bulk of the material, it is possible to show that the emission intensity is given by
(5' - A)(S - B)
-
exp( ? ) e r f c ( B G b ( S (A
CQ
45.1-
-4.5L
3
0.0
v,
E -13.0L r
-
+ A)
1
+ B)(S - B)(A - B)
This result has the proper behavior in the limits where B = 0 and S = 0, and it is well behaved when S = B, S = A, and B = A. Also this result agrees with that given by A l ~ e n k i e l . ~ ~ . ~ ~ Equation 10 includes the effect of photon recycling through the valm of the reduced variables S,A, and B and the inner filter effect through the B dependence of the decay. For the case where the photon recycling, surface recombination velocity, and excitation conditions are constant, it is possible to predict how the decay changes as the monitored emission wavelength changes (i.e., changing 4 or B). For the experiments discussed later B is always less than A. If B < S (the case in these experiments), then the decay law is faster when the inner filter effect is included (IsA(t) < Z'(t), where I'(t) is defined as ZsA(t)with B = 0). If B > S, then the decay law is slower when the inner fdter effect is included (Iw(t)> 11~)). It is important to realize that both Z'(t) and Zw(t) as referred to here include effects of photon recycling which lengthens the decay law over that given by Wilson and Pester (eq 4) *
R d t a md Discu~~~ion The studies which follow explore four aspects of the diffusion equation presented above. The influence of bimolecular radiative recombination is evaluated by study of the intensity dependence of the emission. The importance of the space charge field on the emission decay is assessed by study of the emission decay at different bias voltages. The impact of self-absorption is analyzed through a wavelength dependence of the emission. Lastly the influence of trap filling on the surface recombination is analyzed through computer simulations of the diffusion equation. The emission decay was measured for n-CdSe in contact with air and in contact with 0.5 M KOH. The dopant density in this material was determined to be 0.3 X 1OI6(M.2 X 10l6) using Mott-Schottky analysis.*' The flatband potential (determined from these curves by extrapolation) was found to be -0.8 (f0.2) V versus SCE. For the experiments reported here the measured flatband potential was -0.903 V versus SCE. Qmnt&bq thc b y . Figure 3 shows an emission decay curve obtained for a n-CdSe electrode in contact with air, along with a measured experimental response function (or instrument function). Clearly the decay law is not exponential. A fit of the full decay curve to eq 4 is shown as well as a fit of the long time part of the decay to q 6. The quality of the fit at long time is
I
0
I
I
I
I
I
1
15.9 TIME (nsecs)
I
31.8
-
Figure 3. This figure shows a measured fluorescence decay curve and an instrumental response function. The best fits to the full eq 4 (x2 8.52; lower residual plot) and the best fit to the asymptotic form 6 ( x 2 = 1.OR upper residual plot) are also shown.
quite reasonable, and the bulk recombination time obtained from this fit is 20.6 ns. This value for the lifetime appears reasonable and lies in the range of values reported in the literature. Unfortunately the literature values reported for this paramet~range widely (from 0.1 to 100 n ~ ) . ' The ~ fit to the whole decay curve is not very good however (x2 = 8.52) which is evident from the residuals shown in the figure. The parameters obtained from this fit are u, = 4.18 X lo6 cm/s, L = 7.02 pm, and 7 = 9.56 ns. At early times, the deviation of the fitted curve from the measured curve is 13 standard deviations. Clearly important aspects must be missiig from the model and arc needed to accurately treat the conditions of this experiment. For some of the comparisons described below it was necaary to quantify the decay accurately. This was accomplished by fitting the observed decay curve to a sum of exponentials and using the parameters from this fit to compute an average relaxation time. If the decay law is
/ .\
m
then the average decay time
(7)
is given by m
m
i= 1
where 7, is the correlation time. Although such a fit may not be justified on theoretical grounds, it serves as a useful method to compare decay curves with one another. Bulk R c c o m b i ~ t i oThe ~ ~ assumption concerning the nature of the bulk recombination was analyzed by performingan intensity dependence of the emission. An intensity dependence was measured for both the time integrated and time-resolved emission. For unimolecular radiative decay the intensity dependence should be linear and for bimolecular radiative decay the intensity dependence should be quadratic. The nonradiative decay processes also contribute to the observed intensity dependence. The order of the n d t i v e recumbination kineti- can be quite complicated
10376 The Journal of Physical Chemistry, Vol. 96, No. 25,
Shumaker et al.
0
+
3
:
lo3? : w :
0
:lo'( 0
. I
0
-
2
:.,a
C
w 510':
I"
1o
-~
IO-'
1oo
INTENSITY (mW)
TABLE I: Intensity Dependence of the Fluorescence at Differeat Times after Excitation correlation std timea (ns) slope coeff deviation 0.00 1.74 0.9896 0.1132 1 .oo 1.69 0.9888 0.1139 2.00 1.65 0.9899 0.1055 5.00 1.62 0.9915 0.0951 10.00 1.61 0.9917 0.0933 25.00 1.61 0.9924 0.0890 40.00 1.59 0.9907 0.0977 Peak channel is defined as time zero.
IO-'
1oo
INTENSITY (mW)
Figure 4. (a) A log-log plot of the observed steady-state fluorescence emission versus laser intensity reveals noninteger s l o p at both low and high intensitia for n-CdSc in contact with air (A). (b) This figure shows a log-log plot of the observed steady-state fluorescence emission versus laser intensity in 0.5 M KOH solution under flatband conditions (-0.903 V vs SCE) (A) and with a depletion layer (-0,203V vs SCE) (0).
if multiple trap states are present and if the populations of these traps can be ~ a t u r a t e d . " ~ Figure 4a shows a plot of the time integrated emission (at 725 nm) for n-CdSe in air. The fluorescence intensity is supralinear in the laser intensity, indicating that the bulk recombination is neither strictly unimolecular nor bimolecular. Such intensity dependencies have been observed previously for n-CdSe under excitation by both continuous and pulsed light sourcesand at other suprabandgap energie~.~'~~* The exponent varies from a value of 1.35 f 0.10 for the lowest five laser intensities to a value of 2.24 f 0.32 for the highest five laser intensities. (Note: the errors given are 2 standard deviations.) Figure 4b shows an intensity dependence for an electrode in contact with 0.5 M KOH solution near the flatband condition (-0.903 vs SCE) where the slope is 2.15 f 0.1 and under reverse bias (-0.203 vs SCE) where the slope is 1.85 0.1. Clearly the intensity dependence is sensitive to the space charge field. If the space charge field separates the photogenerated electron/hole pairs, it would reduce the emission intensity. Furthermore, as the laser intensity increases, the electron-hole pair density increases and the space charge field is reduced. These phenomena would explain the trends observed here, namely higher slopes at higher laser intensities and a dependence on bias voltage. The noninteger power may also reflect a mixture of unimolecular and bimolecular decay processes; however, this experiment does not distinguish whether the nonlinear recombination is a result of bimolecular radiative decay or higher order nonradiative processes. Analysis of the full time-resolved decay at different laser intensities allows these latter two possibilities to be distinguished. The radiative emission should have the mo8t bimolecular character at early time and become unimolecular as the minority carrier population decreases. If bimolecular radiative decay were im-
*
portant, then the intensity dependence of the fluorescence would decrease through the decay (as the minority carrier population decreases). A variety of higher order nonradiative processes OCCUT in semiconductors; however, with this material at this temperature (298K)and under these excitation conditions trapping in moderate to deep levels is probably the dominant mechanism. The highest excitation densities are 3 orders of magnitude below those where Auger recombination becomes important as well as those needed for e h plasma formation. At these high temperatures the initially prepared quasiparticles, such as excitons, dissociate within a few picoseconds and the detrapping from shallow traps (CO.1 eV binding energies) is very rapid. Whether or not the nonradiative processes change with minority carrier population through the decay profile will depend on the relative trapping and detrapping rates."' Figure 5 shows the fluorescence decay curves collected in air for two different excitation intensities. From this figure where the curves are normalized to the same number of counts in the peak channel, it is clear that the relaxation rate increases significantly with laser intensity. Using decay curves of this sort in which the relative intensities of the fluorescence emission is accounted for quantitatively, the intensity dependence of fluorescence at different times after excitation can be determined. The results of such an analysis are given in Table I for average incident laser power ranging from 5.0 X lW-5.0 mW. The initial photogenerated carrier density is estimated to be 2 X lo1*cm-3 at the highest excitation power and 2 X lOI4 at the lowest excitation power. Although a trend may be present, the values for the slope do not change significantly throughout the decay profile. Clearly, bimolecular radiative recombination is not the dominant contributor to the nonlinear intensity dependence, but rather this dependence is caused by nonradiative relaxation processes. Space Charge Field. The influence of the space charge field near the interface on the carrier population and the emission decay law can be a h by studying the emission as a function of bias voltage. Figure 6 shows two emission decays measured for the same electrode held at the flatband potential (-0.903 V vs SCE) and under reverse bias conditions (-0.203 V vs SCE). Clearly the form of the decay law changes with the bias voltage, and the
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10377
Carrier Relaxation at Semiconductor Interfaces
lOOL**O' '
"
'
"
"
"
'
'
"
'
"
'
,,
'
5 10 TIME (nsecs)
"
"
'
. , . ., ,
,
Figure 6. The two decay curves shown are for a n-CdSe electrode under reverse bias (-0.203 V vs SCE) (X) and under flat band conditions (-0.903 V vs SCE) (0).
TABLE Ik Decay Law as a Function of Bias Potential and C o ~ t i o aTime in 0.5 M KOH Solutiona trial
bias potential (V vs SCE)
collection time (h)
1 2 3 4 5 6 7
-0,903 -0,203 -0.903 0.203 -0,903 -0.903 -0.203
4.79 11.56 4.81 7.46 4.68 4.96 7.88
av lifetime, ( 4
(T)
33.6632 0.7889 7.0275 0.6704 8.5474 7.7944 0.7962
l O 0 ' * 0 ''
"
"
" '
15
correlation time, r, (ns)
"
5
"
'
"
"
'
10 ' "
"
"
"
"
15
TIME (nsecs)
Figure 7. The two decays curves shown in this plot were taken under identical excitation conditions. Only the wavelength at which the emission was monitored varied ((X) 670 nm, (0) 725 nm). v)
5.81-
-5.8L
1.6428 0.1431 1.4385 0.1389 1.9543 1.9086 0.2139
a Flatband is defined as -0.903 V vs SCE. Reversed bias is defined as -0,203 V vs SCE.
static field has an important effect on the dynamics. The two curves shown in Figure 6 represent the "asymptotic" response of the electrode. It was found that under flatband conditions the form of the decay law evolved with time to an asymptotic value, as if the electrode were being photoetched. In contrast under reverse bias conditions it was found that the emission of the electrode did not change appreciably even though the electrodes properties were changing under flatband conditions. Table I1 shows these effects clearly. The trials in this table correspond to a sequential set of decay curves taken for a single electrode. In trial 1 under flatband conditions the electrode has a long relaxation time. However under reverse bias the emission decay law is much shorter presumably as a result of the separation of charge carriers by the applied field and subsequent reduction in the probability of recombination, In trial 3 the same electrode has changed, and the decay law is shorter than in trial 1 despite it being under the same conditions. Clearly the surface of the electrode must have changed. Despite this evolution however trial 4 is similar to trial 2 for which the conditions are the same. In trial 5 the electrode has reached its asymptotic response under flatband conditions. Trial 6 demonstrates the reproducibility of the response. Trial 7, identical to trial 2 and trial 4, demonstrates the continued reproducibility of the reverse bias response. Despite the time evolution of the electrode surface during these voltage studies, it is clear that the emission decay under flatband conditions is different from that under reverse bias. Clearly a proper quantitativetreatment of the electrode will have to include the space charge field unleas the electrode is held at the flatband potential. Self-Abrorptioa. As discussed previously the effects of selfabsorption are 2-fold, photon recycling and the inner filter effect. Figure 7 shows plots of two fluorescence decay curves for n-CdSe in contact with air and excited at 596 nm. The emission wavelengths are 725 and 670 nm for these curves and clearly the decay is longer for the longer wavelength emission. This trend is exactly that expected when effects of self-absorption are important in the decay law. The effect of photon recycling should be relatively constant throughout this series because the excitation conditions and the spectrum of the material is fued. What these curves show
.... -_
0.0
0.0
I
I
I
1
12.4
I
I
I
I
24.7
TIME (nsecs) Figure 8. This figure shows a fit of the fluorcsccnct decay of n-CdSe under flatband conditions to eq 11. The emission wavelength is 700 nm, and the electrode is in contact with 0.5 M KOH.
is the influence of the inner filtering on the observed fluorescence decay. Fits of t h m data reflect this expectation qualitatively but are quantitatively unsatisfying. Specifically, the best fits of this data to eq 11 have x2 values ranging from 5.9 to 18.8. When the semiamductor is in amtact with air, a space charge field is present, and this likely results in poor fits. Figure 8 shows a best fit of a fluorescence decay curve with emission wavelength of 700 nm and at the flatband potential to eq 10. Clearly the quality of the fit is excellent (x2 = 1-22), and the optimized fitting parameters are not unrealistic (A = 88.2, B = 13.3, S = 83.3, and T = 250 ns or greater). The ratio of A:B is 13:2 which is about a factor of 3 higher than that predicted from direct measurement of the semiconductors absorption Using the m e a d absorption coefficient of n-CdSe at 596 nm (8 X 10, cm-')this value of A gives a diffusion length of 10.9 pm. This length and the decay time above gives a dimLsion coefficient of 4.73 cm2/s. Qualitatively the trends in the fluorescence decay with the emission wavelength is in reasonable agreement with the predictions of eq 10. Unfortunately the quantitative comparison between the fitted parameters for the absorption coefficients and those known from the literature are not very good. An alternative explanation for the wavelength dependence obecrved here would be spectral diffusion. For example, the redder emission could involve transitions between holes in the valence band and a distribution (in energy) of donor states near the conduction band edge. As time evolves and the system relaxes the lower energy donor states in the space charge region take longer to depopulate (either by thermal emission or radiative recombination) than do
10378 The Journal of Physical Chemistry, Vol. 94, No. 25, 1992 '
I
'
I
----. I
......
I
I
T' = 3 1 5 T' = 0 008 T'=O,O
--..... ..........a
"
0.0
0.047 REDUCED TIME
0.094
Figure 9. Three simulated decay curvca are shown where the occupation time of the interfacial traps is varied ( T is the bulk recombination time and f is the population time of the traps).
the higher energy states. Trapping dynamics of this sort and its impact on the emission spectrum with time could also explain the wavelength dependence. SIlrtiee State Dyudc& The details of the trapping processes at the interface can play an important role in the carrier relaxation. Random walk simulations of the d i h i o n equation with interfacial trapping processes included in an explicit manner were used to study these processes. When the simulation is run with no interfacial traps or with traps which do not have a finite occupation time (Le., always available for carriers), very good agreement is found between the simulation results and eq 4.23 Figure 9 shows a plot of the decay profile for the case where the traps have a finite occupation time, expressed as a percentage of the bulk recombination time of the material. Clearly the effect of trap filling depends on the relative concentration of the traps, the number of carriers, and on the population time of a trap. In these studies the ratio of traps to the initial number of carriers is constant. The trap filling has the overall effect of increasing the average carrier lifetime and has its most pronounced effects at long time. Once the trap occupation time becomes similar to the bulk recombination time of the material the effect of the trap filling saturates, kcause the carriers decay by bulk recombination. An attempt to verify the presence of this effect experimentally was unsuccc88fu1. If the trap population indeed saturates it should be possible to modulate the threshold to saturation through the use of subbandgap excitation. The fluorescence decay was collected with no subbandgap excitation and with subbandgap excitation at wavelsagths longer than 900-nm incident on the sample (from a 1ooO-Wtungsten halogen.lamp). No reproducible change in the decay profile was found. However the failure of this experiment might be a result of the subbandgap excitation intensity being too weak or conversely the excitation light being too intense.
This study has probed the assumptions made in the modeling of charge camer relaxation at the semiconductor/electrolyte interface. The model used as a foil in these studies has received much theoretical attention and has begl used in the modeling of experimental results previously. These effects may appear somewhat subtle; however, they have important implications for understanding the dynamics of charge carrier relaxation. This study has found important limitations in using the d i h i o n eq 1. (1) The space charge fidd in the ektrode can play an important role in the charge carrier dynamics and is strongly reflected in the-lf decay law. Although this result is not unexpected, it has been dkcgdcd by some workers. With experimentsbeing performed under flatband conditions in an electrochemical cell, the space charge field can be largely eliminated. Recently the authors have obtained an analytical result for a model diffusion equation which includes a space charge and it should be
Shumaker et al. possible to measure surface recombination velocities with a space charge field present. (2) The bulk recombination is not first order in this material, Over carrier densities ranging from 2 X lo1*to 2 X 1014an-', and even at quite low photocarrier densities (less than the dopant density) may not be linear. The breakdown in this assumption appears to be dominated by nonradiative relaxation processes and not by bimolecular radiative relaxation. RoseMand others have treated the kinetics of bulk trapping p r o " quite extensively, and nonlinear intensity dependencies are well-known in studies of photoconductivity. (3) The influence of self-absorption can be quite sisnifcant and its importame only partially eliminated by observation of emission to the red of the bandgap. The overlap of the absorption and emission spectra of the material leads to photon recycling effects which distort the observed decay parameters, regardless of the observation wavelength. Although these studies are suggestive of self-absorption being important, the quantitative comparison between theory and expehent is not exact. workers have treated this effect previously. (4) The kinetics of surface trapping and the occupation times of the surface states can have important consequences for the o k e d carrier population kinetics. As shown here, the saturation of the surface state population can turn off the interfacial recombination and modify the form of the decay law quite dramatically. Such saturation effects could be quite hard to model analytically. Although it may not be so difficult to write the diffusion equation which would d y incorporate these effects, it is quite another matter to arrive at a general solution of this equation. Furthermore other possibly important effects have not been diricua9ad,for example, the time evolving electric field a r i s i i from the nonequilibrium carriers which drives the initial stages of the minority carrier relaxation. The trap filling effects related to the kinetics of surface state relaxation and the time evolving electric field of the carriers are both many body effects. It seems likely that the modeling of these processes will be quite involved. Previous ~ o r has k used ~ fluorescence ~ ~ ~ ~decay measurements under high injection conditions to measure surface recombination velocities. It is useful here to contrast the high injection approach with the low injection approach-both have their respective advantages and disadvantages. The primary advantages of the low injection approach are that the system's response is reasonably modeled as and the relaxation parameters are directly relevant to practical solar devices. Disadvantages of the approach are the necessity to operate under flatband conditions, although this limitation may not be required in the near and one's inability to externally define the potential in the material, caused by breakdown in the electroneutrality assumption.lg Another disadvantage is degradation of the electrode material in these studies because of the bias potential and the method of ufl" decay measurement; i.e. the material is irradiated for a long amount of time, and long time (minutes to hours) changes in the material may be evident in the decay law (saTable 11). Primary advantages of the high injection method are that the space charge field need not be explicitly treated, at least in the initial stages of the decay, and that the fluorescence yield from a single laser pulse may be enough to collect an entire transient. The major disadvantageis that the relaxation parametus and other pmperties of the material under these cunditions may not be directly relevant to those d e r linearconditions and solar conditions. For example it is well-known that the fmt-order bulk recombination is different under high injection conditions than low injection conditions. Also trap in the material may be saturated under high injection and hence absent from the kinetics. If this were the case,then the interfacial recombination velocity may be underestimated in such studies. However experimental agreement has been found between such high injection studies and steadystate measurements when the recombination velocity is highsSb Other effects become important at very high excitation densities, namely, intensity dependent changes in the ahrption spectrum of the solid, creation of an electron-hole plasma with different
Carrier Relaxation at Semiconductor Interfaces relaxation properties (e.g. anomalously high diffusion coefficient~~'), and Auger recombination processes. The result is a limitation on the useable range of carrier densities, preferably less ~ , limits the dythan loMcm-3 and greater than 1OI8 ~ m - which namic range of the measurement and the ability to test models of the relaxation. At present the preferred method remains unclear, the low injection studies require repetitive excitation and long collection times under which the sample may change, and the high injection studies are not analytically tractable and may result in a sample whose response is not representative of that under low injection conditions. The fluorescence decay curves do show a strong dependence on the properties of the surface. Although it may not be possible to obtain a surface recombination velocity in all cases, it may be possible to determine an "effective" recombination velocity, which includes trap filling effects. It will be necessary to correctly model the bulk recombination of the material or to prepare the material with no surface recombination and use the fluorescencedecay in that material as a reference. If trap filling is not important in the recombination process, then it may be possible to obtain a surface recombination velocity. If however trap saturation is important it is likely that only a time integrated value for the surface recombination can be determined. This limitation is were at a fundamental level, but not so important at a practical level. Fundamentally it means that the dynamics of the recombination are not well understood. Since solar cells and photocatalytic devices operate on long time scales, the time integrated value of the surface recombination velocity is quite useful. Acknowledgment. This work was supported by the Department of Energy, Office of Basic Energy Sciences and by the University of Pittsburgh. We thank David Burdelski for technical assistance with the simulation studies. Registry No. CdSe, 1306-24-7; KOH, 1310-58-3.
References and Notes (1) Orton, J. W.; Blood, P. The Electrical Characterization of Semiconductors: Measuremen?of Minority Carrier Properties; Academic Press: New York, 1990; references therein. (2) Aspnes, D. E. Surf.Sci. 1983, 132, 406. (3) Peter, L. M. Chem. Rev. 1990, 90, 753. (4) Lewis, N. S. Annu. Rev. Phys. Chem. 1991,42, 543. (5) (a) Willig, F. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1312. (b) Gottesfeld, S . Ber. Bumen-Ces. Phys. Chem. 1987,91, 362. (c) Tafalla, D.; Selvador, P.; Benito, R. M. J. Electrochem. Soc. 1990,137, 1810. (d) Norton, A. P.; Bernasek, S. L.; Bocarsly, A. B. J. Phys. Chem. 1988, 92, 6009. (6) Bessler-Podorowski, P.; Huppert, D.; Rosenwaks, Y.; Shapira, Y. J . Phys. Chem. 1991, 95, 4370 and references therein. (7) Nelson, R. J.; Williams, J. S.; Leamy, H. J.; Miller, B.; Casey, H. C., Jr.; Parkinson, B. A.; Heller, A. Appl. Phys. Lett. 1980,36,76 and references
therein. (8) Marvin, D. C.; Beck, S. M.; Wessel, J. E.; Rollins, J. G. I.E.E.E. J .
Quant. Elec. 1989, 25, 1064. (9) Bensaid, B.; Raymond, F.;Leroux, M.; Veri€, C.; Fofana, B. J . Appl. Phvs. 1989. 66. 5542. 110) Ahrenkel, R. K. Current Topics in Photovoltaics; Academic Press: New York, 1988; Vol. 3, Chapter 1. (1 1) Hoffman, C. A.; Jadionas, K.; Gerritsen, H.; Nurmikko, A. V. Appl. Phys. Lett. 1978, 33, 536. (12) (a) Gomez-Jahn, L. A.; Dwayne Miller, R. J. J. Chem. Phys. 1992, 96. 3981 and references therein. (13) (a) Wilson, T.; Pcstcr, P. D. J. Appl. Phys. 1988,63,871. (b) Wilson, T.; Pester, P. D. Phys. Star. Sol. A 1986, 97, 323. (14) Ioannou, D. E.;Gledhill, R. J. J . Appl. Phys. 1984, 56, 1797.
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10379 (15) 't Hooft, G. W.; van Opdrop, C. J . Appl. Phys. 1986, 60, 1065. (16) von Roos, 0. J. Appl. Phys. 1985,57, 2196. (17) (a) Bartlett, P. N.; Hamnett, A.; Dare-Edwards, M. P. J. Electre chem. Soc. 1981, 128, 1492. (b) Reichman, J. Appl. Phys. Lett. 1980,36, 574. (c) El-Guibaly, F.; Colbow, K.; Funt, B. L. J. Appl. Phys. 1981, 52, 3480. (18) (a) Laser, D.; Bard, A. J. J. Electrochem. Soc. 1976,123,1828. (b) Laser, D.; Bard, A. J. J. Electrochem. Soc. 1976, 123, 1837. (19) Gottesfeld, S.; Feldberg. S. W. J. EkctroOM/. Chem. 1983, 146,47. (20) (a) Vaitkus, J. Phys. St. Solidi 1976, A34, 769. (b) Asbeck, P. J. Appl. Phys. 1977, 48, 820. (c) Nelson, R. J. J. Vac. Sci. Technol. 1978,15, 1475. (21) Pankove, J. I. Optical Processes in Semiconductors; Dover, NY., 1971. (22) (a) Leung, L. K.; Komplin, N. J.; E b , A. B.; Tabatabaie, N. J. Phys. Chem. 1991,95,5918. (b) Zhang, J. Z.; Ellis, A. B. J. Phys. Chem. 1992, 96, 2700. (c) Ellis, A. B.; Chemistry and Structure at Interfaces; Hall, R. B., Ellis, A. B., Eds.; VCH: Deerfield Beach, FL, 1986. (23) Shumaker, M. L.; Burdelski, D.; Waldeck, D. H. Picosecond and Femtosecond Spectroscopy from Laboratory to Real World; SPIE: Bellingham, WA, 1990; Vol. 1209, p 109. (24) (a) Tenne, R.; Mariette, H.; Levy-Clement, C.; Jager-Waldau, R. Phys. Rev. E 1987,36, 1204. (b) Tenne, R. et al. J. Cryst. Growth 1988,86, 826. (25) Zeglinski, D. M.; Waldeck, D. H. J. Phys. Chem. 1988, 92, 692. (26) Palik, E. D. Handbook of the Optical Constants of Solids II; Academic Press: New York, 1991. (27) Shumaker, M. L.; Dollard, W. J.; Zeglinski, D. M.; Waldeck, D. H. IS & T 44th Annual Conference Proceedings, 1991. (28) Pleskov, Y. V.; Gurevich, Y. Y. Semiconductor Photoelectrochemistry; Plenum: New York, 1986. (29) Rosen, D. L.; Li, Q. X.;AIPano, R. R. Phys. Rev. B 1985,31,2396. (30) Smith, R. A. Semiconductors; Cambridge, New York, 1978. (31) (a) Mora, S.; Romeo, N.; Tarricone, L. I1 Nuavo Cimnro 1980,608, 97. (b) Moore, A. R.; Lin, Hong-sheng J. Appl. Phys. 1987,61, 5366. (c) Pandey, R. K.; Gore, R. B.; Rooz, A. J. N. J. Phys. D. Appl. Phys. 1987,20, 1059. (32) Dollard, W. J.; Shumaker, M. L.; Waldeck, D. H. Submitted for publication in J . Phys. Chem. (33) (a) Halas, N. J.; Bokor, J. Phys. Rev. Lett. 1989,62, 1679. (b) Reea, G. J. Solid State Elect. 1985. 28. 517. (c) Kauffman.. J. F.:. Balko. B. A.: Richmond, G. L. J. Phys. Chem.'1992,9& 6371. (34) Holloway, H. J . Appl. Phys. 1987, 62, 3241. (35) The coefficients given by Ahrenkiel in eq 3.27 are in error due to a
misprint. To correct his expression let A = - 2a
p2 - u2
E=
+
E/D ( S / D - a)(a - 8)
E / D (S/D-S/D)
Ahrenkiel, R., private communication. (36) (a) Storr, G. L.; Haneman, D. J. Appl. Phys. 1985, 58, 1677. (b) Moore, A. R.; Lm,Hong-sheng J. Appl. Phys. 1987,61,5366. (c) Rosenwah, Yossi Ph.D. Thesis. Tel-Aviv University, 1991. (37) Junnarkar, M. R.; Alfano, R. R. Phys. Rev. B 1986,34, 7045. (38) (a) Bube, R. H.; Cardon, F. J. Appl. Phys. 1964, 35, 2712. (b) Cardon, F.; Bube, R. H. J . Appl. Phys. 1964,35, 3344. (c) Klasens, H. A,; Ramsden, W.; Quantie, C. J. Opt. Soc. Am. 1947, 38, 60. (39) Klasens, H. A. J. Phys. Chem. Solids 1958, 7, 175. (40) Rose, A. Concepts in Photoconductivity and Allied Problems; Interscience, New York, 1963. (41) (a) Haak, R.; Tench, D. J. Electrochem. Soc. 1984, 131,275. (b) Haak. R.: Tench. D. J. Electrochem. Soc. 1984.131. 1442. (42) Shumaker, M. L.; Dollard, W. J.; Waideck; D. H. Unpublished results. (43) (a) Evenor, M.; Gottesfeld, S.; Harzion, 2.; Huppert, D.; Feldberg, S. W.J . Phys. Chem. 1984.88, 6213. (b) Harzion, 2.; Huppert, D.; Gottesfeld, S.;Croitoru, N. J. Electroanal. Chem. 1983, 150, 571.
