3908
J. Phys. Chem. B 1999, 103, 3908-3915
Simulations of the Steady-State Current Density vs Potential Characteristics of Semiconducting Electrodes Samir J. Anz and Nathan S. Lewis* DiVision of Chemistry and Chemical Engineering, 127-72, Noyes Laboratory, California Institute of Technology, Pasadena, California 91125 ReceiVed: NoVember 30, 1998; In Final Form: March 3, 1999
A series of digital simulations has been performed to obtain insight into the steady-state current density vs potential behavior of semiconductor/liquid interfaces. The ToSCA program, incorporating all of the key kinetic parameters involved with the generation, transport, and recombination of charge carriers both in the semiconductor and across the semiconductor/liquid interface, has been used for this purpose. The simulations confirmed conclusions obtained previously from a simplified analytical model, which state that for ideal behavior of a nondegenerately doped semiconducting electrode the photovoltage of an n-type semiconductor/ liquid interface should not change if the concentration of the reduced form of the redox species, A-, is held constant but the concentration of the oxidized form of the redox species, A, is varied. The simplified analytical model also predicts that the photovoltage will be independent of variation in [A-] if [A] is held constant. In contrast, recent work has asserted that ideal junction behavior implies that the photocurrent should exhibit shifts in potential that are linearly dependent on the concentration of the minority carrier acceptor species in the solution, with a magnitude of 59 mV per decade change in the acceptor concentration at 300 K. In accord with the predictions of the simplified analytical model, such shifts are not apparent in the simulations presented in this work. Finally, ToSCA simulations have been applied to analyze literature data on the steady-state current density vs potential behavior of p-InP/Fe(CN)63-/4-(aq) contacts. Such simulations have established an upper bound for the interfacial charge-transfer rate constant of ket≈10-20 cm4 s-1 in this system.
I. Introduction The steady-state current density (J) vs potential (E) behavior of a solid/liquid contact is perhaps the most frequently measured property of a semiconducting electrode.1-3 The J-E behavior is important in characterizing the corrosion, passivation, surface condition, and electrical stability of a semiconducting electrode and also provides the key figures of merit in evaluating the performance of semiconductor electrodes for photoelectrochemical energy conversion applications.4 Despite the use of J-E properties for over 40 years in characterizing the behavior of various semiconductor electrodes, a quantitative understanding of the chemical kinetic factors that control the steady-state J-E behavior is still lacking. Under certain limiting conditions, analytical expressions can be formulated to describe the J-E behavior of semiconductor/liquid contacts.5 Under other conditions, however, the presence of mixed kinetic processes, potential-dependent effects such as surface-state charging, and other important features of a real electrode system preclude obtaining solutions in a simple analytical form that can describe the steady-state J-E properties of semiconductor/liquid junctions.6-8 To address these issues, we have performed a series of digital simulations of the steady-state J-E behavior of semiconductor/ liquid interfaces using the ToSCA program. ToSCA is a twodimensional simulation code incorporating all of the key kinetic parameters involved with the generation, transport, and recombination of charge carriers both in the semiconductor and across the semiconductor/liquid interface.9,10 ToSCA has been used previously to provide insight into the steady-state and transient behavior of Si electrodes under high-level injection conditions11-13
and to evaluate conditions under which interfacial chargetransfer rate constants can be evaluated from the transient photoluminescence signals of semiconductor/liquid contacts.14 The goal of this work is to utilize ToSCA to provide a set of J-E curves to allow analysis of the steady-state J-E behavior of semiconductor/liquid contacts in situations which cannot be treated analytically. Several aspects of the J-E behavior were of specific concern in this study. It has been predicted that for an “ideally” behaving semiconductor electrode,1,2,15-19 in which interfacial chargetransfer processes dominate the kinetics and band-edge movement is absent, the photovoltage should not change if the concentration of the reduced form of the redox species, [A-], is held constant but the concentration of the oxidized form of the redox species, [A], varies.5 This prediction arises because variation of [A] changes both the electron acceptor concentration in the solution and the equilibrium concentration of electrons at the surface of the semiconductor. In an ideal system, these two changes should exactly cancel each other, and therefore no change should be observed in the open-circuit voltage, Voc, of such interfaces. It is clear, however, that this prediction cannot be valid for arbitrarily large variations in the acceptor concentration, because eventually the band edges will start to move, an accumulation or inversion layer will form, and/or other factors will prevent the ideal dependence from dominating the actual interfacial kinetics. These complex kinetics cannot be treated analytically and are thus a focus of our digital simulations. Second, recent work has claimed that the photovoltage of an ideally behaving junction should shift in direct proportion to the concentration of the minority carrier acceptor species in the
10.1021/jp9845571 CCC: $18.00 © 1999 American Chemical Society Published on Web 04/27/1999
Current Density vs Potential Simulations SCHEME 1. Various Carrier Recombination Mechanisms and Quantitative Values for the Parameters Required for the Simulationsa
a The sample’s band-gap energy, Eg, was 1.35 eV; the intrinsic carrier concentration, ni, was 1.5 × 107 cm-3; is the dielectric constant of the electrode and o is the dielectric permittivity of free space. The junction’s barrier height was 0.8 V, i.e., the solution’s redox potential was 0.8 V below the conduction band-edge energy at the surface, and the reorganization energy of the redox species in solution was held constant at λ ) 0.5 eV. The resultant flat-band potential, Vfb, was -0.677 V vs the redox potential of the cell.
solution, with a value of 59 mV per decade change in the acceptor concentration at 300 K.20,21 However, under illumination, the fate of minority carriers is determined by the competition between interfacial charge transfer and various other recombination processes.7,16 If the minority carrier interfacial charge-transfer rate constant were sufficiently large to allow collection of all the photogenerated charge carriers, then any increase in acceptor concentration should not change the J-E behavior. Thus it is not clear in what kinetic regimes, if any, the photocurrent should be linearly dependent on the concentration of the minority carrier acceptor species. Exploration of this regime of behavior is a second focus of the ToSCA simulations presented in this work. A final area of focus involves the relationship between the transient and steady-state J-E properties of semiconductor/ liquid contacts. A variety of time-dependent experiments have been developed to probe the carrier concentration decay dynamics at semiconductor/liquid interfaces.22-29 These experiments often are subject to various interpretations and the resulting data are typically compatible with a wide range of surface recombination velocities and interfacial charge-transfer rate constants. However, ambiguities from analysis of the transient data alone often can be resolved through reference to the steady-state J-E data for the system of interest. The steadystate simulations described herein allow assessment of the selfconsistency of a kinetic analysis of both the transient and steadystate J-E data of a semiconductor/liquid contact and provide insight into the allowed values of the charge-transfer and chargerecombination rate constants for a particular semiconductor electrode. II. Simulation Procedures and Parameters The ToSCA code itself has been described in some detail previously.14,30,31 In this section, we describe only certain features of the basic structure and capabilities of ToSCA and describe some additions to the code that have been made to account specifically for the treatment of certain steady-state properties of concern in this work. ToSCA self-consistently
J. Phys. Chem. B, Vol. 103, No. 19, 1999 3909 solves Poisson’s equations subject to the van Roosbroeck drift/ diffusion model.9,10 Scheme 1 depicts the important energetic and kinetic parameters incorporated into the ToSCA code as used in this work. Input parameters include the bulk radiative recombination rate constant, kr, the bulk trap density, Nt, the surface state trap density, Nt,s, and interfacial charge-transfer rate constants, ket and kht, for electrons and holes, respectively. The traps were assumed to be located in the middle of the band gap of the semiconductor, and their capture cross-sections for electrons, kn, and holes, kp, were both set to 10-8 cm3 s-1 in the bulk and at the surface of the solid.32,33 To maintain generality for both inner-sphere and outer-sphere redox reactions, a Marcus-Gerischer formalism was not used to predict the energy dependence of ket or kht relative to the band-edge energies of the semiconductor.12,34,35 Instead, both ket and kht were treated as independent input parameters of the simulations.12,14 Detailed balance and microscopic reversibility principles were used to relate the rate constants for charge transfer into each band to the rate constants for charge-transfer processes out of each band.15,36-38 Surface states were allowed to undergo interfacial charge transfer with the redox species in solution, and these kinetic processes had electron and hole charge-transfer rate constants of ket,s and kht,s, respectively. Microscopic reversibility and detailed balance principles were used to relate ket,s to kht,s such that, at equilibrium,15,36-38
kht,s Nt,s cred (1 - fo) ) ket,s Nt,s cox fo
(1)
The resultant net current from surface states into solution, js, both at and away from equilibrium is thus15,36-38
[
js/q ) kmax,ht,s Nt,s cred exp
]( )
-(Es - E°′redox - λ)2 f -1 4kTλ fo
(2) In these expressions, the rate constants kmax,ht,s and kmax,et,s are the anodic and cathodic rate constants for charge transfer at optimal exoergicity from surface states into solution. The constants cred and cox represent the concentration of dissolved electron donors and acceptors in the electrolyte, respectively. The fraction of occupied surface states is defined as f. Es is the energy of the surface state and E°′redox is the formal electrochemical potential of the redox species having a reorganization energy denoted by λ.2,34,35 The subscript “o” is used to indicate the value of a parameter in the dark at equilibrium. When f < fo, the net current is negative and hence electrons transfer from surface states into the solution. The rate constants for electron exchange with the redox species were not varied during the course of the simulations, so the Marcus-Gerischer formalism, as specified in eq 2, was used to provide an approximate estimate of these rate constants for the specified reorganization energy of the redox couple. Only the density of surface states was varied in order to investigate the effects of increased electrical trap density on the J-E properties of the semiconductor/liquid contact. In contrast, the rate constants for electron exchange from the conduction and valence bands of the semiconductor were varied independently and were not confined to a specific kinetic relationship with respect to the reorganization energy of the redox couple. The unconstrained variation in kht and ket was employed because the primary focus of the simulations was to elucidate the effects that variations in these two rate constants would produce on the J-E behavior of the semiconductor/liquid contact.
3910 J. Phys. Chem. B, Vol. 103, No. 19, 1999
Anz and Lewis
ToSCA accounted for surface-state charging and any associated shifts in the band edges. It also accounted for any shift in the band edges resulting from the presence of excess mobile charge either in an accumulation or an inversion layer in the semiconductor. The ToSCA code was originally designed to simulate the behavior of solid-state electronic devices. Hence it was necessary to modify the code to the situation for a semiconductor/liquid contact by using a modified (leaky) gate contact with the gate “oxide” having the properties of a Helmholtz double layer with a thickness of 0.3 nm and a dielectric constant of H ) 6o.39 The light intensity was uniform across the electrode surface, and the Beer-Lambert law was used to create injected carrier profiles with a penetration depth R assumed to be 5.4 × 104 cm-1.32 The light intensity was set to provide an input photon flux, Γ0, of 1.25 × 1017 cm-2 s-1. This photon flux produces a light-limited photocurrent density of 20 mA cm-2 if all of the photogenerated charge carriers are separated with an internal quantum yield for charge separation, Φ, of 1.0. For the purpose of these simulations, InP was chosen as a representative semiconductor material and hence all parameter values were associated with InP. The second-order radiative recombination coefficient, kr, was taken to be 2 × 10-10 cm3 s-1, and the intrinsic carrier concentration, ni, was taken to be 1.5 × 107 cm-3.32,40 The effective densities of states in the conduction and valence bands, Nc and Nv, respectively, were set to 5.5 × 1017 cm-3 and 1.8 × 1019 cm-3, respectively, as reported in the literature for InP.32,33,40 The semiconductor was assumed to be doped n-type, with a bulk free carrier concentration, no, of 5 × 1015 cm-3. The mobility of holes, µo,p was taken to be 150 cm2 V-1 s-1 at room temperature.14,33 To account for field-dependent mobilities of electrons in InP, the electron mobility, µn, was calculated using:14
µo,n + Vsat,n µn )
1+
E3 E 4crit
( ) E Ecrit
4
(3)
In eq 3, E denotes the electric field strength, Ecrit is the critical field strength, µo,n denotes the zero-field mobility of the electrons, and Vsat,n represents the saturation velocity of electrons in the presence of large electric fields.41-43 Although ToSCA can be used to compute the dependence of the mobilities of carriers on temperature and doping concentration, the experimental value for the room-temperature low-field electron mobility published by Majerfeld et al. of µo,n ) 3600 cm2 V-1 s-1 for a carrier concentration no ) 9.7 × 1014 cm-3 was input explicitly in the simulation routine.44 A peak velocity of 2.7 × 107 cm s-1 at an electric field of 10 kV cm-1 was obtained using Vsat,n ) 8.5 × 106 cm s-1 and Ecrit ) 12 kV cm-1. This peak velocity agrees well with experimental measurements as well as with recent simulations.32,43-46 Experimentally, electric fields of sufficient magnitude to cause electron velocities to saturate in n-InP can be realized for moderately doped samples at bias potentials as small as 0.15 V positive of the flat-band potential. The simulated sample was 200 µm thick in the dimension normal to the surface plane (x-dimension), 0.1 µm wide (ydimension), and 100 µm in height (z-direction). The electrode surface (the y-z plane) was thus 0.1 µm × 100 µm in area. Because ToSCA is a two-dimensional simulation program, the sample was considered to be uniform in the y-direction. The boundary conditions at the edges of the 0.1 µm wide simulated
Figure 1. Simulated J-E curves. The influence of ket on Voc is depicted for the fixed values of kht ) 10-16 cm4 s-1 and cox ) cred ) 0.1 M. Surface state mediated recombination was made negligible by setting Nt,s ) 101 cm-2 (Sp ) 10-7 cm s-1).
slice were set such that zero flux and zero recombination occurred at the boundaries. Thus, the simulations describe the situation for a sample that is 200 µm thick and is of infinite width and height (i.e., a quasi one-dimensional simulation). Parameters related to the properties of the semiconductor/ solution interface were taken to be those appropriate for the n-InP/CH3OH-1,1′-dimethylferrocene+/0 system. Thus, for cox ) cred ) 0.1 M, the formal potential of the redox couple was set to 0.8 V from the conduction band47 and the reorganization energy was λ ) 0.5 eV.48 III. Results A. Ideal Model Simulations. In most experimental situations, changing the concentration of a redox species in the solution affects: (a) the equilibrium energetics of the solid/liquid contact and (b) the capture rates of carriers across the semiconductor/ liquid interface.5,49 To deal with the interplay between these two parameters, we first present the results of simulations in which only one key kinetic or energetic variable has been changed at a time, even though some of these situations represent hypothetical scenarios. In the subsequent section we assess the combined effects of these individual changes as they pertain to actual experimental situations encountered in semiconductor electrochemistry. 1. Dependence of Voc on ket . Figure 1 displays the simulated J-E curves for a constant value of the minority carrier (hole) charge-transfer rate constant, kht, (10-16 cm4 s-1) as a function of the majority carrier (electron) charge-transfer rate constant, ket. In these simulations, Nt,s is very small, such that the effects of the interfacial kinetics can be investigated without the complications of surface recombination at the semiconductor/ liquid contact. As can be seen in Figure 1, Voc increases as ket decreases. This occurs because, with only interfacial recombination present, the constraint that must be satisfied at Voc is:4,5,49
ΦΓ0 ) ket (ns - nso)
(4)
where ΦΓ0 is the flux of photogenerated minority carriers that enter the space charge region of the semiconductor, ns is the electron concentration at the surface of the solid, and nso is the value of ns at equilibrium (i.e., in the absence of illumination). Thus, at open circuit the interfacial majority carrier flux (ket(ns
Current Density vs Potential Simulations
J. Phys. Chem. B, Vol. 103, No. 19, 1999 3911
- nso)) offsets the interfacial minority carrier flux (ΦΓ0) to produce no net current across the semiconductor/liquid contact. The equation relating ns to the electron quasi-Fermi level, Ef,n, is given as36-38,40
ns ) Nc exp(Ecb - Ef,n)/kT
(5)
with k equaling Boltzmann’s constant and T equaling the temperature. Thus Ef,n must become more negative as ket decreases with (1/q)∂Ef,n/∂(log ket) ) -59 mV at 300 K.5 Because the value of Ef,n decreases while the value of E(A/ A-) remains unchanged, and because Voc ) (1/q){E(A/A-) Ef,n} (keeping Voc as a positive quantity, by convention), the value of Voc therefore initially exhibits a Nernstian dependence on ket, such that -∂Voc/∂(log ket) ) 59 mV at 300 K, as depicted in Figure 1 for ket g 10-17 cm4 s-1. At sufficiently small values of ket, -∂Voc/∂(log ket) < 59 mV, and eventually -∂Voc/∂(log ket) tends to zero. When surface recombination is minimal, this transition from -∂Voc/∂(log ket) ) 59 mV to -∂Voc/∂(log ket) < 59 mV occurs when bulk radiative and/or nonradiative recombination processes start to compete with interfacial charge-transfer events. For the simulations of Figure 1, this occurs when ket< 10-17 cm4 s-1. In this regime, decreasing the value of ket produces an increase in ns, but the resulting increase in the rate of bulk recombination begins to dominate the carrier recombination processes of the system. The quantitative analysis of Figure 1 shows that this bulk recombination limit restricts the range over which the ideal behavior of -∂Voc/∂(log ket) ) 59 mV can be observed experimentally. The range over which ideal behavior is observed depends on the value of kht. For a set of simulations similar to those depicted in Figure 1 but with kht held at 10-14 cm4 s-1, the requirement for ideal behavior was extended to ket > 10-20 cm4 s-1. The trends in Voc apparent in the simulations of Figure 1 are essentially independent of the barrier height, φb ) (1/q){E(A/ A-) - Ecb}, of the solid/liquid contact over the range 0.62 V < φb < 0.92 V. This lack of dependence on φb occurs because, at the light intensity chosen, ns is so much larger than nso that the value of Ef,n does not depend on the initial energetics of the semiconductor/liquid contact (i.e., at Voc, ns . nso). This situation is maintained until the value of E(A/A-) becomes so close to Ecb that degeneracy of the orbital occupancy, as well as band-edge movement, needs to be considered. Under these very low barrier height conditions, the value of -∂Voc/∂(log ket) is always less than the ideal value of 59 mV at 300 K, regardless of the value of ket. 2. Dependence of Voc on kht. Figure 2 displays the simulated steady-state J-E data obtained when the minority carrier chargetransfer rate constant, kht, is decreased and all other kinetic parameters are held constant. In contrast to the situation for varying ket (Figure 1), Voc initially does not change as kht is varied. In fact, for 10-18 e kht e 10-14 cm4 s-1, Voc did not vary by more than 1 mV. For kht < 10-18 cm4 s-1, Voc decreased slightly as kht decreased, independent of the value of φb for the range 0.62 V < φb < 0.92 V. This behavior can be understood within the ideal model described previously. With only interfacial recombination present, the continuity condition for minority carriers at the interface is5
ΦΓ0 ) kht (ps - pso)
(6)
By analogy to eq 5, the minority carrier quasi-Fermi level can be defined as36,40
Figure 2. Simulated J-E curves. The influence of kht on Voc is depicted for the fixed values of ket ) 10-16 cm4 s-1 and cox ) cred ) 0.1 M. Surface state mediated recombination was made negligible by setting Nt,s ) 101 cm-2 (Sp ) 10-7 cm s-1).
ps ) Nv exp(Ef,p - Evb)/kT
(7)
where ps is the hole concentration at the semiconductor surface and Evb is the energy of the top of the valence band at the solid/ liquid contact. For large kht values, the minority carrier concentration at the surface of the semiconductor deviates very little from its equilibrium value in order to sustain the required photocurrent flux across the solid/liquid contact. Thus, Ef,p ≈ E(A/A-). As kht decreases, ps must increase in order to produce the required minority carrier flux across the solid/liquid contact. However, Voc is determined by the value of E(A/A-) - Ef,n. Thus, changes in Ef,p are of no consequence to Voc unless they also affect the value of Ef,n. This lack of change in Ef,n as kht is varied clearly occurs for kht > 10-19 cm4 s-1 in the simulations shown in Figure 2. At sufficiently low kht values, ps will become large enough that electron-hole recombination events will increase in rate relative to interfacial charge-transfer processes. In this regime, the J-E data will show a dependence on the value of kht. However, in this regime, the simulations shown in Figure 2 illustrate that the fill factor is somewhat more sensitive to the value of kht than is Voc. This observation is readily understood because increases in band bending for V > Voc suppress the electron-hole recombination process (by lowering the majority carrier concentration at the surface while increasing the rate of the minority carrier charge-transfer process). The shape of the J-E properties clearly reveals that superposition does not hold in this scenario, but this is due to the dependence of ΦΓ0 on the injected carrier density, when the minority carrier flux to the surface is a function of the amount of electron-hole recombination in the bulk of the solid and not to any band edge shifts, as shown quantitatively by the ToSCA simulations. B. The Effect of Surface States on the Open-Circuit Voltage. Figure 3 illustrates some of the effects of adding surface states to the ideal solid/liquid contact simulated above. For ket g 10-17 cm4 s-1, when Nt,s< 1013 cm-2, little change was observed in Voc or in the dependence of Voc on kht. This occurs because interfacial electron transfer competes successfully with surface recombination for these values of Nt,s and kht. With Nt,s set to any value between 1011 cm-2 and 1013 cm-2 and when ket is in the range of 10-20 cm4 s-1< ket 10 cm s-1, the simulations show that these dependencies exactly offset each other, such that ∂Voc/∂(log [A]) ) 0. For ket[A] < 10 cm s-1, bulk recombination in the sample competes with interfacial charge-transfer events and Ef,n does not decrease as [A] decreases. However, the energetic relationship based on the Nernst equation, ∂φb/∂(log [A]) ) 59 mV, is still valid. Thus, in this kinetic regime, the simulations show that Voc changes as [A] is varied. These effects are exactly those predicted in a qualitative analysis.5 The quantitative analysis described herein confirms this effect for ket[A] > 10 cm s-1, but further demonstrates that Voc will start to vary with decreases in ket for ket[A] < 10 cm s-1 and that this variation will occur even in the absence of surface states or band-edge shifts. For ket,max values of 10-16 cm4 s-1 (ket ) 1.8 × 10-17 cm4 s-1) and the given bulk recombination and barrier height parameters, this implies that the region over which Voc is independent of [A] is limited to [A] g 0.001M (6 × 1017 cm-3). 2. Dependence of Voc on [A-]. A similar situation occurs for the dependence of Voc on [A-] when [A] is held constant. The simulations show that the dependence of eq 2, (1/q)∂Ef,p/∂(log [A-]) ≈ 0, holds for kht[A-] > 500 cm s-1. However, from the Nernst equation, ∂φb/∂(log [A-]) ) -59 mV. Thus, the combination of these two effects produces ∂Voc/∂(log [A-]) ) -59 mV for kht[A-] > 500 cm s-1. Hence, for kht,max ) 10-16 cm4 s-1 (kht ) 9.5 × 10-17 cm4 s-1), Voc decreases as [A-] is decreased until [A-] < 0.01 M (6 × 1018 cm-3). At this point, bulk recombination starts to scavenge carriers and competes with surface-state-mediated recombination for photogenerated carriers, so ∂Voc/∂(log [A-]) < 59 mV is observed. 3. Dependence of Voc on Solution Dilution at Constant [A]/[A-] Ratio. When the redox solution is diluted by a factor ζ through the addition of a solution that contains solvent and electrolyte but no redox species, the Nernst redox potential E(A/ A-) is essentially unchanged, so that -∂φb/∂(log [ζ]) ) 0. However, the dependencies of the minority- and majority-carrier currents on the values of [A] and [A-] are still present. From eqs 5 and 7, it can be seen that (1/q)∂Ef,p/∂(log [A-]) ≈ 0, while -(1/q)∂Ef,n/∂(log [A]) ) 59 mV. In the simulations, this
Current Density vs Potential Simulations
J. Phys. Chem. B, Vol. 103, No. 19, 1999 3913
Figure 5. Simulated J-E curves. The influence of changing cox on Voc is depicted at the fixed values of kct,max ) 10-16 cm4 s-1 (kht ) 9.5 × 10-17 cm4 s-1 and ket ) 1.8 × 10-17 cm4 s-1) and cred ) 0.1 M for Nt,s (Sp) set to (a) 101 (10-7) and (b) 1011 (103) cm-2 (cm s-1).
Figure 4. The percentage of photogenerated carriers recombining through radiative (solid line), bulk nonradiative (dotted line) and surface state (dashed line) recombination is depicted as a function of applied voltage for Nt,s (Sp) set to (a) 1011 (103); (b) 1012 (104); and (c) 1013 (105) cm-2 (cm s-1).
behavior is valid for kht[A-] > 3 × 103 cm s-1 (for Ef,p) and ket[A] > 600 cm s-1 (for Ef,n), respectively. Thus, in this regime, ∂Voc/∂(log [A-]) ) 59 mV. For either kht[A-] or ket[A] outside of these limits, bulk recombination competes with interfacial recombination such that ∂Voc/∂(log [A-]) 10 cm s-1 and kht[A-] > 500 cm s-1; for values of ket,max and kht,max of ≈ 10-16 cm4 s-1, this implies that deviations from this ideal behavior should occur whenever [A] < 0.001 M (6 × 1017 cm-3) or [A-] < 0.01 M (6 × 1018 cm-3). Analysis of the deviations from ideal behavior requires a quantitative treatment of the regime in which interfacial charge transfer can dominate the kinetic processes of the entire semiconductor/liquid contact without a significant contribution from surface or bulk kinetic processes. B. Ideal Dependence of the J-E Curves on the Concentration of the Minority Carrier Redox Acceptor. The simulations above have clearly shown only two limiting behaviors as the minority carrier acceptor, [A-], is varied: ∂Voc/ ∂(log [A-]) ) -59 mV for kht[A-] > 500 cm s-1 and ∂Voc/ ∂(log [A-]) ≈ 0 mV for kht[A-] e 0.5 cm s-1. In contrast, a recent paper has asserted that ideal behavior of a semiconductor/ liquid contact produces ∂Voc/∂(log [A-]) e -59 mV.20 This behavior was not observed in our simulations for any combination of parameters judged to be representative of any reasonable experimental situation. Ideal behavior in this context refers to the conventional ideal thermodynamic conditions of fixed band edge positions at the various potentials applied to the semiconductor/liquid interface, combined with the conventional ideal kinetic conditions for which second-order interfacial chargetransfer kinetic processes, and not surface state capture or exchange events, dominate the J-E behavior the semiconductor/ liquid contact.1,2,15-19 The experimental study of the J-E behavior of (30-50 Å) GaInP2/p-GaAs electrodes in fact indicated that the photocurrent-potential characteristic depended strongly on the acceptor concentration between 5 and 20 mM; above 20 mM the electrode potential for photocurrent onset remained constant.20,21 Even when we explored shifts in band-edge positions and/or changes in electron surface recombination velocities, no simulation showed a trend for which ∂Voc/∂(log [A-]) ≈ -150 to -180 mV. This experimentally observed dependence is therefore attributed primarily to the concentration dependence of the threshold voltage for recombination of photogenerated carriers in the bulk and to other complicating factors at this semiconductor/liquid contact and does not provide evidence for an ideal dependence of Voc on the interfacial charge-transfer kinetic processes. C. Correlation Between Steady-State and Transient Measurements: Use of Steady-State Simulations to Analyze the J-E Behavior of the p-InP/Fe(CN)63-/4-(aq) Interface. The plots in Figure 2 demonstrate that above some value of kht, the steady-state J-E curves of the n-InP/CH3OH-1,1′-dimethylferrocene+/0 contact are insensitive to further increases in kht. Similar simulations have also corroborated this trend for surface recombination velocities as high as 104 cm s-1 (Nt,s ) 1012 cm-2). Thus, it is not possible to observe changes in the J-E curves, or to evaluate the value of kht, above this asymptotic limit on the minority carrier collection velocity. In previous work, the behavior of the p-InP/Fe(CN6)3-/4-(aq) contact has been interpreted to indicate a field-dependent minority carrier charge-transfer rate constant.29 The value of ket[A] (for this p-type semiconductor/liquid contact) extracted from a digital simulation model of the transient photoluminescence data was deduced to vary from 103 cm s-1 to 107 cm s-1
Anz and Lewis
Figure 6. A simulated J-E curve for the p-InP/Fe(CN)63-/4-(aq) contact with ket ) 1 × 10-20 cm4 s-1, S ) 104 cm s-1, and a lightlimited photocurrent density of 35 mA cm-2 (dotted line) is compared to experimental data on this system (ref 25). The simulated J-E curve that includes the reported band-edge movement is also shown (dashed line).
as the electrode potential was varied. The surface recombination velocity was determined to be approximately 104 cm s-1.29 In this section, we predict the steady-state J-E curves that should result from these parameters and attempt to assess the selfconsistency between the observed steady-state data and the interpretation of the transient J-E behavior of this system. As displayed in Figure 2, even for ket[A] ) 103 cm s-1 (the minimum rate constant at the weakest fields), the steady-state J-E curves would have reached the asymptote above which further increases in ket would not have affected the J-E data. In addition, the expected J-E behavior for this lowest value of ket is shifted far more positively than the actual data (Figure 6). This shift in the actual data to negative potentials relative to the flat-band potential clearly indicates the presence of a much smaller charge-transfer rate constant than that extracted from the digital simulation model of the transient photoluminescence data. Even including the observed band-edge movement, the plots in Figure 6 indicate a value of ket[A] < 102 cm s-1. This contrasts with the published interpretation of the transient photoluminescence data which claimed that ket values in this system are extremely large (≈10-12 cm4 s-1).29 This discrepancy can be resolved because recent observations have shown that the transient photoluminescence data are complicated by an artifact arising from light-intensity-dependent damage to the p-InP/Fe(CN)63-/4-(aq) contact.51-53 In this scenario, rapid potential-dependent surface trapping leads to a decay of the photoluminescence signal that was interpreted instead as potential-dependent interfacial charge transfer.53 In fact, the low quantum yields for photocurrent collection at steady state serve to rule out the high branching ratio of charge transfer to surface recombination implied by values of S ) 103 cm s-1 and ket[A] g 103 cm s-1. Furthermore, steady-state J-E experiments on this contact have indicated the growth of an oxide layer and irreversible surface etching and damage in contact with this redox couple.51,54 Thus, steady-state data for both n-type and p-type InP that have been reported in this electrolyte are consistent with the low value of ket evident in the simulation. Instead of suggesting unusually rapid interfacial charge transfer,29,55-58 we therefore conclude that this particular semiconductor/liquid contact illustrates the very slow interfacial charge-transfer kinetics and poor J-E behavior expected for unstable semiconductor surfaces having insulating surface oxide layers through which carriers must tunnel to the redox species.
Current Density vs Potential Simulations Another use of the present simulations is therefore to analyze parameters obtained from transient decay dynamic experiments and to evaluate the self-consistency of these analyses with regard to their ability to predict the steady-state J-E behavior of the semiconductor/liquid contact under study. V. Summary and Conclusions The ToSCA simulations described herein confirmed conclusions obtained previously from a simplified analytical model which state that ideal behavior of a semiconductor electrode implies that the photovoltage of an n-type semiconductor/liquid interface should not change if the concentration of the reduced form of the redox species, A-, is held constant but the concentration of the oxidized form of the redox species, A, is varied, over a significant range of [A] and [A-] in the solution phase. The other prediction of the simplified analytical ideal model, that the photovoltage will be independent of [A-] over a significant range of donor concentration in the solution as [A] is held constant, was also validated by the numerical simulations. Experimental observations of a shift of Voc by 150 mV in response to a factor of 3 change in [A-] are not consistent with ideal interfacial behavior, but likely indicate complications such as band-edge movement; competition between surface, interfacial, and bulk recombination events; and other unresolved kinetic processes. Finally, the simulations allow the establishment of an upper bound for the interfacial charge-transfer rate constant of ket ≈ 10-20 cm4 s-1 from the steady-state current density vs potential behavior of p-InP/Fe(CN)63-/4-(aq) contacts, and the shape of the J-E data confirm the presence of significant surface instabilities that confound a straightforward interpretation of the transient luminescence decay data of this semiconductor/liquid contact. Acknowledgment. We acknowledge Dr. Olaf Kru¨ger for his insightful discussions and Herbert Gajewski and Reiner Nu¨rnberg from the Institute fu¨r Angewandte Analysis und Stochastik in Berlin for their help with ToSCA. We also thank the U.S. Department of Energy, Office of Basic Energy Sciences, DE-FG-03-88ER13932, for their generous support of this work. References and Notes (1) Gerischer, H. In Physical Chemistry: An AdVanced Treatise; Eyring, H., Henderson, D., Yost, W. Eds.; Academic: New York, 1970; Vol. 9A, p 463. (2) Koval, C. A.; Howard, J. N. Chem. ReV. 1992, 92, 411. (3) Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal Electrodes; Plenum: New York, 1980. (4) Tan, M. X.; Laibinis, P. E.; Nguyen, S. T.; Kesselman, J. M.; Stanton, C. E.; Lewis, N. S. Prog. Inorg. Chem. 1994, 41, 21. (5) Rosenbluth, M. L.; Lewis, N. S. J. Phys. Chem. 1989, 93, 3735. (6) Shreve, G. A.; Karp, C. D.; Pomykal, K. E.; Lewis, N. S. J. Phys. Chem. 1995, 99, 5575. (7) Lewis, N. S. Annu. ReV. Phys. Chem. 1991, 42, 543. (8) Reiss, H. J. Electrochem. Soc. 1978, 125, 937. (9) van Roosbroeck, W. Bell Syst. Technol. J. 1950, 29, 560. (10) Gajewski, H. GAMM (Gesellschaft fu¨ r Angewandte Mathematik und Mechanik) Mitteilungen 1993, 16, 35. (11) Tan, M. X.; Kenyon, C. N.; Kru¨ger, O.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2830. (12) Kru¨ger, O.; Kenyon, C. N.; Tan, M. X.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2840. (13) Kenyon, C. N.; Tan, M. X.; Kruger, O.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2850.
J. Phys. Chem. B, Vol. 103, No. 19, 1999 3915 (14) Anz, S. J.; Kruger, O.; Lewis, N. S.; Gajewski, H. J. Phys. Chem. B 1998, 102, 5625. (15) Shreve, G. A.; Lewis, N. S. J. Electrochem. Soc. 1995, 142, 112. (16) Fonash, S. J. Solar Cell DeVice Physics; Academic: New York, 1981. (17) Peter, L. M. Chem. ReV. 1990, 90, 753. (18) Memming, R. Electrochemistry II; Springer-Verlag: New York, 1988; Vol. 143. (19) Bard, A. J.; Bocarsly, A. B.; Fan, F. R. F.; Walton, E. G.; Wrighton, M. S. J. Am. Chem. Soc. 1980, 102, 3671. (20) Rosenwaks, Y.; Thacker, B. R.; Nozik, A. J. Appl. Surf. Sci. 1996, 106, 396. (21) Rosenwaks, Y.; Thacker, B. R.; Bertness, K.; Nozik, A. J. J. Phys. Chem. 1995, 99, 7871. (22) Ahrenkiel, R. K.; Lundstrom, M. S. Minority Carriers in III-V Semiconductors: Physics and Applications; Academic Press Inc.: San Diego, 1993; Vol. 39. (23) Pankove, J. I. Optical Processes in Semiconductors; Dover Publications: New York, 1975. (24) Kauffman, J. F.; Balko, B. A.; Richmond, G. L. J. Phys. Chem. 1992, 96, 6371. (25) Harzion, Z.; Huppert, D.; Gottesfeld, S.; Croituru, N. J. Electroanal. Chem. 1983, 150, 571. (26) Sanders, A.; Kunst, M. Solid-State Electron. 1991, 34, 1007. (27) Boulou, M. B., D. J. Appl. Phys. 1977, 48, 4713. (28) Rosenwaks, Y.; Shapira, Y.; Huppert, D. Phys. ReV. B 1992, 45, 9108. (29) Rosenwaks, Y.; Thacker, B. R.; Nozik, A. J.; Ellingson, R. J.; Burr, K. C.; Tang, C. L. J. Phys. Chem. 1994, 98, 2739. (30) Gajewski, H.; Glitzky, A.; Griepentrog, J.; Hu¨nlich, R.; Kaiser, H.C.; Rehberg, J.; Stephan, H.; Ro¨pke, W.; Wenzel, H. Modellierung und Simulation Von Bauelementen der Nano- und Optoelektronik; Weierstrass Institute for Applied Analysis and Stochastics: Berlin (Germany), 1996. (31) Kru¨ger, O.; Jung, C.; Gajewski, H. J. Phys. Chem. 1994, 98, 12653. (32) Adachi, S. Physical Properties of III-V Semiconductor Compounds; John Wiley & Sons: New York, 1992. (33) Siegel, W.; Ku¨hnel, G.; Koi, H.; Gerlach, W. Phys. Status Solidi A 1986, 95, 309. (34) Gerischer, H. AdV. Electrochem. Electrochem. Eng. 1961, 1, 139. (35) Memming, R. In Electron-Transfer I; Mattay, J. Ed.; SpringerVerlag: Berlin, 1994; Vol. 169. (36) Blakemore, J. S. Semiconductor Statistics; Dover Publications: New York, 1987. (37) Shockley, W.; Read, W. T. Phys. ReV. 1952, 87, 835. (38) Hall, R. N. Phys. ReV. 1952, 87, 387. (39) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; John Wiley & Sons: New York, 1980. (40) Sze, S. M. The Physics of Semiconductor DeVices, 2nd ed.; Wiley: New York, 1981. (41) Selberherr, S. Analysis and Simulation of Semiconductor DeVices; Springer-Verlag: Wien, 1984. (42) Yamaguchi, K.; Toyabe, T.; Kodera, H. Jpn. J. Appl. Phys. 1975, 14, 1069. (43) Fischetti, M. V. IEEE Trans. Electron DeVices 1991, 38, 634. (44) Majerfeld, A.; Potter, K. E.; Robson, P. N. J. Appl. Phys. 1974, 45, 3681. (45) Windhorn, T. H.; Cook, L. W.; Haase, M. A.; Stillman, G. E. Appl. Phys. Lett. 1983, 42, 725. (46) Robson, P. N.; Potter, K. E.; Majerfeld, A. IEEE Trans. Electron DeVices 1975, 22, 569. (47) Pomykal, K. E.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2476. (48) Weaver, M. J.; Yee, E. L. Inorg. Chem. 1980, 19, 1936. (49) Lewis, N. S. J. Electrochem. Soc. 1984, 131, 2496. (50) Schroder, D. K. Semiconductor Material and DeVice Characterization; John Wiley & Sons: New York, 1990. (51) Pomykal, K. E.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. 1996, 100, 3652. (52) Bansal, A.; Tan, M. X.; Tufts, B. J.; Lewis, N. S. J. Phys. Chem. 1993, 97, 7309. (53) Nozik, A. J. Presented at the 211th National Meeting of the American Chemical Society, New Orleans, LA, March 1996. (54) Lewis, N. S. J. Phys. Chem. B 1998, 102, 4843. (55) Smith, B. B.; Halley, J. W.; Nozik, A. J. Chem. Phys. 1996, 205, 245. (56) Smith, B. B.; Nozik, A. J. 19th D. O. E. Solar Photochemistry Research Conference; Washington, D.C., 1995; p 103. (57) Rosenwaks, Y.; Thacker, B. R.; Ahrenkiel, R. K.; Nozik, A. J.; Yavneh, I. Phys. ReV. B 1994, 50, 1746. (58) Nozik, A. J.; Parsons, C. A.; Dunlavy, D. J.; Keyes, B. M.; Ahrenkiel, R. K. Solid State Commun. 1990, 75, 297.
