Progress in Understanding Electron-Transfer Reactions at

© Copyright 1998 by the American Chemical Society

VOLUME 102, NUMBER 25, JUNE 18, 1998

FEATURE ARTICLE Progress in Understanding Electron-Transfer Reactions at Semiconductor/Liquid Interfaces Nathan S. Lewis DiVision of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 ReceiVed: December 1, 1997; In Final Form: March 26, 1998

This article describes theoretical treatments and experimental data focused on the rates of interfacial electrontransfer processes at semiconductor/liquid contacts. These systems are of practical interest because such electron transfers are a critical factor in understanding the behavior of photoelectrochemical cells as energy conversion devices. These processes are of theoretical interest because the description of how a delocalized charge carrier in a semiconducting solid reacts with a localized redox acceptor that is dissolved in the liquid electrolyte is a relatively undeveloped area of electron-transfer theory. The general principles of these processes, a discussion of past and present experimental data, and a comparison between theoretical expectations and experimental observations on a variety of semiconducting electrode systems are the main focus of this article.

I. Introduction Semiconductor/liquid junctions provide the most efficient wet chemical method presently known for converting sunlight into chemical or electrical energy.1-4 In such systems, sunlight is absorbed by a semiconducting photoelectrode, and the electronhole pair produced by the light absorption is converted into a photocurrent (Figure 1). These systems can convert sunlight into electrical energy, chemical energy, or a combination thereof. Quantum yields for photocurrent production can approach unity for wavelengths of light spanning the near-infrared and visible regions of the spectrum, and the overall solar optical-to-electrical energy conversion efficiencies of such devices can be in excess of 15%.1-4 Although these systems have been studied continuously for over 40 years,5 significant gaps still exist in our understanding of several of the fundamental operating principles of semiconductor photoelectrodes. For example, until recently, relatively little was known either theoretically or experimentally about the factors that control the transfer of photoexcited charge carriers from a semiconducting solid into redox-active donors or acceptors dissolved in the electrolyte solution. These interfacial kinetic processes are obviously crucial to the opera-

tion of a photoelectrochemical cell, because they control the forward and reverse electron-transfer processes that ultimately produce the photocurrent through the device. At present, the value for the maximum rate constant for the transfer of charge carriers in a semiconducting electrode to randomly dissolved, outer sphere redox species in the solution phase is controversial, with proposed values for such processes spanning approximately 7 orders of magnitude.6-9 Such interfacial kinetic processes at semiconductor electrodes, which have been the topic of much recent work in the field, are the main focus of this article. A. Basic Principles. The basic processes of light absorption and charge separation at semiconductor/liquid contacts are wellunderstood.2,10 Absorption of photons above the band gap energy of the semiconductor leads to creation of electron-hole pairs in the solid. A strong electric field, present at the solid/ liquid interface, then separates the electron and the hole, typically in W)

(1b)

qNd (W - x) s

(0 e x e W)

(2a)

E(x) ) 0

(x > W)

(2b)

Q)0 E(x) )

( )

( ) [

V(x) ) -

x

qNd (W - x)2 2s

(0 e x e W)

2s E(A/A-) - Ecb - kT ln

W)

q2Nd

]

Nc Nd

(3)

(4)

Here, q is the electronic charge, Nd is the dopant density, s is the static dielectric constant of the semiconductor, W is the depletion width, and Nc is the effective density of states in the conduction band of the semiconductor. Equation 1 is reasonable because the dopants are present in relatively low concentration in the solid (perhaps 1 ppm or less), so essentially all of the dopants are ionized until a sufficient distance has been reached that the required electrical charge has been transferred across the solid/liquid interface. Equations 2 and 3 then follow directly from the charge density profile of eq 1, once the value of W is known from the amount of charge transferred (eq 4). Analogous equations can be obtained for p-type semiconductor/liquid contacts to relate the energetics of the valence band, Evb, to E(A/A-).

For a typical barrier height [(E(A/A-) - Ecb)/q] of 1 V, as well as for a dopant density of 1015 cm-3, eqs 2 and 4 show that the electric field near the surface of the semiconductor is ≈105 V cm-1. Charge carriers in crystalline semiconducting solids typically have mobilities, µ, of 102-103 cm2 V-1 s-1,11 so photogenerated electrons or holes that experience this interfacial electric field are rapidly separated due to drift (µE) effects. Charge carriers can thus be separated by distances of >10-5 cm in ≈1 ps with a relatively minimal loss of potential energy. This is a significant advantage compared to molecularbased charge separation systems, where the charge separation distance is much smaller, typically 1-10 Å in the first few picoseconds, and where much more energy must typically be sacrificed to separate the (less mobile) charge carriers in such systems. At an n-type semiconductor/liquid contact (shown in Figure 1), the electric field drives the minority carriers (i.e., the holes) toward the solid/liquid contact and drives the majority carriers (i.e., the electrons) into the bulk of the semiconductor. The energy storage process is completed if these holes undergo interfacial charge transfer to the donors in the electrolyte prior to the occurrence of any recombination events. Similar processes occur at a p-type semiconductor/liquid contact, except that the sign of the electric field is reversed. In this case, electrons are driven to the solid/liquid contact and holes are driven into the bulk. Obviously, regardless of the type of semiconductor doping that is used, understanding the interfacial charge-transfer events is critical to understanding the operation of the entire photoelectrochemical cell. The interfacial kinetic processes control both the “forward” minority carrier reaction that leads to photocurrent as well as the undesirable “back” electron transfer (majority carriers crossing the interface) that instead merely results in recombination. B. Topics To Be Discussed. The remainder of this article is broken down into several sections. The first section discusses several theoretical treatments that describe, at various levels of sophistication, expectations for rate constants for interfacial charge transfer at semiconductor/liquid contacts. The second section describes historical difficulties that have been encountered in measuring these rate constants. The third section describes experimental data, emphasizing the results of recent work performed at Caltech, that have allowed determination of charge-transfer rate constants for several semiconductor/liquid interfaces. The fourth section summarizes these findings and relates these rate constant values to other aspects of the behavior of photoelectrochemical cells as energy conversion devices. In the discussion that follows, we emphasize results for electrontransfer processes involving one-electron, nonadsorbing, outersphere redox reagents, because they are the simplest systems to treat theoretically and to evaluate experimentally. II. Theoretical Expectations A. Rate Laws, Rate Constants, and Units. At a metal electrode, the current density J for reduction of an electron acceptor, A, located at a fixed distance from the electrode surface is

J ) -qkm,rA(E)CA

(5)

where E is the electrode potential relative to the formal potential, E°′(A/A-), of the redox couple A/A-, CA is the concentration (in molecules cm-2) of A at a distance rA from the electrode surface, and by convention a cathodic current is negative in sign.12 The units of km,rA(E) must be s-1 so that J has units of

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J. Phys. Chem. B, Vol. 102, No. 25, 1998 4845

A cm-2.13 The inverse of the rate constant is thus simply the characteristic decay time for electron transfer from the occupied electronic states in the metal to the acceptor ions in the solution.13 The rate constant km,rA(E) obviously depends on the electrode potential because the driving force for interfacial charge transfer is changed as the electrode potential is varied. The maximum rate constant for such events is typically estimated to be ≈1013 s-1, due to the time scale involved with nuclear reorganization processes that are required in order to obtain product formation in electron-transfer reactions.14 For reduction of a randomly dissolved acceptor species at a metal electrode, the appropriate rate law can be obtained by integration, with respect to all possible electrode-ion distances, r, of the fixed-distance rate law given above.6,15 This integration yields

∫0∞ km(r,E) dr ) -qkm(E) [A]

J ) -q [A]

(6)

where [A] is the concentration (in molecules cm-3) of A in the solution and the rate constant km(E) has units of cm s-1. The maximum rate constant km,max for these interfacial events can be estimated by a variety of methods. A statistical mechanics approach shows that the collision frequency of molecules striking a plane is (kT/2πµm)1/2, where µm is approximately the mass of the acceptor.16 Assuming that every collision leads to a successful reaction event, this relationship yields a value of km,max ) 104-105 cm s-1 at 300 K.17 An alternate approach is to realize that km(r,E) typically has an exponential dependence on distance with an attenuation coefficient, β, of 1 × 108 cm-1. Thus, km(r,E) ) km(ro,E) exp [-β(r-ro)], where km(ro,E) is the value of the rate constant at the closest approach distance ro between the electrode and the acceptor ion. Assuming a maximum value of km(ro,E) of 1013 s-1 (corresponding to a vibrational frequency, νn, that is associated with a typical molecular mode of energy 1000 cm-1), integration of km(r,E) from contact to infinite electron-ion distances produces a value of km,max ) νn/β ≈ 105 cm s-1.15,18 In the treatment above, the concentration of one of the reactants, the electrons in the metal, never appears explicitly in the rate expression. This occurs because the electron concentration in the solid is a constant that is incorporated into km(r,E) or km(E) in a fashion similar to the implicit incorporation of the concentration of a solvent into the rate constant expression for a solvolysis reaction. For a semiconductor electrode, however, the electron concentration at the surface of the solid ns is a variable that can be measured and that can be controlled experimentally (by changing the electrode potential). This surface electron concentration must thus be explicitly represented in the corresponding rate expressions for a semiconductor electrode. For a random concentration of acceptor in the solution phase in contact with a semiconductor electrode,

J ) -qketns[A]

(7)

The rate constant ket therefore has units of cm4 s-1.2,6 As the potential is changed at a semiconductor/liquid contact, the value of ket does not change, because essentially all of the potential drops across the semiconductor. Instead the current density varies with potential because a Boltzmann-type relationship describes the concentration of electrons in the bulk of the semiconductor nb relative to the concentration of electrons at the electrode surface ns:

{

ns ) nb exp

}

-q(Vbi + E) kT

(8)

where E is the applied potential and Vbi, the built-in voltage, equals (1/q)[E(A/A-) - Ecb - kT ln(Nc/Nd)]. The goal of the treatments that follow is to establish both intuitive and analytical estimates for the maximum values of ket that should be observable at optimal exoergicity of this charge-transfer event. Because the Marcus treatment expresses ket as a product of the rate constant at optimal exoergicity, ket,max, multiplied by the well-known expression for the Franck-Condon factors associated with nuclear reorganization for the charge-transfer event,14 the essence of the problem thus reduces to describing the factors that determine ket,max at a semiconductor/liquid contact. B. Models for the Charge-Transfer Process. 1. Approximate Collisional Model. In the simplest approximation, the value of km,max ) 105 cm s-1 can be divided by the electron concentration at the surface of the metal, because the electron concentration in the electrode is implicit in the rate constant expression that defines km,max, but the surface electron concentration is explicitly present in the rate law expression for a semiconductor electrode. The electron concentration at the Fermi level of various metals is typically 1022 cm-3,19 so the collision-based model yields ket,max ≈ (105 cm s-1)/(1022 cm-3) ≈ 10-17 cm4 s-1. 2. Semiclassical Collisional Model. In another simple model, the interfacial charge-transfer event can be considered to involve two reactants, the electron in the semiconductor and the acceptor ion in the solution, in a classical collision event. This type of approach was used in the 1960s by Morrison,20 and has recently been developed somewhat further by our group at Caltech.6 The maximum rate constant in this formalism is given by ket,max ) Vrel σδ, where Vrel is the relative velocity of the reactants, σ is the cross section for capture of the electron by the ion, and δ is the effective distance over which the interfacial capture process can occur (Figure 2). Because electrons in crystalline semiconductors move at a thermal velocity of ≈107 cm s-1, the ions contribute little to the relative velocity of the reactants, so Vrel ≈ 107 cm s-1. Taking δ ≈ 10-7 cm as a reasonable reaction zone over which an electron-transfer event can occur through a solvent and using the geometric cross section for the acceptor ion of σ ) 10-15 cm2 (as is appropriate for an off-resonance event like an electron-transfer process, for an acceptor having an actual geometric radius of ≈ 3 × 10-8 cm) produces ket,max ) 10-15 cm4 s-1. However, this value must be corrected by the ratio of a nuclear motion time scale to an electronic motion time scale, because if the nuclei do not rearrange when the collision occurs, the collision will be elastic and no products will result. This ratio is ≈(1013 s-1)/(1015 s-1), which produces ket,max ≈ 10-17 cm4 s-1 from this treatment as well. 3. An Alternate Simple Collisional Model. Gerischer has proposed another semiclassical model for estimating ket,max.21 In this model, the redox acceptors are situated at a distance dA from the solid/liquid interface, and the wave function of an electron in the semiconductor extends a distance, F, into the lattice. The rate of reaction depends on the number of pairs of electrons and acceptors that are situated within electron-transfer distance of each other, and this number of available pairs, P, at the interfacial plane is calculated assuming a hemispherical distribution of electrons around each acceptor. Estimating the volume in the semiconductor for which charge carriers can participate in a surface reaction introduces a factor of F3, yielding P ) ns[A]dA(2π/3)F3. The second-order rate constant is therefore ket,max ) dA(2π/3)F3νn. Substitution of dA ) 3 × 10-8 cm and F ) 3 × 10-8 cm into this expression yields an estimate of ket,max ) 2 × 10-17 cm4 s-1.

4846 J. Phys. Chem. B, Vol. 102, No. 25, 1998

Lewis

Figure 2. A classical collisional model for electron transfer from an electron in a semiconductor to a nonadsorbing, outer-sphere redox couple in the electrolyte. In this formalism, the reaction is treated as a two-body collision, with re the effective radius of the electron in the solid, rA the radius of the solution acceptor, δ the distance over which electron transfer can occur, and Vrel is the relative velocity of the reactants. Since the thermal velocity of electrons in the solid Vel is much larger than the thermal velocity of the acceptors in the electrolyte, Vrel ≈ Vel.

4. Electronic Coupling Model: Adaptation of Marcus’ Treatment of Charge Transfer at Liquid/Liquid Interfaces. Yet another model, developed in my group at Caltech, uses an adaptation of Marcus' treatment of electron transfer between ions at an interface between two immiscible liquids. In this approach, integration over all possible reactant pairs produces an expression for the donor/acceptor charge-transfer rate constant, kD-A,max, of kD-A,max ) 2πνn(rA + rD) β-3, where rA and rD are the radii of the acceptor and donor ions, respectively, and β is the distance dependence of the attenuation of the electronic coupling between A and D through the solvents of concern.22 A simple adaptation of this treatment to a semiconductor electrode approximates the delocalized electron as a particle having an effective radius of ≈10-7 cm-1 (approximately the experimentally measured Bohr radius of electrons in typically used semiconductor electrodes), and again takes νn ) 1013 s-1 and β ) 1 × 108 cm-1. This treatment therefore also yields a value of ket,max ≈ 10-17 cm4 s-1.6,23 Kathy Pomykal at Caltech has further developed this model to account qualitatively for effects including adiabaticity and finite approach distances of ions to the electrode, and she finds that ket,max ≈ 10-17-10-16 cm4 s-1.24 5. Electronic Coupling Model: Use of the Fermi Golden Rule To Relate the Electronic Coupling to the Nonadiabatic Rate Constant. A final model that will be discussed is based upon use of the Fermi Golden Rule to evaluate the rate of a process in terms of the value of the matrix elements that couple the reactant and product states of the system. This approach has been developed in some detail by William Royea, Arnel Fajardo, and myself, at Caltech, to describe the kinetics of electron-transfer events at semiconductor/liquid contacts.7 This treatment is satisfying in that it uses the same formalism that has been developed to treat nonadiabatic charge-transfer events between molecular donor and molecular acceptor reagents,14 as well as to treat charge transfer between metal electrodes and acceptors in a solution.13,25 In this approach, the charge-transfer rate is given by

is assumed to fall off exponentially from the electrode with an attenuation coefficient β. The terms in this equation can be clearly broken down into the density of occupied electronic states in the electrode, the electronic coupling matrix element, and the Franck-Condon term for the electron-transfer event. A similar equation has been used by B. Smith of the National Renewable Energy Laboratory (NREL) in his recent, quantummechanical evaluation of rate constants at the semiconductor/ liquid interface.26 Referring to the rate law for a semiconductor electrode, rate ) ket ns [A], and removing terms that produce values for [A] and ns from eq 9, it can be shown7 that the resulting expression for ket,max is

ket,max )

lsc 2 4π2 (10) {(4πλkT)-1/2}{HoAB}{β-1} 2/3 h (dsc) (6/π)1/3

where lsc is the effective coupling length into the semiconductor and dsc is the atomic density of the semiconductor. Note that since the maximum rate constant is the value of ket at optimal exoergicity, the nuclear term in eq 9 equals unity when ket ) ket.max. Assuming that the charge-transfer process is nearly adiabatic 2

at contact (i.e., HoAB ≈ 10-2 eV2 state-1), that the rate of this process falls off with a typical attenuation of β ) 1 × 108 cm-1 through the solvent,27 and that the effective coupling length in the semiconductor is lsc ≈ 3 Å, this treatment produces a value of ket,max ) 10-17-10-16 cm4 s-1. This type of analysis can also be used to compare the data obtained at metal electrodes to that predicted for semiconductor electrodes, when the differences in the densities of states are accounted for at each type of electrode system but other 2

parameters (e.g., λ, HoAB) are assumed to be comparable in magnitude between metal and semiconductor electrode electrontransfer events. Using rate constant relationships derived by Royea, Fajardo, and me,7 the rate constant data of Chidsey for (η5-C5H5)Fe(η5-C5H4)CO2(CH2)16SH on Au13 can be used to predict ket,max ) 5.1 × 10-17 cm4 s-1 for a semiconductor electrode, while the rate constant data of C. Miller and coworkers28 can be used to predict ket,max ) 8.9 × 10-17 cm4 s-1 for a semiconductor electrode. All of these quantum mechanical treatments thus agree with the simpler and/or classically based models described above in predicting ket,max ≈ 10-17-10-16 cm4 s-1 for the transfer of charge at optimal exoergicity from the delocalized electrons in a semiconductor to nonadsorbing outersphere electron acceptors randomly dissolved in the electrolyte solution. III. Historical Perspective

ratesc ) 2 ∞ 4π2 {(4πλkT)-1/2}[A]{HoAB}{β-1} -∞ F (E,E)Feff(E) × h -[(E - qE°′(A/A-)) + λ]2 exp dE (9) 4λkT

{

∫

}

where h is Planck’s constant, λ is the reorganization energy of 2

the redox species, HoAB is the square of the matrix element that couples reactant and product states at the distance of closest approach of the redox species to the electrode surface, F (E,E) is the Fermi occupancy of the semiconductor electronic states in the solid as a function of energy E, Feff(E) is the effective state density (in units of states eV-1), and the electronic coupling

A. Use of Photoluminescence Decay Transients to Probe Surface Nonradiative Recombination. Given these theoretical expectations, it is useful to obtain some perspective on why an extensive body of information on rate constants at semiconductor/liquid contacts was not collected decades ago. Modern laser methods designed to probe interfacial charge-transfer dynamics at semiconductor electrodes were initiated approximately 15 years ago by S. Gottesfeld, D. Huppert, and co-workers who showed that, in the n-CdS/H2O-sulfide-polysulfide system, the radiative luminescence due to charge carrier recombination in the semiconductor was quenched on the 10-9 s time scale by formation of the solid/liquid contact.29 These were among the first dynamic measurements of interfacial charge carrier capture processes in semiconductor electrochemistry.

Feature Article Unfortunately these measurements are not simple to interpret because a transport, generation, and recombination model, along with the appropriate boundary conditions, is required in order to extract the surface nonradiative decay rate constants from the observed luminescence decay signals. Such models, of varying sophistication, have been developed and described in the literature.29-35 The model used in the early work of Huppert, Gottesfeld, and co-workers produces interfacial minority carrier capture velocities, Smin, at the solid/liquid interface of 105-106 cm s-1.29 This quantity can be physically interpreted as the effective velocity at which charge carriers that strike the plane of the solid/liquid contact appear to be collected by recombination events and/or by charge-transfer processes that occur at, or across, the solid/liquid boundary plane. If these Smin values indeed represented charge transfer to nonadsorbed redox species in the solution phase, then the interfacial charge-transfer rate constant is readily computed from the relationship ket ) Smin/ [A] ) (105 cm s-1)/(3 × 1020 cm-3) ) 3 × 10-16 cm4 s-1. The authors did not, in fact, conclude that the measurements were due to outer-sphere charge-transfer events, but instead assigned the decays to redox-active species that were adsorbed onto the electrode surface.29 This might be expected because sulfide ion is known to chemisorb to the Cd sites on the CdS surface, and similar interactions were observed for the other systems that exhibited rapid luminescence quenching. Subsequent simulations of current-potential curves provided further evidence supporting this interpretation.36 In contrast, decay dynamics on another system measured by these workers, n-CdS/ H2O-NaOH, where adsorption was not obviously evident, were too slow to measure.29 In these latter systems, other processes in the solid, including diffusion and drift-induced charge separation, set a lower limit on the experimentally accessible time scale for the measurements. After this pioneering work had been reported, Gail Ryba initiated similar luminescence decay measurements at Caltech on the n-GaAs/KOH-Se-/2-(aq) junction. This semiconductor/ liquid contact had been shown previously by A. Heller, B. Miller, and B. Parkinson, then at Bell Laboratories, to provide a stable, efficient, GaAs-based photoelectrochemical cell in aqueous solvents.37 The luminescence data obtained by Ryba mainly served to verify that adsorbed metal ions acted primarily as electrocatalysts for minority carrier capture at the solid/liquid interface, as opposed to acting to reduce surface recombination by passivating surface states.38 The data, however, also yielded evidence for relatively rapid charge carrier quenching at the n-GaAs/KOH-Se-/2- contact. The data in Ryba’s measurements were obtained under high level injection conditions, so that either electron or hole quenching would produce a reduction in the observed luminescence signal. The measurements implied an interfacial charge carrier collection velocity of S ) 104-105 cm s-1 in these systems.38 Transient grating experiments by R. J. D. Miller and co-workers were performed concurrently with this study and are in good agreement with it, having yielded minority carrier capture times of approximately 1-2 ps in these systems.39 If one were to assume a second-order rate law and to divide these S values by the concentration of redox species in the solution, one would compute kct ) 3 × 10-16 cm4 s-1. However, it is clear that the charge carrier capture processes in these systems are not due to transfer of charge in the solid directly into a distribution of randomly adsorbed, outer-sphere redox reagents, so this computation is not valid. n-Type GaAs anodes are well-known to be unstable in aqueous solution, except in the presence of high concentrations of Se-/2-, Te-/2-,

J. Phys. Chem. B, Vol. 102, No. 25, 1998 4847 and I2/I-.37,40,41 Numerous measurements including electrode stability studies, flat-band potential determinations, and J-E data indicate that these species stabilize the photoanode through strong adsorption, which facilitates the rapid capture of photogenerated holes that in turn imparts stability to the photoanode. Although these surface carrier capture velocities are relatively rapid (out of necessity to obtain stable photoelectrochemical cells), these measurements therefore do not allow comparison with the outer-sphere redox processes for which the electrontransfer theories to date have been developed. A similar issue confounds the more recent photoluminescence decay measurements of charge carrier dynamics at sulfidetreated p-GaAs/CH3CN-ferrocene (Fc)+/0 junctions42 and at p-InP/Fe(CN)63-/4-(aq) contacts.8 Rapid photoluminescence decay dynamics have been observed in these systems by workers at NREL, and in this respect the decays are similar to those reported earlier on the n-GaAs/KOH-Se-/2- electrolyte.8,42 After this work, detailed electrochemical and surface spectroscopic studies have revealed large adsorption-induced changes in surface recombination in these systems. For example, Kathy Pomykal at Caltech found that irreversible, light-induced surface damage, along with electrode corrosion processes, are artificial causes of the very rapid experimental photoluminescence decay transients that have been observed in the p-InP/Fe(CN)63-/4-system.24 Laser-induced surface damage has also been identified as a confounding variable in the experimental protocol used in the original rate constant determinations on the p-InP/Fe(CN)63-/4- junction.43 Similarly, studies of GaAs/ CH3CN contacts by a number of workers have revealed that the energetics and kinetics of these junctions are dominated by surface recombination processes, so that the direct transfer of carriers in the semiconductor to redox species in the solution phase is generally not rate-limiting.24,44-46 These systems thus serve to underscore the rapid charge carrier trapping that can be produced by pathways involving adsorbed ions and surface defects. Unfortunately, it does not appear that any photoluminescence-based transient measurements performed to date have determined directly the dynamics of charge transfer to outersphere redox reagents at a semiconductor/liquid contact. This is not an inherent limitation of the method, but instead reflects the difficulty of obtaining a stable, defect-free, and wellcharacterized semiconductor/liquid contact which possesses sufficiently slow decay dynamics such that the electron-transfer time scale to outersphere redox reagents is accessible using this technique. B. Microwave Conductivity Data To Probe Interfacial Charge-Transfer Processes at Semiconductor/Liquid Contacts. In an attempt to utilize outer-sphere redox couples and to take advantage of the stability and other well-behaved properties of Si/CH3OH-ferrocene contacts, Malcolm Forbes at Caltech attempted to employ transient microwave conductivity methods to investigate the interfacial charge-transfer kinetics in this system. Silicon has an indirect (i.e., not optically allowed) band gap,11 so its radiative luminesence signal is very weak. This makes detection of the charge carrier decay dynamics through monitoring luminescence signals quite challenging. The use of microwave conductivity methods, however, allows such detection with significant signal-to-noise levels.47-49 Monitoring carrier decay dynamics using microwave conductivity techniques is also valuable because the amplitude of the microwave conductivity signal is related to the product of the charge carrier concentrations and their individual mobilities.47-49 Thus, even if all of the holes were quenched nonradiatively in Si, for example, 75% of the microwave signal would remain

4848 J. Phys. Chem. B, Vol. 102, No. 25, 1998 until the photogenerated electrons were also quenched, because µn ) 3µp for Si under most carrier injection levels. This contrasts with a luminescence decay measurement of interfacial kinetics, for which quenching of either the electron or hole eliminates the luminescence signal. To perform a microwave conductivity experiment, thin Si samples, having long bulk lifetimes and passivated surfaces, were required. We obtained such samples from R. Swanson and R. Sinton of Sunpower, Inc., who were interested in using similar samples to construct novel Si solid-state photovoltaic devices.50 In a 100 µm thick Si sample with a charge carrier lifetime of >1 ms, carriers can collide into the surfaces hundreds of times before they will nonradiatively decay in the bulk of the solid. Thus, surface recombination processes should dominate the decay dynamics of such systems. In addition, if passivated back and front surfaces are initially used, then etching the front surface of the sample and immersion into the electrolyte should produce a significant decrease in the microwave decay time if interfacial charge-transfer events are rapid. The most rapid charge-transfer events would result in decay dynamics that are only limited by diffusion of carriers to the solid/liquid interface, and would produce a limiting (nonexponential) decay of ≈1 µs for a 100 µm thick Si sample. Instead of providing evidence for such rapid charge-transfer events, Forbes’ measurements showed that a slow charge capture velocity was present at the Si/CH3OH-1,1′-dimethylferrocene (Me2Fc)+/0 contact even in the presence of relatively high concentrations of Me2Fc+/0. In fact, this system exhibited a decay time constant on the order of 200 µs, implying that the carriers could undergo many hundreds of collisions with the solid/liquid interface before undergoing a fruitful carrier capture and/or recombination event.51 The decay dynamics were so slow that residual nonradiative surface recombination, not interfacial charge transfer, likely dominated the observed decay time. Thus, even at the highest concentrations of redox couple used in these studies, the experiments only allowed establishment of an upper bound on ksc of < 10-16 cm4 s-1 in this system. C. Transient Photocurrent Decay Measurements: An Apparent Contradiction. This relatively slow interfacial charge transfer quenching time seemed curious because transient photocurrent measurements, in addition to transient photoluminescence decay measurements, had been performed previously on other semiconductor/liquid contacts. These transient photocurrent measurements had shown relatively rapid relaxation time constants and had been thought to provide evidence for rapid interfacial charge-transfer events.52-57 In fact, transient photocurrent measurements were believed, by many workers, to be limited by the time constants of the electrochemical cells and not by the more rapid interfacial charge-transfer processes. At first glance, it thus seemed that the long-lived microwave conductivity decays implied that the charge-transfer dynamics in the n-Si/CH3OH-Me2Fc+/0 system could be readily resolved using conventional time-resolved photocurrent transient methods. In view of this possibility, Chris Kenyon at Caltech set out to investigate the n-Si/CH3OH-Me2Fc+/0 contact by timeresolved photocurrent techniques. To his surprise, instead of a relatively slow (>100 µs) decay, Kenyon observed transient photocurrent decays which were so rapid that they were limited by the ≈1 µs RC time constant of the electrochemical cell used in the experiment.58 So an apparent contradiction was present between the microwave conductivity decay measurements and the transient photocurrent decay data, if indeed the two measurements had both probed interfacial kinetics. Kenyon

Lewis

Figure 3. An equivalent circuit that describes photocurrent transients. Csc is the differential space-charge capacitance of the semiconductor, Rrec is the resistance for recombination in the space-charge region of the solid, CH is the differential capacitance of the Helmholtz layer, Rf is the resistance to Faradaic charge transfer across the solid-liquid interface, Rcell is the series resistance of the cell (associated largely with ion transport through the solution), Rmeas is the measuring resistor across which the potential is monitored, and Rload is the variable resistor that allows access to various potentials.

therefore attempted to resolve this dilemma by designing experiments to elucidate which circuit elements actually were being measured in the photocurrent transient decay experiment. Due to the stability of the nonaqueous n-Si/CH3OH-Me2Fc+/0 contact, it was possible for Kenyon to vary the concentration of the redox reagent and to perform other experiments that were not possible on most other, previously studied, systems. These earlier interfaces demanded a fixed, as well as high, concentration of the redox reagent in aqueous solution in order to impart stability to the electrode surface. Because the nonaqueous solvent allowed variation in the redox concentration while still maintaining electrode stability, Kenyon was able to show that acceleration of the charge-transfer event, performed by coating the Si surface with a continuous overlayer of metal, did not change the photocurrent decay dynamics. Even more compellingly, Kenyon found that eliminating the redox couple, and therefore eliminating the interfacial charge-transfer process completely, did not affect the observed decay time of the photocurrent transient!58 This latter result conclusively demonstrated that the photocurrent transients are not sensitive to the interfacial charge-transfer events but instead are dominated by other circuit element in the system. The equivalent circuit that appears to capture the essence of the situation is depicted in Figure 3. In this circuit, the optical pulse that induces the photocurrent creates electron-hole pairs that charge the capacitor Csc, which physically represents the space-charge capacitance of the semiconductor. Charge separation occurs rapidly across this capacitor due to the interfacial electric field in the space-charge region of the solid. Recombination in the solid back across this charged capacitor must have a high equivalent resistance (Rrec) because the electric field suppresses such back reactions. After this charge separation process, the minority carriers may, or may not, cross the interface, depending on the time scale (and the equivalent resistance Rf) for the charge-transfer process. However, regardless of this time scale, the majority charge carriers will act independently, because they have been separated from minority carriers by the electric field in the semiconductor. Majority carriers will be driven through the

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J. Phys. Chem. B, Vol. 102, No. 25, 1998 4849 light and for sufficiently rapid interfacial charge-transfer rate constants, this competition between diffusion and drift should be observable as a decline in the internal quantum yield for majority carrier collection at very short wavelengths. A quantitative analysis of the situation has been performed by Reichman60 and yields an expression for the internal quantum yield of majority carriers crossing the interface, ηmaj(λphot):

Figure 4. A schematic illustration of the two processes that affect the collection of carriers generated by short-wavelength light at a semiconductor/liquid junction. Electrons and holes are collected with interfacial charge-transfer rate constants of kn and kp, respectively. Concentration gradients drive both carriers toward the solid/liquid contact. In contrast, drift effects (µE, where µ is the mobility and E is the electric field) due to the electric field in the space-charge region of the n-type semiconductor drive holes toward the surface but drive electrons away from the surface. Because drift and diffusion both act to drive minority carriers toward the surface, such carriers are collected efficiently at all wavelengths of illumination. However, drift and diffusion act in opposite directions on majority carriers, so such carriers will be lost to interfacial charge transfer if they are created very near the interface. The net quantum yield for charge collection at the back of the solid should thus decrease at very short wavelengths due to the increased contribution of this majority carrier loss process.

external circuit by this electric field, and current will flow so that the highest capacitance in the system is charged (thus producing the lowest voltage drop in the system). Majority charge carriers will thus flow through the measuring load resistance and onto the metal/liquid contact of the counter electrode, and will produce a photocurrent transient that is detected through the measuring instrumentation connected across the load resistor, regardless of the fate of the minority carriers. From this analysis, it is clear that no direct information on minority carrier interfacial charge-transfer processes can be obtained through measurement of the majority carrier-based transient photocurrents in the external circuit. This conclusion, proposed by Kenyon to explain his set of data,58 in fact agrees with that of Willig, who proposed a similar equivalent circuit to explain the photocurrent transients observed for the n-GaAs/ KCl(aq)-Fe(CN)63-/4- system.59 Thus, the dilemma had been apparently resolved: the interpretation of the relatively slow microwave conductivity kinetic decays was still presumably valid, and the transient photocurrent measurements could not probe the desired interfacial charge-transfer processes. Unfortunately, however, because of the relatively slow decays observed in the microwave conductivity experiments, only upper bounds on the interfacial charge-transfer rate constant of the n-Si/CH3OH-Me2Fc+/0 contact were available at this point in time, and the reason for the relatively slow interfacial carrier decay times was not yet fully understood. D. Upper Limits on Charge-Transfer Rate Constants from Short-Wavelength Quantum Yields. In parallel with the work of Kenyon, Amit Kumar of Caltech performed another set of experiments that provided partial information on chargetransfer rate constants at semiconductor/liquid contacts. Kumar performed quantum yield measurements using very short optical wavelength illumination of a semiconductor/liquid contact. For photoexcited electrons at an n-type semiconductor/liquid contact, the presence of redox acceptors in the solution establishes a concentration gradient that tends to drive electrons toward the surface of the solid. This chemical potential gradient acts to oppose the electric potential gradient that tends to drive electrons into the bulk (Figure 4). For sufficiently short wavelengths of

ηmaj(λphot) )

{( )( 1+

)}

Vd qEo 1+ Vc kTR(λphot)

-1

(11)

where Vd is the majority carrier drift velocity (Vd ) µEo, where µ is the majority carrier mobility), Eo is the maximum electric field strength at the interface, and R(λphot) is the absorption coefficient at wavelength λphot. The quantities Vd and Eo are positive in sign regardless of whether eq 11 is used to describe the majority carrier processes for n-type or for p-type semiconductors. The majority carrier collection velocity Vc is a direct measure of the interfacial collection kinetics. The majority carrier photocurrent opposes the minority carrier photocurrent, and thus the net internal quantum yield for photocurrent flow across the interface is the difference between the Ga¨rtner equation,61 which describes the quantum yield for collection of photogenerated minority carriers, and eq 11:

{

ηnet(λphot) ) 1 -

} {( )(

exp[-R(λphot)W]

1 + R(λphot)Lp qEo Vd 1+ Vc kTR(λphot)

)}

-1

(12)

To verify this dependence on penetration depth, Kumar studied over 15 different semiconductor/liquid contacts. He was able to obtain excellent agreement between experimental data and the prediction of eq 12 for all of the systems he studied.62 As expected for the situation in which carrier collection at the surface is very rapid (in fact, limited by the collision rate of carriers into the metal), metal/semiconductor contacts showed observable declines in quantum yield at very short wavelength illumination.62 However, none of the semiconductor/liquid contacts that Kumar investigated showed such behavior. In fact, all of the semiconductor/liquid contacts studied, including those using adsorbing redox couples, showed excellent majority carrier flow into the solid even at the shortest wavelengths that could be obtained experimentally.62 The short-wavelength spectral response data again only provided an upper bound on ket. Because the concentration of acceptor species had to be relatively low to allow efficient optical excitation of the solid in the near-ultraviolet region of the spectrum without excessive absorption by the solution, the upper bound on ket that could be established experimentally was not very stringent. For example, the n-Si/CH3OH-Me2Fc+/0 interface with an acceptor concentration of [A] ) 0.1 mM displayed a majority carrier collection velocity of Smaj e 106 cm s-1, and therefore this system produced an upper limit of ket < (106 cm s-1)/(6 × 1016 cm-3) ) 1 × 10-11 cm4 s-1. These measurements did, nevertheless, provide bounds on ket that are consistent with the expectations from the theoretical treatments developed above. They also show why slow electron transfer for majority carrier processes can be advantageous, because less photocurrent is lost due to majority carriers moving the “wrongway” toward the solid/liquid interface.63 Another method would, however, have to be exploited in order to obtain direct information on ket,max at semiconductor/liquid interfaces.

4850 J. Phys. Chem. B, Vol. 102, No. 25, 1998

Lewis

IV. Rate Constant Measurements A. Constraints. In the early 1960s, Morrison and coworkers used steady-state current density-potential methods in the absence of illumination to attempt to determine rate constants for several species at n-type ZnO electrodes.20,64,65 In many of these cases, the n-ZnO/liquid contacts exhibited the expected first-order dependence on ns, but only some systems displayed the first-order dependence on the concentration of acceptor [A]. Measurements of heterogeneous electron-transfer rates at the n-ZnO/H2O-Fe(CN)63-/4- junction yielded a ket of 10-18 cm4 s-1.65 This value agrees qualitatively with theory, but it is not clear that Fe(CN)63-/4- is an outer-sphere, nonadsorbing redox couple. Attempts to fit the data for n-ZnO obtained with several redox couples to deduce any free-energy dependence of ket were thwarted by the obvious lack of a constant reorganization energy for the various, simple metal-ion-based, redox systems that were available at the time.20,65 In addition, many of these ions are well-known to adsorb onto oxide/hydroxide surfaces, and there is a significant possibility of inner-sphere electron-transfer processes between the ion and the electrode for many of the systems studied. Finally, some systems, such as Fe(CN)63-/4-, displayed rate constants that changed significantly as the pH of the solution was varied, even though the changes in driving force due to the changes in pΗ were estimated to be sufficiently small that little change in ket should have been observed.64 Little follow-up of these measurements was performed during the ensuing three decades. The systems that did not show the expected first-order dependence on [A] do not allow direct determination of ket, because the rate law of eq 7 is not obeyed experimentally. Morrison and co-workers hypothesized that another process, first involving carrier capture by surface states, then followed by electron transfer from the occupied surface states into the acceptor, dominated the interfacial kinetics.64 If the rate constants for electron capture and emission by surface states with the conduction band are kn and kn′, respectively, and the rate constants for exchange between the surface states and the redox species are kss and kss′, respectively, it is straightforward to show that, at steady state, the rate of electron transfer from the semiconductor to the solution is

ratef )

{

ketns[A] + kssNt[A]

}

knns + kss[A]

knns + kss′[A] + kn′Nc + kss[A]

(13)

where Nt is the density of surface electrical traps. Under conditions when electron transfer to the surface states is slow compared to exchange with the redox species, this expression simplifies to

{

ratef ) ketns[A] + Nt[A]

}

ksskss′ kss + kss′

(14)

Thus, if kss were only 10-14 cm3 s-1, a surface state density of 1012 cm-2, or one defect in every 1000 surface atoms, would be sufficient to prevent measurement of ket and would allow surface-state carrier capture processes to dominate the observed interfacial current density. More recently, using the collisional model developed in section II.B.3. above, Gerischer derived a similar expression between the rates of surface-state exchange with the semiconductor and the redox species and reached the same conclusion (i.e., that a surface state density of 1012 cm-2 would likely eliminate the possibility of observing the electrontransfer process from the conduction band edge).21 Gerischer

moreover proposed that, if the electronic coupling between surface states and the redox species were larger than that between conduction band states and redox species, then the surface-state-mediated route would dominate the kinetics even with Nt < 1012 cm-2. Since the original work of Freund and Morrison on ZnO, the lack of a dependence of J on [A] has been observed in a variety of systems. This lack of adherence to the rate law of eq 7 prevents measurement of ket and has generally thwarted comparison between theory and experiment of kinetics for charge-transfer processes with outer-sphere redox reagents at semiconductor electrodes. This situation has been discussed in further detail in an excellent review of this topic in 1992 by Jason Howard and Carl Koval of the University of Colorado, Boulder.66 B. Rate Constant Measurements at n-Si/CH3OH Interfaces. As discussed above, to obtain reliable rate constant measurements from steady-state J-E data at an unilluminated semiconductor/liquid contact, two fairly stringent conditions must be met. First, the rate law of eq 7 must be obeyed, and first-order kinetic dependences must be observed on both the concentration of acceptors in the liquid and on the concentration of electrons in the solid. Second, a reliable method for determining the value of ns must be available so that the observed current density can be used to solve directly for the desired value of ket. We have recently discovered two semiconductor/liquid contacts that obey both of these criteria.67-69 Both of these systems have allowed determination of ket values directly from simple, steady-state J-E methods. The following sections summarize the findings of these studies, and then we compare these results to the theoretical expectations of section II. n-Si/CH3OH contacts with ferrocenes, studied as early as 1982 by Chris Gronet in my research group,70 have been shown to be extremely efficient, stable, and well-behaved semiconductor/ liquid contacts. In subsequent mechanistic studies of the n-Si/ CH3OH-Me2Fc+/0 system, Mary Rosenbluth and I found that these junctions were so efficient that they were limited only by electron-hole pair recombination in the bulk of the solid, as opposed to electron transfer across the solid/liquid contact.71,72 Apparently the barrier height was so large and the interfacial kinetics sufficiently slow that another mechanism (in which minority carriers are injected into the solid, diffuse against the space-charge barrier to the bulk of the Si, and then recombine with electrons) dominated the total current flow across this solid/ liquid contact. Although this type of behavior is desirable to construct the most efficient photoelectrochemical energy conversion devices from such systems, no direct information on ket can be obtained from the steady-state J-E properties of this system. In subsequent work, Rosenbluth observed that redox couples with more negative electrochemical potentials exhibited lower open circuit voltages than the n-Si/Me2Fc+/0 contact.73 Fajardo and I reasoned that this behavior likely indicated that another mechanism, involving interfacial kinetic processes, had begun to control the current. Fajardo thus initiated an effort to investigate thoroughly the steady-state J-E properties of Si/ CH3OH contacts that operated in this kinetically controlled regime. These systems consisted of viologens as the redox species, because their electrochemical potentials were not so negative that direct contact was made with the conduction band of the Si (yielding ohmic behavior), and were not so positive that the bulk recombination/diffusion mechanism, which had been observed with Me2Fc+/0, would dominate the interfacial

Feature Article kinetics of the device. The hope was that these experiments would provide a direct measurement of ket for a series of homologous, outer-sphere, one electron, redox couples in contact with a common, well-controlled semiconductor surface. These studies have recently been reported in the literature.67,69 The n-Si/CH3OH-viologen2+/+ system does indeed seem to exhibit ideal energetic and kinetic behavior, allowing for a direct, steady-state measurement of ket for various viologens. For each viologen studied, Fajardo observed that the current density increased linearly over at least a factor of 10 variation in the acceptor concentration. The reaction is thus first-order kinetically in [A], as expected from the rate law of eq 7. Similarly, the current density increased exponentially with more negative electrode potentials, as expected from the increasing value of the electron concentration at the electrode surface as the electrode potential is made more negative. In fact, the magnitude of this increase is very close to the ideally expected value, verifying that the observed interfacial flux is first-order in the concentration of electrons at the surface of the solid. The rate law of eq 7 was thus established experimentally for the interfacial kinetic process of concern. The remaining quantity needed to evaluate ket is the value of ns at any electrode potential. This can be determined if the potential drop across the electrode Vbi can be determined, because then the Boltzmann relationship of eq 8 can be used to solve for ns. A conventional method for determining the potential drop across a semiconductor is to determine the differential capacitance of the solid/liquid contact, and to plot this value vs the applied electrode potential.11,12 For a semiconductor electrode in depletion (i.e., when the dopants are ionized and eqs 1-4 describe the electrode properties), the differential capacitance of the semiconductor space-charge layer Csc can be computed in a straightforward fashion.11,12 Because Csc is in series with, but is so much smaller than the Helmholtz (double layer) differential capacitance CH, the observed differential capacitance should be dominated by the value of Csc.12 It can be shown that, unlike CH, Csc varies significantly with electrode potential and has a functional form such that Csc-2 should depend linearly on E. In addition, the slope of a plot of Csc-2 vs E can be predicted from knowledge of the dopant density of the semiconductor. Because Nd can be determined independently by electrical resistivity measurements, agreement between the computed and expected slopes would provide further evidence that the desired quantity, Csc, had been probed experimentally. This conventional method has, however, been problematic to apply in a straightforward fashion to semiconductor/liquid interfaces. In principle, the value of Csc should be independent of frequency until the dielectric relaxation time of the carriers in the semiconductor is approached experimentally. However, the observed differential capacitance values for semiconductor/ liquid contacts are typically dependent on the measurement frequency even for modest frequencies of 100 Hz-100 kHz.74-77 The simple equivalent circuit, involving only Csc and CH and a cell resistance, is thus generally not adequate to describe the observed experimental behavior of these semiconducting electrodes. To deal with this complication, various other circuit elements, including parallel and series resistances and capacitances, representing various physical situations of surface states, faradaic charge-transfer processes, and other possibilities have been proposed.74-77 Under such circumstances, however, determination of Csc is dependent on the validity of the equivalent circuit used to represent the physical situation at the semiconductor/

J. Phys. Chem. B, Vol. 102, No. 25, 1998 4851

Figure 5. A plot of the electron-transfer rate constant as a function of the effective driving force for interfacial charge transfer. The ∆G°′ values were taken to be the barrier heights measured at [A] ) [A-] ) 10 mM. The error in ∆G°′ was the standard deviation of the barrier height measured at these concentrations, while the error in ket was the standard deviation of twelve to sixteen rate constant determinations for each interface. Electron-transfer rate constants from n-Si electrodes were measured using CH3OH-1.0 M LiCl solutions with the following redox couples: 7,8-dihydro-6H-dipyrido[1,2-a:2′,1′-c]-[1,4]-diazepinium2+/+ (I), 6,7-dihydro-2,11-dimethyldipyrido[1,2-a:2′,1′-c]pyrazinium2+/+ (II), 6,7-dihydrodipyrido[1,2-a:2′,1′-c]pyrazinium2+/+ (III), methyl viologen2+/+ (IV), benzyl viologen2+/+ (V), 1,1′-bis(ethoxycarbonylmethyl)-4,4′-bipyridinium2+/+ (VI), and 1,1′-bis(diethoxycarbonylmethyl)-4,4′-bipyridinium2+/+ (VII). The fitted parabolic curve has parameters of ket,max ) (6.4 ( 1.7) × 10-17 cm4 s-1 and λ ) 0.67 ( 0.12 eV.

liquid contact, and typically it is not possible to obtain independent support for the validity of the model used to analyze the data. Contradictions also often arise between the energetics of the solid/liquid contact deduced from analysis of an equivalent circuit model and other measurements, such as steady-state J-E data, of these semiconductor/liquid contacts.78 As a key part of his study, Fajardo was able to perfect the impedance methods such that the frequency dispersion of the measured differential capacitance was minimal. In doing so, he eliminated most of the obvious ambiguities in the interpretation of the experimental impedance data for the n-Si/CH3OHviologen2+/+ series of contacts. Fajardo found that essentially all of the frequency dispersion in these systems could be traced to measurement-related phenomena, such as small electrode areas, counter electrode size, unaccounted-for impedance transfer functions of the potentiostat, etc., and that by careful attention to these factors it was possible to obtain measurements of Csc that did not, over a fairly wide frequency range, depend on the measurement frequency. In addition, plots of Csc-2 vs E were linear, had slopes that agreed to within 10-20% of those predicted by theory, and varied in the correct fashion when the dopant density of the electrode was varied. These data have thus been used to obtain the values of Vbi, and therefore values of ns, that are required to compute values of ket, using eq 7, for the various n-Si/CH3OH-viologen2+/+ interfaces. Figure 5 displays the compilation of these measurements, in the form of a plot of ket vs the effective driving force for interfacial charge transfer, ∆G°′, for each of the n-Si/CH3OHviologen2+/+ contacts studied to date.67,69 Several conclusions are obvious from this plot. First, the maximum rate constant values are clearly in the range of ket,max ) 10-17-10-16 cm4 s-1. Second, the rate constants are reasonably well-fit by the expected dependence of ket on ∆G°′ as given by the classical Marcus expression for electron-transfer processes. Both of these observations are in satisfying agreement with theoretical expectations. The data also underscore the degree of electrical perfection that has apparently been achieved, and is apparently necessary,

4852 J. Phys. Chem. B, Vol. 102, No. 25, 1998 to determine reliably the desired electron-transfer rate constants from these steady-state J-E and C-2-E methods. The ideal energetic behavior, with a lack of movement of the band edge of Si at the various CH3OH-viologen2+/+ contacts, implies a surface state density of less than 1012 cm-2. Otherwise, surface state charging would result in some of the applied potential being dropped across the Helmholtz layer and would complicate analysis of the dependence of ket on ∆G°′ and also would complicate extraction of ket from the steady-state J-E data. The presence of a high density of surface states on semiconductors like Si, InP, and GaAs had previously been postulated to be inherent to such surfaces and to lead to extreme nonidealities, resulting in a situation denoted as Fermi level pinning.1,44-46,76,79-81 However, Fajardo showed that control over etching and handling of the n-Si surfaces was able to minimize these nonidealities. In fact, the band edge position of Si/CH3OH contacts was constant, within experimental error, for the entire series of n-Si/CH3OH-viologen2+/+ contacts.69 This set of redox systems represents a variation in the redox potential of >400 mV with a change in band edge position of 400 mV in the redox potential of the electrolyte. The C-2-E behavior was also essentially independent of frequency, and the slopes were in good agreement with values expected theoretically. These observations indicated that ns could be determined using eq 8. Pomykal’s J-E data also showed a first-order dependence of the current density on the concentration of the acceptor in the solution phase. In addition, the current density depended exponentially on the applied potential, although the slope of a plot of ln J vs E deviated somewhat from theoretical expectations for a reaction that is purely first-order kinetically in the concentration of electrons at the surface of the semiconductor. Use of eq 7 led to estimation of ket for this system, and values in the range of 10-17-10-16 cm4 s-1 were observed.68 This is satisfying because these rate constants are very similar in

Lewis magnitude to those observed for the n-Si/CH3OH interface even though different redox couples and a different semiconductor were used in this set of experiments. Unfortunately, the slight, but significant, deviation in the slope of ln J vs E from ideal expectations precluded determination of ket with sufficient precision to allow formulation of a meaningful Marcus-type plot of ket vs ∆G°′ for the n-InP/CH3OH interfaces. Nevertheless, it seems clear that the value of ket,max is well-defined by the data for this system to be in the theoretically expected range of ket,max ) 10-17-10-16 cm4 s-1. D. Rate Constant Data on Other Semiconductor/Liquid Contacts. Although the n-Si/CH3OH-viologen2+/+ and n-InP/ CH3OH systems provide, to our knowledge, the only rate constant vs driving force data for a semiconductor electrode to a series of nonadsorbing outer-sphere redox couples that are available to date, a few isolated rate constant measurements have also been reported recently on other semiconductor/liquid contacts. In a very interesting experiment, Bard and co-workers showed that defects originating from edges of exposed van der Waals planes on macroscopic p-WSe2 samples were responsible for the nonideality of p-WSe2/H2O contacts.83 They demonstrated this hypothesis through use of the scanning electrochemical microscope to probe only local regions of the semiconductor/liquid interface in which no step defects were present. Even though macroscopic J-E measurements obtained using the entire exposed electrode surface displayed nonideal kinetic behavior, measurements performed in these defect-free regions produced first-order kinetic behavior for the majority carrier concentration at the electrode surface (holes, in this case of a p-type semiconductor) and for the concentration of the electron donor in the solution phase. This technique yielded an interfacial charge-transfer rate constant of 5.7 × 10-17 cm4 s-1 for the p-WSe2/H2O-Ru(NH3)63+/2+ interface. Estimates of the nuclear reorganization terms for this reaction result in a rate constant at optimal exoergicity of 2 × 10-16 cm4 s-1.69 Curiously, this ideal behavior was only observed for Ru(NH3)63+/2+, and could not be observed for redox couples such as Fe3+/2+, Fe(CN)63-/4-, and Co(phenanthroline)33+/2+, so that a systematic study of the rate constant vs driving force is not available in this system to date.83 In a separate study, steady-state J-E and C-2-E measurements have yielded a rate constant of 10-18 cm4 s-1 for the p-GaAs/HCl(aq)-Cu2+/+ interface, which when corrected for the nuclear terms, also resulted in an estimate of the rate constant at optimal exoergicity of 10-16 cm4 s-1 in this system.84 It is not clear that this interface consists of an outer-sphere, nonadsorbing redox system, so the agreement in this case between theory and experiment might be fortuitous. Meier and co-workers have used steady-state J-E and impedance vs potential measurements on both GaInP2-coated n-GaAs surfaces and on etched n-GaAs surfaces to perform rate constant determinations in contact with CH3CN-cobaltocene+/0 solutions.85 The rate constant for GaInP2-coated n-GaAs surfaces was found to be ≈10-17 cm4 s-1, which is in good agreement with the theoretical expectations discussed above. Rate constants as high as 10-10 cm4 s-1 were computed for n-GaAs/CH3CN-cobaltocene+/0 contacts, but more recent studies of the n-GaAs/CH3CN-cobaltocene+/0 system have indicated that adsorption of the redox acceptor plays a critical role in determining the J-E properties of this solid/liquid contact.86 In several other systems, it is not possible to obtain direct measurements of the interfacial charge-transfer rate constants because the kinetics do not obey the second-order rate law of eq 7. Typically, in such systems, the current does not depend on the concentration of acceptor species in the solution phase. Under such conditions, one can only obtain an upper bound on

Feature Article the value of ket, because larger values of the charge-transfer rate constant would have made this process rate-determining and thus would have produced the first-order kinetic dependence on [A] expected theoretically. To obtain such bounds on ket, it is of course necessary to measure ns at any given electrode potential. Fajardo and Pomykal at Caltech have made such measurements, using C-2E data, for n-Si/CH3OH-Me2Fc+/0, n-GaAs/CH3CN-Fc+/0, and p-InP/CH3CN-cobaltocene+/0 interfaces.24 In the case of n-Si/ CH3OH-Me2Fc+/0 contacts, the electron-transfer process from the conduction band is so slow that only bulk diffusion/ recombination of holes in the Si is the rate-limiting recombination step.71 In the case of n-GaAs/CH3CN-Fc+/0 contacts, surface-state recombination has been assigned to be the ratedetermining recombination process.44 In both cases, no dependence of J on [A] is observed, and therefore only upper bounds on ket can be obtained from J-E and C-2-E data. These values are ket < 10-15 cm4 s-1 and ket < 10-14 cm4 s-1, respectively, and are thus consistent with the theoretical expectations, and experimental data on related contacts, described above. E. Implications for Performance of Photoelectrochemical Energy Conversion Devices. These rate constant measurements provide insight into numerous aspects of the behavior of photoelectrochemical cells as energy conversion devices. It is well-documented that the photovoltages produced by illuminated semiconductor/liquid contacts can be much higher than those obtained from semiconductor/metal contacts using the same types of semiconducting substrates.2 Ashish Bansal and Lou Casagrande have observed this behavior for n-GaAs/liquid vs n-GaAs/metal contacts,44,87 Heben, Kumar, and Zheng observed similar behavior in their studies of n-InP,78 Kumar noted this behavior for n-Si contacts,88 and Charles Lieber, Gronet, and I made similar observations as early as 1984 during investigations of the behavior of p-Si contacts.89 This observation seems to be generally true for other semiconductor electrodes as well, when quantitative comparisons have been made between the two types of contacts. This behavior can now be readily understood, because with a ket,max ) 10-16 cm4 s-1 and a concentration of redox species of 0.1 M ) 6 × 1019 cm-3, the effective interfacial carrier collection velocity is only (10-16 cm4 s-1)(6 × 1019 cm-3) ) 6 × 103 cm s-1. In contrast, the collection velocity at a semiconductor/metal contact is typically 107 cm s-1, because the metal has a continuum of states available to collect essentially every electron that reaches the solid/metal contact and because these electrons arrive at the interface at a thermal velocity of ≈107 cm s-1. Thus, even identical barrier heights would produce less interfacial majority carrier current, and therefore less recombination, at a semiconductor/liquid contact than that at a semiconductor/metal contact. This produces higher open circuit voltages, and higher energy conversion efficiencies, for the semiconductor/liquid contacts relative to semiconductor/metal systems. This value of ket,max also explains how it is possible to observe bulk recombination-limited minority carrier injection currents at the n-Si/CH3OH-Me2Fc+/0 contact.71 Even for a barrier height of 1.0 V, thermionic emission into a metal would lead to a large collection velocity, and thus a large recombinaton current, at a semiconductor/metal contact. Because the band gap of Si is only 1.1 eV, it is not possible to achieve larger barrier heights than 1.0 ( 0.1 V in any practical Si/metal system. It would thus not be possible for any such semiconductor/metal contact to reach the desirable situation in which the surface recombination rate was so low that only the bulk recombination/ diffusion process dominated the total recombination current in the device. This desirable situation is, however, observed

J. Phys. Chem. B, Vol. 102, No. 25, 1998 4853 experimentally at n-Si/CH3OH-Me2Fc+/0 interfaces, which is consistent with the lower recombination predicted at a 1.0 V barrier height with a liquid having a ket,max ) 10-16 cm4 s-1. Similarly, for Si/metal contacts, it should not be possible to reach high-level injection conditions at 1 Sun illumination levels, due to rapid interfacial recombination processes, if ket,max . 10-16 cm4 s-1. However, as shown by Ming Tan at Caltech, such high-level injection conditions are readily achieved using Si/ CH3OH contacts.90 The low surface recombination velocities exhibited by these systems under high level injection, as measured by Tan, Kenyon and Olaf Kru¨ger using both steadystate and transient photovoltage techniques,91-93 are also consistent with the relatively slow charge-transfer rates predicted by values of ket,max ≈ 10-17-10-16 cm4 s-1. This value of ket,max ) 10-17-10-16 cm4 s-1 is consistent with the slow time decays of the Si/CH3OH-Me2Fc+/0 microwave conductivity experiments,51 and readily explains the behavior observed in that system. It is also consistent with the differences in spectral response data at semiconductor/liquid and semiconductor/metal contacts, and it explains the absence of observable recombination at short wavelength for the semiconductor/liquid contacts studied by Kumar.62 Additionally, it explains why no outer-sphere redox systems have been able to stabilize small band gap semiconductors effectively in contact with aqueous electrolyes: compared to the 55 M concentration of water that can participate in anodic oxidation or corrosion processes, it is not possible to scavenge effectively all of the photogenerated minority carriers with a ket,max of 10-16 cm4 s-1, even at hole acceptor concentrations of >1 M. Hence, electrode stability is only observed for high concentrations of adsorbing redox species such as Se2- or Te2- on GaAs, S2- on CdS, Ion n-GaAs, and similar systems.2 Outer-sphere redox couples would only be expected to be effective in competing with very slow photocorrosion processes in aqueous solution, but these redox species can, of course, be exploited to produce a variety of stable photoelectrochemical cells in dry nonaqueous solvents.2 Again these predictions are in excellent accord with experimental observations. Although much progress has clearly been made in this area in recent years, this field still seems to hold promise for future important developments. Experiments such as those being performed by D. Waldeck and co-workers, focused on elucidating the distance dependence of electron transfer at semiconductor/liquid contacts,94 are obviously significant and important. Conclusive experimental observation of the Marcus inverted region for tethered or dissolved outer-sphere redox reagents has not yet been reported, to our knowledge,95 nor have data been reported that demonstrate unambiguously the predicted effects of changing the reorganization energy of the ion on the measured interfacial charge-transfer rate constant. Some of the reorganization energy in these interfacial electron-transfer events might well be due to solvation of electrons near the surface of the electrode and also possibly to some nuclear reorganization of the bonds in the near-surface region of the solid itself. No clearcut experimental evidence is available to address these points at the present time. Refinements in the theoretical description of such processes, including more accurate descriptions of the band structure of the electrode, of the distance dependence of the reorganization energy of the acceptor, and of the possible role of quantum vibrational modes in such systems, also await a thorough description. Thus, although one of the goals in preparation of this feature article was to acquaint the reader with the advances that have been made in this area in recent years, another goal is to stimulate even further interest in these problems within the electron transfer community, so that for

4854 J. Phys. Chem. B, Vol. 102, No. 25, 1998 the forseeable future this research field will continue to be intimately connected with work at the forefront of physical chemistry. Acknowledgment. The National Science Foundation and the Department of Energy, Office of Basic Energy Sciences, are acknowledged for providing the funding for most of the work from Caltech that is summarized in this article. Acknowledgment is also made to a very talented group of graduate student and postdoctoral co-workers, only some of whom have been acknowledged by name, but all of whom have made essential contributions to developing this work to its current level of understanding. I also wish to acknowledge invaluable discussions with colleagues including F. Anson, A. Bard, C. Chidsey, S. Feldberg, H. Finklea, A. Fujishima, H. Gerischer, S. Gottesfeld, A. Heller, J. Hupp, C. Koval, R. Marcus, R. Memming, B. Miller, C. Miller, R. J. D. Miller, T. Mallouk, A. Nozik, B. Parkinson, H. Tributsch, D. Waldeck, and M. Wrighton, among others, that have immensely contributed to advancing my personal understanding of this interesting field of science. References and Notes (1) Wrighton, M. S. Acc. Chem. Res. 1979, 12, 303. (2) Tan, M. X.; Laibinis, P. E.; Nguyen, S. T.; Kesselman, J. M.; Stanton, C. E.; Lewis, N. S. Prog. Inorg. Chem. 1994, 41, 21. (3) Heller, A. Acc. Chem. Res. 1981, 14, 154. (4) Rajeshwar, K.; Singh, P.; DuBow, J. Electrochim. Acta 1978, 23, 1117. (5) Gerischer, H.; Beck, F. Z. Phys. Chem., N. F. 1957, 13, 389. (6) Lewis, N. S. Annu. ReV. Phys. Chem. 1991, 42, 543. (7) Royea, W. J.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 11152. (8) Rosenwaks, Y.; Thacker, B. R.; Nozik, A. J.; Ellingson, R. J.; Burr, K. C.; Tang, C. L. J. Phys. Chem. 1994, 98, 2739. (9) Miller, R. J. D.; McLendon, G.; Nozik, A. J.; Schmickler, W.; Willig, F. Surface Electron-Transfer Processes; VCH Publishers: New York, 1995. (10) Gerischer, H. In Solar Energy ConVersion. Solid-State Physics Aspects; B. O. Seraphin, Ed.; Springer-Verlag: Berlin, 1979; Vol. 31. (11) Sze, S. M. Physics of Semiconductor DeVices, 2nd ed.; Wiley: New York, 1981. (12) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; Wiley: New York, 1980. (13) Chidsey, C. E. D. Science 1991, 251, 919. (14) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (15) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (16) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155. (17) Marcus, R. A. J. Phys. Chem. 1963, 67, 853. (18) Marcus, R. A. J. Phys. Chem. 1991, 95, 2010. (19) Kittel, C. Introduction to Solid State Physics, 6th ed.; Wiley: New York, 1986. (20) Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal Electrodes; Plenum: New York, 1980. (21) Gerischer, H. J. Phys. Chem. 1991, 95, 1356. (22) Marcus, R. A. J. Phys. Chem. 1990, 94, 4152. (23) An alternate, but perhaps more sound physical interpretation would take the de Broglie wavelength of an electron in the conduction band of a semiconductor to be equal to the effective radius of the electron in these formulae. The same numerical results, i.e., an effective radius of re ) (310) × 10-8 cm and ket, max ) 10-17 cm4 s-1, are obtained from either method. (24) Pomykal, K. E.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. 1996, 100, 3652. (25) Smalley, J. F.; Feldberg, S. W.; Chidsey, C. E. D.; Linford, M. R.; Newton, M. D.; Liu, Y. P. J. Phys. Chem. 1995, 99, 13141. (26) Smith, B. B.; Nozik, A. J. J. Phys. Chem. B 1997, 101, 2459. (27) Bowler, B. E.; Raphael, A. L.; Gray, H. B. Prog. Inorg. Chem. 1990, 38, 259. (28) Terrettaz, S.; Becka, A. M.; Traub, M. J.; Fettinger, J. C.; Miller, C. J. J. Phys. Chem. 1995, 99, 11216. (29) Evenor, M.; Gottesfeld, S.; Harzion, Z.; Huppert, D.; Feldberg, S. W. J. Phys. Chem. 1984, 88, 6213. (30) Hollingsworth, R. E.; Sites, J. R. J. Appl. Phys. 1982, 53, 5357. (31) Smandek, B.; Chmiel, G.; Gerischer, H. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 1094. (32) Rosenwaks, Y.; Shapira, Y.; Huppert, D. Phys. ReV. B 1992, 45, 9108.

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J. Phys. Chem. B, Vol. 102, No. 25, 1998 4855 (93) Kenyon, C. N.; Tan, M. X.; Kru¨ger, O.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2850. (94) Gu, Y.; Waldeck, D. H. J. Phys. Chem. 1996, 100, 9573. (95) Although there appear to have been no direct measurements to date of the behavior in the Marcus inverted region for nonadsorbing, outer-sphere redox couples, several interesting sets of experiments do seem to suggest that this type of behavior will apply to the semiconductor/liquid contact: (a) Nakabayashi, S.; Itoh, K.; Fujishima, A.; Honda, K. J. Phys. Chem. 1983, 87, 5301. (b) Moser, J. E.; Gra¨tzel, M. Chem. Phys. 1993, 176, 493. (c) Lu, H.; Prieskorn, J. N.; Hupp, J. T. J. Am. Chem. Soc. 1993, 115, 4927. (d) Heimer, T. A.; Bignozzi, C. A.; Meyer, G. J. J. Phys. Chem. 1993, 97, 11987.