J. Phys. Chem. 1993,97, 13441-13443
13441
Existence of a Light Intensity Threshold for Photoconversion Processes Brian A. Gregg' and Arthur J. Nozik' National Renewable Energy Laboratory, I61 7 Cole Blvd., Golden. Colorado 80401-3393 Received: October 20, 1993'
Two models of the mechanism of photoinduced electron transfer a t semiconductor surfaces have long been differentiated by their prediction, or their denial, of the existence of a light intensity threshold for fuel-forming photoconversion proteoses. We attempt to clarify this problem by making a distinction between two possible types of thresholds: a threshold for incipient product formation and a threshold for product formation in a specified state, such as its standard state. A light intensity threshold for incipient product formation appears to be forbidden by molecular electron-transfer theory and has apparently never been observed. Conversely, a light intensity threshold for product formation in its standard state must always occur, simply because the product concentration must first build up from its equilibrium value to its standard-state value. Since the former threshold is forbidden, while the latter is unavoidable, the existence of a threshold cannot be used to distinguish between the models.
Introduction The presence or absence of a light intensity threshold for photoconversion processes has been controversial for a number of years.l-I0 In essence the question was, must the intensity of the light first exceed some threshold value before the photoconversion process (photoinduced production of chemical fuels) becomes spontaneous? In one model, due to Gerischer, it is assumed that theenergeticsofthechargecarriersinan illuminated semiconductor can best be approximated by a treatment based on (equilibrium) statistical mechanic~.IJ.~J~ This results in the conclusion that the difference in the quasi-Fermi levels, AEr, in an illuminated semiconductorelectrode is equal to the free energy of the electron-hole pairs:
where n andp are the total concentrations of electrons and holes, respectively, Ncand N, are the densitiesof states in the conduction and valence bands, respectively, U, is the bandgap of the semiconductor, and the other symbols have their usual meaning. In this model, the available energy from the illuminated semiconductor is logarithmicallyproportionalto the light intensity (Le., tothenpproduct). qAEris thus "theremainingfreeenergy ... [that] is availablefor redox reactions."' A light intensity threshold is predicted since the free energy of charge carriers must first become greater than the free energy of products formed in solution before the electron-transfer (ET) reaction can be spontaneous. Thus, Gerischers wrote "A threshold in illumination intensity must be surpassed to reach the driving force for photoelectrolysis of water." Similar theoretical treatments, employing the concept of a free energy of molecular excited states, have been applied to photoconversion processes in homogeneous solutions.11-13 Williams and Nozik4v5 first questioned the applicability of concepts based on statistical detailed balance to nonequilibrium fuel-forming photoconversion processes. They suggested that the ET reaction might be viewed as an irreversibleprocess similar to photoemission into a vacuum. In this model, the ET process is described as individual events, each (thermalized) carrier in the semiconductor having the potential of its respective band edge. This model predicts that only the (light intensityindependent) difference in potential between the semiconductor band edges and the acceptor or donor levels in solution would determine the spontaneity of ET. Therefore, this model predicts Abstract published in Aduance ACS Abstracts. December 1, 1993.
0022-365419312097-13441$04.00/0
that no light intensity threshold for photoconversion processes should be observed. Until recently,6no such threshold had been reported in the literature. Kumar, Santangelo, and Lewis (KSL)recently measured a light intensity threshold for the evolution of Hz and 02 at 1 atm in a cell using a SrTiO3 photoelectrode.6 The data were interpreted as "distinguishing between"6 the statistical model of Gerischerl-3 and the so-called stochastic model of Williams and Nozika4JThe existence of the threshold was believed to be clear evidence in support of the statistical formalism. A kinetic model that predicted the appearance of a light intensity threshold for sustained fuel-forming photoconversion processes was also introduced as further support for their interpretationa6 We believe, however, that both the predicted and the observed light intensity thresholds reported by KSL do not distinguish between the two models, as we describe below. Two Types of Thresholds. We believe that much of the controversy about light intensity thresholds can be resolved by clearly defining what is meant by such a threshold for a photoconversion process. There appear to be two relevant definitions of a light intensity threshold for a photoconversion process: a threshold for incipient product formation and a threshold for product formation in a specific state, such as its standard state. In the following discussion we will assume this specific state is always the standard state. A light intensity threshold for incipient product formation appears to be in conflict with the accepted theories of molecular ET reactions developed by Marcus and others.lcl9 The basic concepts of molecular ET theories have been thoroughly tested and confirmed in the past 30 or so year^.'^.*^ We discuss an example in homogeneous solution as the simplest case. If the excited-statestandard oxidation potential of a photoexcited molecule, S*,in solution is more negative than the standard reduction potential E ' A , ~of an acceptor, A, electron transfer from S* to A will be a spontaneous process. The driving force for the photoinduced ET reaction, S* A -.,S+ A-, is the standard free energy of reaction,14 AGO, = AGOrsaetpnts = FEOS.,,,~- F E O A ,< ~ 0 (F is Faraday's constant). There is no dependence of the reaction driving force on the concentration of excited species and therefore no dependence on the light intensity or the free energy of the excited states. At equilibrium (i.e., in the dark) detailed balance requires that the rate of the back-reaction, S+ A- S* + A, be exactly q u a l to the rate of the forward reaction. Upon illumination of arbitrarily low intensity, the concentration of S* will increase
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13442 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
above its equilibrium value, and hence the rate of forward ET will increase above its equilibrium rate, leading to a greater than equilibrium concentration of product A-. No light intensity threshold for incipient product formation occurs; if the photoinduced ET reaction is spontaneous at any light intensity, it will be spontaneous at all light intensities. Molecular ET theory leads unambiguously to this conclusion. The tremendous amount of research done on photoinduced ET reactions]’ has apparently never resulted in the observation of a light intensity threshold for reaction. Photoinduced ET at semiconductor electrodes leading to a photoelectrolysis current, Le., to product formation, is exactly analogous to the solutioncase: thespontaneity of the photoinduced ET reaction is related to the difference in standard potentials of the donor (say an electron at the conduction band edge), and the acceptor (in solution) and is therefore independent of concentration, light intensity, and free energy of electron-hole pairs. We conclude that a light intensity threshold for photocurrent (incipient product formation through photoelectrolysis) should not occur. On the other hand, a light intensity threshold for product formation in its standard state must occur, quite independent of the mechanism of the photoinduced ET reaction. This threshold results simply because either the concentrations of photoproducts must build up to their standard-state values (when starting from equilibrium) or the photoprocess must oppose the reverse dark reaction if the photoproducts are initially present in their standard state (see below). Below the light intensity threshold, the concentration of a product will be governed by the rate of its formation (proportional to the light intensity) and the rate of its destruction (proportional to its concentration). Therefore, the steady-state product concentration will be usually logarithmically proportional to the light intensity until it reaches its solubility limit, which frequently is its standard state. Above this threshold light intensity, in the case of solutions, the product may continuously separate from solution at a rate linearly proportional to the light intensity minus the threshold light intensity. Thus, a light intensity threshold for product formation in its standard state should always occur; however, its existence contains no information about the mechanism of ET at semiconductor electrodes or in homogeneous solution. The KSL Paper. We return briefly to KSL’s paper.6 In our view, the essential element of both the theoretical and experimental aspects of this work may be summarized as follows: when a bias is maintained across a photoelectrochemical cell such that a steadystatedarkcurrent flows ina directionopposite to the photocurrent, a threshold light intensity must be surpassed before the photocurrent will overcome this applied bias. Seen from this perspective, the observation of a threshold in the KSL experiments is expected and is independent of any theoretical model of ET at semiconductor electrodes. Apparently in order to maintain definite redox potentials in solution, KSL bubbled 1 atm of H2 over the platinum counter electrode and 1 atm of 02 over the SrTiOJ working electrode. Thus, in the dark a fuel cell was created in which a chemical potential gradient was maintained across the cell, resulting in a constant dark current of opposite polarity to the photocurrent. A threshold light intensity was required to counteract the fuel cell and reduce the dark current to zero (see their Figure 4a); above this threshold a net anodic current was produced a t the semiconductor electrode. In the context of our discussion, it seems clear that the observed threshold was a threshold for the formation and separation of Hz and 02 a t 1 atm (standard conditions) and was thus unrelated to the issue of statistical us stochastic (or molecular) mechanisms of ET. Therefore, we disagree with their claim to have distinguished between these two mechanisms. If no external H2 and 02 were used to bias the cell, that is, if the experiment were started from equilibrium in the dark where no
Letters net dark current can flow, the light intensity threshold for photoelectrolysis current, Le., for incipient product formation, would be zero. The several experiments reported by KSL, as well as their kinetic model, seem to agree entirely with this interpretation.
Discussion We believe that once the distinction between the two types of light intensity thresholds is made, much oftheexistingcontroversy, with respect to threshold light intensities, between the two models for photoinduced ET a t semiconductor electrodes should be resolved. There can be no threshold for incipient product formation or photoelectrolysis current, but there must always be a threshold for product separation or formation in standard-state conditions. Therefore, no distinction between the models can be madeon the basis of a threshold; this is contrary to the implication in ref 5 that such a distinction is possible. In the context of our discussion above, it appears that an important distinction was never clearly formulated in the statistical model: for a photoconversion reaction to occur, is it required that the free energy of electron-hole pairs be greater than the standard free energy of products or only greater than the free energy of products? The previous discussions of the problem seem always to have tacitly assumed the former-otherwise there apparently would have been no controversy. Only the former case would result in a light intensity threshold for photoelectrolysis current, the focus of most of the previous arguments and experiments. And it is this interpretation that we haveaddtessed in this paper, arguing that a threshold for incipient product formation (photoelectrolysis current) is forbidden by molecular ET theory. The second interpretation does not predict a light intensity threshold for photoelectrolysis current since, at equilibrium, the free energy of electron-hole pairs must be equal to the free energy of products. Then, as in the kinetic example described above, the absorption of an arbitrarily low light intensity by the semiconductor will cause an increase in the concentration (free energy) of charge carriers which will lead to a corresponding increase in the concentration (free energy) of products. Thus, no threshold for incipient product formation is predicted. Therefore, this second interpretation of the statistical model leads to a prediction identical to that of the molecular, or stochastic, model with regard to a light intensity threshold. Conclusions A light intensity threshold for incipient product formation would be a violation of molecular electron-transfer theories and has apparently never been observed. On the other hand, a light intensity threshold for product formation in the standard state, or for product separation from solution, should always be observed. Since the latter threshold is unavoidable, its existence contains no information about the mechanism of the electron-transfer process, and it cannot be used to distinguish between the molecular and statistical models of photoinduced electron transfer at semiconductor surfaces. Acknowledgment. This work was supported by the U.S. Department of Energy, Office of Energy Research, Division of Basic Energy Sciences, Chemical Sciences Division. References and Notes (1) Gerischer, H. In Semiconductor-Liquid Junction Solar Cells, Proceedings of the Conjerence on the Electrochemistry and Physics of Semiconductor-Liquid Interfaces Under Illumination; Heller, A., Ed.; Electrochemical Society: Princeton, NJ, 1977; pp 1-19. ( 2 ) Gerischer, H. InSolar PowerandFuels; Bolton, J. R.,Ed.;Academic Press: New York, 1977; pp 77-117. (3) Gerischer, H. In Topics in Applied Physics; Seraphin, B. 0..Ed.; Springer: Berlin, 1979; Vol. 31, pp 115-172. (4) Williams, F.; Nozik, A. J. Nature 1978, 272, 137-139. (5) Nozik, A. J. Annu. Rev. Phys. Chem. 1978, 29, 189-222.
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Mikkelsen, K. V.; Ratner, M. A. Chem. Reu. 1987, 87, 113-153. Photoinduced Electron Transfer; Fox, M. A., Chanon, M., Eds.; Elsevier: Amsterdam, 1988. (18) Bolton, J. R.; Archer, M. D. In Electron Transfer in Inorganic, Organic,and BiologicalSysrems; Bolton, J . R., Mataga, N., McLendon, G., Eds.; American Chemical Society: Washington, DC, 1991. (19) Balzani, V.; Scandola, F. In Energy Resources Through Phorochemistryand Catalysis;Grltzel,M., Ed.; Academic Press: New York, 1983; pp 1 4 8 . (20) Electron Transfer in Inorganic, Organic, and Biological Systems; Bolton, J . R., Mataga, N., McLendon, G., Eds.; American Chemical Society: Washington, DC, 1991; Vol. 228.
