J. Phys. Chem. 1994, 98, 11020- 11033
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FEATURE ARTICLE Ultrafast Nonlinear Optical Studies of Surface Reaction Dynamics: Mapping the Electron Trajectory J. M. Lanzafame, S. Palese, D. Wang, and R. J. D. Miller* Department of Chemistry and the Institute of Optics, University of Rochester, Rochester, New York 14627
A. A. Muenter Eastman Kodak Imaging Research and Advanced Development, Rochester, New York 14652 Received: May 5, 1994; In Final Form: August 15, 1994@
Heterogeneous electron transfer involves the coupling of a dense manifold of highly delocalized electronic levels of the solid state to a discrete molecular state as well as an abrupt change in phase in the reaction coordinate. These features make this problem unique relative to homogeneous solution phase or gas phase reaction mechanisms which involve coupling between discrete states within a uniform medium. Recent advances in time domain optical methods are discussed in the context of studying interfacial charge transfer processes at single crystal semiconductor surfaces as a means to probe the primary processes governing heterogeneous electron transfer. Two distinct boundary conditions are discussed: charge injection into a semiconductor from an adsorbate and charge emission from a semiconductor to an acceptor. The reaction dynamics are investigated using a combination of nonlinear spectroscopies with an emphasis on mapping the electron transport and transfer and investigating the role of nuclear vs electronic relaxation mechanisms in the bamer crossing dynamics. A fundamental understanding at this level seeks to determine the criteria for fully optimizing charge separation at surfaces.
I. Introduction Interfacial electron transfer between a discrete molecular state and a conducting surface is the simplest of all surface reactions: it involves only the exchange of an electron, and no bonds are broken. This simplicity offers the best opportunity for coming to a detailed understanding of a surface reaction coordinate. Research in this area is strongly motivated by both its fundamental importance and the large number of practical applications. For example, electrochemistry, lithographic processes, many surface catalytic mechanisms, and almost all modem imaging (photography and xerography) are derived from interfacial electronic Interfacial charge transfer also harbors great potential as an efficient mechanism for solar energy c o n ~ e r s i o n . ~This - ~ latter application is primarily due to much more efficient charge separation in these solid state materials relative to an all molecular approach. Despite the numerous applications of this process, our understanding of charge trausfer at surfaces is lagging well behind the analogous problem of homogeneous electron transfer due to the greater inherent experimental difficulties in studying surfaces. In order to fully understand interfacial charge transfer processes, one needs information on the electronic coupling between the two resonant electronic states undergoing electron exchange, on the nuclear activation barrier to attaining this condition of electronic resonance (the energetics), and on the barrier crossing dynamics. These three features are common to all charge transfer reactions, both heterogeneous and homogeneous, and each plays a role in determining the rate of the reaction. @Abstract published in Advance ACS Abstracts, September 15, 1994.
0022-3654/94/2098-11020$04.50/0
For the majority of electron transfer reactions, the kinetics are dominated by the energetics of the reaction coordinate. While the details of these energetics can be difficult to obtain, once the information is known, the reaction rate dependence is well described by a thermal equilibrium distribution for the nuclear modes that are coupled to the reaction coordinate. Because of this dominant role of the energetics, finding a regime where barrier crossing dynamics can be accessed is a distinct challenge. For carefully chosen cases, the reacting system can be prepared at the adiabatic crossing point. Here, the barrier crossing rate is just the rate of nuclear passage along the reaction coordinate and is controlled by nuclear relaxation of the intramolecular and intermolecular modes of the acceptor and the solvent. At this level of generality, the dependence of charge transfer on the energetics and barrier crossing dynamics is the same for both homogeneous and heterogeneous charge transfer reactions. In contrast, a marked distinction between the homogeneous and heterogeneous cases is expected for the electronic coupling. The effect of the solid state band on this coupling is the key, current fundamental issue: the degree of electronic coupling determines the level of theoretical approximation needed to correctly model the problem. In addition, the most unique feature of electron transfer processes at surfaces is that it is one of the few reaction mechanisms in which there is a dense manifold of electronic levels intimately connected to the reaction coordinate. In this regard, the conduction band of the solid state represents a large number of source terms for electron donor or acceptor levels in the reaction coordinate. This effect is expected to significantly enhance the degree of mixing between states relative to the discrete two state coupling in the homogeneous electron transfer problem. In certain cases, the 0 1994 American Chemical Society
Feature Article electronic continuum component to the reaction coordinate can also alter the conditions of the barrier crossing dynamics, and this point needs to be considered explicitly. Historically, most of the work on interfacial charge transfer has been conducted using electrochemical approaches to the problem. These approaches involve experiments which are steady state in nature and, therefore, determine the rate-limiting step in the reaction mechanism. To access information on the fastest dynamical processes occurring at the surface, i.e., the primary processes, a “jump” experiment of some kind is required. These experiments entail a rapid change in one of the reaction variables (temperature, pressure, electronic state, etc.) on a time scale faster than the dynamics of interest. By probing the subsequent relaxation of the system from this nonequilibrium point, information on the fastest processes involved in a particular reaction mechanism can be obtained. Recent studies of metal electrodes have explored the use of rapid laser heating to provide a T jump in the surface electron distribution and follow the reaction by changes in surface potential.* This approach gets around transport limits in measuring heterogeneous rate constants and makes it possible to study adsorbed states. In order to access information on the barrier crossing dynamics and issues of electronic coupling for surface electron transfer, it is necessary to study the reaction dynamics on time scales shorter than nuclear relaxation (approximately 100 fs). This time scale requires an all optical approach to the problem with spectroscopies capable of submonolayer sensitivities. This review will focus on the use of time domain optical spectroscopies to directly probe the dynamics of charge transfer at semiconductor surfaces. In this line of study, the semiconductor serves as the optical switch for turning on interfacial charge transfer processes with high quantum yield (-1). The impulsive nonequilibrium condition (the “jump”) is the photogeneration of electron-hole pairs in the surface region. With current laser technology, the surface can be optically prepared near or at the adiabatic crossing point in the electron transfer coordinate on time scales faster than all the relevant relaxation dynamics. In this scenario, the subsequent time evolution of the electron distribution across the interface directly accesses information on the degree of electronic coupling and the barrier crossing dynamics. The interfacial charge transfer processes will be explored for two different t = 0 boundary conditions: (1) charge carriers will be optically prepared in the solid state and allowed to cross the interface to a molecular acceptor (electron emission case) and (2) optical excitation of a molecular species at the surface in which electron transfer occurs into the electronic band of the solid (electron injection case). These different t = 0 boundary conditions can be attained by using the appropriate excitation wavelengths. As will be detailed below, these two different experimental conditions provide critical information concerning the role that both the electronic continuum and nuclear continuum play in localizing free carriers on discrete molecular states at surfaces. For the reactive carriers optically prepared within the solid state, the experiments exploit semiconductorfliquidjunctions to photoinitiate the charge transfer processes. The key photophysical processes governing the electron transfer step for these systems are shown in Figure 1A. In order to understand the interfacial charge transfer dynamics, the dynamics affecting the carrier distribution at the surface need to be characterized, and these will be summarized below. With respect to the surface photochemistry, the most important feature of liquid junctions is the space charge field which develops at the interface due to the difference in chemical potential between the two phases. 1-7
J. Phys. Chem., Vol. 98, No. 43, 1994 11021
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Figure 1. (A) Electron emission case. The surface reaction dynamics consist of a field-assisted spatial transport component (l),a subsequent band to acceptor interfacial charge transfer step (2) which is in competition with nonradiative relaxation in the solid state (3), and surface state trapping ( 5 ) . Process (4) refers to interfacial charge transfer of thermalized carriers. The upper limit for the charge transfer dynamics in this case is limited by the nuclear relaxation of the solvent This time scale defines modes coupled to the reaction coordinate the adiabatic or strong coupling limit for the charge transfer process. (B) Electron injection case. There is a charge transfer step from the optically prepared excited state level (l), followed by nonradiative relaxation (2) and trapping in midgap levels (4). The injected electron must have enough kinetic energy to escape the attractive Coulombic field of the product parent cation which imposes certain limitations on the energetics. The overall efficiency of charge separation depends on the field-assisted charge separation of the thermalized electron from the parent cation ( 5 ) and the competing back electron transfer (3) and hole transfer (6). Information on all these dynamical processes is needed to fully understand the surface photochemistry and is acquired by exploiting optical probes of the electron transfer coordinate.
Due to the immobile nature of the ionized dopants in the semiconductor, this field exists primarily within the semiconductor and spatially extends between 100 and 1000 8, from the surface for typical doping densities. This region is delineated in Figure 1A as the quadratic curvature in the bands near the surface. This curvature reflects the spatial dependence of the carrier potential in the presence of the surface field. The magnitude of the space charge field can be enormous (lo5lo6 Vkm) and acts to very efficiently separate the electronhole pairs that are either photogenerated within or migrate into the space charge region. The large Coulombically driven separation of the electron and hole carrier wave functions prevents recombination. In addition, the field polarity is such that the minority canier is selectively confined to the surface reaction plane within wave function overlap of the molecular acceptor distribution at the interface. The ability of the space charge field to spatially confine the minority carrier and reduce carrier recombination leads to the high photocurrent quantum yields in these systems. The overall dynamics of the charge carrier lifetimes at the surface are related to the electronic coupling across the interface and the relaxation dynamics on
11022 J. Phys. Chem., Vol. 98, No. 43, 1994 both the solid and liquid state sides of the surface. Hence, this process must be viewed with an eye on both of these physical processes. This part of the review will concentrate on charge transfer processes at the GaAs(100) surface, which serves as a model interface in that it has well-characterizedelectronic, optical, and electrochemical properties relative to other surface^.^ For this interface, the question concerning the degree of electronic coupling is related to current issues behind the hot carrier model vs thermalized band edge or surface state mediated models for charge transfer at semiconductor liquid junctions. In the strong coupling or adiabatic limit for the electronic coupling, the charge transfer processes can become competitive with nonradiative relaxation in the solid state half-space of the reaction coordinate. This opens up the possibility of hot electron channels for the surface chemistry. If this limit can be achieved, it would avoid energy loss through lattice heating and would have important implications for solar energy conversion strategies. The highly efficient energy conversion processes at GaAs interfaces with strongly adsorbed acceptors forms an important test case for this concept. The other t = 0 boundary condition, pertaining to the electron injection case, involves the optical preparation of molecular excited states resonant with conduction band states at dyesensitized semiconductor surfaces. Electron injection under these conditions probes the complementary process of electron transfer into an electronic continuum, the general features of which are shown in Figure 1B. The main difference between this t = 0 boundary condition and that depicted in Figure 1A is that the electron is initially localized on a discrete molecular state. In the electron emission case (Figure lA), the reactive charge carrier is initially prepared within the solid state and only becomes localized by the surface space charge field along the surface normal. Displacement of the system into the product channel requires nuclear repolarization (relaxation) to stabilize the charge-separated product state. For the energetics shown in Figure lB, the electron transfer process is barrierless, and the reaction coordinate involves coupling of the discrete state to the field-assisted k states that would appear as traveling waves moving away from the surface. The propagation of the electron away from the initial site and into the solid state does not require nuclear relaxation. In this case, the time evolution of the excited state (SI) to the free carrier state provides a direct probe of the electronic coupling or wave function overlap between a discrete molecular state and the delocalized solid state electronic levels. The information gained from these studies provides the closest analogy to the time evolution of transition states at electrode surfaces. For this t = 0 boundary condition, the main problem is to come up with a system which is both well-defined and experimentally accessible. Dye sensitization of semiconductor surfaces is the underlying principle of photography, and a great deal of work has been done in this area. Silver halides, however, suffer from the very photochemical processes that form the basis of the photographic process. Other more robust metal oxide surfaces have been explored for extending the spectral sensitivity of semiconductor solar cells. In general, these metal oxide surfaces exhibit extremely low quantum yields for electron transfer, the reasons for which are still not well understood.ls10 This part of the review will focus on the recent development and characterizationof SnS2 surfaces for studying dye-sensitized charge transfer. This semiconductor represents a class of 2-dimensional layered semiconductorswhich are the only single crystal surfaces to date which are photostable and exhibit high quantum efficiencies for photoinduced electron transfer
Lanzafame et al. (>80%).11312Furthermore, the energetics depicted in Figure 1B correspond to experimentally achievable conditions at this surface. The high quantum yield, chemically inert nature of the surface, and fortuitous optical and energetic properties of SnS2 make it a nearly ideal system for studying the effects of an electronic continuum on the electron transfer reaction coordinate. A fundamental understanding of electron transfer at this surface should provide the general principles for maximizing the efficiency of charge separation at interfaces using dye sensitization.
11. Models for Surface Electron Transfer A. Electron Emission Case: Weak Electronic Coupling Limit. The emphasis in electron transfer theory has been in correlating the electron transfer rates to nuclear and electronic factors of the reaction coordinate. Early works by Marcus,13 Levich,14 and Gerischer15 have led to relatively simple expressions amenable to experimental tests. The rate equation, as formulated for the problem of electron transfer at electrode surfaces, assumes harmonic displacementsfor the solvent modes involved in solvent reorganization along the reaction coordinate. These theories are in the weak electronic coupling limit: the electron transition probability is largely determined by the Boltzmann statistics of overcoming the nuclear activation barrier to attaining electronic resonance. For a semiconductor surface, the problem of charge transfer is simplified by assuming it involves thermalized charge carriers at the valence or conduction band edges.15 With these assumptions, the factors controlling the electron transfer rate constant ( k ) for an electron of energy E at the surface are embodied in the relation
where ilis the medium reorganization energy, ErdOx is the redox potential of the acceptor/donor redox couple in solution, veff is the effective frequency of the nuclear coordinate or reorganization, and K ( I ) is the transmission coefficient which is related to the square of the electronic coupling in the weak (nonadiabatic) coupling limit. This is essentially a Fermi golden rule expression where the transition probability is weighted by the FranckCondon factors contained in the Boltzmann statistics of attaining resonance. The nuclear activation barrier to achieving electronic resonance usually dominates the reaction kinetics. This barrier is estimated in eq 1 assuming the energy scales quadratically with displacement along the nuclear coordinates. This aspect of analytical treatments of electron transfer rates may be the most significant understanding gained from the solution phase studies of homogeneous electron transfer. It is not evident a priori that the simplest approximation (i.e., linear response) should capture the essential details of the nuclear coordinate. However, the dependence of electron transfer rate constants on the energetics has been found to scale quadratically for homogeneous electron transferL6,l7and more recently at surfaces by Miller and GratzelIs and Chidsey19 using well-defined, chemically modified electrodes. This observation reflects the large number of nuclear degrees of freedom in the repolarization volume that are part of the reaction coordinate. All the displacements are small and correspond to linear regions of the intermolecular potentials. The nuclear barrier and relaxation can then be approximated quite well by assuming a dielectric continuum to calculate il and v-as generally done for eq 1. B. Electron Emission Case: Strong Coupling Limit and the Hot Carrier Model. To achieve the maximum rate of
Feature Article charge transfer from the semiconductor to a surface acceptor, it is desirable to have as small a nuclear activation barrier as possible; this can be achieved by proper choice of redox potentials. The real issue then is the degree of electronic coupling occumng between molecular states and extended band states, Le., the degree of adiabaticity. If the electronic coupling is large enough, the K term also becomes unity, and the electron transfer occurs adiabatically for a range of K state energy levels within the crystal band structure; i.e., the electronic coordinate equilibrates faster than the nuclear coordinate. In this event, the electron transfer process would occur on the same time scale as the fluctuationsin nuclear coordinatesthat lead to stabilization of charge on the molecular acceptor (v,ff1). This time scale is approximately equal (but not identical) to the longitudinal relaxation time of the solution phase20,21adjacent to the surface or the intramolecular vibrational relaxation time22-24for reactions with appreciable intramolecularreorganization. This point will be discussed further below. The time scale of this effective bath mode is on the order of 10-12-10-13 s for many molecular systems and will serve as the lower limit for the electron transfer time and the yardstick for measuring the adiabatic limit. The electronic coupling between the molecular potential and the periodic lattice potential is the fundamental issue. At surfaces, the coupling is complicated by any intervening solvent layer or contamination that would act as an insulating barrier to the electronic overlap. The electronic coupling has been estimated by using a triangular potential for the surface space charge region and a square well model for the molecular potential. This model illustrated that, even with an intervening barrier the thickness of the solvation shell, the electron tunneling time is on the 100 fs time scale, demonstrating appreciable electronic coupling. This one-dimensionalmodel was originally used by Boudreaux et al.25to demonstrate the feasibility of hot carrier transitions at surfaces: for 100 fs electron transfer times, the transfer step would be competitive or faster than the carrier thermalization dynamics. The estimated time scale is compatible with solvent relaxation processes needed to stabilize the charge separation and represents a viable charge transfer channel. This one-dimensional analysis ignores the mismatch in the electronic density of states between the molecular acceptor and the semiconductor which would act to statistically reduce the transmission probability to the molecular acceptor. Thus, the estimates of the electron transfer dynamics based on this model should be considered upper limits. However, the estimates are consistent within the adiabatic limits for charge transfer. This work was conceptually very important in that it pointed out the possibility of using hot electron channels for energy storage. Efficient charge separation through this reaction channel avoids energy loss due to carrier thermalization26and could lead to solar energy conversion efficiencies as high as 66%, which is twice that of thermalized carriers and is the theoretical limit of solid state devices. The hot carrier model is described within the framework of the Marcus-Gerischer model for weakly adiabatic conditions. From a different standpoint, Schmickler has recently reworked the problem for variable electronic coupling at an electrode surface in which the associated Hamiltonian is treated to higher order and explicitly includes the bath phonons.27 Using typical electronic couplings observed for homogeneous electron transfer, solutions were found in which the electronic coupling is so enhanced at the surface that it falls in the strong coupling limit. In comparison to homogeneous electron transfer, a molecule at a surface has many more electronic levels acting as electron sources. In this case, the process is activationless, and the energy distribution, rather than showing the above
J. Phys. Chem., Vol. 98, No. 43, 1994 11023 Gaussian distribution (eq 1), is strongly perturbed by the electronic coupling. From Schmickler’s analysis and the one-dimensional calculations of the electronic coupling, there are well-founded reasons to believe that electron transfer at surfaces can occur in the strong coupling limit on very fast time scales. Since the early prediction of the possibility of unthermalized electron transfer, there have been a number of experiments which support the concept of extremely fast electron transfer at surface^.*^-^^ Without measuring the dynamics directly, however, these steady state experiments alone are always subject to speculation over other kinetic mechanisms that would give the same experimental result^.^ It is clearly desirable to observe the carrier dynamics directly, which is the major focus of this review. Furthermore, measurements of the various photophysical processes, as shown in Figure 1, will enable a kinetic rationalization of the photochemistry observed at semiconductor interfaces. C. Electron Injection Case: Incorporating the Electronic Continuum. For charge injection from a localized molecular state into a semiconductor band, the process is the complementary problem to the charge transfer discussed above. There is, however, a distinction in that the huge phase space of the acceptor semiconductor states creates the possibility of localizing the charge in the acceptor half-space without any need to invoke nuclear relaxation. The effect of the electronic continuum on the reaction coordinate and the role of surface fields will become apparent in this discussion. Traditionally, electron transfer theory follows the MarcusLevich described above which postulates that the electronic levels of the donor and acceptor can be modeled as single discrete levels in a potential well. The nuclear potentials of the reactants and products are assumed to be parabolic with a splitting due to coupling between sites. Relaxation within this nuclear continuum is necessary to stabilize the charge transfer, and the rate of this relaxation provides the upper limit for the electron transfer rate. The semiclassical expression derived for the transfer rate constant is very similar to eq 1. For electron transfer reactions between molecular species which have a simple electronic structure, this treatment has been quite successful. In the case of semiconductor interfaces, the charge donor/acceptor at the surface is still a molecular species; however, the semiconductorhas a quasi-continuum of electronic states with considerable delocalizationof the wave function over many electronic states. A schematic representation of the potential energy surface and electronic degeneracy at the adiabatic crossing point of the reaction coordinate is shown in Figure 2. The semantics of the discussion are critical. A charge placed in the conduction band coherently samples all other states coupled to that initial state. This places the carrier in a mixed state of the coupled electronic states on an incredibly short time scale. This delocalization is analogous to electronic dephasing in the band and includes no actual population renormalization; the carrier is delocalized due solely to the volume of phase space the mixed state occupies. Subsequent relaxation processes remove the carrier population from any coherence with the initial site. We define the complete charge transfer event to include the population relaxation dynamics, thus removing the ambiguity between a carrier propagating in a mixed state and a carrier actually changing states. For a carrier transferring across the interface, the delocalization in the band will be much faster than the interfacial transfer time. The sheer volume of phase space available in the band significantly lengthens the recurrence time on the
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Figure 2. A Marcus-type potential energy diagram for charge transfer processes at semiconductor interfaces showing the interaction between the reaction coordinate and the manifold of k states in the semiconductor. S refers to the semiconductor surface, and M is the molecular state. The axis labeled the reaction coordinate(R") represents a cross section of a multidimensional nuclear potential energy surface. This diagram schematically illustrates the effect of the large manifold of k states on the transition probability at the adiabatic crossing point. For the electron injection case (Figure IB), M is the reactant surface and S is the product surface. In this case. the large number of electmnic levels coupled to the product surface assists the development of a charge-separated state. For the electron emission case (Figure IA) discussed in sections IIA and IIB, the surfaces switch roles with S now the reactant surface and M that of the product. For this case, the localization of the electron on the discrete molecular state (out of resonance with the band) looks like a pinhole on the surface. The space charge field helps to confine the reaction phase space. increasing the probability that the pinhole will be sampled.
donor. In the absence of any relaxation, the wave function propagation remains coherent; hence, some kind of relaxation process (electronic scattering or nuclear relaxation in the solid phase) is required to localize the carrier in the substrate half space. Since the lengthened recurrence time tends to keep the carrier in the hand states and since the relaxation processes for the carrier in these states can he significantly faster than the nuclear relaxation of the molecular species, this molecular relaxation is no longer the rate-limiting step for the overall charge transfer event. The earliest theoretical work on charge injection at interfaces centered around a diffusional based on the Onsager model.39." In these models the carrier motion is taken to be diffusive in the hand, once separation from the parent ion has occurred. and is influenced by the applied field and the induced image potential at the surface. These models are essentially classical in nature and attempt to determine the current generated in materials after charge injection and do not attempt to answer any questions about the actual carrier dynamics. In our model, the electron resonantly tunnels across the interface to a single electronic state of the semiconductor which is mixed with the entire quasi-continuum of conduction band states. This transfer rate depends only on the electronic density of states and the electronic coupling. The electron now exists in a mixed state of coupled hand states. The very large number of these states prevents any recurrence on the donor site before scattering and relaxation in the solid phase (electronic dephasing and carrier thermalization) processes break the coherence. These conditions are sufficient for the propagation of the
Lanzafame et al. electron into the semiconductor and separation from the initial molecular site. In a normal Marcus approach, the relaxation of the nuclear degrees of freedom. both intramolecular and intermolecular, is the rate-limiting step in an electron transfer event-on the order of 100 fs for most mokcular The delocalizatiofdrelaxationmodel described above allows electron transfer to occur on the same time scale as the electron dephasing in the semiconductor (as fast as 10 fs), and it is not necessary to invoke the bath mode (ye*). To make these concepts as transparent as possible, a simulation of the electron wave function propagation was conducted. The model used to simulate the conduction hand is a rigid cubic lattice of N identical atoms. This system can be treated quantum mechanically within the Dirac interaction picture and the time dependence of the system calculated using the time-dependent Schroedinger equation; the Hamiltonian can he written
The unperturbed Hamiltonian is Ho,.with eigenvalues E. and eigenvector &, and V., represents the coupling of state m to state n. The summation is performed over all of the states in the cluster, and the eigenvectors are taken as a basis set for the entire space. Within the Dirac interaction picture, the time evolution of the state vector is determined by calculation of the eigenvalues and eigenvectors of the coupling matrix. Initially, the cluster is taken to consist of neutral elements only, so there are no time-dependent Coulombic fields to consider. The electron is initially localized at one site. This calculation follows standard p r o c e d u r e ~ P ' and ~ ~ only the salient details will be given here. To model a charge transfer process, the system is treated as a 512-site cubic lattice with a single external site representing a donor/acceptor species. The coupling between lattice sites is taken to decrease exponentially with distance with a l/e point at the nearest neighbors; the edge atoms are fitted with quasi-periodic h o u n d q conditions to soften the edge effects. For the results shown in Figure 3A. the external site is weakly coupled to one site on the face of the cube, and wave function is initially localized on this external donor site. The figure plots the time evolution of the probability of the electron's localization on the external site as the site coupling is varied from approximately 1 to 5% the rmt-meansquare (rms) coupling of the lattice atoms to each other. The rate of loss of the electron from the external site (i.e.. the transfer rate) is found to he approximately quadratic in coupling as one would expect from a Golden Rule formulation. For the larger values of the coupling strength, the recurrence on the initial site is minimal even for a cluster this small and the recurrence time becomes longer than all normal solid state dephasing processes (electron scattering and hand thermalization). For the complementary process of charge transfer from the lattice to an adsorbed acceptor on the surface [charge emission], the same simulation is run hut with the electron wave function initially localized at a lattice site two sites removed from the external acceptor. All other couplings are as described above except that the acceptor is much more strongly coupled to the lattice with a coupling equal to one-third that of the rms coupling of the lattice atoms. The time evolution of the probability of finding the electron on the external acceptor is then calculated. The results are shown in Figure 3B. There exist frequent occurrences of the wave function on the adsorbate. They remain, however, on the order of 5% at a maximum and are transient in nature. Since there is no localization process and
J. Phys. Chem., Vol. 98, No. 43, 1994 11025
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Figure 3. Model calculation of electron propagation acmss an interface. The blackened sites in the representativecubic reaction volume give the initial electron position. The upper figure (A) shows the time evolution of the survival probability of an electron on an external site (the donor) coupled to a 512-atom cubic lattice (8 x 8 x 8). The probability is shown for various values of the donor-lattice coupling. The solid line (top curve) corresponds to a donor-lanice coupling 1/72 that of therms lattice-lattice coupling. The dashed line (middle curve) comsponds to a donor-lattice coupling twice that while the dotted line (lower curve) shows a donor-lattice coupling 4 times that of the top curve. Note the quadratic dependence on coupling. The lower figure (B) illustrates the complementary back-transfer, the probability for being on an external acceptor as a function of time for an electron initially localized in the 512-atom lanice coupled with one-third the rms coupling to the acceptor. Note the !kquent but transient appearances of the electron on this acceptor in the absence of any nuclear relaxation Io localize the electron here, the phase space volume of the lattice is over 500 times that of the adsorbate, the lattice is far too inviting to the wave function for it to stay on the adsorbate for long. This clearly illustrates the need for a Marcus-type deactivation for this electron emission process as discussed in the previous subsection. Some kind of nuclear relaxation in the acceptor half-space is required to localize the carrier on the adsorbed acceptor. The rate of this nuclear relaxation process becomes the fundamental limit of the charge emission rate.
III. Experimental Approaches to a Real Time View of Electron Transfer In terms of dynamics, the effect of the activation barrier is well described by Boltmann statistics. Thus, from an experimental point of view, one would like to determine the spatial propagation of an electron across an interface under zero barrier conditions, Le., at the adiabatic crossing point of the reaction surface. At this critical point, the electron transfer time is determined by the barrier crossing dynamics and the degree of electronic coupling. This condition defines the upper limits to the electron transfer rate for a given surface. Optical preparation of nonequilibrium electrons that are resonant with both band and molecular states access this condition. For metals, the
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