Redox Processes at Semiconductors-Gerischer Model and Beyond

Introduction

In this article we describe ideas and experimental results that are fundamental to electron transfer between molecules (redox ions) and the surface of semiconductor (SC) electrodes. We do not make any attempt here of covering the extensive literature on electrochemistry at semiconductor electrodes. Rather, experimental data are shown to illustrate relevant results. We consider only the transfer of one electron between a molecular monomer (redox ion) and the electrode. We do not consider electron transfer from dimers and higher aggregates and also not the more complicated processes like corrosion, etching, and tunneling through barriers. In the case of ultrafast injection from an excited dye molecule, we show results where the system is exposed to ultrahigh vacuum since the solvent environment would obscure the most interesting results obtained from time-resolved measurements of electron transfer. The effects arising from the addition of a solvent environment are mentioned.

Electron Transfer Between a SC Electrode and Redox Ions (Molecules) in Solution

The free energy of charge carriers in the semiconductor electrode (SC) is characterized by the Fermi energy and that of the charge exchanging molecules (redox ions) in solution by their redox potential. When the two subsystems are brought into contact with their free energy levels not too far apart, the two subsystems will exchange charges until a common electrochemical potential is established throughout the whole system [1] (Fig. 1).

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 1

Scheme illustrating equilibrium between a semiconductor electrode and a redox electrolyte. The density of states D(E) in the valence band and conduction band on the left (labeled occupied and unoccupied, respectively) and one Gaussians each for the density of states D(E) of the reduced redox ions and the oxidized redox ions on the right (labeled occupied and unoccupied, respectively). The free energy is the same throughout the system, with the Fermi energy E _F,SC at the same energy as the redox potential E _{F, redox} (Reproduced from Fig. 23, Ref. [1])

The corresponding equilibrium situation is established via setting up a space-charge region with a corresponding band bending in the near-surface region of the semiconductor. Figure 2 illustrates the band bending for the lower edge of the conduction band and the upper edge of the valence band versus distance. Note that the applied potential (η) drops over the space-charge layer in the semiconductor and only a negligible fraction of the voltage drop occurs at the electrode surface. Thus, the rate constant of electron transfer remains virtually unchanged at the surface of the semiconductor when the applied voltage is changed. This is very different from a metal electrode. The arrow in Fig. 2 indicates the reduction in the electron concentration at the surface due to applied voltage because the latter enhances the barrier height for electrons moving from the bulk of the semiconductor to the surface.

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 2

Band bending due to the space-charge layer near the surface of the semiconductor. The ordinate is energy and the abscissa distance. The Fermi level (E _F ) is downshifted against the redox potential (E _F,el ) by the applied potential η (multiplied by the elementary charge e). The SC is n-doped (Fermi level lies close to the conduction band edge in the bulk). The applied potential enhances the barrier for electrons moving from the bulk of the semiconductor to the surface and thus reduces the electron concentration at the surface (2) (Reproduced from Fig. 2.4, Ref. [2])

Gerischer [1, 3] postulates a density of states function in the form of a Gaussian distribution for the reduced species in solution and a corresponding Gaussian distribution shifted toward the vacuum level for the oxidized species. The two distributions are labeled occupied and unoccupied in Fig. 1. Solvent configurations with the highest probability give rise to the two peaks, and the Gaussian distribution arises from different solvent configurations formed around the reduced and the oxidized redox ions, respectively. The two peaks and the corresponding distributions for reduced and oxidized redox ions are shifted against each other on the energy axis due to the fact that the system has a slow and a fast polarization response to a sudden change in the charge of the redox ion. The polarization response of the electronic subsystem to a change in the charge is fast, instantaneous for the time scales considered here, and the polarization response from the change in the spatial coordinates of the solvent molecules is slow to a sudden change in the charge on the redox ion. Removing an electron from the reduced redox ion has to overcome the attraction of a more positively charged environment than is present when the electron is returned to the oxidized redox ion after the slow polarization response is already completed and the system is completely relaxed. Completing the slow polarization response after removal of the electron means that the effective positive charge in the environment has decreased around the now oxidized redox ion that carries less negative charge than the reduced redox ion. Thus, less energy is gained from the return of the electron to the oxidized redox ion after the environment has relaxed in response to the lowered negative charge than had to be spent in separating the electron from the reduced redox ion where the environment forms a higher positive charge around the redox ion with a higher negative charge, i.e., the reduced redox ion. Marcus [4] has presented a quantitative measure for the change in polarization energy for a system where the redox ions are conducting spheres carrying a different charge and the environment is a dielectric medium with two different dielectric constants, one for the fast response and the other for the slow response. Other authors (e.g., [5]) have added an additional slow polarization energy that arises from a change in the equilibrium coordinates of the atoms making up the molecule (redox ion) when the charge is changed on the molecule. The polarization energy has thus an outer (solvent) and an inner (atoms of the molecule) contribution. Classical Marcus theory of the rate constant of electron transfer of a redox ion in a polarizable medium [4] predicts also a Gaussian energy dependence of the rate constant. As long as the rate constant of electron transfer is described with the tools of only classical physics assuming a fast and a slow polarization response, the predictions of classical Marcus theory for the rate constant of electron transfer are identical with those derived from the assumption of a density of states for the redox ions in the form of two Gaussian energy distributions, one for the reduced and the other for the oxidized species as illustrated in Fig. 1. The peaks of the two distributions are separated on the energy axis in Fig 1 by twice the amount of the above polarization energy. Classical Marcus theory of electron transfer and the Gerischer scenario (Fig. 1) both fail when quantum effects become important. The latter arises in the dynamics of electron transfer when strong coupling between the electronic states and high-energy vibrational modes in the molecules (redox ions) are incorporated into the model [5, 6]. This strong coupling is a common feature of different classes of molecules and redox ions [7–9]. The additional qualitative effect arising from high-energy (quantum) vibrational modes in the molecule (redox ion) is faster electron transfer compared to the classical model when electron transfer occurs to a low-lying acceptor, i.e., electron transfer becomes faster in the downhill high-energy wing of the energy distribution than is predicted by the classical calculation of the rate constant, means the shape deviates here from a Gaussian. Such quantum effects are automatic ingredients if quantum theory is employed for calculating rates of electron transfer [5, 6]. For a long period of time, quantum theory was used in the form of perturbation theory, where the maximum permissible strength of electronic interaction had an upper limit and thus the electron transfer time had a lower limit. Recently, fully quantum mechanical calculations of electron transfer have been presented without the restrictions of perturbation theory that can address ultrafast electron transfer at semiconductor electrodes, as will be described below. Experimental results will be shown below where the peak of the distribution curves shown in Fig. 1 corresponds to electron transfer in the range of a few femtoseconds. Conventional experimental measurements in electrochemistry can only access the time window of nanoseconds, at the most picoseconds. The corresponding rate constants correspond to the tails of the distribution curves, many order of magnitude smaller than at the peak. Rate constants in the wings can be visualized if a logarithmic plot of the rate constant versus energy is used instead of the linear plot shown in Fig. 1. In the wings the rate constants decrease about exponentially with increasing energy difference. Therefore, the energy range with sufficient overlap between electronic donor and acceptor states making a significant contribution to charge exchange across the interface is usually comprising only an energy interval twice the mean thermal energy (2k_BT with T = temperature and k_B = Boltzmann’s constant) above the respective band edge of the semiconductor.

A systematic variation in the type of redox ion and measurements of the corresponding exchange currents have been performed at metal electrodes [10] and at insulator electrodes, where charge can be injected by redox ions only into the valence band [11]. Electron transfer reactions at semiconductor/liquid interfaces were studied by the Marcus group with a Fermi golden rule approach [12, 13] and agreed reasonable well with experimental results by the Lewis group [14, 15]. Such data displays considerable uncertainty in the value of the reorganization energy for a specific redox ion even if the measurements were carried out in an identical ionic environment. Shifts in the redox energy at the electrode surface compared to the value measured against a reference electrode can also occur. Using such compiled data of the reorganization energies of redox ions, one can arrive at rough qualitative prediction concerning the value of the rate constant with a logarithmic plot instead of Fig. 1. Quantum effects arising in the downhill energy wing can make the prediction even less reliable. The uncertainty margin for the thus estimated rate constant should be expected in the range of a factor of 10–100. Depending on the type of measurement, the interfacial rate constant can be obtained with different dimensions, i.e., s⁻¹, cms⁻¹, cm³s⁻¹, and cm⁴s⁻¹. Making plausible assumptions about the reaction distance and reaction volume such values can be converted with an uncertainty margin.

Photocurrent Transient Due to Light Absorption in the Bulk of the SC and Interfacial Electron Transfer

The photocurrent due to the photo-generation of minority carriers is controlled by the discharge of the minority carriers from the surface of the electrode into the electrolyte and by competing recombination reactions of the minority charge carriers with the majority charge carriers. Of course, there are also competing side reactions of the minority carriers at the crystal surface, e.g., those leading to the corrosion of the electrode surface. Several attempts can be found in the literature of obtaining the rate constant of interfacial electron transfer in the SC/redox electrolyte system from the time-resolved photocurrent response to optical bulk excitation of the SC electrode. This is not possible, however, since the photocurrent transient contains the influence of electron transfer to the redox ions in the electrolyte only indirectly as a reaction channel competing with recombination between minority and majority carriers near the surface of the electrode. Figure 3 illustrates the different processes in the energy versus distance diagram along with an equivalent circuit augmented by two current sources where the relevant physical processes can be introduced in the form of appropriate equations describing transport and reactions of the charge carriers [16]. Superimposed on the actual dynamics is the response of the system to a change in voltage arising from the passive elements like capacitors and resistors in the circuit as illustrated on top of Fig. 3.

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 3

Light is impinging from the left and is absorbed in the SC electrode generating electron–hole pairs with function g(t). Holes are the minority carriers in the n-Si electrode. Screened minority carriers generated in the bulk (right-hand side) diffuse toward the depletion layer where they are separated from the screening charge (rate constant kq) and move from there to the electrode surface driven by the electric field of the depletion layer. The time-dependent concentration of holes at the electrode surface is p_S(t). Bimolecular recombination of holes and electrons occurs with rate constant k_r. A reduced redox ion transfers an electron to the hole at the surface with rate constant k_h; an oxidized redox ion accepts an electron from the surface of the SC electrode with rate constant k_e. The simplest equivalent circuit with two current sources for the equations describing reactions and transport of charge carriers is shown on top of the diagram. The electric response is measured as time-dependent voltage across the external resistor R_M (Reproduced from Fig. 1, Ref. [16])

The electrical photo response to excitation with a weak laser pulse of 10 ps duration absorbed in the bulk of Si is measured as time-dependent voltage drop across the external resistor R_M. If the time elapsed after the laser pulse is short compared to the RC constant of the circuit, the measured signal can be interpreted as photovoltage. It can be interpreted as photocurrent if the elapsed time is long compared to the RC constant. Figure 4 shows the response of an n-Si electrode in the ns time window to the absorption of a laser pulse of 10 ps duration. The black shaded area in Fig. 3 illustrates the generation of electron–hole pairs by the incident light inside of the Si material.

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 4

Calculated and measured photovoltage across the external resistor R_M. A weak laser pulse (10⁷ photons/mm²) of 10 ps duration with 590 nm central wavelength impinged on the n-Si electrode. The thin curve is the calculated response including the current sources, whereas the dashed curve is the response without the current sources (so-called RC response). Time resolved is the diffusional flux of screened minority carriers arriving at the edge of the depletion layer convoluted with the RC response. Holes are separated from the screening charge and driven to the surface by the electric field in the depletion layer (Reproduced from Fig. 3, Ref. [16])

The apparent instantaneous initial rise of the signal shown in Fig. 4 arises from the separation of photo-generated electron–hole pairs that are generated inside the depletion layer. The ensuing slower rise is the diffusional flux of screened minority carriers arriving from the bulk at the edge of the depletion layer, convoluted with the RC response of the circuit. At the edge of the depletion layer, the holes are separated from the screening charge. This process is described by the phenomenological rate constant k_q. The signal is convoluted with the RC response, and the latter controls in particular the decay. The smooth solid curve is the total calculated response which includes the current sources describing transport and reactions of the charge carriers. The dashed curves are calculated with the current sources omitted. They represent the so-called RC response arising from the passive elements in the equivalent circuit shown on top of Fig. 3. The parameter in the inset is the band bending. Calculations have shown that the measured photovoltage is due to the displacement current flowing through the capacitor C_S formed by the depletion layer. The measured photovoltage is not sensitive to the faradaic current at the interface because the capacitor C_H is much larger than C_S making the contribution from the current source in C_H very small [16, 17]. Thus, the measured signal is not sensitive to interfacial electron transfer, specifically not to the rate constant k_h in Fig. 3 for electron transfer from reduced redox ions in solution to the holes at the surface of the n-Si electrode.

Recombination between the photo-generated holes (minority carriers) and electrons (majority carriers in the n-Si electrode) near the surface gives rise to a time-dependent dip in the photocurrent measured in response to a rectangular illumination of 1 ms duration (Fig. 5). The shape of the photocurrent signal in dependence on the bias voltage can be simulated by introducing a corresponding rate for bimolecular recombination between electrons and holes into the current source for the depletion layer (upper part of Fig. 3). Negative bias voltage lowers the band bending depicted in Fig. 3 and increases the dip, i.e., decreases the stationary photocurrent. The flat-band situation is reached at a bias of – 0.95 V as indicated by the disappearance of the initial photocurrent peak, labeled peak (filled squares in Fig. 5). Both the stationary photocurrent, labeled final (filled circles in Fig. 5), and the corresponding stationary recombination loss are linked to the stationary accumulation of holes at the interface. The stationary photocurrent disappears already at finite band bending of about 0.3 V with respect to the flat-band potential at – 0.95 V (Fig. 5).

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 5

Peak value (filled square) and stationary plateau value (filled circle) of the photocurrent transient in response to a rectangular illumination pulse of 1 ms duration in dependence on the voltage bias. The inbuilt depletion layer voltage is decreased with the bias shifting in negative direction. The flat-band potential is at – 0.95 V. The reduced redox ions that can discharge the holes at the surface are 0.005 M 1,1′-dimethylferrocene (Reproduced from upper part of Fig. 4, Ref. [16])

The concentration of electrons in the depletion layer increases exponentially with the bias voltage going into negative direction. This leads to an increased recombination and a deeper dip in the photocurrent transient. Both the rate of electron transfer from the reduced redox ions to the holes at the electrode surface and the concentration of reduced redox ions must have an influence on the measured photocurrent of the holes since discharge into the electrolyte reduces the concentration of holes at the surface of the electrode and competes with recombination. A high concentration of the reduced redox ions combined with a reasonably high value of the rate constant k_h will keep the concentration of photo-generated holes fairly small at the surface of the n-Si electrode and thus reduce the recombination loss. On the other hand, the concentration of holes will increase at the surface if the concentration of the reduced redox ions is made very small slowing down discharge into the electrolyte. With a higher concentration of holes, there will be enhanced bimolecular recombination with the electrons. Bimolecular recombination of the holes with the electrons will dominate once a sufficient concentration of holes has accumulated at the surface. With only a slow discharge in the equations for the current source compared to bimolecular recombination, the photocurrent transient is predicted to develop an asymmetric shape with respect to switching on and off the illumination (Fig. 6). Moreover, the recombination dip should appear now as S-shaped time dependence of the initial photocurrent transient. Both features are very different when the photocurrent transient is measured in the presence of a high concentration of reduced redox ions with a fast discharge of the holes (Fig. 5). There is almost perfect agreement of the experimental data (noisy curve in Fig. 6) with the predicted time dependence of the photocurrent transient (thin solid curve in Fig. 6). In summary, the photocurrent transient does not show the rate constant of electron transfer at the interface. From its shape one can obtain qualitative information on whether recombination or discharge is dominating the fate of the photo-generated minority carriers at the surface.

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 6

(Reproduced from upper part of Fig. 5, Ref. [16])

Photocurrent transient (voltage across R_M) in response to a rectangular illumination pulse. The signal is strongly asymmetric with respect to beginning and end of the illumination. The concentration of reduced redox ions is negligible, and recombination of the holes with electrons dominates their fate. The calculated thin curve virtually merges with the measured noisy curve except for some deviation at the right-hand side. The voltage calculated for the Helmholtz layer U_H and for the depletion layer U_S cannot be measured.

Electron Injection into a SC Electrode from Photoexcited Adsorbed Molecules

Several interesting effects can arise in dye-sensitized charge injection, for example, spin-dependent recombination kinetics that has been studied with organic insulator crystals functioning as electrodes [18]. There are several ways of light-induced charge injection, e.g., injection from the locally excited electronic dye molecule into a semiconductor or direct optical electron transfer from the ground state of an adsorbed molecule to states in the empty conduction band of a semiconductor. Experiments have shown that the latter process is less efficient than the first one [19]. The most impressive progress has been made in this field with the recent feasibility of measurements on the femtosecond time scale. The most important tool for studying the latter is a frequency tunable laser generating pulses of a few femtosecond duration. The most recent progress is the direct measurement of the energy distribution of the injected electron in the electronic acceptor states of the semiconductor [20, 21]. The data is collected as femtosecond two-photon photoemission signal [20–22]. The energy distribution of the corresponding excited molecular donor state has been obtained from the stationary absorption spectrum of the adsorbed molecule [22, 23]. Time-dependent interfacial electron transfer in the above system is probed either by femtosecond transient absorption spectroscopy [24–26], mostly applied in the case of a nanostructured electrode, or by femtosecond two-photon photoemission (fs-2PPE) spectroscopy [20, 21] which is more sensitive and able to time-resolve the reaction on a well-prepared planar surface in ultrahigh vacuum. Since the solvent environment of a traditional electrochemical system has been omitted from these systems, the investigations focus on the role of the high-energy (quantum) molecular vibrations in the dynamics. We note here that adding a solvent environment would cause a downward energy shift of the electronic levels in the adsorbed molecule, introduce additional inhomogeneous broadening of the electronic levels, increase the reorganization energy of the reaction, and would make transient absorption measurement extremely difficult in the most relevant time window shorter than 100 fs. The solvent when excited by an ultrashort laser pulse generates the so-called coherent artifact which obscures the actual signal. The ET reaction would still occur on the same time scale as in the absence of the solvent if the downshift of the electronic level due to the solvent environment is compensated and the molecular donor level still positioned high enough above the lower conduction band edge. Vibrational peaks would be broadened or completely obscured due to additional inhomogeneous broadening caused by the solvent environment. To avoid these difficulties and for employing fs-2PPE, the experimental data shown below was collected in ultrahigh vacuum.

The availability of time-resolved [20, 26] and frequency-resolved experimental data [22, 23] for the same systems has motivated several different theory groups to model ultrafast heterogeneous electron transfer with advanced theoretical tools [23, 27–34]. Recent quantum mechanical calculations of the injection dynamics [23, 27–30, 34] are not any more based on a perturbation approach as was customary for earlier model calculations but permit an arbitrarily high value for the electronic interaction energy between the excited state of the molecular donor and the electronic acceptor states of the solid. Thus, the new theoretical calculations can address the experimentally observed ultrashort electron transfer times of a few femtoseconds. The simplest assumption about the electronic coupling is a constant matrix element across the whole conduction band of the semiconductor. By choosing perylene as the molecular donor with its excited donor level located high above the lower conduction band edge of the wide-gap semiconductor TiO₂ [20], the most general case of a heterogeneous electron transfer reaction can be realized. This case is referred to as the wide-band limit [35], where the complete electron transfer spectrum is mapped as energy distribution of the injected electron onto the continuum of empty electronic states of the electrode [20]. Another advantage of using perylene as the donor is vibrational structure in the optical spectra being dominated by just one high-energy vibrational mode. Therefore, low-resolution spectra can be fitted by considering only this 0.17 eV skeletal stretch mode [23]. The injection dynamics predicted by the fully quantum mechanical calculations for the perylene/TiO₂ system is illustrated in Fig. 7.

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 7

Scheme illustrating laser pulse induced ultrafast nonadiabatic heterogeneous electron transfer according to fully quantum mechanical model calculations [27, 28, 30, 31, 34]. The energy distribution of the excited molecular donor state (upper red curve) is much narrower than the energy distribution of the injected electron. The latter spreads from the donor state to lower energies in the electronic acceptor states and shows vibrational structure. For details see text (Reproduced from Fig. 1 Ref. [36])

The dynamics [27, 28, 37] is characterized by two different energy distributions, i.e., that of the donor state and that of the injected electron, as illustrated in Fig. 7. The corresponding electron injection from an excited donor state high above the lower edge of the conduction band occurs in the wide-band limit [35] and gives rise to an exponential decay in the population of the donor state due to electron transfer [37]. This general case is realized by the perylene dye/TiO₂ systems [20, 26]. Strong coupling of the electronic states to high-energy vibrational modes [7–9] is a characteristic feature of all the molecules that have been used as visible light sensitizers of wide bandgap semiconductors. This strong coupling leads to the wide energy spread and vibrational structure in the energy distribution of the injected electron. Just one high-energy vibrational mode is considered in Fig. 7 since this is sufficient for describing low-resolution spectra of perylene dye/TiO₂ systems [23, 38] used in the experiments. The energetic position of the donor state with respect to the conduction band of TiO₂ is known from UPS data combined with optical absorption spectra and independently from two-photon photoemission experiments [20]. Excitation by an ultrashort laser pulse creates the excited donor state at the respective photon energy above the ground state [20, 28]. When generated by an ultrashort laser pulse, the energy distribution of the donor state consists firstly of the width of this laser pulse, secondly of a Lorentzian which is controlled by the ET time of the system [29, 35], and thirdly by any inhomogeneous distribution of the states involved in the electronic transition [20]. The excited donor state shifts with higher photon energy to a higher vibrationally excited state (not shown in Fig. 1). Redistribution of vibrational excitation energy to other modes is much slower than ultrafast electron injection from the initially excited vibrational state in this system [39]. The dynamics in this system is very different from earlier injection scenarios where it was assumed that redistribution of vibrational excitation energy occurs either faster than electron transfer or at least on a comparable time scale. Figure 7 shows the energy distribution of the injected electron to spread from the excited donor state to acceptor states at much lower energies. The energy lost by the injected electron is used for exciting a high-energy (quantum) vibrational mode in the ionized molecule. Specific advantages of using the perylene/TiO₂ system for testing the theoretical predictions illustrated in Fig. 7 are firstly the high-lying excited singlet state which functions as the donor state and realizes the wide-band limit [35], secondly slow redistribution of vibrational excitation energy on the picosecond time scale [39] and much slower singlet–triplet conversion, and thirdly the fact that low-resolution spectra [23, 37] are dominated by just one high-energy vibrational mode, i.e., the 0.17 eV skeletal stretch mode.

The electron transfer spectrum is obtained as a cross section along the energy axis [20, 22] through the complete set of the fs-2PPE data [23, 39], in this case for the perylene/TiO₂ system with the –COOH anchor–bridge group (Fig. 8a). The injected electron spreads over a wide energy range reaching electronic acceptor levels more than 0.5 eV below the donor state. Convoluting a spectrum structured by the dominant 0.17 eV stretch mode of perylene with a Gaussian of 80 meV width (FWHM), which accounts for the instrumental response of the used time-of-flight (TOF) detector [20], results in the fit to the data points. Fully quantum mechanical model calculations have shown that the femtosecond two-photon photoemission (fs-2PPE) signal preserves both energy spread and vibrational structure of the energy distribution of the injected electron [39]. The energetic width of the donor state has been extracted from the stationary absorption spectrum of the adsorbed molecules [22]. The scheme in Fig. 7 considers also additional broadening due to the short laser pulse. The energy distribution of the injected electron covers a much wider range than the excited donor state. The injected electron retains its energy distribution unchanged for a sufficiently long time in this system, here 200 fs [20] allowing for the measurement of the initial distribution. The electron is injected into surface states, probably created by the anchor groups where energy relaxation is much slower than the 40 fs time scale measured for energy relaxation in bulk states of TiO₂ [40]. By inserting different anchor–bridge groups between the perylene chromophore and the surface atoms of TiO₂ the injection time has been varied from 10 fs to one picosecond [20]. With perylene attached via the anchor–bridge group –CH₂–SH to an Ag electrode electron transfer from the excited state of the molecule occurs on the same time scale of a few fs [21] as for perylene attached via the –COOH group to the TiO₂ electrode. At the metal electrode it would be extremely difficult, however, to measure the initial energy distribution of the injected electron. The latter is immediately distorted by inelastic scattering processes occurring with the high density of thermal electrons in the metal electrode. From Figs. 7 and 8a, it is clear that strong coupling of the electronic states to high-energy vibrational modes controls the dynamics of electron transfer from a typical molecular donor like perylene to the semiconductor TiO₂. Figure 8b explains the choice of the delay time td = 40 fs at which the cross section along the energy axis has been taken to obtain the data points of Fig. 8a. Figure 8b shows a cross section along the time axis through the complete set of fs-2PPE data [20]. The transient contains an early peak arising from photoemission from the excited singlet state of perylene and a second peak due to photoemission from the injected electrons at the surface of TiO₂. The latter population is generated from the first population. Thus, the first peak decays with the same time constant that is controlling the rise of the second peak; it is the electron transfer time, i.e., 9 fs for the –COOH anchor–bridge group [20, 26]. The delay time td = 40 fs is chosen such that the peak due to photoemission from the injected electron is probed after the peak due to photoemission from the excited singlet state has decayed. The ultrafast injection time of 9 fs and the excitation of the high-energy vibrational mode in the ionized molecule demonstrate ultrafast nonadiabatic electron transfer as the dominant ET mechanism. Recent experimental results (Fig. 8) and recent quantum mechanical model calculations are in agreement, but both differ strongly from early quantum mechanical perturbation theory models where it was automatically assumed that nonadiabatic electron transfer is slow. The fully quantum mechanical model of the injection dynamics illustrated in Fig. 7 has the virtue of incorporating strong coupling of the electronic states to high-energy vibrational modes and allowing at the same time for ultrafast ET [27, 28, 30, 31, 34]. Theoretical model calculations of the above type need the input of parameter values which are taken either from experimental data or from specific theoretical calculations.

Redox Processes at Semiconductors-Gerischer Model and Beyond, Fig. 8

(a) Experimental data points are obtained from a cross section along the energy axis of the complete set of femtosecond two-photon photoemission data [20]. The data points are collected at a delay time of 40 fs determined from the data shown in (b). The energy distribution of the data points can be fitted using a structured spectrum predicted by a theoretical model for the perylene/TiO₂ system [39] and by convoluting with the response function of the used time-of-flight detector. (b) Experimental data points are obtained from a cross section along the time axis of the complete set of femtosecond two-photon photoemission data [20]. The transient contains two contributions; photoemission from the excited molecular donor state is followed by photoemission from the electronic acceptor states filled via electron transfer from the donor state. The time constant for the decay of the first peak is identical with that for the rise of the second peak, i.e., it is the electron transfer time, here 9 fs (Reproduced from Fig. 2 Ref. [36])

It is important to note here that the ultrafast injection dynamics are very different from Gerischer’s much earlier intuitive injection scenario [41, 42]. The latter assumes strong thermal broadening of the excited donor state which is not borne out by the experimental data [22, 26]. Moreover, a thermally relaxed donor state is assumed in Gerischer’s scenario which would imply for the perylene/TiO₂ system that electron transfer must be slowed down by a factor of 1,000 compared to the measured electron transfer times [20, 26] because intermolecular relaxation of vibrational energy has been measured for perylene in the time range of several ps [39]. A correspondingly slow electron injection in the picosecond time domain is realized with an excited donor state located below the conduction band edge rendering electron transfer thermally activated. An experimental example for this case has been found with electron injection from a dye J-aggregate into AgBr [43]. The latter represents a rather special situation, and the general case shown in Fig. 7 is not described by the Gerischer’s scenario for dye-sensitized injection.

The recent fully quantum mechanical model of dye-sensitized electron injection has addressed also the specific border case where the excited donor state lies above but fairly close to the lower edge of the conduction band. For the latter case a slowdown in the electron transfer time is predicted compared to the case of the wide-band limit. Moreover, if the excited donor state is generated with a sufficiently short laser pulse, the population is predicted to show time-dependent oscillations in this border case. Time-dependent oscillations are also predicted for the populations of the injected electron and of vibrational states in the ionized molecule [37, 44]. The electronic level of a donor molecule can be shifted through a chemical modification of the chromophore; however, the chemical change might bring about additional changes in the system. A better defined energy shift of the donor level can be realized with a semiconductor quantum dot where the electronic levels shift with the size of the particle [45]. Recently, a low-lying excited donor state has been realized in the form of an exciton in a PbSe nanocrystal attached to the surface of TiO₂, and strong oscillations have been reported for a second harmonic signal probing this interface [46]. The latter signal could indeed indicate that the theoretically predicted electron transfer dynamics for the border case of a low-lying donor state [37, 44] has been realized in the system, but this has to be checked in more detail.