First-principles calculations for the temporal characteristics of hole-phonon relaxation in the valence band of titanium dioxide and zinc oxide have been performed. A first-principles method for the calculations of the quasistationary distribution function of holes has been developed. The results show that the quasistationary distribution of the holes in TiO2 extends to an energy level approximately 1 eV below the top of the valence band. This conclusion in turn helps to elucidate the origin of the spectral dependence of the photocatalytic activity of TiO2. Analysis of the analogous data for ZnO shows that in this material spectral dependence of photocatalytic activity in the oxidative reactions is unlikely.
The relaxation of excited holes in the photocatalytic oxides is manifested in the transfer of holes to the highest valence band states and in trapping them on defects or surface states, the processes that profoundly affect the properties of the oxides [
In photocatalytic oxides, the thermalization of excited electrons via electron-phonon interaction occurs very rapidly, within several hundreds of femtoseconds; see references in [
The dependence of photocatalytic activity on the energy of excitation has been observed for the first time by Grela et al. [
Emeline et al. [
Although the authors of the cited papers emphasized the important role of the hole-phonon relaxation, they did not evaluate the temporal characteristics of these processes. To our knowledge, only Morishita et al. [
In the remaining papers cited at the onset [
Therefore, we have carried out a first-principles study for the processes of relaxation of the nonequilibrated holes in TiO2 and ZnO. In what follows we outline the physical foundations of the method and the details of the calculations and discuss the results and conclusions important for a better understanding of the photocatalytic properties of these oxides.
In Figure
The scheme of the processes that occur after the excitation of the electron-hole pairs near the surface of a photocatalyzer. The notations have the following sense: VB is the band of valence states, CB is the band of conduction states, and OMO and UMO are the occupied and unoccupied molecular states, respectively.
Our evaluation of the hole energy loss time is based on the “golden Fermi rule” of the perturbation theory. Consistent with this rule, the probability, per unit of time, of an elementary transition of a single electron from the state with the wave-vector
Having calculated the rate of hole-phonon relaxation, we can evaluate the relaxation time of the hole defined as the time from the moment of its emergence to the moment of transition to a higher level, owing to the filling of the hole-containing state with an electron that emits phonon:
Initially, we consider the temporal evolution of the mean population
Looking for an approximation for averaged transition probability
The IDF-function can be obtained from the electronic band structure calculations. In order to calculate this, one should summarize the probabilities of all possible direct excitations from the states at the
In order to calculate the probabilities of excitations, we apply the atomic sphere approximation [
The numerical evaluations for the electron-phonon coupling have been done using the density-functional perturbation theory [
The electron excitation probabilities
In Figures
The mean characteristics of the hole-phonon (empty rhombus) and electron-phonon (black rhombus) relaxation for anatase as functions of the excess energy
The characteristics of the hole-phonon and electron-phonon relaxation for rutile depending on the excess energy
The characteristics of the hole-phonon and electron-phonon relaxation for ZnO depending on the excess energy
In Figure
The time of energy loss of a hole (solid lines) depending on the final value of the excess energy
Results of our calculations for the distribution functions of holes are shown in Figure
The calculated distribution function of holes in anatase (a), rutile (b), and zinc oxide (c). In solid lines the data are shown for the photon energy of 3.5 eV, in dashed lines the data are shown for 4.0 eV, and in point lines the data are shown for 4.5 eV. It is assumed that
It has been shown [
Because the specimens studied in experiments were mixtures of the anatase and rutile phases, the question did not arise concerning which phase was responsible for the spectral dependence of the quantum yield. We can now turn our attention to the difference in the distribution functions of holes in the energy interval 0.2–0.7 eV. Here the value of this function is for rutile substantially higher than that for anatase. This difference indicates that rutile is the phase mostly responsible for the spectral dependence. To explain the differences in the distribution functions of anatase and rutile, let us examine the total densities of the states near the top of the valence band; see Figure
The total density of states near the top of the valence band versus the excess energy. The solid line is for anatase, the dashed line is for rutile, and the dotted line is for zinc oxide.
Our data indicate that zinc oxide is least favorable for the manifestation of spectral dependence of photocatalysis. The distribution function of holes in ZnO has a peak only below 0.2 eV, with no tail at a higher excess energy. The origin of the difference in the distribution functions of TiO2 and ZnO is concealed in the details of the IDF-function
Instantaneous distribution function for anatase (a), rutile (b), and zinc oxide (c) depending on the excess energy of hole
In contrast, in the case of ZnO, the holes are produced only in states near the top of the valence band. The IDF dependence has only one peak, which does not appear beyond 0.2 eV. In accordance with (
The results of our calculations make it possible to estimate the timescale of hole transfer from the bulk of the crystals to absorbed molecules. The experimental data [
In order to compare this result with the experimental data of Morishita et al. [
Unfortunately, the experimental data on the hole relaxation rate in ZnO are absent. Also the absence of experimental data on the spectral dependence of the photocatalytic yields makes the evaluation of the hole transfer time in ZnO impossible now.
Note that first-principles calculations for the hole transfer time have not been performed.
We have performed first-principles calculations for characteristics of electron-phonon relaxation of excited holes in the valence band of TiO2 and ZnO. These values are the constant of electron-phonon coupling, the rate of energy loss, the mean energy of the emitted phonon, and the time of energy loss of a hole. In the case of rutile we find that the time of energy loss of a hole is about 30% higher than that of an electron. For anatase the time of energy loss of a hole is higher than that of an electron only when excess energy exceeds 0.4 eV. Since these data do not directly correlate with experimental results on the rate of oxidative reactions, we conclude that the time of energy loss of a hole is not a major factor in the emergence of the spectral dependence of photocatalytic activity.
In order to elucidate the origin of the spectral dependence of oxidative reactions, we calculated the instantaneous and quasistationary distribution functions of the holes. These functions differ significantly from the analogous functions of excited electrons. The electron distribution functions demonstrate the concentration of excited electrons near zero excess energy that is in the states near the bottom of the conduction band. The distribution functions of holes in anatase reveal the presence of holes in the states more than ~0.8 eV below the top of the valence band. The distribution of holes in rutile shows a significant increase at excess energy above 0.4 eV, thus reflecting a phenomenon favorable for photocatalytic activity, as this energy increases the oxidative potential of TiO2. The presence of holes in the states below the top of the valence band is consistent with the available experimental data which indicate an increase in the photocatalytic activity at excess energy above 0.2 eV. We thus conclude that the major factor responsible for the spectral dependence of photocatalytic activity in TiO2 is a high probability of excitations from the states below the top of the valence band.
We obtained different results for ZnO which make us conclude that the probability of finding spectral dependence of the photocatalytic activity is less for ZnO than for TiO2. The major argument in favor of this conclusion is that the calculated distribution function of holes in ZnO does not extend above the excess energy of 0.2 eV.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge financial support from the Spanish MICINN (Grant no. FIS2010-19609-C02-01), the Departamento de Educacíon del Gobierno Vasco, the University of the Basque Country (Grant no. GIC07-IT-366-07), and the Presidium of the Ural Branch of Russian Academy of Sciences (Grant no. 12-U-3-1001). The help of Professor L. Baker in the preparation of the paper is also greatly acknowledged. The calculations were performed using the URAN cluster of the Institute of Mathematics and Mechanics of the Russian Academy of Sciences, Yekaterinburg.