Spectroscopic and kinetic characterization of photogenerated charge carriers in photocatalysts

Abstract

The catastrophic consequences of increased power consumption, such as drastically rising CO₂ levels, natural disasters, environmental pollution and dependence on fossil fuels supplied by countries with totalitarian regimes, illustrate the urge to develop sustainable technologies for energy generation. Photocatalysis presents eco-friendly means for fuels production via solar-to-chemical energy conversion. The conversion efficiency of a photocatalyst critically depends on charge carrier processes taking place in the ultrafast time regime. Transient absorption spectroscopy (TAS) serves as a perfect tool to track those processes. The spectral and kinetic characterization of charge carriers is indispensable for the elucidation of photocatalytic mechanisms and for the development of new materials. Hence, in this review, we will first present the basics of TAS and subsequently discuss the procedure required for the interpretation of the transient absorption spectra and transient kinetics. The discussion will include specific examples for charge carrier processes occurring in conventional and plasmonic semiconductors.

Graphical abstract

1 Introduction

The first photocatalytic reactions were reported in the second decade of the twentieth century (1910–1920) [1]. The oxidative degradation of organic molecules (e.g. dye and pigments) was observed on the illuminated surfaces of metal oxides, such as zinc oxide, ZnO, and titanium dioxide, TiO₂. Subsequent research focused on the development of photocatalytic materials for the cleaning of surfaces, air, and water. After Honda and Fujishima discovered in 1972, the photosensitizing effect of a TiO₂ electrode for the electrolytic splitting of water, photocatalysis gained additional significance as a potential method to convert solar into chemical energy [2]. In a typical photocatalytic reaction, a photocatalyst is exposed to light, whose energy excites electrons from the filled valence band (VB) to the conduction band (CB). The charge carriers thus generated, that is, the electron in the CB (a reducing species) and the hole in the VB (an oxidizing species), migrate to the surface where they can undergo redox reactions with adsorbed molecules. Herein, the photocatalytic activity represents the ability of a photocatalyst to convert a certain amount of absorbed photons into the respective redox products. Many semiconducting materials show the ability to directly convert light energy into chemical or electrical energy; however, their photocatalytic activity is determined by the character of the photogenerated charge carriers.

Just 12 years after the discovery made by Honda and Fujishima, the active species responsible for TiO₂ photocatalysis, namely electrons and holes, were characterized. Bahnemann et al. [3] reported in 1984 for the first time the spectral identification of the charge carriers formed in colloidal TiO₂ by means of nanosecond transient absorption spectroscopy (TAS, denoted at this time as flash photolysis). Herein, the photocatalyst was excited by a pulsed laser in the nanosecond range resulting in the photogeneration of the electron–hole pairs, which were subsequently monitored through their respective transient optical absorption signature. Nowadays, TAS is a standard characterization method and enables the explanation of material-dependent photocatalytic efficiencies. The decay time or rather the lifetime of the charge carriers varies depending on the morphological and physical properties of the photocatalyst, thus determining its photocatalytic activity.

Transient absorption spectroscopy has significantly advanced over the years, allowing the generation and probing of charge carriers from THz frequencies to hard X-rays in time regimes down to femtosecond time scales and beyond [4]. Several reviews and books provide excellent descriptions of most time-resolved techniques [4,5,6,7]. Recently published reviews with a focus on the use of TAS for in situ and operando studies for advanced photoelectrochemical and photocatalytic applications are also available [8,9,10].

This review is devoted to the interpretation and understanding of the physical properties of charge carriers as observed by time-resolved optical spectroscopy, which still remains an important and frequently far-from-trivial scientific task. The review starts with the basic principle of TAS. The transient absorption features of the electrons and holes photogenerated in organic and inorganic semiconductors are summarized and discussed, followed by the characterization of charge carrier processes in recently developed plasmonic materials. The second part of the review offers a discussion of the mathematical and physical meaning of the transient decay kinetics and its importance for the mechanistic understanding of photocatalytic processes. This overview is intended to be helpful to those working in the field and should stimulate new research activities aiming to elucidate fundamental charge carrier processes occurring in a photocatalyst under illumination, to correlate those processes with the structural and optoelectronic properties and thus to establish a solid platform for the design of novel efficient photocatalysts.

2 Working principle of transient absorption spectroscopy

Transient absorption spectroscopy is a well-established method to study dynamic processes in a wide range of solar energy conversion materials. Time-resolved spectroscopy as we know it today was developed in the 1950s and early 1960s as flash photolysis and relaxation methods, a development that culminated with the Nobel Prize in Chemistry 1967 to Porter, Norrish, and Eigen [11]. The fundamental idea of the method is to use a pulsed excitation source, often denoted as pump source, to excite a sample from the ground to the excited state. The optical absorption of the excited state or other photogenerated intermediates is then spectroscopically monitored by a probe beam at certain time delay after its generation through a second excitation source. In particular, the probe beam intensity before ${I}_{0}(\lambda )$ and after the excitation ${I}^{*}(\lambda ,\tau )$ of the sample is detected and passed to the transient recorder, oscilloscope, which allows the temporal resolution of the signal (see Fig. 1). Hence, the quantity of interest is $\Delta A$, the difference between the ground state absorption ${A}_{0}(\lambda )$ and the excited state absorption ${A}^{*}(\lambda ,\tau )$.

Fig. 1

Principle of transient absorption spectroscopy. Excitation of a sample by a laser pulse leads to the formation of an excited state, the electronic transitions of which are induced by a probe beam and are spectroscopically monitored at a certain time delay after the laser pulse by a detector. Here, the probe beam intensity is measured before ${{\varvec{I}}}_{0}({\varvec{\lambda}})$ and after excitation ${{\varvec{I}}}^{\boldsymbol{*}}({\varvec{\lambda}},{\varvec{\tau}})$ and are applied in Eq. (1) to determine $\Delta {\varvec{A}}$. The transient signal, $\Delta {\varvec{A}}$, represents the difference between ground state A⁰(λ) and excited state A^*(λ, τ) absorption

If the experiment is conducted in transmission mode, $\Delta A$ is directly proportional to the concentration $c$ of the photogenerated species in accordance with the Beer-Lambert law:

$$\begin{array}{c}\Delta A={A}^{*}\left(\lambda , \tau \right)-{A}^{0}\left(\lambda \right)={\rm log}\left(\frac{{I}_{0}^{*}}{{I}^{*}}\right)-{\rm log}\left(\frac{{I}_{0}}{I}\right)=log\left(\frac{I}{{I}^{*}}\right)=\varepsilon cd,\end{array}$$

(1)

where $\varepsilon$ is the extinction coefficient of the formed intermediate, $d$ the beam path length, and ${I}_{0}$ = ${I}_{0}^{*}$.

To obtain quantitative information from TA experiments, it is necessary to measure the extinction coefficients of the species formed in the semiconductor upon excitation. For example, Durrant et al. [12] applied a spectro-electrochemical method to determine the extinction coefficient of the electron and holes in TiO₂.

If the experiments are performed in diffuse reflectance mode (time-resolved diffuse reflectance spectroscopy, TRDRS), then for the quantification of the intermediate’s concentrations, several considerations must be taken into account. For an optically dense sample, the reflected light is measured. The reflectance, $R$, is defined as the quotient of the incident intensity of the analyzing light, ${I}_{0}(\lambda )$, and the diffusely reflected light, ${J}_{0}(\lambda )$: ${R}_{0}(\lambda )={J}_{0}(\lambda )/{I}_{0}(\lambda )$. If the laser excitation generates a transient species that has an absorption at wavelength $\lambda$, this will result in the decrease of ${J}_{0}(\lambda )$ to ${J}^{*}(\lambda , \tau )$, while ${I}_{0}(\lambda )$ remains constant. Hence, the reflectance will be reduced to ${R}^{*}(\lambda ,\tau )$. Mathematically, this can be summarized as: [13]:

$$\begin{array}{c}\Delta R=\frac{{R}_{0}(\lambda )-{R}^{*}(\lambda , \tau )}{{R}_{0}(\lambda )}=1-{R}_{T}^{*}\left(\lambda ,\tau \right),\end{array}$$

(2)

where ${R}_{T}^{*}\left(\lambda ,\tau \right)={R}^{*}(\lambda , \tau )/{R}_{0}(\lambda )$. More detailed descriptions of the TRDRS working principle are available elsewhere [3, 13, 14].

In most TRDRS experiments, the concentration of excited states as a function of the distance from the sample surface follows an exponential decay [13]. This is because the experiments are mostly performed at conditions in which the ratio of the number of absorber units to the number of the exciting photons is high; thus, a low conversion percentage of the ground state to the transient state is expected. For the description of the exponentially falling-off concentration, the system can be divided into a series of “thin slices”, for which the Kubelka–Munk model is valid. Numerical solutions for this approach predict that a linear relationship exists between the reflectance change ($1-{R}_{T}^{*}\left(\lambda ,\tau \right)$) and the total transient species concentration, as long as ($1-{R}_{T}^{*}\left(\lambda ,\tau \right)$) remains below 0.1 [15].

Most time-resolved studies on the reaction dynamics of photogenerated charge carriers in semiconductors have been performed in transmission mode, on transparent colloidal dispersions or on transparent films [3, 16,17,18,19,20,21,22]. TRDRS allows the observation of the charge carrier dynamics of photocatalyst powders or opaque suspensions [23,24,25]. A major advantage is that the obtained kinetic data can be directly correlated with the outcome of photocatalytic tests with these samples, since they are used in powdered form in both cases.

Independently from the mode in which the TAS measurements were conducted, the time-resolved absorption changes can be recorded as a function of the probe wavelength. The transient absorption spectra are obtained by plotting $\Delta A$, monitored at a fixed time after the laser pulse excitation, as a function of the wavelength. The obtained transient absorption spectra normally contain three main contributions, the ground state bleach (GSB), stimulated emission (SE) and the so-called photoinduced absorption (PIA) arising from the absorption of the intermediates formed after the excitation, such as the excited states, positive or negative polarons, and trapped states (see Fig. 2).

Fig. 2

A plot of ∆A vs. wavelength involving three main contributions to the transient absorption spectra

In transient spectra, GSB appears as a negative band due to the depletion of the ground state occupancy (caused by laser excitation). Accordingly, more light can be transmitted after the sample was excited: ${I}^{*}(\lambda ,\tau )$ > ${I}_{0}(\lambda )$. In luminescent samples, after laser excitation, the probing light can induce the stimulated emission (working principle of lasers) which results in higher light intensity reaching the detector, ${I}^{*}(\lambda ,\tau )$ > ${I}_{0}\left(\lambda \right),$ and thus in a negative TA band. PIA is a positive TA signal: here ${I}^{*}(\lambda ,\tau )$ < ${I}_{0}\left(\lambda \right)$, since the formed intermediates harvest the probing light. PIA bands occur in such systems where successful charge separation has occurred; hence, these are transient bands which are relevant for the discussion of photocatalytic processes. The detailed discussion on the origin of PIA bands reported for organic and inorganic semiconductors will be thus presented in the next section, where we will denote PIA as a transient absorption.

3 Transient absorption spectra of photogenerated electrons and holes

3.1 Conventional organic and inorganic semiconductors

The electrons and holes photogenerated in a semiconductor drive the photocatalytic redox reactions. The spectroscopic tracking of these charge carriers may provide important knowledge to elucidate the photocatalytic reaction mechanisms and to reveal factors influencing the efficiency of the initial steps in photocatalysis. Since the photogenerated charge carriers are transient species, their spectroscopic characterization requires techniques such as TAS. The analysis and interpretation of the electron’s and hole’s transient absorption spectra are demanding and involve the separation of their transient absorption bands, knowledge of the intermediate’s chemical identity, and the assignment of the corresponding electronic transitions.

Under inert atmosphere, both electrons and holes are present after the excitation of the sample, and the recorded TA spectra are typically broad and featureless. The addition of an electron donor or an acceptor enables the separation of the individual wavelength regions where photogenerated electrons and holes absorb. By reacting with electron donors or acceptors, holes or electrons (respectively) are consumed, and thus their contribution to the transient absorption spectra is reduced, in addition to the suppression of undesired recombination reactions.

Alcohols act as perfect electron donors to scavenge photogenerated holes in inorganic semiconductors [26, 27]. For example, Tamaki et al. [27] investigated the reaction dynamics of trapped holes in TiO₂ with different alcohols, and observed a rapid decay of their transient absorption. The lifetime of the trapped holes in methanol, ethanol, and 2-propanol was found to be 300, 1000, and 3000 ps, respectively. Generally, it is assumed that the hole-induced alcohol oxidation includes the cleavage of the C–H bond, resulting in the formation of the respective α-hydroxyalkyl radicals, while the formation of the respective aldehyde occurs in a second step after injection of an electron into the CB of the semiconductor (“current doubling”) [28]. For organic polymers used as photocatalysts, alcohols were found to be less active for hole scavenging. Here, basic amines, such as trimethylamine or tri-ethanolamine, are most commonly used as electron donors [29].

Recently, combined experimental and theoretical studies have demonstrated the hole scavenging ability of water [30]. In spherical anatase TiO₂ nanoparticles, water adsorbates trap photogenerated holes. On the contrary, holes formed in faceted TiO₂ are unaffected by the presence of water. The low-coordinated Ti sites, present on nanoparticles, but lacking on faceted surfaces, were found to play a decisive role in water dissociation, which contributes to hole trapping at low-coordinated Ti–OH sites. Hence, water molecules were identified as active species in realistic aqueous operation conditions.

A prominent example of electron scavengers is molecular oxygen, which forms upon reaction with photogenerated electrons a superoxide radical [31, 32]. The reduction of molecular oxygen by electrons trapped in TiO₂ has been reported to occur within less than 100 ns [21], while the reduction process with the free CB electrons occurs 10–100 times slower [32, 33]. These different reaction dynamics can be explained by the fact that the trapped electrons are mostly localized at the surface and thus can be faster transferred to the surface bound oxygen atoms than the bulk located free CB electrons, although the former exhibit a lower reduction potential than the latter. However, to be an efficient electron scavenger, the reaction of the photogenerated electrons with the electron acceptor should be faster than the recombination. Platinum is an alternative electron scavenger, capturing photogenerated electrons within a few picoseconds. A homogeneous deposition of Pt on the semiconductor surface is challenging; typically, there is an unequal distribution of Pt islands [3, 34]. Their formation and optical absorption might affect the transient absorption spectra of the electrons, causing misleading interpretations of the TA results [34]. Silver cations, Ag⁺, have also been applied as electron scavengers. Upon reaction with photogenerated electrons, they are reduced to Ag [35, 36]. However, the silver nanoparticles exhibit a strong plasmon absorption at around 430 nm, i.e., in the wavelength region where photogenerated holes absorb in TiO₂ [34, 37]. Cu²⁺ ions have as well the ability to scavenge electrons, forming Cu⁺ ions, which have the advantage that no light is absorbed at wavelengths longer than 350 nm [35].

In addition to the chemical scavenging of charge carriers, the photogenerated electrons or holes can be extracted potentiostatically. Under applied cathodic bias the holes are scavenged, while anodic bias removes the electrons from the system. However, applying a bias to an electrode causes band bending, which might affect the trapping behavior of charge carriers [38]. A recently published review provides a broad overview on the use of transient absorption spectroscopy for in situ and operando studies of photoelectrodes [9].

In summary, it is obvious that there is no optimal scavenger for the photogenerated electrons and holes. For the identification of the spectroscopic regions of electron and holes, a scavenger is indispensable. However, it must always be taken into account that the scavenger might cause chemical or optical changes of the studied material, and thus leads to wrong conclusions [39]. Table 1 summarizes the spectroscopic regions of photogenerated electrons (free or trapped) and holes obtained for various semiconductors in the presence of electron/hole scavengers.

Table 1 Transient absorption band maxima for holes and electrons recorded in the presence of scavengers in different semiconducting materials

After identifying the spectral region in which electrons and holes absorb, their chemical identity has to be determined to assign the transient absorption spectra to the corresponding transitions. Electron paramagnetic resonance spectroscopy (EPR), an experimental technique to monitor the formation of paramagnetic species, is commonly applied to chemically characterize the electrons and holes in a photocatalyst [90,91,92]. For example, in TiO₂, the electrons are localized at Ti^IV centers converting it to a paramagnetic Ti^III species, characterized by g tensor components ranging between 1.9640 and 2.0025. In metal oxides, photogenerated holes are trapped at oxygen centers forming either surface adsorbed ^·OH radicals or lattice bound O^·− radicals. The latter species show a strong EPR signal as well [93]. However, except for a few examples [94], most EPR measurements are conducted at very low temperatures, cooling either with liquid nitrogen or liquid helium. These conditions are very different to the TA experiments, so a direct correlation of the results is sometimes challenging.

Today, with the availability of large-scale facilities specializing in the generation of ultrashort pulses in a wide spectral range, like free electron lasers, or high-power laser facilities, femtosecond pulses can be generated from THz frequencies to hard X-rays [4]. This enables the probing of light–matter interactions over a broad range of time-, energy-, and length-scales. Hence, Baker et al. [95] applied femtosecond extreme ultraviolet (XUV) spectroscopy in conjunction with X-ray photoelectron spectroscopy (XPS) to study ultrafast surface electron dynamics in NiO. XUV spectroscopy probes core-to-valence transitions, which are element-specific and provide detailed electronic structure information, including the transient oxidation state of the Ni metal centers. The findings from this study resolved important questions related to the mechanisms of carrier trapping and subsequent recombination in NiO, and provided parameters for the design of efficient materials.

After the chemical identity of the charge carriers was clarified, the corresponding electronic transitions must be assigned. In general, free electrons are delocalized within the CB [21, 96]. After the absorption of light with certain energies, the free electrons can be excited from a lower state in the CB to a higher state in the CB, known as an inter-band transition. These transitions are usually observed in the infrared range [21]. The energy levels of the trapped charge carriers are normally located within the band gap [21, 96]. For trapped electrons, their optical transitions correspond to the excitation from the trap state to the CB. For TiO₂, the localized Ti³⁺ electrons undergo d–d transitions upon excitation [97]. The possible transitions of trapped holes  are still controversial. Trapped holes represent defect electrons (missing electrons at a certain energy state). In the case of metal oxides, holes correspond to oxygen centered radicals, which could originate either from terminal hydroxyl groups or lattice oxygen [98]. The TA of these radicals can be attributed either to transitions from the trap states to the CB, forming unstable O atoms in the lattice, or from the VB to the trap states. Henderson et al. assumed that the latter transition may not be optically allowed [96]. However, recently published DFT (density functional theory) calculations on TiO₂ demonstrate that the TA of trapped holes is due to the transition of electrons from the VB to trap states [97]. Fig. 3 shows the comparison of experiments and theoretical calculations, with the corresponding electronic transitions responsible for the TA spectra of electrons and holes trapped in TiO₂.

Fig. 3

a Experimental transient absorption spectra after high intensity excitation, reprinted from reference [99]. Copyright 2010 Elsevier. b Computed absorption spectra of the (TiO₂)^38– and (TiO₂)³⁸⁺ clusters, representing an excess electron and excess hole, respectively, in the relaxed D_1,e/D_1,h (orange solid and green dashed curves) and unrelaxed S₀ geometries (blue dashed and green dotted curves). The solid blue line is the sum of the trapped electron, trapped hole, and CB electron contributions. Inset: Electronic transitions of the trapped holes, from VB to trap state and of the trapped electrons from the trap state to the CB, reprinted from reference [97]. Copyright 2016 American Chemical Society

In summary, we can conclude that TAS is an excellent tool to track the short-lived photoactive species. However, it should be considered that the observed transient spectra might be affected by photothermal effects [100] and irreversible changes of the semiconductor caused through laser excitation. For example, Schneider et al. [51, 101] observed irreversible changes of TiO₂ powders induced by the laser excitation in TRDRS, related to the formation of nonreactive trapped electrons accompanied by the release of oxygen atoms from the TiO₂ matrix. The irreversible changes were identified by means of UV–Vis and electron paramagnetic resonance spectroscopies. Moreover, in the case of pure anatase samples, some TiO₂ nanoparticles located in the inner region showed a phase transition to rutile. Furthermore, the laser-induced irreversible changes drastically affected the transient signals. In case of TiO₂, a considerable deceleration of the decay kinetics and a strong increase of the TA signal recorded in the wavelength region where the trapped holes absorb were found. Although for anatase samples, such changes disappear at weak excitation conditions, in the case of rutile, they cannot be avoided.

3.2 Plasmonic semiconductors

With the rapid development of semiconductor-related nanoscience, a new class of low-cost plasmonic semiconducting materials for solar fuel generation has emerged. Plasmonic semiconductors offer new opportunities to overcome some limitations of conventional semiconductors and plasmonic metals. These are heavily doped semiconductors, with a high concentration of free charge carriers. Similar to metals, these free charge carriers can be excited upon illumination, a process known as localized surface plasmon resonance (LSPR). The LSPR phenomenon has been observed in many heavily doped semiconductor nanostructures and quantum dots, including Cu_2−xS, B-/P-doped Si, Cu_2−xSe, Sn–In₂O₃, MoO_3−x, and Bi₂O_3−x [102,103,104,105,106,107,108]. Unlike metals, plasmonic semiconductors show broadly tunable plasmon frequencies (from visible to infrared) and rich surface chemistry. Especially, the infrared activity of plasmonic semiconductors is of high interest as it allows to capture an untapped energy source that accounts for 49% of solar irradiation, and thus opens a new avenue for green fuel generation by fully exploiting solar light.

By means of TAS, Zhou et al. [109] demonstrated for the first time plasmon-driven hot electron generation, and transfer from plasmonic metal oxide nanocrystals to surface adsorbed molecules. Here, F and In co-doped CdO nanocrystals were excited with a 1650 nm laser pulse, and the hot electron injection to the adsorbed electron acceptor (Rhodamine B, RhB) was probed in the visible wavelength region (400–700 nm). A strong ground state bleach band matching exactly with RhB absorption, and transient absorption at ~ 412 nm arising from the RhB radical anion, were observed (see Fig. 4a). The analysis of RhB GSB kinetics revealed an electron transfer time of < 50 fs (rise) and 407 fs for a subsequent back electron transfer process (decay). Together with the excitation wavelength- and power-dependent studies, the authors provided a mechanistic picture of the process occurring upon infrared light illumination (Fig. 4c). After photoexcitation, the plasmon resonance damps its energy and excites electrons from the states near Fermi level, E_f, thus depopulating those states in < 10 fs through Landau damping. Subsequent indirect hot electron transfer from the plasmonic semiconductor occurs before electron thermalization. The number of hot electrons above the RhB lowest unoccupied molecular orbital (LUMO) was found to be tunable via the excitation fluence (energy per excited area).

Fig. 4

Copyright 2020 Springer Nature

a Transient absorption spectrum of F and In co-doped CdO and RhB showing RhB ground state bleach (GSB) and anion radical induced transient absorption. b Transient absorption kinetics of RhB GSB (empty circles) and the exponential fitting (blue line) in F and In co-doped CdO–RhB complex. c) Scheme of hot electron transfer with two excitation energies (0.75 and 0.85 eV) after Landau damping before electron thermalization. The gray region shows the undisturbed filled states in the conduction band. The orange and blue regions represent the excited electrons and empty states left, respectively, after Landau damping under different excitation photon energies. Higher excitation energy leads to more electrons above the barrier height. Reprinted from reference [109].

The rich surface chemistry of plasmonic semiconductors enables the coupling not only with molecules as shown in the aforementioned example but also with other photoactive semiconductors. Plasmonic semiconductor nanostructures can be easily assembled onto active supports through a facile wet chemical method. Zhang et al. [110] have fabricated W₁₈O₄₉ nanowires (as branches) onto TiO₂ electrospun nanofibers (as backbones). These materials exhibited plasmon enhanced photocatalytic activity for hydrogen generation from ammonia borane upon excitation by low-energy infrared photons. For the elucidation of the photoinduced processes, TAS was performed in combination with theoretical calculations. The 800 nm excitation of pure W₁₈O₄₉ resulted in a broad transient absorption band with a maximum at 1045 nm. This transient absorption was assigned to the LSPR, by which free electrons around the Fermi level could reach a virtual high-energy surface plasmon (SP) state to form energetic hot electrons. When W₁₈O₄₉ was coupled with TiO₂ to form a branched heterostructure, the transient absorption disappeared. This observation indicated the LSPR-excited hot electron transfer from W₁₈O₄₉ branches to the adherent TiO₂ backbones. A thorough kinetic analysis of the transients monitored at various wavelengths confirmed the hot electron transfer process, with fast rate constants of 3.8 × 10¹² to 5.5 × 10¹² s⁻¹. Such an ultrafast transfer process should hinder the relaxation of hot electrons to low energy levels in plasmonic W₁₈O₄₉ and therefore boost the generation of active electrons for executing the catalytic reaction.

Among the development of photocatalysts based on the plasmon-induced hot electron transfer, materials conducting plasmon-induced hole transfer were also studied. A novel series of p-type semiconductors such as copper chalcogenide nanocrystals, which show excellent tunable hole-based LSPR absorption in the near-infrared (NIR) region, have attracted much attention as candidates for infrared-responsive photocatalysts [111, 112]. For example, time-resolved infrared spectroscopy enabled the direct observation of hot hole transfer in a LSPR-excited CdS/CuS system [113]. Figure 5a, b show the transient absorption signal obtained for CuS and CdS/CuS upon excitation with 1200 nm laser pulse. The observed bleaches in both systems were attributed to the LSPR bleaching caused through LSPR-excitation-induced sequential events such as hole dephasing, hole–hole scattering, hole–phonon coupling, and lattice heat dissipation. In both systems, a blue shift of the bleaching signals in comparison to the LSPR band occurred. The authors have attributed this phenomenon to the change in the concentration of holes induced either by trapping processes or by hole transfer from CuS to CdS in the heterostructure. A further evidence for the hole transfer from CuS to CdS was a broad and featureless absorption derived from the trapped holes in the CdS phases, detected as a positive signal in the visible region, see Fig. 5b. The kinetic analysis of the bleaching kinetics revealed that hot holes were not directly injected into CdS but instead transferred stepwise via the carrier trapping state. The process was named plasmon-induced transit charge transfer (PITCT). The occurrence of this PITCT process enabled a long-lived charge separation (9.2 μs) with high quantum yields (19%). The PITCT mechanism is summarized in Fig. 5c.

Fig. 5

Copyright 2018 Springer Nature

Time-resolved infrared spectroscopy (TR-IR) spectral changes for CuS (a) and CdS/CuS (b) nanocrystals from visible to near-infrared regions in the microseconds (μs) time scale. c Schematic illustration of the LSPR-induced stepwise hole transfer process. The pink arrows mean plasmon excitation by near-infrared (NIR) light. For CuS NCs, the generated hot holes decayed via hole–hole and phonon–hole scattering (1) or ultrafast hole trapping to the shallow (2) or deep trapping state (3), followed by relaxation to the intrinsic hole state. In CdS/CuS heterostructured nanocrystals, the holes in the deep trapping state transferred to the valence band of the CdS phases (4, PITCT) and the holes in the CdS phases moved to the trapping state, showing structureless absorption in the visible region and recombination to the initial state. PITCT: plasmon-induced transit carrier transfer. Reprinted from reference [113].

As the aforementioned examples demonstrate, plasmonic photocatalysts exhibit promising potential for enhancing the efficiency and solar light harvesting of important chemical transformations. Transient absorption spectroscopy has served as a perfect tool to elucidate the mechanism of the underlying reaction processes through the analysis of the transient absorption bands and decays. However, the mechanism of plasmonic photocatalysis is still poorly understood and requires more investigation. Especially, the contributions of non-photothermal (i.e., hot carrier) and photothermal pathways remain a question of intense debate [114].

4 Charge carrier kinetics

4.1 Overview

The assignment of TA signals to specific species, as described in Sect. 3, allows extracting important qualitative information from a photocatalytic system. As a further step, the analysis of the kinetic profiles provides quantitative information on the timescale of the processes. This is generally done by fitting the profiles to a mathematical function that (ideally) captures the main physical characteristics of the system [10, 115]. Thus, the functional form of the profiles already provides physical insights. Moreover, the fitting yields kinetic parameters (e.g., rate constants) that can be used to characterize samples and compare different materials.

When approaching kinetic data, it is important to have in mind the approximate time scale of the different processes. As summarized in Fig. 6 for TiO₂, while excitation is practically instantaneous, charge carrier trapping at the surface (shallow traps) occurs in a few hundred fs, while trapping in the bulk is considerably slower (~ 50 ps) [116]. Recombination starts as soon as 1 ps, and can reach the ns range for bulk-trapped species. Finally, interfacial charge transfer is strongly dependent on the specific reaction, with e.g., electron transfer to Pt islands occurring in ~ 10 ps, and molecular oxygen reduction occurring in ns to µs time scales. An important consequence is that, except for ultrafast experiments or under strong biases, recombination will most likely be the main observed process, as could be expected from the low quantum yields of photocatalytic reactions (typically below a few percent [117]).

Fig. 6

Copyright 2014 American Chemical Society

Scheme of different photoinduced processes observed in transient absorption studies on TiO₂ photocatalysts, and approximate time scales. Adapted with permission from reference [116].

Although the physics of charge carrier dynamics are a complicated subject, a simple picture can provide a good initial overview. Excitation of a photocatalyst, thus, leads to the formation of a bound electron—hole pair: an exciton. The high dielectric constant of typical inorganic photocatalysts (e.g., TiO₂) shields their mutual attraction, and thus thermal energy is enough to dissociate the pair in two separate entities, each with a characteristic mobility. Subsequent encounters of the electron and hole results in recombination, which can then be considered as a bimolecular process. From chemical intuition, it can be expected that the kinetics of the process follows a second-order rate law:

$$\begin{array}{c}\left[c\right]=\frac{{\left[c\right]}_{t=0}}{{\left[c\right]}_{t=0}{k}_{\mathrm{r}}t+1},\end{array}$$

(3)

where $[c]$ represents the charge carrier (electron or hole) concentration at time $t$, ${[c]}_{t=0}$ its concentration at $t=0$ (i.e., right after excitation), and ${k}_{\mathrm{r}}$ the second-order rate constant. Indeed, second-order kinetic profiles are commonly observed in TA studies of different photocatalysts [20, 51, 118,119,120].

This model, however, predicts that the charge carrier concentration reaches zero at long times, in contrast with typical observations. This long-lived absorption has thus been accounted for by including an additional constant term $"f"$ [120]. This results in a functional behavior termed “second-order with a baseline”, that is, possibly, the most widely applied in the analysis of TA decays:

$$\begin{array}{c}\left[c\right]=\frac{{\left[c\right]}_{t=0}}{{\left[c\right]}_{t=0}{k}_{\mathrm{r}}t+1}+f.\end{array}$$

(4)

Illustrating the complexity of recombination processes, it has been observed that the kinetic profiles usually depend on experimental parameters such as the laser pulse intensity or the semiconductor’s particle size. These parameters define the average number of photogenerated charge carriers per particle. Serpone et al. [20] analyzed colloidal TiO₂ sols and observed that, when the average number of electron–hole pairs per particle was lower than ~ 0.5, the decays followed first-order kinetics:

$$\begin{array}{c}\left[c\right]={\left[c\right]}_{t=0} {e}^{-{k}_{\mathrm{r}}t}.\end{array}$$

(5)

Contrarily, when the average number of pairs per particle was greater than ~ 30, the decays were fitted by the “second-order with a baseline”, Eq. (4).

To understand this behavior, it is necessary to recall that the derivation of classical kinetic laws implies that the species of interest are present in numbers large enough to consider their concentrations as continuous variables. In TA experiments, however, the photocatalyst particles act as micro-reactors, where the number of electron–hole pairs is usually much smaller than that limit, thus rendering the basic assumptions of classical kinetic models invalid. The correct treatment then involves stochastic kinetics tools, as was realized by Grätzel and co-workers in 1985 [42]. By assuming a Poisson distribution of electron–hole pairs in colloidal TiO₂ particles, and an exponentially decaying survival probability for single electron–hole pairs, it was possible to obtain an equation that reproduced the charge carrier decays, and correctly reduced to first-order and second-order rate laws under very low or very high initial charge carrier numbers, respectively.

A related stochastic approach was also used to explain the baseline behavior (Eq. (4)). Grela and Colussi performed numerical modeling of charge carrier recombination at the surface of colloidal TiO₂ particles, and found that the “baseline” is the consequence of recombination taking place in a 2-dimensional space, i.e., the particles surface [121]. Electron–hole pairs which are initially close together recombine quickly, while those far apart live much longer, giving rise to a phenomenon known as fractal kinetics, typical of low-dimensional media [122]. According to this analysis, recombination at the surface never follows second-order kinetics: single electron–hole pairs decay exponentially, while multiple pairs decay in a second-order fashion where the rate constant is actually time-dependent (and thus should be called rate coefficient).

Processes in which the rate coefficient ${k}_{\mathrm{r}}$ are time-dependent are said to follow dispersive kinetics, since, equivalently, it can be considered that there is not a unique value for (time-independent) ${k}_{\mathrm{r}}$ but rather a distribution of them [123]. A well-known example is the model of Kohlrausch–Williams–Watts (KWW model) [124, 125]. Although it was initially employed in 1854 by Kohlrausch to describe the discharge of a capacitor [124], it can be applied to chemical reactions with dispersive kinetics, by assuming a Lévy distribution (approximately, an asymmetric Gaussian-like distribution with a ‘heavy’ tail) for ${k}_{\mathrm{r}}$ [126]. The KWW function can then be seen as the superposition of many first-order reactions, each with a unique rate constant. The concentration of charge carriers thus follows a so-called stretched exponential behavior:

$$\begin{array}{c}\left[c\right]={\left[c\right]}_{t=0} {e}^{-{\left({k}_{r}t\right)}^{\beta }},\end{array}$$

(6)

where $\beta$ represents the distribution width ($0<\beta \le 1$). In the $\beta =1$ limit, the expression reduces to the classical first-order rate law.

A related model is that by Albery and co-workers, initially used to fit interfacial electron transfer kinetics on semiconductor nanoparticles, in which the values of ${k}_{r}$ follow a log-normal distribution [127]:

$$\begin{array}{c}\left[c\right]\propto {\int }_{0}^{\infty }\frac{{e}^{-{\left(\mathrm{ln}\left({k}_{\mathrm{r}}\right)-\mathrm{ln}\left({k}_{0}\right)\right)}^{2}/{\gamma }^{2}}}{{k}_{r}} {e}^{-{k}_{\mathrm{r}}t}\mathrm{d}{k}_{\mathrm{r}}.\end{array}$$

(7)

Here $\gamma$ characterizes the distribution width and ${k}_{0}$ is the mean rate coefficient. As $\gamma$ approaches zero, the dispersion diminishes, and the model approaches a first-order exponential decay.

Sieland et al., while analyzing charge carrier decays in TiO₂ powders, observed that the application of the second-order scheme leads to different kinetic parameters when employing different time windows [128]. To solve this problem, they utilized a fractal kinetic equation that can be understood as a second-order law, modified to account for dispersion in the rate constant [129]:

$$\begin{array}{c}\left[c\right]=\frac{{\left[c\right]}_{t=0}\left(1-h\right)}{{\left[c\right]}_{t=0}{k}_{\mathrm{r}}{t}^{1-h}+\left(1-h\right)},\end{array}$$

(8)

where $h$ ($0\le h\le h$) is the parameter dictating the width of the distribution. By setting $h=0$ the equation reduces to the classical second-order behavior (Eq. (3)). It is noteworthy that this function has correctly described transient absorption decays of TiO₂ powders spanning the 50 ns to 1 ms time range.

At long times, the right-hand side of the divisor in the fractal model (Eq. (8)) becomes negligible, reducing the expression to a power law dependence [123]:

$$\begin{array}{c}\left[c\right]\propto {t}^{-\alpha },\end{array}$$

(9)

where $\alpha$ could be associated with $1-h$ in the previous equation. This model was, for instance, used by Cowan et al. [12] to describe charge carrier decays in the microsecond to second time range in nanocrystalline TiO₂ films under applied bias. From a physical point of view, the function has been rationalized on the basis of the so-called trapping—detrapping model of charge carrier transport, which, as continuous-time random walk simulations show [130], yields the same kinetic profile. Briefly, the model is based on an energetic distribution of trap states, in which electrons are continuously trapped and later de-trapped with help from thermal energy. The assumption of an exponential density of trap states leads to a power law behavior for the recombination kinetics [130, 131].

In the following, we give a more detailed view and a few applications for each of these models.

4.2 Classical kinetics models

We start with the simplest case, i.e., a first-order reaction representing the decay of a charge carrier $c$ (an electron or a hole), with a rate constant ${k}_{\mathrm{r}}$:

$$\begin{array}{c}c\stackrel{{k}_{\mathrm{r}}}{\to }\left(\mathrm{recombination}\right).\end{array}$$

(10)

The rate $r$ for a first-order reaction in a homogeneous medium is:

$$\begin{array}{c}r=-\frac{\mathrm{d}\left[c\right]}{\mathrm{d}t}={k}_{\mathrm{r}}\left[c\right].\end{array}$$

(11)

The derivative here implies that $\left[c\right]$ is a real variable, a valid assumption when e.g., dealing with an Avogadro’s number of molecules. However, this is not true for photogenerated charge carriers in transient absorption experiments, since laser excitation produces at most a couple hundred per particle. Omitting this issue for the moment, the equation can be integrated to yield the familiar first-order equation, or exponential decay:

$$\begin{array}{c}\left[c\right]={\left[c\right]}_{t=0} {e}^{-{k}_{\mathrm{r}}t}.\end{array}$$

(12)

From the physical point of view, this simple model could be related to primary geminate recombination, that is, before charge carriers migrate on separate paths. This is relevant for semiconductors with a low dielectric constant (e.g., organic), or in low temperature experiments. Exponential decays are also observed in experiments with low average numbers of photogenerated charge carriers [20, 42, 132]. For example, Tamaki et al. [132] analyzed transparent nanocrystalline TiO₂ films at different excitation intensities, and determined that pulses with energies below 160 nJ led to average numbers of photogenerated pairs of one or less per particle (Fig. 7). An important point is that the profiles measured under these conditions fall on top of each other upon normalization, a signature of true mono-exponential decays [10].

Fig. 7

Transient absorption decays for nanocrystalline TiO₂ films. Excitation was performed at 355 nm and detection at 2500 nm. The intensity of the excitation pulse varied from 590 to 40 nJ pulse⁻¹, from top to bottom. Republished with permission of the Royal Society of Chemistry, from reference [132]; permission conveyed through Copyright Clearance Center, Inc.

When analyzing TA kinetics, it is often found that they can be fitted with a linear combination of exponential decays. This could be the result of consecutive reactions. An example is the situation in which charge carriers recombine only in their trapped state:

$$\begin{array}{c}{c}_{\mathrm{free}}\stackrel{{k}_{\mathrm{r}}}{\to }{c}_{\mathrm{trapped}}\stackrel{{k}_{\mathrm{r},2}}{\to }\left(\mathrm{recombination}\right).\end{array}$$

(13)

In this case we have:

$$\begin{array}{c}\frac{\mathrm{d}\left[{c}_{\mathrm{free}}\right]}{\mathrm{d}t}={-k}_{\mathrm{r}}\left[{c}_{\mathrm{free}}\right],\end{array}$$

(14)

$$\begin{array}{c}\frac{\mathrm{d}\left[{c}_{\mathrm{trapped}}\right]}{\mathrm{d}t}={{k}_{\mathrm{r}}\left[{c}_{\mathrm{free}}\right]-k}_{\mathrm{r},2}\left[{c}_{\mathrm{trapped}}\right].\end{array}$$

(15)

Equation (14) is solved in the same way as Eq. (11):

$$\begin{array}{c}\left[{c}_{\mathrm{free}}\right]={\left[{c}_{\mathrm{free}}\right]}_{t=0} {e}^{-{k}_{\mathrm{r}}t}.\end{array}$$

(16)

For Eq. (15), assuming ${\left[{c}_{\mathrm{trapped}}\right]}_{t=0}=0$ and ${k}_{\mathrm{r}}\ne {k}_{\mathrm{r},2}$, after some algebra [133] we get:

$$\begin{array}{c}\left[{c}_{\mathrm{trapped}}\right]=\frac{{k}_{\mathrm{r}}{\left[{c}_{\mathrm{free}}\right]}_{t=0}}{{k}_{\mathrm{r},2}-{k}_{\mathrm{r}}} \left({e}^{-{k}_{\mathrm{r}}t}-{e}^{-{k}_{\mathrm{r},2}t}\right).\end{array}$$

(17)

If both ${c}_{\mathrm{free}}$ and ${c}_{\mathrm{trapped}}$ give rise to transient absorption, the observed signal $\Delta A$ will be a linear combination of both concentrations (accounting for possibly different absorption coefficients, $A$ and $B$):

$$\begin{array}{c}\Delta A=A\left[{c}_{\mathrm{free}}\right]+B\left[{c}_{\mathrm{trapped}}\right].\end{array}$$

(18)

After rearranging and grouping constants together, we get a functional dependence known as double-exponential or biphasic decay:

$$\begin{array}{c}\Delta A=C{e}^{-{k}_{\mathrm{r}}t}+D{e}^{-{k}_{\mathrm{r},2}t}.\end{array}$$

(19)

An alternative mechanism that results in a multi-exponential decay originates when similar species decay by different pathways. For instance, we could consider that charge carriers trapped in the surface or in the bulk, while resulting in similar transient absorption spectra, could decay with different rates:

$$\begin{array}{c}{c}_{\mathrm{surf trapped}}\stackrel{{k}_{\mathrm{r},\mathrm{surf}}}{\to }\left(\mathrm{recombination}\right),\end{array}$$

(20)

$$\begin{array}{c}{c}_{\mathrm{bulk trapped}}\stackrel{{k}_{\mathrm{r},\mathrm{bulk}}}{\to }\left(\mathrm{recombination}\right).\end{array}$$

(21)

Both species decay exponentially:

$$\begin{array}{c}\left[{c}_{\mathrm{surf trapped}}\right]={\left[{c}_{\mathrm{surf trapped}}\right]}_{t=0} {e}^{-{k}_{\mathrm{r},\mathrm{surf}}t},\end{array}$$

(22)

$$\begin{array}{c}\left[{c}_{\mathrm{bulk trapped}}\right]={\left[{c}_{\mathrm{bulk trapped}}\right]}_{t=0} {e}^{-{k}_{\mathrm{r},\mathrm{bulk}}t}.\end{array}$$

(23)

If both species present transient absorption, then the total signal will be given by:

$$\begin{array}{c}\Delta A=A\left[{c}_{\mathrm{surf trapped}}\right]+B\left[{c}_{\mathrm{bulk trapped}}\right].\end{array}$$

(24)

Combined with Eqs. (22) and (23), and grouping constants, it again results in a double exponential decay:

$$\begin{array}{c}\Delta A=C{e}^{-{k}_{\mathrm{r},\mathrm{surf}}t}+D{e}^{-{k}_{\mathrm{r},\mathrm{bulk}}t}\end{array}$$

(25)

This type of behavior has been observed, for instance, by Zhang et al. for colloidal CdS suspensions [134], and by Horikoshi et al. for oxygen-vacancy rich TiO₂ powders [135]. In the latter case, illustrated in Fig. 8, the authors analyzed the decays in the presence or absence of a microwave field. For dry samples, it made no difference, but in wet pastes, recombination significantly slowed down under microwave irradiation. Moreover, the observation of a double-exponential decay was explained in terms of the recombination mechanism: the fast component was attributed to recombination of free or shallowly-trapped electrons with holes, while the slow decay was related to recombination of deeply trapped electrons with (free or trapped) holes [135].

Fig. 8

Transient absorption decays for oxygen-vacancy rich TiO₂ powders, in dry or wet conditions (top and bottom, respectively). Excitation was performed at 532 nm, and detection at 550 nm. Blue and red circles correspond to data points in absence or presence of microwave irradiation, respectively. Lines show double-exponential fittings. The material was prepared by heat-treating commercial TiO₂ in the presence of molecular hydrogen. Republished with permission of the Royal Society of Chemistry, from reference [135]; permission conveyed through Copyright Clearance Center, Inc.

A mechanism that does not result in multi-exponential profiles is that of parallel decay channels. For instance, we could assume that charge carriers recombine through two separate pathways (e.g., radiative and non-radiative):

$$\begin{array}{c}c\stackrel{{k}_{\mathrm{r}}}{\to }\left(\mathrm{recombination}\right) \quad {\rm and} \quad c\stackrel{{k}_{\mathrm{r},2}}{\to }\left(\mathrm{recombination}\right)\end{array}$$

(26)

The corresponding differential equation is now:

$$\begin{array}{c}-\frac{\mathrm{d}\left[c\right]}{\mathrm{d}t}={k}_{\mathrm{r}}\left[c\right]+{k}_{\mathrm{r},2}\left[c\right]={(k}_{\mathrm{r}}+{k}_{\mathrm{r},2})\left[c\right]={k}_{\mathrm{r}}^{^{\prime}}\left[c\right].\end{array}$$

(27)

which can then be integrated to obtain a mono-exponential decay, with a rate constant ${k}_{\mathrm{r}}^{^{\prime}}={k}_{\mathrm{r}}+{k}_{\mathrm{r},2}$. It is clear then that a parallel decay channel does not change the kinetic profile but only its apparent rate constant.

In cases where the electron—hole pairs dissociate into its components, is perhaps more meaningful to consider a second-order mechanism. In the equal-concentration case, the differential equation reads:

$$\begin{array}{c}\frac{\mathrm{d}\left[c\right]}{\mathrm{d}t}={-k}_{\mathrm{r}}{\left[c\right]}^{2}.\end{array}$$

(28)

After rearrangement and integration, we get:

$$\begin{array}{c}\left[c\right]=\frac{{\left[c\right]}_{t=0}}{{\left[c\right]}_{t=0}{k}_{\mathrm{r}}t+1}.\end{array}$$

(29)

There are many examples of its use from femtosecond to microsecond time windows [20, 51, 118,119,120]. An important observation is that the derived second-order kinetic constants for recombination are frequently correlated with the photonic efficiency of photocatalytic processes [26, 49, 136], justifying its wide application despite the simplicity of the physical model.

4.3 Dispersive kinetics models

The previous section deals with classical first-order or second-order mechanisms, with a distinct rate constant for each elementary step. However, in reaction media that are not spatially and energetically homogeneous, the rate constants are not unique but instead follow a distribution. A typical example is the hydrogen abstraction reaction by alkyl radicals in organic glasses [137], which instead of following (pseudo) first-order kinetics, is empirically described by a function termed stretched exponential or Kohlrausch–Williams–Watts (KWW) function [124, 125]:

$$\begin{array}{c}\left[A\right]={\left[A\right]}_{t=0}{e}^{-{\left({k}_{\mathrm{r}}t\right)}^{\beta }},\end{array}$$

(30)

where $0<\beta \le 1$. In the $\beta =1$ limit, the equation is reduced to an exponential decay (Eq. (10)). One way to look at this equation is by assuming that ${k}_{\mathrm{r}}$ is not a constant but instead changes over time, ${k}_{\mathrm{r}}={k}_{\mathrm{r}}^{^{\prime}}(t)$. Thus, the exponential factor in the last equation can be rewritten as:

$$\begin{array}{c}-{\left({k}_{\mathrm{r}}t\right)}^{\beta }=-{{k}_{\mathrm{r}}}^{\beta }{t}^{\beta }=-{{k}_{\mathrm{r}}}^{\beta }{t}^{\beta -1}t=-{k}_{\mathrm{r}}^{^{\prime}}\left(t\right) t.\end{array}$$

(31)

This implies that the reaction follows a first-order equation, where the time-dependent rate coefficient ${k}_{\mathrm{r}}^{^{\prime}}\left(t\right)$ equals ${{k}_{\mathrm{r}}}^{\beta }{t}^{\beta -1}$, i.e., the apparent rate constant gets progressively smaller over time.

An alternative interpretation assumes that the rate coefficient is time-independent, and instead of a unique value, there is a distribution of them. In nanocrystalline photocatalysts, this could be caused e.g., by particle polydispersity. Under this view, stretched exponential function can be understood as the summation of mono-exponential decays with different ${k}_{\mathrm{r}}$ values, representing an ensemble of particles where recombination occurs at distinct rates [126, 138]:

$$\begin{array}{c}\left[c\right]\propto {e}^{-{\left({k}_{\mathrm{r}}t\right)}^{\beta }}={\int }_{0}^{\infty }{g}_{\mathrm{KWW}}\left({k}_{\mathrm{r}},\beta \right) {e}^{-{k}_{\mathrm{r}}t}\mathrm{d}{k}_{\mathrm{r}}.\end{array}$$

(32)

Here ${g}_{\mathrm{KWW}}({k}_{\mathrm{r}},\beta )$ is the probability distribution describing the possible values of ${k}_{\mathrm{r}}$ and their associated probability. The corresponding function for the KWW model is called a Lévy positive alpha-stable distribution. There are closed-form expressions only for a small set of $\beta$ values, such as $\beta =1/2$ [138]. More generally, it can be calculated as [138]:

$$\begin{array}{c}{g}_{\mathrm{KWW}}\left({k}_{\mathrm{r}},\beta \right)=\frac{1}{\pi }{\int }_{0}^{\infty }{e}^{-{u}^{\beta }\mathrm{cos}\left(\pi \beta /2\right)}\mathrm{cos}\left[{k}_{\mathrm{r}}u-{u}^{\beta }\mathrm{sin}\left(\frac{\pi \beta }{2}\right)\right]\mathrm{d}u.\end{array}$$

(33)

From a qualitative point of view, the distribution function shows an asymmetric, approximately Gaussian shape, with a ‘heavy’ tail.

The KWW model was applied, for instance, by Kamat et al. [139] to describe electron injection from CdSe quantum dots to TiO₂ nanoparticles, as observed in femtosecond transient absorption experiments, or by Durrant et al. [32], who studied interfacial electron transfer from TiO₂ films to molecular oxygen in the presence of ethanol (hole scavenger). Another interesting example is that by Castellano et al. [140], who applied time-resolved photoluminescence and transient absorption spectroscopies to pyrenyl-functionalized CdSe quantum dots, and found stretched-exponential decays (with similar kinetics) for both sets of experiments (Fig. 9).

Fig. 9

Transient absorption decay of 2.4 nm CdSe quantum dots functionalized with 1-pyrene-carboxylic acid (PCA). Excitation was performed at 488 nm with a 2 mJ pulse, and detection at 430 nm. The red lines show the fit to a stretched-exponential function. The inset compares the fitted function to that obtained from the fitting of photoluminescence data. Reprinted by permission from Springer Nature: Nature Chemistry, reference [140], Copyright 2017

Other rate constant distributions lead to different models. For instance, the one from Albery and co-workers [127] assumes a log-normal distribution of the first-order rate constant. The physical meaning is that instead of a unique activation energy $\Delta G^{\dag }$ there is a (Gaussian) distribution of them [126]:

$$\Delta G^{\dag } = \Delta G_{0}^{\dag } + \gamma xRT.$$

(34)

Here $\gamma$ determines the width in energy dispersion (with $\gamma =0$ leading to the first-order case), while the possible values of $x$ follow a Gaussian distribution function ($p(x)\propto {e}^{-{x}^{2}}$). By substituting into the Arrhenius equation, we get [126]:

$$k_{{\text{r}}} \propto e^{{ - \frac{{\Delta G_{0}^{\dag } }}{RT} - \gamma x}} = k_{0} e^{ - \gamma x} ,$$

(35)

where ${k}_{0}$ represents the mean rate constant. The rate law is obtained by summing over all possible values of ${k}_{\mathrm{r}}$, or, equivalently, all possible values of $x$ [126]:

$$\begin{array}{c}\left[c\right]\propto {\int }_{-\infty }^{\infty }{e}^{-{x}^{2}}{e}^{-{k}_{0}t {e}^{-\gamma x}}\mathrm{d}x.\end{array}$$

(36)

Changing variables, this equation can be expressed in terms of the explicit form of the distribution function ${{g}_{\mathrm{Albery}}(k}_{\mathrm{r}})$, in a similar way as for the KWW model [126]:

$$\begin{array}{c}\left[c\right]\propto {\int }_{0}^{\infty }{{g}_{\mathrm{Albery}}(k}_{\mathrm{r}}) {e}^{-{k}_{\mathrm{r}}t}\mathrm{d}{k}_{\mathrm{r}}.\end{array}$$

(37)

With:

$$\begin{array}{c}{g}_{\mathrm{Albery}}\left({k}_{\mathrm{r}}\right)=\frac{{e}^{-\frac{{\left(\mathrm{ln}\left({k}_{\mathrm{r}}\right)-\mathrm{ln}\left({k}_{0}\right)\right)}^{2}}{{\gamma }^{2}}}}{{k}_{\mathrm{r}}}.\end{array}$$

(38)

The equation shows that ${k}_{\mathrm{r}}$ follows a log-normal distribution. To fit experimental data, Eq. (36) must be integrated. This can be done using the extended Simpson’s rule, resulting in [127]:

$$\begin{array}{c}\frac{\left[c\right]}{{\left[c\right]}_{t=0}}=\left(\frac{0.2}{3{\pi }^\frac{1}{2}}\right)\begin{array}{c}\{2\left[f\left(0.1\right)+f\left(0.3\right)+f\left(0.5\right)+f\left(0.7\right)+f\left(0.9\right)\right]\\ +f\left(0.2\right)+f\left(0.4\right)+f\left(0.6\right)+f\left(0.8\right)+{e}^{-{k}_{0}t}\}\end{array},\end{array}$$

(39)

where $f\left(\lambda \right)={\lambda }^{-1}{e}^{-{(\mathrm{ln}\lambda )}^{2}}({e}^{-{k}_{0}t{\lambda }^{\gamma }}+{e}^{-{k}_{0}t{\lambda }^{-\gamma }})$. Both $\gamma$ and ${k}_{0}$ are used as fitting parameters.

This model was applied, for instance, by Draper and Fox to analyze the transient absorption profiles of photoinduced reactions over aqueous suspensions of TiO₂ powders (particularly, photooxidation reactions) [141, 142]. The distribution in rate constant values was attributed to the dispersion in particle radii. Among other examples [143], the Albery model was also employed by Peek et al. [144] to analyze the emission decay of Cr(VI) ions supported on silica. Interestingly, they found a significant change in the dispersion width values $\gamma$ when reducing the temperature to 77 K, concluding that at low temperature there is a wider distribution of emitting sites, caused by a larger number of emitting sites.

The Albery and KKW models, as described above, are ultimately based on first-order kinetics. Other models are derived instead from dispersive second-order kinetics. The fractal model of Sieland et al. [128] can be understood in these terms by assuming a time-dependent rate coefficient ${k}_{r}^{^{\prime}}$ for the classical second-order expression (Eq. (29)):

$$\begin{array}{c}{k}_{\mathrm{r}}^{^{\prime}}= {k}_{\mathrm{r}}{t}^{-h}.\end{array}$$

(40)

The equation for the fractal model is thus:

$$\begin{array}{c}\left[c\right]=\frac{{\left[c\right]}_{t=0}\left(1-h\right)}{{\left[c\right]}_{t=0}{k}_{\mathrm{r}}{t}^{1-h}+\left(1-h\right)}.\end{array}$$

(41)

Similarly to the previous models, $h$ ($0\le h<1$) describes the dispersion in the recombination rates, related by the authors to spatial heterogeneity [128]. This model was applied to transient absorption measurements of TiO₂-based powders, performed in reflectance mode on a microsecond time scale (Fig. 10). Although these decays could be fitted with the “second-order with a baseline” classical model, the authors noted that this only worked for short time windows, and the kinetic parameters depended on the chosen windows. This is in line with observations from other authors [145], and was analyzed by Grela and Colussi in their stochastic model [121]. The fractal model, on the contrary, was successfully applied to different time windows and laser excitation energies. Moreover, the calculated ${k}_{r}$ values were strongly correlated with the photonic efficiencies for NO degradation over different TiO₂ samples [146], although this correlation was not observed for other photocatalytic reactions (e.g., acetaldehyde degradation).

Fig. 10

Copyright 2017 American Chemical Society

Transient absorption decays observed at 500 nm for commercial anatase TiO₂ powders (Top: Kronos1001, bottom: PC500). The orange lines show the fits to the fractal model. Excitation was performed at 355 nm. Reprinted with permission from reference [128].

As described above, the Albery model assumes a Gaussian distribution of activation energies. Alternatively, the assumption of an exponential distribution ($p(x)\propto {e}^{-x}$, with $x\ge 0$), leads, in analogy with Eq. (37), to [126]:

$$\begin{array}{c}\left[c\right]\propto {\int }_{0}^{\infty }{e}^{-x}{e}^{-{k}_{0}t {e}^{-\gamma x}}\mathrm{d}x.\end{array}$$

(42)

In the limit ${k}_{0}t\gg 1$, the integral can be solved to get the power law time dependence [147]:

$$\begin{array}{c}\left[c\right]\propto {t}^{-\alpha }.\end{array}$$

(43)

With $\alpha =1/\gamma$ and $0<\alpha \le 1$ determining the width in energy dispersion. In the same way as for the KWW and Albery models, the power law decay can be interpreted as a superposition of first-order processes with unique rate constants [126]:

$$\begin{array}{c}\left[c\right]\propto {t}^{-\alpha }={\int }_{0}^{\infty }{g}_{\mathrm{Power \; Law}}\left({k}_{r}\right) {e}^{-{k}_{\mathrm{r}}t}\mathrm{d}{k}_{\mathrm{r}}.\end{array}$$

(44)

With:

$$\begin{array}{c}{g}_{\mathrm{Power \; Law}}\left({k}_{\mathrm{r}}\right)={\Gamma \left(\alpha \right)}^{-1}{\left(\frac{{k}_{\mathrm{r}}}{{k}_{0}}\right)}^{\alpha -1},\end{array}$$

(45)

where $\Gamma$ is the gamma function. An interesting observation is that, at long times, the fractal model (Eq. (41)) reduces to a power law time dependence for $\left[c\right]$.

Cowan et al. observed [12] power law decays on TA measurements of nanocrystalline TiO₂ electrodes under applied potentials. By means of the bias, the authors manipulated holes lifetimes; without bias, recombination was the main process, and it followed power law kinetics. However, at positive potentials the lifetime of holes was long enough for water oxidation to compete with recombination. Under these conditions, fittings were only successful when adding a second decay component given by a stretched exponential function (Eq. (30)), and related to the consumption of holes by the water oxidation reaction.

4.4 Stochastic models

In both classical and dispersive kinetic models, an underlying assumption is that charge carrier concentration $[c]$ is a continuous real-valued function of time. However, in nanocrystalline materials, each particle acts as a micro-reactor where the number of photogenerated charge carriers is at most a couple hundred, and thus $[c]$ is integer-valued. As an example, if 10 electron–hole pairs were generated in 100 particles, the classical view would wrongly assume that all particles would be populated with 0.1 pairs. The correct description of 90% empty particles and 10% particles with one electron–hole pair (neglecting the small number of particles populated with two or more pairs) thus calls for stochastic kinetics to be employed.

Indeed, Rothenberger et al. [42] derived a stochastic rate equation from simple assumptions, the most important being that the survival probability of a single electron–hole pair decreases exponentially with time, and that the initial distribution of pairs follows Poisson statistics. Using this expression, the authors successfully described the kinetic profiles, including the limiting cases of very low and very high initial occupancies, which result in first-order and second-order behaviors, respectively. These approximations were determined to be valid for an average number of pairs lower than 0.5 and larger than 30 per particle, respectively.

Another good example of stochastic treatments is that by Grela and Colussi, who performed numerical modeling of electron–hole recombination (and reactive processes) on the surface of colloidal TiO₂ particles [121]. In this simple but elegant model, the surface is modeled as a square 2D periodic lattice, and electron and holes are initially generated at random positions. Moreover, holes perform a random walk on the surface, while electrons are fixed (thus assuming that they are deeply trapped in colloidal TiO₂). Under these conditions, recombination in particles with multiple pairs follows dispersive second-order kinetics, with the rate coefficient approaching a ${t}^{-1/2}$ dependence. This behavior is explained by the inhomogeneity in the initial distribution of charge carriers on the surface: while electron and hole pairs generated at close positions promptly recombine, those farther apart survive for longer times. Electron–hole pairs with very long lifetimes are then responsible for the “baseline” behavior previously mentioned (Eq. (4)). Under low average numbers of photogenerated electron–hole per particle, the simulations correctly reproduce the experimentally observed [42] mono-exponential decays.

A remarkable success of this work is that, using a single set of kinetic parameters, it can correctly fit transient absorption decays for TiO₂ colloids of different sizes, in the presence or absence of molecular oxygen, and at low and high laser irradiances. On the other hand, the model is built on assumptions (mobile holes, immobile electrons) that may be valid only for colloids, and not, for instance, for powders such as Evonik P25 TiO₂, where electrons freely diffuse [148].

Related models based on charge carriers random walk were also developed, by Nelson and others [130, 131, 149, 150], to understand recombination processes in dye-sensitized solar cells. They pay special attention to electron traps, as they have been observed to reduce electronic conductivity by several orders of magnitude [149]. The specific charge transport model is indeed central to these methods. In the trapping–detrapping model [150], electrons are assumed to move through delocalized states in the CB, sporadically getting trapped and subsequently released by thermal fluctuations. In both rutile and anatase TiO₂, the charge of photogenerated electrons produces a distortion of the lattice that leads to self-trapping at Ti^IV sites (also called small polaron formation) [151,152,153]. There is thus a potential energy barrier for electron migration to another Ti^IV site. Random walk simulations employing this model and an exponential distribution of trap states nicely result in power law kinetics for recombination (Eq. (43)) [130], thus providing a physical basis for the application of this model. In addition, they can explain the strong dependence of the recombination rate in dye-sensitized systems on the applied bias, and have in some cases shown a remarkable agreement with experimental data using just one fitting parameter (Fig. 11) [130].

Fig. 11

Copyright 2001 by the American Physical Society

Normalized transient absorption decays for TiO₂ electrodes under different applied electrical potentials. Black dots show experimental data and gray lines are the results of continuous-time random walk simulations. Reprinted figure with permission from reference [130].

4.5 Practical considerations

From the above discussion, it may be concluded that the proper description of charge carrier kinetics requires a stochastic analysis. However, the development and application of such models is a big task on itself, and thus it is often outside the scope of TAS investigations. Moreover, fitting experimental data with these models is not straightforward, complicating the analysis. On the other hand, fitting to either classical or dispersive kinetic schemes tends to be much easier, and thus these models are more widespread, even while they are not physically as sound as stochastic ones.

Looking at their functional form, these different kinetic models may seem unrelated to each other. However, as described above, they are closely interrelated. Employing classical kinetics, and depending on the specific mechanism, mono-, multi-exponential, and second-order decays can be easily obtained (e.g., Equations (11), (18), and (28)). The “second-order with a baseline” function (Eq. (4)) is simply an empirical correction on the latter. As a step further in complexity, dispersive kinetic schemes are based on two alternative interpretations: the rate coefficient is time-dependent, or equivalently, it is constant but there is a distribution on its values. Thus, a superposition of first-order decays in which the rate constant follows a Lévy distribution leads to the KWW model. If the rate constants follows instead a log-normal distribution, we obtain the Albery model. The assumption of a distribution in the rate constants for a second-order decay leads to the fractal model. This model, in turn, is reduced to a power law decay at long times.

Depending on the system and processes of interest, any of these models could result in an adequate fitting to the experimental data. However, as in any reaction kinetics problem, a good fit to a specific model does not allow to establish a mechanism. Often, kinetic schemes with different physical justifications result in similar functional forms [154], a problem exacerbated by the often noisy profiles obtained from TA. To illustrate this point, in Fig. 12, we show the transient absorption decay of a commercial TiO₂ photocatalyst powder; six different models yield a satisfactory fitting. In other situations, the simultaneous occurrence of multiple processes requires the combination of several models to adequately describe the kinetic profiles [12].

Fig. 12

Transient absorption decay at 500 nm for a Millennium PC105 anatase TiO₂ powder (circles), together with fittings to six different kinetic models (colored lines). Excitation was performed at 355 nm and at a 5.1 mJ cm⁻² intensity, and detection was done in reflectance mode

Transient absorption spectroscopy is often used as a characterization tool in sets of photocatalytic materials, where correlations are sought between, e.g., decay lifetimes and photocatalytic rates. Therefore, if a particular model adequately describes the kinetics of the entire series, its application may be useful to obtain kinetic parameters and correlate with reactions rates, even though the physical basis of the model may not be clear. Nevertheless, in this case, it may be worth considering the use of a more transparent parameter, such as the time needed for the transient signal to reach 50% of its initial value [57, 155]. The same metric can be used in situations where the kinetic profiles are not easily described by these models, such as in the presence of electron donors or acceptors [156].

A prerequisite to apply these kinetic models is information on the spectral signatures on the charge carriers of interest (Sect. 3.1). TA signals are thus typically analyzed at a wavelength where contributions are mainly related to a specific species. Alternatively, an interesting approach that combines signal assignment and kinetic modeling is the so-called global target analysis [157]. Briefly, the full spectral data (“global”) are fitted as a function of time, on the basis of a pre-established kinetic model (“target”). Although not commonly employed for photocatalytic systems, there are some recent interesting examples, such as the ultrafast studies on graphitic carbon nitrides by Corp and Schlenker [87] or the work of Larsen et al. on CdSe/CdS core/shell quantum dots [158].

5 Conclusion

In this review, we have provided an overview of the application of the transient absorption spectroscopy to characterize charge carrier processes in various semiconductor nanomaterials serving a photocatalyst or photoelectrode. The analysis of photocatalysts by TAS starts with the assignment of the transient spectrum features to photogenerated electrons and holes. The main strategy to differentiate their contributions is to employ scavenger compounds that selectively react either with electrons or holes. For instance, the observed transient spectrum in presence of a hole scavenger can be attributed to photogenerated electrons. It is important to bear in mind, however, that the presence of the scavenger will most likely have a strong influence on the physicochemical and optical properties of the studied system. In general, there are no “innocuous” scavengers, and one should consider how the chosen scavenger may affect the system beyond their main function. As an alternative to chemical scavengers, if the semiconductor is configured as an electrode, it is also possible to apply bias to selectively remove charge carriers. For deeper characterization of the charge carriers, analysis with additional methods is required, such as EPR or TAS with advanced spectroscopic resolution.

The analysis of charge carrier kinetic profiles commonly entails the fitting of experimental data with an appropriate function. It is important to bear in mind that, in chemical kinetics, a good fit to a model cannot exclude the occurrence of other, perhaps more complex ones. In a best-case scenario, the strongest conclusion that could be reached is that the chosen model is adequate to describe the data. It is also important to recognize that, not rarely, kinetic schemes with different physical justifications can result in similar or even indistinguishable functional shapes. This problem may be exacerbated by the rather noisy profiles normally obtained in TA experiments.

The interpretation of the TAS data presented in this review should be helpful to characterize the reaction dynamics of charge carriers photogenerated in different photocatalysts, and thus to understand the properties of different photocatalytic systems and to specifically develop new photocatalysts with higher activities, longer charge carrier lifetimes or other improved properties.