Effective dielectric function of laser-pumped anatase nanoparticles: influence of free carriers trapping and depletion of valence band

1. Introduction

In the past three decades, electrical and optical properties of oxide- and nitride-based semiconductor nanomaterials have become an object for extensive theoretical and experimental studies. This is particularly due to the fact that in a number of cases these materials exhibit rather specific properties in the resonant or nonlinear interactions with light. In principle, this specificity allows us to consider them as the basis for constructing new material platforms in photonics, electronics, catalytic chemistry, etc. Among these semiconductor systems, the titania-based nanostructured materials attract significant interest due to peculiar electronic and optical properties of titanium dioxide as a wide-zone semiconductor. Because of a large amount of oxygen vacancies, this is a n-type semiconductor with the bandgap approximately equal to 3.0 eV or 3.2 eV, in the dependence on the titania modification (rutile or anatase) [1–3]. Anatase modification is characterized by an indirect type of the interband transition, whereas the direct transition is specific for rutile. This causes dramatic differences in the photo-induced behavior of these materials (in particular, a significantly slower decay of the photoconductivity in anatase compared to rutile [4]). At the macroscopic level, titanium dioxide exhibits high values of the refractive index n in the visible and the near UV regions; in accordance with the databases (see, e.g., [5]), the n value can reach 5.2 – 5.4 in the vicinity of the edge of the fundamental absorption band. Moreover, the real part of the permittivity${\epsilon }^{\prime }=n^2-k^2$ of TiO₂ due to large values of the absorption index$k$ is negative at the wavelengths < 300 nm. This causes excitation of the localized surface modes in low-dimensional titania-based nanoparticles (“the plasmonic behavior”) in the fundamental absorption band and manifests itself in the existence of relatively narrow extinction and depolarization degree peaks for diluted suspensions of these nanoparticles [6–8]. The high scattering efficiency of submicrometer-sized TiO₂ particles in combination with the relatively low absorption efficiency outside the fundamental absorption band give the reason to use compositions of densely packed particles as the high-performance light diffusers (for example, in the random lasing systems [9,10] or solar energy cells [11–14]).

It should be noted that a significant part of experimental and theoretical studies of optical and electronic properties of bulk and nanostructured titanium dioxide is related to its photocatalytic activity (see, e.g., [15–20]). However, despite the numerous works carried out in the past two decades, knowledge of the basic mechanisms of photo-induced charge transfer in the titania nanophase is still inadequate. In particular, intense laser pumping of TiO₂nanoparticles in the fundamental absorption band should lead to the expressed charge transfer between the valence and conduction bands and can sometimes cause dramatically different behavior of optical properties of nanoparticle ensembles (beginning from an increasing extinction of the ensemble with an increasing pump intensity and ending by “bleaching” of the ensemble due to partial depletion of the valence band [6]). Depending on the intensity and duration of laser pumping, the effect of charge carrier localization at the surface or bulk traps can have a key impact on the non-linear optical properties of titania nanoparticles. It can be assumed that transition between various scenarios of non-linear optical behavior of this material depending on irradiation conditions is governed by the relationship between the rates of the basic charge transfer processes (the interband transition, recombination, and trapping). Development of optical probes for quantitative characterization of the mechanisms relating the photo-induced charge transfer in the bulk and nanostructured semiconductor materials can be widely applied in photonics and catalytic chemistry. At present, the most popular techniques for such probes are the time-resolved photoluminescence, the light induced transient gating techniques, the angle-resolved photoemission, etc [21–25].

In this work we consider an approach to the recovery of the laser-mediated non-linear effective permittivity of semiconductor nanoparticles on the basis of measurements of the non-linear extinction and non-linear Rayleigh scattering.The measurements were carried out using the modified closed-aperture z-scan technique [26] with simultaneous detection of scattered light intensity under condition of pumping nanoparticle suspensions by the sequences of nanosecond laser pulses with various intensities and sequence durations. The diluted water suspensions of TiO₂ (anatase) nanoparticles were used as the probed medium.The diagrammatic Cole-Cole technique [27] was applied for interpretation of the recovered data on the non-linear behavior of effective permittivity of nanoparticles. Evolution of the imaginary part of effective permittivity with an increase in the pump intensity and duration of pulse sequences was considered in the framework of the recurrent kinetic model for the charge carrier populations in the valence and conduction bands.

2. Experimental technique and results

We used the water suspensions of titanium dioxide (anatase) nanoparticles as the probed samples. The suspensions were prepared on the basis of the Sigma Aldrich product # 637254 (spheroidal particles with an average diameter ≤25 nm) in the deionized water. The weight fraction of the particles in the suspension was 10⁻³. 10-mm-thick quartz cuvettes were filled by the suspensions and probed using the z-scan technique with simultaneous measurements of the Rayleigh scattering at the direct angle to the probe beam axis (Fig. 1). In our case, the closed aperture modification of the z-scan technique was applied. The pulsed YAG:Nd laser with the frequency conversion (LS-2134 from the Lotis TII company, the operating wavelengths are 355 nm and 532 nm, the pulse duration is 10 ns, the pulse repetition rate is 15 Hz, the maximal energy of the pulse is 2mJ at 355 nm and 8mJ at 532 nm) was applied as a source of probe light. The laser beam was focused using a quartz lens with the focal length equal to 110 mm; the sample cuvette was translated along the beam axis using a 1-D motorized linear translation stage Standa 8MT167-100 (the product of the Standa company). The pulse energy of laser light transmitted through the cuvette was measured using the light energy/power meter Gentec Maestro with a sensor Gentec Q12MF1.

 

Fig. 1 The scheme of the experimental setup. 1 – the pulsed YAG:Nd laser with the tripled frequency output; 2 – the quartz lens; 3 – the cuvette with the suspension of nanoparticles; 4 – the collimator; 5 – the spectrometer QE6500; 6,7 – the energy-power meter. The electric field in the laser beam is oriented perpendicularly to the scattering plane.

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Intensity of the scattered light was measured through the side wall of the cuvette using a unit consisting of a spectrometer Ocean Optics QE65000 and a collimator Ocean Optics 74DA fixed directly on the input fiber connector of the spectrometer. This arrangement provided a collection of the scattered light in the narrow angle and its focusing onto the entrance slit of the spectrometer, which was placed on the translation stage with the cuvette. The collimator axis was aligned perpendicularly to the probe beam axis and separation between the collimator lens and the probed volume inside the cuvette was approximately equal to 6 mm. Direction of the electric field in the probe beam was chosen perpendicular to the scattering plane (see Fig. 1) for maximization of the intensity of scattered light. The experimental setup was calibrated before measurements in order to evaluate the average intensity of the probe light in the central part of the probed volume inside the cuvette for its various positions along the beam axis. The calibration was made using gradual eclipsing of the beam by the sharp edge moving in the lateral direction with simultaneous measurements of the transmitted energy. It was found that the intensity profiles for the various cross-sections of the focused beam can be fitted with the appropriate accuracy by the Gaussian distributions. The estimated Rayleigh range of the probe beam and the distance between the waist plane and the focusing lens were used to calculate the average intensity in the probed volume depending on the cuvette position with respect to the waist plane. The corresponding calibration curve for 355 nm radiation is displayed in Fig. 2.

 

Fig. 2 2D plots of $I_{sc}^{norm}$(a) and $I_{tr}^{norm}$(b) against the number of acting pulses $N_p$ and the cuvette position with respect to the waist plane $z$ together with the calibration curve $I_p=f\left(z\right)$. The wavelength is 355 nm; the duration of laser pulses is 10 ns.

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The measurements were carried out in the following way: the cuvette placed at a given position along the z-axis was filled with the examined suspension and probed by a pulse sequence with the given number of pulses. Both detector units (5 and 6 in Fig. 1) captured the signals corresponding to the last pulse in the sequence).The integration time of the spectrometer QE65000 was established equal to the time interval between the sequential pulses. Synchronization was provided using programming of the control units of the detector and laser. After each probe, the probed suspension was removed from the cuvette and the cuvette was filled with a fresh amount of suspension. This procedure allowed us to exclude the influence of irradiation-induced degradation of the samples on the results. The refreshed samples were probed 5 times with the given number of pulses in the sequence at the given cuvette position with follow-up averaging of intensity value of the transmitted and scattered light. After this, the cuvette was translated to a new position with the step equal to 100 µm.

A reasonable question is how strong the translational Brownian dynamics and sedimentation of anatase nanoparticles can influence the results of our z-scan experiments. Both processes cause a transfer of laser-treated particles from the waist zone and untreated particles to the zone. Assessment of the translational diffusion coefficient for 25 nm-sized particles in the water at the room temperature using the Einstein-Stokes formula$D_t=kT/6\pi r\eta$ ($T$is the absolute temperature, $r$ is the particle radius, and $\eta$ is the dynamic viscosity of the liquid) gives the value ≈1.75·10⁻¹¹ m²/s. Consequently, the characteristic time of the particle displacement from the central part of the waist zone to its edge can be approximately evaluated as${\tau }_d\approx r_w^2/4D_t$≈3 s ($r_w$ is the radius of the beam waist). Note that the observed dramatic changes in the optical properties of laser-treated particles occur at sufficiently shorter times of the laser treatment (the sequence durations, see below). Moreover, our estimates of the sedimentation rate show a negligible contribution of this process to the particle transfer between the waist zone and surrounding space in the course of the laser treatment. Therefore, we can consider with an appropriate accuracy the ensembles of laser-treated particles as “frozen” stable systems.

Another question is related to a possible formation of vapor envelopes around the nanoparticles due to thermal treatment of the laser pulses; these envelopes can significantly affect the scattering and absorption of the laser light by particles. Our previous estimates of this effect carried out using the model of the spherical TiO₂ particle with a gaseous shell in the water and analysis of the behavior of the scattered and transmitted light depending on the pump intensity [28] showed a negligible role of this effect. This is presumably due to sufficiently lower efficiency of the light-to-heat transformation in anatase nanoparticles compared to metallic or carbon nanoparticles.

Figure 2 displays the collected experimental data for the 355 nm probe light as the 2D plots of normalized averaged intensity of the transmitted and scattered light for the last pulse in the sequence depending on the cuvette position (i.e., pump intensity) and the number of pulses in the sequence. The obtained values of the transmitted and scattered intensity in the linear regime (at the cuvette positions far from the waist zone) were used as the normalization factors. Figure 3 shows the dependencies $I_{sc}^{norm}\left(I_p,N_p=1\right)$ and $I_{tr}^{norm}\left(I_p,N_p=1\right)$ for the case of single-pulse probing at 532 nm. In this case the increase of the number of pulses in the acting sequence does not cause any remarkable changes in the $I_{sc}^{norm}\left(I_p,N_p\right)$ and$I_{tr}^{norm}\left(I_p,N_p\right)$ dependencies.

 

Fig. 3 The dependencies $I_{sc}^{norm}\left(I_p,N_p=1\right)$ and $I_{tr}^{norm}\left(I_p,N_p=1\right)$; the wavelength is 532 nm. The duration of laser pulses is 10 ns. Selectively shown error bars correspond to a significance level equal to 0.9.

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3. Interpretation of the experimental data

The obtained dependencies $I_{sc}^{norm}\left(I_p,N_p\right)$ and $I_{tr}^{norm}\left(I_p,N_p\right)$ were interpreted using the concept of the effective scattering system [28]. In the framework of this concept, the laser-pumped random ensemble of titania nanoparticles undergoing action of the last pulse in the sequence is considered as a random system of non-pumped nanoparticles consisting of a specific material with the certain values of the real and imaginary parts of dielectric function at the frequency of pumping light. The given effective system is characterized by the same values of the average diameter and the volume fraction of specific particles in the surrounding medium as the examined system. Besides, the effective system exhibits the same behavior in scattering and transmittance as the examined system. This means that the real and imaginary parts of the dielectric function of the specific material are dependent on the intensity of pumping light and duration of the pulse sequence.

Therefore, the ensemble-averaged scattering and absorption cross-sections of the specific spherical particles can be presented as [29]:

The intensity of the detected scattered light $I_{sc}\left(I_p,N_p\right)$is proportional to $〈{\sigma }_{sc}\left(I_p,N_p\right)〉N{I}_p$, where $N$ is the number of particles in the probed volume. On the other hand, the values $I_p$ and $N$ can be expressed as $I_p\approx E_p/S{\tau }_p$and $N~\rho S$, where $E_p$ and ${\tau }_p$ are energy and duration of a single laser pulse, $S$ is the cross-sectional area of the probe beam in the probed volume, and $\rho$ is particle concentration. Thus, the normalized intensity of the scattered light $I_{sc}^{norm}\left(I_p,N_p\right)$depends only on the values${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)$and${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$. Validity of this assumption is confirmed by the behavior of the dependencies$I_{sc}^{norm}\left(I_p,N_p\right)$; the normalized intensity of the scattered light does not depend on the cuvette position and equals 1 in range of the cuvette positions with $\left|z\right|\ge$20 mm (Fig. 2).This allows us to express the normalized intensity of the detected scattered light as

Considering the intensity of light transmitted through the cuvette, we can express it using the generalized Bougier law

Using the estimate for $1+〈{\sigma }_{abs}^{lin}〉/〈{\sigma }_{sc}^{lin}〉\approx$5.73 and the measured value $I_p^{lin}/I_{tr}^{lin}\approx$4.484 for the examined system, we can introduce the parameter $\Gamma \left(I_p,N_p\right)$ as $\Gamma \left(I_p,N_p\right)\approx 5.73\left\{\frac{\ln \left[4.484/I_{tr}^{norm}\left(I_p,N_p\right)\right]}{1.501I_{sc}^{norm}\left(I_p,N_p\right)}\right\}-1.$

Finally, we obtain the following equation for ${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)$and ${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$:

The system of Eqs. (4) and (7) with free terms in the right-hand sides defined by the measured values $I_{sc}^{norm}\left(I_p,N_p\right)$ and $I_{tr}^{norm}\left(I_p,N_p\right)$can be solved numerically to recover ${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)$and ${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$. We used the Newton algorithm for recovery; the sequential evaluation of${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)$ and ${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$was carried out using the set of values$I_{sc}^{norm}\left(I_p,N_p\right)$ and $I_{tr}^{norm}\left(I_p,N_p\right)$ for the given number of pulses $N_p$ in the sequence. The starting values ${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)$ and ${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$ for the given pump intensity at each recovery cycle were taken as the finally recovered values at a previous cycle.

4. Discussion of results

4.1 Pump-intensity-dependent Cole-Cole diagrams of anatase nanoparticles

In the further analysis, we applied the diagrammatic technique pioneered by K. S. Cole and R. H. Cole [27]. This technique is based on parametric presentation of the real and imaginary parts of permittivity of an examined material in the$\left({\epsilon }^{\prime },{\epsilon }^{″}\right)$plane. The Cole-Cole diagrams are widely applied for interpretation of the frequency dependencies of permittivity in the low-frequency region. Consequently, this diagrammatic technique is one of the basic approaches in the impedance spectroscopy of dielectric materials within the frequency range from 10² to 10¹⁰ Hz, which is used to identify the mechanisms of dielectric relaxation. Recovery of the Cole-Cole diagrams from the frequency dependencies${\epsilon }^{\prime }\left(\omega \right)$ and ${\epsilon }^{″}\left(\omega \right)$ is not widely used for the material characterization in the optical domain (at the frequencies 10¹⁴ ÷ 10¹⁷ Hz). However, applications of this approach to some bulk metals in the optical range are presented, for example, in [29,30].

We applied the Cole-Cole technique to the retrieved values$\widetilde{{\epsilon }^{\prime }}\left(I_p,N_p\right)$and $\widetilde{{\epsilon }^{″}}\left(I_p,N_p\right)$; Fig. 4(a) displays a family of the recovered intensity-dependent Cole-Cole diagrams in the case of small durations of the pumping pulse sequences ($N_p\le$ 16) at $\lambda =$355 nm. To compare, Fig. 4(b) shows the photon-energy-dependent Cole-Cole diagram recovered using the data on the optical constants of bulk anatase [5]. An increase in duration of the pulse sequences causes dramatic changes in the shapes of intensity-dependent Cole-Cole diagrams, illustrated by Fig. 5(a). These changes allow us to suggest that the non-linear interaction of the light with anatase nanoparticles in the fundamental absorption band is mainly controlled by fundamental changes in the density of local states of charge carriers in the nanoparticles. These changes are presumably due to the depletion of the ground state (the valence band) and trapping of mobile charge carriers in the conduction and valence bands. In other words, in this case the rate of upward interband transitions and population of the conduction band fall into small values at large durations of the pulse sequences (the “insulator-like” behavior of pumped material).

 

Fig. 4 The recovered Cole-Cole diagrams for anatase; a – the family of the intensity-dependent diagrams recovered from $I_{sc}^{norm}\left(I_p,N_p\right),I_{tr}^{norm}\left(I_p,N_p\right)$ for examined anatase nanoparticles in the case of “short” durations of the pumping pulse sequences; the wavelength of pumping light equals 355 nm; the pump intensities: i – 1.0·10⁶ W/cm²; ii – 1.0·10⁸ W/cm²; iii – 1.0·10⁹ W/cm²; iv– 1.0·10¹⁰ W/cm²; v– 1.08·10¹¹ W/cm²; the sequence durations $N_p$: 1 - 1; 2 - 4; 3 - 8; 4 - 10; 5 - 12; 6 - 16; selectively shown error bars correspond to a significance level equal to 0.9; b – the comparison of the photon-energy-dependent diagram (1) for bulk anatase (linear regime of light-material interaction) and the intensity-dependent Cole-Cole diagram (2) for examined anatase nanoparticles (corresponds to curve 1 in Fig. 4(a)); the arrows indicate the increase of the control parameters ($I_p$ and $h\nu$); $h\nu$ varies from 2.25 eV (i) to 5.65 eV (v).

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Fig. 5 a – the same as in Fig. 4(a), in the case of “long” durations of the acting pulse sequences; the sequence durations $N_p$: 1 - 16; 2 - 24; 3 - 31; 4 - 38; b – the intensity dependent Cole-Cole diagram for examined anatase nanoparticles in the case of the single pulse pumping at 532 nm; the pump intensities: i – 1.0·10⁷ W/cm²; ii – 1.0·10⁸ W/cm²; iii – 1.0·10⁹ W/cm²; iv– 1.0·10¹⁰ W/cm²; v– 1.0·10¹¹ W/cm²; vi – 1.0·10¹² W/cm²; selectively shown error bars correspond to a significance level equal to 0.9.

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To compare, Fig. 5(b) shows the intensity-dependent Cole-Cole diagram for the examined material in the case of single pulse pumping outside the fundamental absorption band (at 532 nm).

The dependencies ${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)=f\left\{{\tilde{\epsilon }}^{″}\left(I_p,N_p\right)\right\}$recovered for small sequence durations (Fig. 4(a)) can be fitted with good accuracy within the used range of $I_p$ by the opened elliptical lines. This behavior indicates a resonant character of the pumping light interaction with the anatase nanophase. The obtained Cole-Cole diagrams were compared to parametric dependencies ${\tilde{\epsilon }}^{\prime }\left(\omega \right)=f\left\{{\tilde{\epsilon }}^{″}\left(\omega \right)\right\}$ for the single-oscillator Lorentz model of the dielectric function (see, e.g., [29]):

 

Fig. 6 An overlay of the recovered intensity- and duration-dependent Cole-Cole diagrams (colored curves 1, 2, 3 with markers) by the parametric dependencies (8) corresponding to the single-oscillator Lorentz model (a set of solid black curves). The coloring of the curves 1-3 corresponds to the coloring used in Fig. 4(a).

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This interpretation allows us to conclude that:

The values of the resonance frequency, plasma frequency, and damping parameter can be estimated for the non-pumped bulk anatase from the dependencies of the optical constants on the photon energy [5] as${\omega }_0\approx$6.24·10¹⁵ Hz, ${\omega }_p\approx$1.18·10¹⁶Hz, and $\gamma \approx$ 1.32·10¹⁵ Hz. Thus, the following relationship${\left({\omega }_0^2-{\omega }^2\right)}^2\ll {\omega }^2{\gamma }^2$ takes place in the case of pumping the examined system at 355 nm. A decrease in${\tilde{\epsilon }}^{\prime }\left(I_p,N_p\right)$ with the simultaneous increase in the imaginary part${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$ for the values of pump intensity exceeding ≈5·10⁸ W/cm²and small $N_p$ values (see Fig. 4(a) under the increasing plasma frequency can be caused only by the diminishing term$\left({\omega }_0^2-{\omega }^2\right)$. Therefore we can suggest the occurrence of redshift of the resonant frequency towards the frequency of pumping light with the increasing pump intensity. Figure 7 displays the dependencies of the “excess” imaginary part of the effective dielectric function on the pump intensity for various durations of the pulse sequences.

 

Fig. 7 The dependencies of the “excess” imaginary part of the effective dielectric function on $I_p$ for examined anatase nanoparticles. Durations of acting pulse sequences $N_p$: 1 – 1; 2 – 4; 3 – 10; 4 – 16; selectively shown error bars correspond to a significance level equal to 0.9.

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The “excess”, or radiation-induced part of ${\tilde{\epsilon }}^{″}\left(I_p,N_p\right)$ was introduced as the difference between the recovered and undisturbed imaginary parts: ${{\tilde{\epsilon }}^{″}}_{excess}\left(I_p,N_p\right)={\tilde{\epsilon }}^{″}\left(I_p,N_p\right)-{\tilde{\epsilon }}^{″}\left(0,0\right)$. These peculiarities of the recovered dependencies ${{\tilde{\epsilon }}^{″}}_{excess}\left(I_p,N_p\right)$ as a non-monotonic behavior and occurrence of the relatively narrow peaks for small durations of pulse sequences ($N_p\le$5 – 6) can be considered in terms of $I_p,N_p$-dependent optical conductivity ${\sigma }_{oc}$ of laser-irradiated semiconductor materials. Indeed, ${\tilde{\epsilon }}^{″}\approx {{\tilde{\epsilon }}^{″}}_{core}+{{\tilde{\epsilon }}^{″}}_{excess}$with ${{\tilde{\epsilon }}^{″}}_{core}\ll {{\tilde{\epsilon }}^{″}}_{excess}~{\sigma }_{oc}$in the case of strong interband transition (${{\tilde{\epsilon }}^{″}}_{core}$is the so-called “core” imaginary part of the dielectric function independent of the radiation conditions).

On the other hand, ${\sigma }_{oc}~P_{oc}$, where $P_{oc}$ is the number of excess mobile carriers in the nanoparticle volume. With the small durations of the pulse sequences, $P_{oc}$ rises with an increasing pump intensity up to the value corresponding to the condition $\eta =$1 ($\omega ={\omega }_0$) in the framework of the considered Lorentz model (the ascending branches of the curves 1, 2). A further increase in $P_{oc}$ at larger pump intensities will cause an increasing deviation of the resonance frequency on the pump frequency (the descending branches). Note that the “critical” value of ${{\tilde{\epsilon }}^{″}}_{excess}$ corresponding to the condition $\omega ={\omega }_0$, approximately equals 53.5 (see the horizontal dashed line in Fig. 7).

At large durations of pulse sequences, $P_{oc}$ in the examined nanoparticles tends to decrease (presumably due to depletion and trapping effects, see the next subsection). Consequently, the real and imaginary parts of the effective dielectric function gradually approach the initial values characteristic for the untreated system (Fig. 5(a)). We can estimate the “critical” duration of the pulse sequence, which corresponds to transition from the accumulation of excess mobile carriers to their vanishing in our case as $N_p\approx$12 (curve 3 in Fig. 7). An evident impossibility to reach the condition ${{\tilde{\epsilon }}^{″}}_{core}\ll {{\tilde{\epsilon }}^{″}}_{excess}{}_{oc}$ in the absence of interband transition is clearly illustrated by the comparison of Fig. 4(a) and Fig. 5(b).

4.2 Recurrent kinetic model for evolution of mobile carrier concentration in anatase nanoparticles under pulsed laser pumping

Let us consider a kinetic model describing the photo-induced changes in the charge carrier density due to laser pumping by the sequence of light pulses. We denote the population of absorbing centers (coupled charge carriers) in the ground state (the valence band), which are ready for photoionization, as $N_g^{(k)}$; the superscript “(k)” means that this population occurs in the beginning of $k$-th pumping pulse. Accordingly, $N_{oc}^{(k)}$ is the population of mobile negative and positive charge carriers in the conduction and valence bands; subscript “oc” means “optical conductivity”. At the initial stage of laser pumping (the single-pulse action):

It should be noted that the discussed model has a sufficiently more generalized form compared to other kinetic models applied for a detailed description of the photoindused charge transfer in semiconductors (see, e.g., [31]). In particular, the factor $\beta$ takes into account all the contributions of various radiative and non-radiative recombination mechanisms leading to restoration of the population of absorbing centers ready for the follow-up photoionization during the next laser pulse action. These mechanisms could be a direct radiative recombination, exciton annihilation, Auger recombination, etc [32–34]. The factor $\gamma$ deals with charge carriers (electrons, holes, e-type and h-type polarons) which irretrievably transit from the free (mobile) state to the arrested state. The corresponding variety of mechanisms causes decay in optical conductivity and depletion of the ground state. Despite the general character, the considered model seems useful for establishing the relationship between the classical and quantum parameters of the examined system (for example, via the link $N_{oc}\to {\sigma }_{oc}\to {\tilde{\epsilon }}^{″}\to {\omega }_p,{\omega }_0$).

Preliminary consideration of the behavior of $N_{oc}^{(k)}$ depending on the pulse sequence duration for various values of the model parameters $\alpha ,\beta ,\gamma$ and comparison of the obtained results with the behavior of the examined system allowed us to assume that contribution of recombination processes in the evolution of the effective dielectric function of anatase nanoparticles is rather negligible ($\beta <\gamma \ll \alpha$). Therefore we have set $\beta \approx$0 and focused on the analysis of competition of photoionization and trapping channels as the main factors controlling the rapid growth of $N_{oc}^{(k)}$ with further decay. The selected modeled data are presented in Fig. 8, where the normalized populations ${\tilde{N}}_{oc}^{(k)}$are shown depending on the sequence duration $N_p$ and photoionization efficiency $\alpha$ for various values of the trapping efficiency $\gamma$ (the initial population of the ground state $N_g^{(0)}$ was applied as the normalization parameter).

 

Fig. 8 2D plots of ${\tilde{N}}_{oc}^{(k)}$ against $\alpha$ and $N_p$ for various values of $\gamma$. a – $\gamma =$ 0.01; b – $\gamma =$ 0.05; c – $\gamma =$ 0.20; d – $\gamma =$ 0.40.

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The photoionization efficiency $\alpha$ significantly affects the ${\tilde{N}}_{oc}^{(k)}$ peak value and, to a much lesser extent, the duration of the growth-decay process; the latter parameter is much more sensitive to variations of the trapping efficiency $\gamma$ (Fig. 8). We introduced the value $N_p^{0.5}$ as duration of the pulse sequence, which corresponds to the twofold decay in ${\tilde{N}}_{oc}^{(k)}$ with respect to its peak value. Further, we considered the value ${〈N_p^{0.5}〉}_{\alpha }$ averaged over all possible values of photoionization efficiency in the range 0 $<\alpha <$1 as the generalized parameter establishing the relationship between the duration of the growth-decay process for ${\tilde{N}}_{oc}^{(k)}$ and the trapping efficiency. The dependence ${〈N_p^{0.5}〉}_{\alpha }=f\left(\gamma \right)$ is presented in Fig. 9; the circle marker corresponds to the examined anatase nanoparticles; respectively, $\gamma \approx$0.15.

 

Fig. 9 Dependence of ${〈N_p^{0.5}〉}_{\alpha }$on the trapping factor $\gamma$. The circle marker corresponds to the examined system.

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4.3 Peculiarities of photoinduced charge transfer in anatase nanoparticles

Based on the obtained experimental and modeled data as well as the previously reported results of photoindused conductivity of anatase particles (see, e.g., [35]), we can postulate a slow decay of the concentration of laser-generated excess mobile carriers. This decay is mainly controlled by arresting the carriers in the deep traps with the depth values significantly exceeding $kT$, and the gradual depletion of the valence band (the decrease in concentration of the coupled carriers ready for photoionization). The role of recombination mechanisms leading to restoration of the ground state population is rather insignificant. The data relating the microwave conductivity of anatase particles after the action of the laser pulse in the fundamental absorption band ($\lambda =$266 nm), which were reported by Schindler and Kunst [35], clearly show the power-law decay in ${\sigma }_{oc}$ at large time scales significantly exceeding the pulse duration: ${\sigma }_{oc}~t^{-\delta }$ with $\delta \approx$ 0.1. This behavior can be considered as an indirect confirmation of the insufficient role of recombination mechanisms; these mechanisms are usually characterized by significantly stronger dependence of the decay rate on the time lapse (see, e.g., [36]). In our case, assuming the similar power-law decay of$N_{oc}$ between the sequential laser pulses, we can introduce the characteristic decay time$t_d$, which corresponds to the decrease in $N_{oc}$ during the time interval between the pulses, which was estimated from the experimental data (Fig. 9).

It allows us to establish the following relationship between $t_d$ and$\gamma$: ${\left(T_p/t_d\right)}^{-0.1}=1-\gamma \approx$0.85 ($T_p$ is the time interval between the laser pulses) and to estimate$t_d$ as ≈13 ms. This value, which can be considered as the characteristic dwell time of a photoinduced mobile carrier in the anatase nanoparticle before it will be captured into the localized state by a deep trap, is many times greater than the usually reported lifetime values for free carriers in the anatase. Therefore we can assume the prevailing hopping conductivity with e-type and h-type polarons as the charge carriers and large arresting times for these carriers between sequential hops. Indeed, anatase in the single crystal form is characterized by a large steepness parameter, which indicates the strong coupling between phonons and charge carriers [37]. Transition from a single crystal to a nanostructured anatase should cause further increase in the steepness parameter associated with the increasing Urbach energy [37]; in this case, the probability the occurring of phonon-coupled charge states increases. In turn, this should lead to a dramatically decreasing charge mobility. It should be noted that the role of such photoinduced charge transfer mechanisms in the high photocatalytic and photovoltaic efficiency of anatase nanostructures is the object of recent intensive studies [38–43].

5. Conclusions

The considered techniques of the analysis and interpretation of the experimental data on the non-linear scattering and absorption of the laser light by nanoparticles (the recovery of the intensity- and duration-dependent effective dielectric function at the frequency of the probing light, the Cole-Cole diagrammatic technique, and the recurrent kinetic model of the photoinduced charge transfer) seem to be a useful addition to the already existing experimental techniques of characterization of light interactions with dispersive nanosystems. Applicability of the Lorentz model for the description of the recovered intensity-dependent effective dielectric functions of anatase nanoparticles at the various durations of pumping pulse sequences seems rather surprising. Usually various modifications of the Drude model (e.g., the Drude-Smith model [42],) were applied to describe the photoinduced conductivity in the similar system. In our opinion, the observed “resonance-like” behavior of the recovered effective dielectric functions can be presumably related to a strong phonon-charge coupling in the examined system. In combination with the low-efficient recombination of photoinduced charges and abundance of bulk and surface traps in anatase nanoparticles, this leads to a slow decay of the photoinduced conductivity and gradual depletion in the ensemble of potential elementary absorbers of pumping radiation. These peculiarities do not contradict this well-established feature of the anatase nanophase as an extremely high photocatalitic activity. These issues will be the object for future extended studies.