Abstract

Anatase nanoparticles were examined using the z-scan technique with simultaneous measurements of the intensity of the scattered laser light. In the course of the z-scan experiments, the average intensity of probing laser pulses varied between 1.0·106 W/cm2 and 1.08·1011 W/cm2 at the wavelength equal to 355 nm and between 1.0·107 W/cm2 and 1.0·1012 W/cm2 at 532 nm. The pulse duration was equal to 10 ns in all cases. A method for recovery of an effective dielectric function of nanoparticles is suggested. It was found that, in the case of the interband transition, the recovered dielectric functions for a short duration of the laser pulse sequences can be fitted by parametric dependencies corresponding to the Lorentz model. A kinetic model describing the changes in the population of mobile carriers is considered. It was found that the efficiency of the charge recombination is considerably less than the efficiency of the trapping. The dwell time of the mobile charge carriers before being captured was estimated as ≈13 ms.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 permittivityε=n2k2 of TiO2 due to large values of the absorption indexk 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.

 figure: Fig. 1

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.

 figure: Fig. 2

Fig. 2 2D plots of Iscnorm(a) and Itrnorm(b) against the number of acting pulses Np and the cuvette position with respect to the waist plane z together with the calibration curve Ip=f(z). 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 formulaDt=kT/6πrη (Tis the absolute temperature, r is the particle radius, and η 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 asτdrw2/4Dt≈3 s (rw 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 Iscnorm(Ip,Np=1) and Itrnorm(Ip,Np=1) 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 Iscnorm(Ip,Np) andItrnorm(Ip,Np) dependencies.

 figure: Fig. 3

Fig. 3 The dependencies Iscnorm(Ip,Np=1) and Itrnorm(Ip,Np=1); 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 Iscnorm(Ip,Np) and Itrnorm(Ip,Np) 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]:

{σsc(Ip,Np)=k4v218π|ε˜(Ip,Np)1|2{27[ε˜(Ip,Np)+2]2+ε˜2(Ip,Np)};σabs(Ip,Np)=kv3{27[ε˜(Ip,Np)+2]2+ε˜2(Ip,Np)}ε˜(Ip,Np),
where ε˜(Ip,Np) and ε˜(Ip,Np)are the normalized intensity- and duration-dependent real and imaginary parts of the dielectric function of particles at the frequency of the pumping light ε˜(Ip,Np)=ε(Ip,Np)/ε0;ε˜(Ip,Np)=ε(Ip,Np)/ε0,where ε0is the permittivity of the surrounding medium), v is the average volume of particle, and k is the wave number of the pumping light in the surrounding medium.

The intensity of the detected scattered light Isc(Ip,Np)is proportional to σsc(Ip,Np)NIp, where N is the number of particles in the probed volume. On the other hand, the values Ip and N can be expressed as IpEp/Sτpand N~ρS, where Ep and τp are energy and duration of a single laser pulse, S is the cross-sectional area of the probe beam in the probed volume, and ρ is particle concentration. Thus, the normalized intensity of the scattered light Iscnorm(Ip,Np)depends only on the valuesε˜(Ip,Np)andε˜(Ip,Np). Validity of this assumption is confirmed by the behavior of the dependenciesIscnorm(Ip,Np); the normalized intensity of the scattered light does not depend on the cuvette position and equals 1 in range of the cuvette positions with |z|20 mm (Fig. 2).This allows us to express the normalized intensity of the detected scattered light as

Iscnorm(Ip,Np)=σsc(Ip,Np)σsclin=[ε˜(Ip,Np)1]2+ε˜2(Ip,Np)[ε˜lin1]2+(ε˜lin)2××[ε˜lin+2]2+(ε˜lin)2[ε˜(Ip,Np)+2]2+ε˜2(Ip,Np),
where the notation “lin” means that the corresponding values satisfy the condition of the linear behavior of the probed system. Let us introduce the parameter Φ(Ip,Np) as
Φ(Ip,Np)=Iscnorm(Ip,Np)[ε˜lin1]2+(ε˜lin)2[ε˜lin+2]2+(ε˜lin)2.
The value Φlin corresponding to Iscnorm = 1 (the linear behavior of the probed system) can be estimated as = 0.567 using the values ε˜lin = 9.67 and ε˜lin = 2.19 expected for the bulk anatase at 355 nm (see, e.g., [5]).Using this parameter, we can derive the following equation forε˜(Ip,Np)andε˜(Ip,Np):

Φ(Ip,Np)[ε˜(Ip,Np)1]2+ε˜2(Ip,Np)[ε˜(Ip,Np)+2]2+ε˜2(Ip,Np).

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

Itr(Ip,Np)Ipexp{ρd[σsc(Ip,Np)+σabs(Ip,Np)]},
whered is the cuvette thickness. This expression can be transformed in the following way
ln[Ip/Itr(Ip,Np)]ln[Iplin/Itrlin]σsc(Ip,Np)+σabs(Ip,Np)σsclin+σabslin.
After transformations of Eq. (6), we arrive at ln{Iplin/[Itrnorm(Ip,Np)Itrlin]}ln[Iplin/Itrlin]σsc(Ip,Np)σsclin1+σabs(Ip,Np)/σsc(Ip,Np)1+σabslin/σsclin withσsc(Ip,Np)/σsclinIscnorm(Ip,Np)and σabs(Ip,Np)σsc(Ip,Np)6πk3vε˜(Ip,Np)[ε˜(Ip,Np)1]2+ε˜2(Ip,Np).

Using the estimate for 1+σabslin/σsclin5.73 and the measured value Iplin/Itrlin4.484 for the examined system, we can introduce the parameter Γ(Ip,Np) as Γ(Ip,Np)5.73{ln[4.484/Itrnorm(Ip,Np)]1.501Iscnorm(Ip,Np)}1.

Finally, we obtain the following equation for ε˜(Ip,Np)and ε˜(Ip,Np):

Γ(Ip,Np)172.7ε˜(Ip,Np)[ε˜(Ip,Np)1]2+ε˜2(Ip,Np).
The numerical coefficient in the right-hand side of Eq. (7) is defined by the parameter 6π/k3v of the examined system at the given wavelength (the above numerical values correspond to 355 nm).

The system of Eqs. (4) and (7) with free terms in the right-hand sides defined by the measured values Iscnorm(Ip,Np) and Itrnorm(Ip,Np)can be solved numerically to recover ε˜(Ip,Np)and ε˜(Ip,Np). We used the Newton algorithm for recovery; the sequential evaluation ofε˜(Ip,Np) and ε˜(Ip,Np)was carried out using the set of valuesIscnorm(Ip,Np) and Itrnorm(Ip,Np) for the given number of pulses Np in the sequence. The starting values ε˜(Ip,Np) and ε˜(Ip,Np) 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(ε,ε)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 dependenciesε(ω) and ε(ω) 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 valuesε˜(Ip,Np)and ε˜(Ip,Np); 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 (Np 16) at λ=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).

 figure: Fig. 4

Fig. 4 The recovered Cole-Cole diagrams for anatase; a – the family of the intensity-dependent diagrams recovered from Iscnorm(Ip,Np),Itrnorm(Ip,Np) 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 Np: 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 (Ip and hν); hν varies from 2.25 eV (i) to 5.65 eV (v).

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 figure: Fig. 5

Fig. 5 a – the same as in Fig. 4(a), in the case of “long” durations of the acting pulse sequences; the sequence durations Np: 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 ε˜(Ip,Np)=f{ε˜(Ip,Np)}recovered for small sequence durations (Fig. 4(a)) can be fitted with good accuracy within the used range of Ip 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 ε˜(ω)=f{ε˜(ω)} for the single-oscillator Lorentz model of the dielectric function (see, e.g., [29]):

{ε˜=1+ζ2(1η2)(1η2)2+μ2η2;ε˜=ζ2μη(1η2)2+μ2η2.
Here η, ζ, and μ are the electromagnetic field frequencyω, the plasma frequencyωp, and the damping parameterγ of the Lorentz system normalized by the resonant frequency ω0 of the system. The results of such comparison are presented in Fig. 6 for various values of ζ2/μ. Note that the Cole-Cole diagrams for the ideal Lorentz system in the whole range 0ω< are also opened due to different asymptotic values of ε˜ forω andω0. Figure 6 displays the overlay effect of the diagramsε˜(Ip,Np)=ψ[ε˜(Ip,Np)] recovered from the experimental data by a set of “Lorentzian” curves corresponding to the variation of theζ2/μparameter within the range from 30 to 58.

 figure: Fig. 6

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:

  • - each pair of valuesε˜(Ip,Np), ε˜(Ip,Np)is uniquely associated with the pair ε˜(ω), ε˜(ω) corresponding to the dielectric function of the ideal Lorentz system calculated for a certain value of ζ2/μ parameter;
  • - the behavior of ε˜(Ip,Np) and ε˜(Ip,Np) with an increasing intensity of the pumping light within the range 1Np10 can be interpreted in the framework of the Lorentz model resulting from an increase in the plasma frequency (due to the growing concentration of mobile charge carriers in the conductivity and valence bands) as well as a decrease in the resonant frequency as. Indeed, the real part of the dielectric function of Lorentz medium is expressed as ε˜(ω)=1+ωp2(ω02ω2)/[(ω02ω2)2+γ2ω2], and the imaginary part is written as ε˜(ω)=ωp2γω/[(ω02ω2)2+γ2ω2].

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] asω06.24·1015 Hz, ωp1.18·1016Hz, and γ 1.32·1015 Hz. Thus, the following relationship(ω02ω2)2ω2γ2 takes place in the case of pumping the examined system at 355 nm. A decrease inε˜(Ip,Np) with the simultaneous increase in the imaginary partε˜(Ip,Np) for the values of pump intensity exceeding ≈5·108 W/cm2and small Np values (see Fig. 4(a) under the increasing plasma frequency can be caused only by the diminishing term(ω02ω2). 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.

 figure: Fig. 7

Fig. 7 The dependencies of the “excess” imaginary part of the effective dielectric function on Ip for examined anatase nanoparticles. Durations of acting pulse sequences Np: 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 ε˜(Ip,Np) was introduced as the difference between the recovered and undisturbed imaginary parts: ε˜excess(Ip,Np)=ε˜(Ip,Np)ε˜(0,0). These peculiarities of the recovered dependencies ε˜excess(Ip,Np) as a non-monotonic behavior and occurrence of the relatively narrow peaks for small durations of pulse sequences (Np5 – 6) can be considered in terms of Ip,Np-dependent optical conductivity σoc of laser-irradiated semiconductor materials. Indeed, ε˜ε˜core+ε˜excesswith ε˜coreε˜excess~σocin the case of strong interband transition (ε˜coreis the so-called “core” imaginary part of the dielectric function independent of the radiation conditions).

On the other hand, σoc~Poc, where Poc is the number of excess mobile carriers in the nanoparticle volume. With the small durations of the pulse sequences, Poc rises with an increasing pump intensity up to the value corresponding to the condition η=1 (ω=ω0) in the framework of the considered Lorentz model (the ascending branches of the curves 1, 2). A further increase in Poc 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 ε˜excess corresponding to the condition ω=ω0, approximately equals 53.5 (see the horizontal dashed line in Fig. 7).

At large durations of pulse sequences, Poc 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 Np12 (curve 3 in Fig. 7). An evident impossibility to reach the condition ε˜coreε˜excessoc 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 Ng(k); the superscript “(k)” means that this population occurs in the beginning of k-th pumping pulse. Accordingly, Noc(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):

{Noc(1)=Ng(0)α(1βγ);Ng(1)=Ng(0)Ng(0)α(1β).
Hereα is the dimensionless parameter quantifying the efficiency of photoionization due to interband transitions under laser pumping;α is dependent of the pulse duration, pump intensity, and quantum yield of photo-ionization.βis the dimensionless parameter quantifying the efficiency of carrier recombination and Ng restoration during the time interval between two sequential pulses. γis the dimensionless parameter quantifying the efficiency of the mobile carriers arresting by deep traps during the same time interval.This channel of the charge transfer causes a gradual decrease in Noc (decay in optical conductivity) and Ng (depletion of the ground state). We can write the following set of equations for Ng(k):
{Ng(2)=Ng(1)Ng(1)α(1β)=Ng(0){1αf(I)(1β)};Ng(3)=Ng(2)Ng(2)α(1β)=Ng(0){1αf(I)(1β)}3;...............;Ng(k)=Ng(0){1αf(I)(1β)}k.
Accordingly, the similar set of recurrent equations for Noc(k) has the following form

{Noc(2)={Ng(0)α(1βγ)+Ng(1)α}(1βγ)==Ng(0)α{(1γ)+(1β)(1α)}(1βγ);Noc(3)=(Noc(2)+Ng(2)α)(1βγ)==Ng(0)α[{(1γ)+(1β)(1α)}(1βγ)2+{1α(1β)}2(1βγ)];Noc(4)=(Noc(3)+Ng(3)α)(1βγ)==Ng(0)αf(I)[{(1γ)+(1β)(1α)}(1βγ)3+{1α(1β)}2(1βγ)2++{1α(1β)}3(1βγ)];.......................;Noc(k)=Ng(0)α[{(1γ)+(1β)(1α)}(1βγ)k1+m=2k1{1α(1β)}m(1βγ)km]

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 β 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 γ 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 Nocσocε˜ωp,ω0).

Preliminary consideration of the behavior of Noc(k) depending on the pulse sequence duration for various values of the model parameters α,β,γ 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 (β<γα). Therefore we have set β0 and focused on the analysis of competition of photoionization and trapping channels as the main factors controlling the rapid growth of Noc(k) with further decay. The selected modeled data are presented in Fig. 8, where the normalized populations N˜oc(k)are shown depending on the sequence duration Np and photoionization efficiency α for various values of the trapping efficiency γ (the initial population of the ground state Ng(0) was applied as the normalization parameter).

 figure: Fig. 8

Fig. 8 2D plots of N˜oc(k) against α and Np for various values of γ. a – γ= 0.01; b – γ= 0.05; c – γ= 0.20; d – γ= 0.40.

Download Full Size | PPT Slide | PDF

The photoionization efficiency α significantly affects the 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 γ (Fig. 8). We introduced the value Np0.5 as duration of the pulse sequence, which corresponds to the twofold decay in N˜oc(k) with respect to its peak value. Further, we considered the value Np0.5α averaged over all possible values of photoionization efficiency in the range 0 <α<1 as the generalized parameter establishing the relationship between the duration of the growth-decay process for N˜oc(k) and the trapping efficiency. The dependence Np0.5α=f(γ) is presented in Fig. 9; the circle marker corresponds to the examined anatase nanoparticles; respectively, γ0.15.

 figure: Fig. 9

Fig. 9 Dependence of Np0.5αon the trapping factor γ. The circle marker corresponds to the examined system.

Download Full Size | PPT Slide | PDF

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 (λ=266 nm), which were reported by Schindler and Kunst [35], clearly show the power-law decay in σoc at large time scales significantly exceeding the pulse duration: σoc~tδ with δ 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 ofNoc between the sequential laser pulses, we can introduce the characteristic decay timetd, which corresponds to the decrease in Noc 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 td andγ: (Tp/td)0.1=1γ0.85 (Tp is the time interval between the laser pulses) and to estimatetd 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.

Funding

Russian Science Foundation (RSF) (16-19-10455); Ministry of Education and Science of the Russian Federation (3.7567.2017).

Acknowledgments

The authors acknowledge the support of this work concerning the experimental study and data interpretation from the Russian Science Foundation. DAZ also acknowledges the support concerning the modeling by the Ministry of Education and Science of the Russian Federation.

Disclosures

The authors declare that there are no conflicts of interest relating this article.

References

1. A. Beltrán, L. Gracia, and J. Andrés, “Density functional theory study of the brookite surfaces and phase transitions between natural titania polymorphs,” J. Phys. Chem. B 110(46), 23417–23423 (2006). [CrossRef]   [PubMed]  

2. N. Daude, C. Gout, and C. Jouanin, “Electronic band structure of titanium dioxide,” Phys. Rev. B. 15(6), 3229–3235 (1977). [CrossRef]  

3. D. Mardare, M. Tasca, M. Delibas, and G. I. Rusu, “On the structural properties and optical transmittance of TiO2r.f. sputtered thin films,” Appl. Surf. Sci. 156(1–4), 200–206 (2000). [CrossRef]  

4. M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011). [CrossRef]   [PubMed]  

5. http://matprop.ru/img/nk/Oxides/tio2b.gif.

6. D. A. Zimnyakov, O. V. Ushakova, A. V. Gorokhovsky, E. V. Tretyachenko, E. A. Isaeva, A. A. Isaeva, and A. B. Pravdin, “Resonant scattering and absorption in the titanate-based nanoplatelet dispersions in near ultraviolet region,” Appl. Opt. 51(16), 3675–3683 (2012). [CrossRef]   [PubMed]  

7. D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012). [CrossRef]  

8. D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015). [CrossRef]  

9. R. G. S. El-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption,” Appl. Phys. Lett. 98(16), 161106 (2011). [CrossRef]  

10. R. C. Polson and Z. V. Vardeny, “Organic random lasers in the weak-scattering regime,” Phys. Rev. B Condens. Matter Mater. Phys. 71(4), 045205 (2005). [CrossRef]  

11. B. O’Regan and M. Grätzel, “A low-cost, high-efficiency solar cell based on dye sensitized colloidal TiO2 films,” Nature 353(6346), 737–740 (1991). [CrossRef]  

12. M. Grätzel, “Photoelectrochemical cells,” Nature 414(6861), 338–344 (2001). [CrossRef]   [PubMed]  

13. A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110(11), 6595–6663 (2010). [CrossRef]   [PubMed]  

14. A. J. Frank, N. Kopidakis, and J. van de Lagemaat, “Electrons in nanostructured TiO2 solar cells: transport, recombination and photovoltaic properties,” Coord. Chem. Rev. 248(13–14), 1165–1179 (2004). [CrossRef]  

15. A. Fujishima and K. Honda, “Electrochemical photolysis of water at a semiconductor electrode,” Nature 238(5358), 37–38 (1972). [CrossRef]   [PubMed]  

16. E. Pelizzetti and C. Minero, “Mechanism of the photo-oxidative degradation of organic pollutants over TiO2 particles,” Electrochim. Acta 38(1), 47–55 (1993). [CrossRef]  

17. Y. Nakaoka and Y. Nosaka, “ESR investigation into the effects of heat treatment and crystal structure on radicals produced over irradiated TiO2 powder,” J. Photochem. Photobiol. Chem. 110(3), 299–305 (1997). [CrossRef]  

18. M. A. Henderson, “A surface science perspective on TiO2 photocatalysis,” Surf. Sci. Rep. 66(6–7), 185–297 (2011). [CrossRef]  

19. M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011). [CrossRef]   [PubMed]  

20. A. Fujishima, X. Zhang, and D. A. Tryk, “TiO2 photocatalysis and related surface phenomena,” Surf. Sci. Rep. 63(12), 515–582 (2008). [CrossRef]  

21. J. E. Kroeze, T. J. Savenije, and J. M. Warman, “Electrodeless determination of the trap density, decay kinetics, and charge separation efficiency of dye-sensitized nanocrystalline TiO2.,” J. Am. Chem. Soc. 126(24), 7608–7618 (2004). [CrossRef]   [PubMed]  

22. P. Ščajev, K. Jarašiūnas, S. Okur, Ü. Özgür, and H. Morkoç, “Carrier dynamics in bulk GaN,” J. Appl. Phys. 111(2), 023702 (2012). [CrossRef]  

23. T. S. Sosnowski, T. B. Norris, H. H. Wang, P. Grenier, J. F. Whitaker, and C. Y. Sung, “High-carrier-density electron dynamics in low-temperature-grown GaAs,” Appl. Phys. Lett. 70(24), 3245–3247 (1997). [CrossRef]  

24. C. K. Yong, H. J. Joyce, J. Lloyd-Hughes, Q. Gao, H. H. Tan, C. Jagadish, M. B. Johnston, and L. M. Herz, “Ultrafast dynamics of exciton formation in semiconductor nanowires,” Small 8(11), 1725–1731 (2012). [CrossRef]   [PubMed]  

25. J. M. Lantz and R. M. Corn, “Time-resolved optical second harmonic generation measurements of picosecond band flattening processes at single crystal TiO2 electrodes,” J. Phys. Chem. 98(38), 9387–9390 (1994). [CrossRef]  

26. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. H. Hagan, and E. W. van Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]  

27. K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics – I. Alternating current characteristics,” J. Chem. Phys. 9(4), 341–3512 (1941). [CrossRef]  

28. D. A. Zimnyakov and S. A. Yuvchenko, “Effective dielectric function of TiO2 nanoparticles under laser pumping in the fundamental absorption band,” Quantum Electron. 47(6), 547–552 (2017). [CrossRef]  

29. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

30. A. E.-H. Seoud and M. Shibab, “Numerical calculation of the wave-vector dependent Cole-Cole diagram of copper,” IJSR 4(11), 424–426 (2015). [CrossRef]  

31. C. Soci, D. Moses, Q.-H. Xu, and A. J. Heeger, “Charge-carrier relaxation dynamics in highly ordered poly(p-phenylenevinylene):Effects of carrier bimolecular recombination and trapping,” Phys. Rev. B. 72(24), 245204 (2005). [CrossRef]  

32. J. Schneider, M. Matsuoka, M. Takeuchi, J. Zhang, Y. Horiuchi, M. Anpo, and D. W. Bahnemann, “Understanding TiO2 photocatalysis: mechanisms and materials,” Chem. Rev. 114(19), 9919–9986 (2014). [CrossRef]   [PubMed]  

33. D. P. Colombo Jr., K. A. Roussel, J. Saeh, D. E. Skinner, J. J. Cavaleri, and R. M. Bowman, “Femtosecond study of the intensity dependence of electron-hole dynamics in TiO2nanoclusters,” Chem. Phys. Lett. 232(3), 207–214 (1995). [CrossRef]  

34. H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014). [CrossRef]  

35. K.-M. Schindler and M. Kunst, “Charge-carrier dynamics in TiO2 powders,” J. Phys. Chem. 94(21), 8222–8226 (1990). [CrossRef]  

36. C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008). [CrossRef]  

37. H. Tang, F. Lévy, H. Berger, and P. E. Schmid, “Urbach tail of anatase TiO2.,” Phys. Rev. B Condens. Matter 52(11), 7771–7774 (1995). [CrossRef]   [PubMed]  

38. C. Spreafico and J. VandeVondele, “The nature of excess electrons in anatase and rutile from hybrid DFT and RPA,” Phys. Chem. Chem. Phys. 16(47), 26144–26152 (2014). [CrossRef]   [PubMed]  

39. S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013). [CrossRef]   [PubMed]  

40. C. Di Valentin and A. Selloni, “Bulk and surface polarons in photoexcited anatase TiO2,” J. Phys. Chem. Lett. 2(17), 2223–2228 (2011). [CrossRef]  

41. E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017). [CrossRef]   [PubMed]  

42. M. C. Fravventura, D. Deligiannis, J. M. Schins, L. D. A. Siebbs, and T. J. Savenije, “What limits photoconductance in anatase TiO2 nanostructures? A real and imaginary microwave conductance study,” J. Phys. Chem. C 117(16), 8032–8040 (2013). [CrossRef]  

43. M. C. Fravventura, L. D. A. Siebbeles, and T. J. Savenije, “Mechanisms of photogeneration and relaxation of excitons and mobile carriers in anatase TiO2,” J. Phys. Chem. C 118(14), 7337–7343 (2014). [CrossRef]  

References

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  • |

  1. A. Beltrán, L. Gracia, and J. Andrés, “Density functional theory study of the brookite surfaces and phase transitions between natural titania polymorphs,” J. Phys. Chem. B 110(46), 23417–23423 (2006).
    [Crossref] [PubMed]
  2. N. Daude, C. Gout, and C. Jouanin, “Electronic band structure of titanium dioxide,” Phys. Rev. B. 15(6), 3229–3235 (1977).
    [Crossref]
  3. D. Mardare, M. Tasca, M. Delibas, and G. I. Rusu, “On the structural properties and optical transmittance of TiO2r.f. sputtered thin films,” Appl. Surf. Sci. 156(1–4), 200–206 (2000).
    [Crossref]
  4. M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
    [Crossref] [PubMed]
  5. http://matprop.ru/img/nk/Oxides/tio2b.gif .
  6. D. A. Zimnyakov, O. V. Ushakova, A. V. Gorokhovsky, E. V. Tretyachenko, E. A. Isaeva, A. A. Isaeva, and A. B. Pravdin, “Resonant scattering and absorption in the titanate-based nanoplatelet dispersions in near ultraviolet region,” Appl. Opt. 51(16), 3675–3683 (2012).
    [Crossref] [PubMed]
  7. D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012).
    [Crossref]
  8. D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
    [Crossref]
  9. R. G. S. El-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption,” Appl. Phys. Lett. 98(16), 161106 (2011).
    [Crossref]
  10. R. C. Polson and Z. V. Vardeny, “Organic random lasers in the weak-scattering regime,” Phys. Rev. B Condens. Matter Mater. Phys. 71(4), 045205 (2005).
    [Crossref]
  11. B. O’Regan and M. Grätzel, “A low-cost, high-efficiency solar cell based on dye sensitized colloidal TiO2 films,” Nature 353(6346), 737–740 (1991).
    [Crossref]
  12. M. Grätzel, “Photoelectrochemical cells,” Nature 414(6861), 338–344 (2001).
    [Crossref] [PubMed]
  13. A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110(11), 6595–6663 (2010).
    [Crossref] [PubMed]
  14. A. J. Frank, N. Kopidakis, and J. van de Lagemaat, “Electrons in nanostructured TiO2 solar cells: transport, recombination and photovoltaic properties,” Coord. Chem. Rev. 248(13–14), 1165–1179 (2004).
    [Crossref]
  15. A. Fujishima and K. Honda, “Electrochemical photolysis of water at a semiconductor electrode,” Nature 238(5358), 37–38 (1972).
    [Crossref] [PubMed]
  16. E. Pelizzetti and C. Minero, “Mechanism of the photo-oxidative degradation of organic pollutants over TiO2 particles,” Electrochim. Acta 38(1), 47–55 (1993).
    [Crossref]
  17. Y. Nakaoka and Y. Nosaka, “ESR investigation into the effects of heat treatment and crystal structure on radicals produced over irradiated TiO2 powder,” J. Photochem. Photobiol. Chem. 110(3), 299–305 (1997).
    [Crossref]
  18. M. A. Henderson, “A surface science perspective on TiO2 photocatalysis,” Surf. Sci. Rep. 66(6–7), 185–297 (2011).
    [Crossref]
  19. M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
    [Crossref] [PubMed]
  20. A. Fujishima, X. Zhang, and D. A. Tryk, “TiO2 photocatalysis and related surface phenomena,” Surf. Sci. Rep. 63(12), 515–582 (2008).
    [Crossref]
  21. J. E. Kroeze, T. J. Savenije, and J. M. Warman, “Electrodeless determination of the trap density, decay kinetics, and charge separation efficiency of dye-sensitized nanocrystalline TiO2.,” J. Am. Chem. Soc. 126(24), 7608–7618 (2004).
    [Crossref] [PubMed]
  22. P. Ščajev, K. Jarašiūnas, S. Okur, Ü. Özgür, and H. Morkoç, “Carrier dynamics in bulk GaN,” J. Appl. Phys. 111(2), 023702 (2012).
    [Crossref]
  23. T. S. Sosnowski, T. B. Norris, H. H. Wang, P. Grenier, J. F. Whitaker, and C. Y. Sung, “High-carrier-density electron dynamics in low-temperature-grown GaAs,” Appl. Phys. Lett. 70(24), 3245–3247 (1997).
    [Crossref]
  24. C. K. Yong, H. J. Joyce, J. Lloyd-Hughes, Q. Gao, H. H. Tan, C. Jagadish, M. B. Johnston, and L. M. Herz, “Ultrafast dynamics of exciton formation in semiconductor nanowires,” Small 8(11), 1725–1731 (2012).
    [Crossref] [PubMed]
  25. J. M. Lantz and R. M. Corn, “Time-resolved optical second harmonic generation measurements of picosecond band flattening processes at single crystal TiO2 electrodes,” J. Phys. Chem. 98(38), 9387–9390 (1994).
    [Crossref]
  26. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. H. Hagan, and E. W. van Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [Crossref]
  27. K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics – I. Alternating current characteristics,” J. Chem. Phys. 9(4), 341–3512 (1941).
    [Crossref]
  28. D. A. Zimnyakov and S. A. Yuvchenko, “Effective dielectric function of TiO2 nanoparticles under laser pumping in the fundamental absorption band,” Quantum Electron. 47(6), 547–552 (2017).
    [Crossref]
  29. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  30. A. E.-H. Seoud and M. Shibab, “Numerical calculation of the wave-vector dependent Cole-Cole diagram of copper,” IJSR 4(11), 424–426 (2015).
    [Crossref]
  31. C. Soci, D. Moses, Q.-H. Xu, and A. J. Heeger, “Charge-carrier relaxation dynamics in highly ordered poly(p-phenylenevinylene):Effects of carrier bimolecular recombination and trapping,” Phys. Rev. B. 72(24), 245204 (2005).
    [Crossref]
  32. J. Schneider, M. Matsuoka, M. Takeuchi, J. Zhang, Y. Horiuchi, M. Anpo, and D. W. Bahnemann, “Understanding TiO2 photocatalysis: mechanisms and materials,” Chem. Rev. 114(19), 9919–9986 (2014).
    [Crossref] [PubMed]
  33. D. P. Colombo, K. A. Roussel, J. Saeh, D. E. Skinner, J. J. Cavaleri, and R. M. Bowman, “Femtosecond study of the intensity dependence of electron-hole dynamics in TiO2nanoclusters,” Chem. Phys. Lett. 232(3), 207–214 (1995).
    [Crossref]
  34. H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014).
    [Crossref]
  35. K.-M. Schindler and M. Kunst, “Charge-carrier dynamics in TiO2 powders,” J. Phys. Chem. 94(21), 8222–8226 (1990).
    [Crossref]
  36. C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008).
    [Crossref]
  37. H. Tang, F. Lévy, H. Berger, and P. E. Schmid, “Urbach tail of anatase TiO2.,” Phys. Rev. B Condens. Matter 52(11), 7771–7774 (1995).
    [Crossref] [PubMed]
  38. C. Spreafico and J. VandeVondele, “The nature of excess electrons in anatase and rutile from hybrid DFT and RPA,” Phys. Chem. Chem. Phys. 16(47), 26144–26152 (2014).
    [Crossref] [PubMed]
  39. S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
    [Crossref] [PubMed]
  40. C. Di Valentin and A. Selloni, “Bulk and surface polarons in photoexcited anatase TiO2,” J. Phys. Chem. Lett. 2(17), 2223–2228 (2011).
    [Crossref]
  41. E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
    [Crossref] [PubMed]
  42. M. C. Fravventura, D. Deligiannis, J. M. Schins, L. D. A. Siebbs, and T. J. Savenije, “What limits photoconductance in anatase TiO2 nanostructures? A real and imaginary microwave conductance study,” J. Phys. Chem. C 117(16), 8032–8040 (2013).
    [Crossref]
  43. M. C. Fravventura, L. D. A. Siebbeles, and T. J. Savenije, “Mechanisms of photogeneration and relaxation of excitons and mobile carriers in anatase TiO2,” J. Phys. Chem. C 118(14), 7337–7343 (2014).
    [Crossref]

2017 (2)

D. A. Zimnyakov and S. A. Yuvchenko, “Effective dielectric function of TiO2 nanoparticles under laser pumping in the fundamental absorption band,” Quantum Electron. 47(6), 547–552 (2017).
[Crossref]

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

2015 (2)

A. E.-H. Seoud and M. Shibab, “Numerical calculation of the wave-vector dependent Cole-Cole diagram of copper,” IJSR 4(11), 424–426 (2015).
[Crossref]

D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

2014 (4)

J. Schneider, M. Matsuoka, M. Takeuchi, J. Zhang, Y. Horiuchi, M. Anpo, and D. W. Bahnemann, “Understanding TiO2 photocatalysis: mechanisms and materials,” Chem. Rev. 114(19), 9919–9986 (2014).
[Crossref] [PubMed]

H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014).
[Crossref]

C. Spreafico and J. VandeVondele, “The nature of excess electrons in anatase and rutile from hybrid DFT and RPA,” Phys. Chem. Chem. Phys. 16(47), 26144–26152 (2014).
[Crossref] [PubMed]

M. C. Fravventura, L. D. A. Siebbeles, and T. J. Savenije, “Mechanisms of photogeneration and relaxation of excitons and mobile carriers in anatase TiO2,” J. Phys. Chem. C 118(14), 7337–7343 (2014).
[Crossref]

2013 (2)

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

M. C. Fravventura, D. Deligiannis, J. M. Schins, L. D. A. Siebbs, and T. J. Savenije, “What limits photoconductance in anatase TiO2 nanostructures? A real and imaginary microwave conductance study,” J. Phys. Chem. C 117(16), 8032–8040 (2013).
[Crossref]

2012 (4)

P. Ščajev, K. Jarašiūnas, S. Okur, Ü. Özgür, and H. Morkoç, “Carrier dynamics in bulk GaN,” J. Appl. Phys. 111(2), 023702 (2012).
[Crossref]

C. K. Yong, H. J. Joyce, J. Lloyd-Hughes, Q. Gao, H. H. Tan, C. Jagadish, M. B. Johnston, and L. M. Herz, “Ultrafast dynamics of exciton formation in semiconductor nanowires,” Small 8(11), 1725–1731 (2012).
[Crossref] [PubMed]

D. A. Zimnyakov, O. V. Ushakova, A. V. Gorokhovsky, E. V. Tretyachenko, E. A. Isaeva, A. A. Isaeva, and A. B. Pravdin, “Resonant scattering and absorption in the titanate-based nanoplatelet dispersions in near ultraviolet region,” Appl. Opt. 51(16), 3675–3683 (2012).
[Crossref] [PubMed]

D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012).
[Crossref]

2011 (5)

M. A. Henderson, “A surface science perspective on TiO2 photocatalysis,” Surf. Sci. Rep. 66(6–7), 185–297 (2011).
[Crossref]

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

R. G. S. El-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption,” Appl. Phys. Lett. 98(16), 161106 (2011).
[Crossref]

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

C. Di Valentin and A. Selloni, “Bulk and surface polarons in photoexcited anatase TiO2,” J. Phys. Chem. Lett. 2(17), 2223–2228 (2011).
[Crossref]

2010 (1)

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110(11), 6595–6663 (2010).
[Crossref] [PubMed]

2008 (2)

A. Fujishima, X. Zhang, and D. A. Tryk, “TiO2 photocatalysis and related surface phenomena,” Surf. Sci. Rep. 63(12), 515–582 (2008).
[Crossref]

C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008).
[Crossref]

2006 (1)

A. Beltrán, L. Gracia, and J. Andrés, “Density functional theory study of the brookite surfaces and phase transitions between natural titania polymorphs,” J. Phys. Chem. B 110(46), 23417–23423 (2006).
[Crossref] [PubMed]

2005 (2)

R. C. Polson and Z. V. Vardeny, “Organic random lasers in the weak-scattering regime,” Phys. Rev. B Condens. Matter Mater. Phys. 71(4), 045205 (2005).
[Crossref]

C. Soci, D. Moses, Q.-H. Xu, and A. J. Heeger, “Charge-carrier relaxation dynamics in highly ordered poly(p-phenylenevinylene):Effects of carrier bimolecular recombination and trapping,” Phys. Rev. B. 72(24), 245204 (2005).
[Crossref]

2004 (2)

J. E. Kroeze, T. J. Savenije, and J. M. Warman, “Electrodeless determination of the trap density, decay kinetics, and charge separation efficiency of dye-sensitized nanocrystalline TiO2.,” J. Am. Chem. Soc. 126(24), 7608–7618 (2004).
[Crossref] [PubMed]

A. J. Frank, N. Kopidakis, and J. van de Lagemaat, “Electrons in nanostructured TiO2 solar cells: transport, recombination and photovoltaic properties,” Coord. Chem. Rev. 248(13–14), 1165–1179 (2004).
[Crossref]

2001 (1)

M. Grätzel, “Photoelectrochemical cells,” Nature 414(6861), 338–344 (2001).
[Crossref] [PubMed]

2000 (1)

D. Mardare, M. Tasca, M. Delibas, and G. I. Rusu, “On the structural properties and optical transmittance of TiO2r.f. sputtered thin films,” Appl. Surf. Sci. 156(1–4), 200–206 (2000).
[Crossref]

1997 (2)

T. S. Sosnowski, T. B. Norris, H. H. Wang, P. Grenier, J. F. Whitaker, and C. Y. Sung, “High-carrier-density electron dynamics in low-temperature-grown GaAs,” Appl. Phys. Lett. 70(24), 3245–3247 (1997).
[Crossref]

Y. Nakaoka and Y. Nosaka, “ESR investigation into the effects of heat treatment and crystal structure on radicals produced over irradiated TiO2 powder,” J. Photochem. Photobiol. Chem. 110(3), 299–305 (1997).
[Crossref]

1995 (2)

D. P. Colombo, K. A. Roussel, J. Saeh, D. E. Skinner, J. J. Cavaleri, and R. M. Bowman, “Femtosecond study of the intensity dependence of electron-hole dynamics in TiO2nanoclusters,” Chem. Phys. Lett. 232(3), 207–214 (1995).
[Crossref]

H. Tang, F. Lévy, H. Berger, and P. E. Schmid, “Urbach tail of anatase TiO2.,” Phys. Rev. B Condens. Matter 52(11), 7771–7774 (1995).
[Crossref] [PubMed]

1994 (1)

J. M. Lantz and R. M. Corn, “Time-resolved optical second harmonic generation measurements of picosecond band flattening processes at single crystal TiO2 electrodes,” J. Phys. Chem. 98(38), 9387–9390 (1994).
[Crossref]

1993 (1)

E. Pelizzetti and C. Minero, “Mechanism of the photo-oxidative degradation of organic pollutants over TiO2 particles,” Electrochim. Acta 38(1), 47–55 (1993).
[Crossref]

1991 (1)

B. O’Regan and M. Grätzel, “A low-cost, high-efficiency solar cell based on dye sensitized colloidal TiO2 films,” Nature 353(6346), 737–740 (1991).
[Crossref]

1990 (2)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. H. Hagan, and E. W. van Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

K.-M. Schindler and M. Kunst, “Charge-carrier dynamics in TiO2 powders,” J. Phys. Chem. 94(21), 8222–8226 (1990).
[Crossref]

1977 (1)

N. Daude, C. Gout, and C. Jouanin, “Electronic band structure of titanium dioxide,” Phys. Rev. B. 15(6), 3229–3235 (1977).
[Crossref]

1972 (1)

A. Fujishima and K. Honda, “Electrochemical photolysis of water at a semiconductor electrode,” Nature 238(5358), 37–38 (1972).
[Crossref] [PubMed]

1941 (1)

K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics – I. Alternating current characteristics,” J. Chem. Phys. 9(4), 341–3512 (1941).
[Crossref]

Andrés, J.

A. Beltrán, L. Gracia, and J. Andrés, “Density functional theory study of the brookite surfaces and phase transitions between natural titania polymorphs,” J. Phys. Chem. B 110(46), 23417–23423 (2006).
[Crossref] [PubMed]

Angelsky, O. V.

D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

Anpo, M.

J. Schneider, M. Matsuoka, M. Takeuchi, J. Zhang, Y. Horiuchi, M. Anpo, and D. W. Bahnemann, “Understanding TiO2 photocatalysis: mechanisms and materials,” Chem. Rev. 114(19), 9919–9986 (2014).
[Crossref] [PubMed]

Auböck, G.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

Bahnemann, D. W.

J. Schneider, M. Matsuoka, M. Takeuchi, J. Zhang, Y. Horiuchi, M. Anpo, and D. W. Bahnemann, “Understanding TiO2 photocatalysis: mechanisms and materials,” Chem. Rev. 114(19), 9919–9986 (2014).
[Crossref] [PubMed]

Baldini, E.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

Ballantyne, A. M.

C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008).
[Crossref]

Barišic, O. S.

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

Beltrán, A.

A. Beltrán, L. Gracia, and J. Andrés, “Density functional theory study of the brookite surfaces and phase transitions between natural titania polymorphs,” J. Phys. Chem. B 110(46), 23417–23423 (2006).
[Crossref] [PubMed]

Berger, H.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

H. Tang, F. Lévy, H. Berger, and P. E. Schmid, “Urbach tail of anatase TiO2.,” Phys. Rev. B Condens. Matter 52(11), 7771–7774 (1995).
[Crossref] [PubMed]

Bernhard, C.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

Boschloo, G.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110(11), 6595–6663 (2010).
[Crossref] [PubMed]

Bostwick, A.

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

Bowman, R. M.

D. P. Colombo, K. A. Roussel, J. Saeh, D. E. Skinner, J. J. Cavaleri, and R. M. Bowman, “Femtosecond study of the intensity dependence of electron-hole dynamics in TiO2nanoclusters,” Chem. Phys. Lett. 232(3), 207–214 (1995).
[Crossref]

Bradley, D. D. C.

C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008).
[Crossref]

Cavaleri, J. J.

D. P. Colombo, K. A. Roussel, J. Saeh, D. E. Skinner, J. J. Cavaleri, and R. M. Bowman, “Femtosecond study of the intensity dependence of electron-hole dynamics in TiO2nanoclusters,” Chem. Phys. Lett. 232(3), 207–214 (1995).
[Crossref]

Chang, Y. J.

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

Chergui, M.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

Chiodo, L.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

Cole, K. S.

K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics – I. Alternating current characteristics,” J. Chem. Phys. 9(4), 341–3512 (1941).
[Crossref]

Cole, R. H.

K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics – I. Alternating current characteristics,” J. Chem. Phys. 9(4), 341–3512 (1941).
[Crossref]

Colombo, D. P.

D. P. Colombo, K. A. Roussel, J. Saeh, D. E. Skinner, J. J. Cavaleri, and R. M. Bowman, “Femtosecond study of the intensity dependence of electron-hole dynamics in TiO2nanoclusters,” Chem. Phys. Lett. 232(3), 207–214 (1995).
[Crossref]

Corn, R. M.

J. M. Lantz and R. M. Corn, “Time-resolved optical second harmonic generation measurements of picosecond band flattening processes at single crystal TiO2 electrodes,” J. Phys. Chem. 98(38), 9387–9390 (1994).
[Crossref]

Daude, N.

N. Daude, C. Gout, and C. Jouanin, “Electronic band structure of titanium dioxide,” Phys. Rev. B. 15(6), 3229–3235 (1977).
[Crossref]

de Mello, J.

C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008).
[Crossref]

Delibas, M.

D. Mardare, M. Tasca, M. Delibas, and G. I. Rusu, “On the structural properties and optical transmittance of TiO2r.f. sputtered thin films,” Appl. Surf. Sci. 156(1–4), 200–206 (2000).
[Crossref]

Deligiannis, D.

M. C. Fravventura, D. Deligiannis, J. M. Schins, L. D. A. Siebbs, and T. J. Savenije, “What limits photoconductance in anatase TiO2 nanostructures? A real and imaginary microwave conductance study,” J. Phys. Chem. C 117(16), 8032–8040 (2013).
[Crossref]

Di Valentin, C.

C. Di Valentin and A. Selloni, “Bulk and surface polarons in photoexcited anatase TiO2,” J. Phys. Chem. Lett. 2(17), 2223–2228 (2011).
[Crossref]

Dominguez, A.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

Durrant, J. R.

C. G. Shuttle, B. O’Regan, A. M. Ballantyne, J. Nelson, D. D. C. Bradley, J. de Mello, and J. R. Durrant, “Experimental determination of the rate law for charge carrier decay in a polythiophene: Fullerene solar cell,” Appl. Phys. Lett. 92(9), 093311 (2008).
[Crossref]

El-Dardiry, R. G. S.

R. G. S. El-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption,” Appl. Phys. Lett. 98(16), 161106 (2011).
[Crossref]

Forró, L.

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

Frank, A. J.

A. J. Frank, N. Kopidakis, and J. van de Lagemaat, “Electrons in nanostructured TiO2 solar cells: transport, recombination and photovoltaic properties,” Coord. Chem. Rev. 248(13–14), 1165–1179 (2004).
[Crossref]

Fravventura, M. C.

M. C. Fravventura, L. D. A. Siebbeles, and T. J. Savenije, “Mechanisms of photogeneration and relaxation of excitons and mobile carriers in anatase TiO2,” J. Phys. Chem. C 118(14), 7337–7343 (2014).
[Crossref]

M. C. Fravventura, D. Deligiannis, J. M. Schins, L. D. A. Siebbs, and T. J. Savenije, “What limits photoconductance in anatase TiO2 nanostructures? A real and imaginary microwave conductance study,” J. Phys. Chem. C 117(16), 8032–8040 (2013).
[Crossref]

Fujishima, A.

A. Fujishima, X. Zhang, and D. A. Tryk, “TiO2 photocatalysis and related surface phenomena,” Surf. Sci. Rep. 63(12), 515–582 (2008).
[Crossref]

A. Fujishima and K. Honda, “Electrochemical photolysis of water at a semiconductor electrode,” Nature 238(5358), 37–38 (1972).
[Crossref] [PubMed]

Gao, Q.

C. K. Yong, H. J. Joyce, J. Lloyd-Hughes, Q. Gao, H. H. Tan, C. Jagadish, M. B. Johnston, and L. M. Herz, “Ultrafast dynamics of exciton formation in semiconductor nanowires,” Small 8(11), 1725–1731 (2012).
[Crossref] [PubMed]

Gao, Y.

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

Gorokhovsky, A. V.

D. A. Zimnyakov, O. V. Ushakova, A. V. Gorokhovsky, E. V. Tretyachenko, E. A. Isaeva, A. A. Isaeva, and A. B. Pravdin, “Resonant scattering and absorption in the titanate-based nanoplatelet dispersions in near ultraviolet region,” Appl. Opt. 51(16), 3675–3683 (2012).
[Crossref] [PubMed]

D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012).
[Crossref]

Gout, C.

N. Daude, C. Gout, and C. Jouanin, “Electronic band structure of titanium dioxide,” Phys. Rev. B. 15(6), 3229–3235 (1977).
[Crossref]

Gracia, L.

A. Beltrán, L. Gracia, and J. Andrés, “Density functional theory study of the brookite surfaces and phase transitions between natural titania polymorphs,” J. Phys. Chem. B 110(46), 23417–23423 (2006).
[Crossref] [PubMed]

Grätzel, M.

M. Grätzel, “Photoelectrochemical cells,” Nature 414(6861), 338–344 (2001).
[Crossref] [PubMed]

B. O’Regan and M. Grätzel, “A low-cost, high-efficiency solar cell based on dye sensitized colloidal TiO2 films,” Nature 353(6346), 737–740 (1991).
[Crossref]

Grenier, P.

T. S. Sosnowski, T. B. Norris, H. H. Wang, P. Grenier, J. F. Whitaker, and C. Y. Sung, “High-carrier-density electron dynamics in low-temperature-grown GaAs,” Appl. Phys. Lett. 70(24), 3245–3247 (1997).
[Crossref]

Grioni, M.

E. Baldini, L. Chiodo, A. Dominguez, M. Palummo, S. Moser, M. Yazdi-Rizi, G. Auböck, B. P. P. Mallett, H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Rubio, and M. Chergui, “Strongly bound excitons in anatase TiO2 single crystals and nanoparticles,” Nat. Commun. 8(1), 13 (2017).
[Crossref] [PubMed]

S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, “Tunable polaronic conduction in anatase TiO2.,” Phys. Rev. Lett. 110(19), 196403 (2013).
[Crossref] [PubMed]

Hagan, D. H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. H. Hagan, and E. W. van Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Hagfeldt, A.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110(11), 6595–6663 (2010).
[Crossref] [PubMed]

Heeger, A. J.

C. Soci, D. Moses, Q.-H. Xu, and A. J. Heeger, “Charge-carrier relaxation dynamics in highly ordered poly(p-phenylenevinylene):Effects of carrier bimolecular recombination and trapping,” Phys. Rev. B. 72(24), 245204 (2005).
[Crossref]

Henderson, M. A.

M. A. Henderson, “A surface science perspective on TiO2 photocatalysis,” Surf. Sci. Rep. 66(6–7), 185–297 (2011).
[Crossref]

Herz, L. M.

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Tasca, M.

D. Mardare, M. Tasca, M. Delibas, and G. I. Rusu, “On the structural properties and optical transmittance of TiO2r.f. sputtered thin films,” Appl. Surf. Sci. 156(1–4), 200–206 (2000).
[Crossref]

Terashige, T.

H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014).
[Crossref]

Tokura, Y.

H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014).
[Crossref]

Tret’yachenko, E. V.

D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012).
[Crossref]

Tretyachenko, E. V.

Tryk, D. A.

A. Fujishima, X. Zhang, and D. A. Tryk, “TiO2 photocatalysis and related surface phenomena,” Surf. Sci. Rep. 63(12), 515–582 (2008).
[Crossref]

Uchida, R.

H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014).
[Crossref]

Ushakova, O. V.

D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012).
[Crossref]

D. A. Zimnyakov, O. V. Ushakova, A. V. Gorokhovsky, E. V. Tretyachenko, E. A. Isaeva, A. A. Isaeva, and A. B. Pravdin, “Resonant scattering and absorption in the titanate-based nanoplatelet dispersions in near ultraviolet region,” Appl. Opt. 51(16), 3675–3683 (2012).
[Crossref] [PubMed]

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A. J. Frank, N. Kopidakis, and J. van de Lagemaat, “Electrons in nanostructured TiO2 solar cells: transport, recombination and photovoltaic properties,” Coord. Chem. Rev. 248(13–14), 1165–1179 (2004).
[Crossref]

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M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. H. Hagan, and E. W. van Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

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[Crossref] [PubMed]

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[Crossref]

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T. S. Sosnowski, T. B. Norris, H. H. Wang, P. Grenier, J. F. Whitaker, and C. Y. Sung, “High-carrier-density electron dynamics in low-temperature-grown GaAs,” Appl. Phys. Lett. 70(24), 3245–3247 (1997).
[Crossref]

Wang, Y.

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

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J. E. Kroeze, T. J. Savenije, and J. M. Warman, “Electrodeless determination of the trap density, decay kinetics, and charge separation efficiency of dye-sensitized nanocrystalline TiO2.,” J. Am. Chem. Soc. 126(24), 7608–7618 (2004).
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T. S. Sosnowski, T. B. Norris, H. H. Wang, P. Grenier, J. F. Whitaker, and C. Y. Sung, “High-carrier-density electron dynamics in low-temperature-grown GaAs,” Appl. Phys. Lett. 70(24), 3245–3247 (1997).
[Crossref]

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M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

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M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

M. Xu, Y. Gao, E. M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, and C. Wöll, “Photocatalytic activity of bulk TiO2 anatase and rutile single crystals using infrared absorption spectroscopy,” Phys. Rev. Lett. 106(13), 138302 (2011).
[Crossref] [PubMed]

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C. Soci, D. Moses, Q.-H. Xu, and A. J. Heeger, “Charge-carrier relaxation dynamics in highly ordered poly(p-phenylenevinylene):Effects of carrier bimolecular recombination and trapping,” Phys. Rev. B. 72(24), 245204 (2005).
[Crossref]

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H. Matsuzaki, Y. Matsui, R. Uchida, H. Yada, T. Terashige, B.-S. Li, A. Sawa, M. Kawasaki, Y. Tokura, and H. Okamoto, “Photocarrier dynamics in anatase TiO2 investigated by pump-probe absorption spectroscopy,” J. Appl. Phys. 115(5), 053514 (2014).
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D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

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C. K. Yong, H. J. Joyce, J. Lloyd-Hughes, Q. Gao, H. H. Tan, C. Jagadish, M. B. Johnston, and L. M. Herz, “Ultrafast dynamics of exciton formation in semiconductor nanowires,” Small 8(11), 1725–1731 (2012).
[Crossref] [PubMed]

Yuvchenko, S. A.

D. A. Zimnyakov and S. A. Yuvchenko, “Effective dielectric function of TiO2 nanoparticles under laser pumping in the fundamental absorption band,” Quantum Electron. 47(6), 547–552 (2017).
[Crossref]

D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

Zdrajevsky, R. A.

D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

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[Crossref] [PubMed]

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A. Fujishima, X. Zhang, and D. A. Tryk, “TiO2 photocatalysis and related surface phenomena,” Surf. Sci. Rep. 63(12), 515–582 (2008).
[Crossref]

Zimnyakov, D. A.

D. A. Zimnyakov and S. A. Yuvchenko, “Effective dielectric function of TiO2 nanoparticles under laser pumping in the fundamental absorption band,” Quantum Electron. 47(6), 547–552 (2017).
[Crossref]

D. A. Zimnyakov, R. A. Zdrajevsky, S. A. Yuvchenko, O. V. Ushakova, O. V. Angelsky, and S. B. Yermolenko, “Enhancement of light depolarization by random ensembles of titania-based low-dimensional nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 152, 37–44 (2015).
[Crossref]

D. A. Zimnyakov, A. V. Gorokhovsky, E. V. Tret’yachenko, O. V. Ushakova, E. A. Isaeva, and A. A. Isaeva, “Surface mode induced extinction of potassium titanate nanoplatelets,” Opt. Mater. 34(11), 1865–1868 (2012).
[Crossref]

D. A. Zimnyakov, O. V. Ushakova, A. V. Gorokhovsky, E. V. Tretyachenko, E. A. Isaeva, A. A. Isaeva, and A. B. Pravdin, “Resonant scattering and absorption in the titanate-based nanoplatelet dispersions in near ultraviolet region,” Appl. Opt. 51(16), 3675–3683 (2012).
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Figures (9)

Fig. 1
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.
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( z ). The wavelength is 355 nm; the duration of laser pulses is 10 ns.
Fig. 3
Fig. 3 The dependencies I sc norm ( I p , N p =1 ) and I tr norm ( I p , N p =1 ); 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.
Fig. 4
Fig. 4 The recovered Cole-Cole diagrams for anatase; a – the family of the intensity-dependent diagrams recovered from I sc norm ( I p , N p ), I tr norm ( I p , N p ) 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 hν); hν varies from 2.25 eV (i) to 5.65 eV (v).
Fig. 5
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.
Fig. 6
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).
Fig. 7
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.
Fig. 8
Fig. 8 2D plots of N ˜ oc (k) against α and N p for various values of γ. a – γ= 0.01; b – γ= 0.05; c – γ= 0.20; d – γ= 0.40.
Fig. 9
Fig. 9 Dependence of N p 0.5 α on the trapping factor γ. The circle marker corresponds to the examined system.

Equations (11)

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{ σ sc ( I p , N p ) = k 4 v 2 18π | ε ˜ ( I p , N p )1 | 2 { 27 [ ε ˜ ( I p , N p )+2 ] 2 + ε ˜ 2 ( I p , N p ) }; σ abs ( I p , N p ) = kv 3 { 27 [ ε ˜ ( I p , N p )+2 ] 2 + ε ˜ 2 ( I p , N p ) } ε ˜ ( I p , N p ),
I sc norm ( I p , N p )= σ sc ( I p , N p ) σ sc lin = [ ε ˜ ( I p , N p )1 ] 2 + ε ˜ 2 ( I p , N p ) [ ε ˜ lin 1 ] 2 + ( ε ˜ lin ) 2 × × [ ε ˜ lin +2 ] 2 + ( ε ˜ lin ) 2 [ ε ˜ ( I p , N p )+2 ] 2 + ε ˜ 2 ( I p , N p ) ,
Φ( I p , N p )= I sc norm ( I p , N p ) [ ε ˜ lin 1 ] 2 + ( ε ˜ lin ) 2 [ ε ˜ lin +2 ] 2 + ( ε ˜ lin ) 2 .
Φ( I p , N p ) [ ε ˜ ( I p , N p )1 ] 2 + ε ˜ 2 ( I p , N p ) [ ε ˜ ( I p , N p )+2 ] 2 + ε ˜ 2 ( I p , N p ) .
I tr ( I p , N p ) I p exp{ ρd[ σ sc ( I p , N p ) + σ abs ( I p , N p ) ] },
ln[ I p / I tr ( I p , N p ) ] ln[ I p lin / I tr lin ] σ sc ( I p , N p ) + σ abs ( I p , N p ) σ sc lin + σ abs lin .
Γ( I p , N p )172.7 ε ˜ ( I p , N p ) [ ε ˜ ( I p , N p )1 ] 2 + ε ˜ 2 ( I p , N p ) .
{ ε ˜ =1+ ζ 2 ( 1 η 2 ) ( 1 η 2 ) 2 + μ 2 η 2 ; ε ˜ = ζ 2 μη ( 1 η 2 ) 2 + μ 2 η 2 .
{ N oc (1) = N g (0) α( 1βγ ); N g (1) = N g (0) N g (0) α( 1β ).
{ N g (2) = N g (1) N g (1) α( 1β )= N g (0) { 1αf( I )( 1β ) }; N g (3) = N g (2) N g (2) α( 1β )= N g (0) { 1αf( I )( 1β ) } 3 ; ...............; N g (k) = N g (0) { 1αf( I )( 1β ) } k .
{ N oc (2) ={ N g (0) α( 1βγ )+ N g (1) α }( 1βγ )= = N g (0) α{ ( 1γ )+( 1β )( 1α ) }( 1βγ ); N oc (3) =( N oc (2) + N g (2) α )( 1βγ )= = N g (0) α[ { ( 1γ )+( 1β )( 1α ) } ( 1βγ ) 2 + { 1α( 1β ) } 2 ( 1βγ ) ]; N oc (4) =( N oc (3) + N g (3) α )( 1βγ )= = N g (0) αf( I )[ { ( 1γ )+( 1β )( 1α ) } ( 1βγ ) 3 + { 1α( 1β ) } 2 ( 1βγ ) 2 + + { 1α( 1β ) } 3 ( 1βγ ) ]; .......................; N oc (k) = N g (0) α[ { ( 1γ )+( 1β )( 1α ) } ( 1βγ ) k1 + m=2 k1 { 1α( 1β ) } m ( 1βγ ) km ]

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