## 1. Introduction

Due to their unique properties, silicon nanocrystals get wide applications in various fields of science and engineering (Bisi et al., 2000). Recently, it has been revealed that photoexcitation of nanocrystals in microporous silicon (micro-PS) layers leads to the generation of singlet oxygen on the surface of the samples (Kovalev et al., 2002). It is known that the ground state of the oxygen molecule is represented by the triplet state (^{3}O_{2}, where the superscript indicates the spin multiplicity; the total spin of the triplet molecule is *S*_{TO}= 1). During energy absorption, the oxygen molecule transforms into an excited singlet state (^{1}O_{2}, *S*_{SO}= 0) (Halliwell & Gutteridge, 1999). In this state, the oxygen molecule exhibits the highest reactivity and enters into oxidation reactions with many substances. This property of singlet oxygen ^{1}O_{2} is widely used in biomedicine and, in particular, in photodynamic therapy of cancer (Halliwell & Gutteridge, 1999). It should be noted that direct excitation of molecular oxygen from the triplet state to the singlet one is forbidden by the selection rules for the orbital and spin quantum numbers. In order to transform molecular oxygen into the singlet state, it is common practice to use organic dyes that serve as photosensitizers (Kumar et al., 2009). An important property of micro-PS lies in the fact that upon photoexcitation of silicon nancrystals excitons are generated at rather high concentrations (the quantum yield of exciton photoluminescence reaches a few percent (Bisi et al., 2000)). It has been demonstrated that the energy can be effectively transferred from excitons to ^{3}O_{2} molecules adsorbed on the surface of silicon nanocrystals with the subsequent transformation of these molecules into an excited state (Gross et al., 2003; Kovalev et al., 2002). Therein lies the essence of the mechanism of photosensitization of molecular oxygen. The energy exchange between excitons and ^{3}O_{2} molecules occurs through the exchange of electrons (Gross et al., 2003) (the Dexter mechanism (Dexter, 1953)). Compared to organic dyes, the use of micro-PS as a photosensitizer of molecular oxygen offers a number of advantages, for example, the relatively simple and available technique for synthesizing this material and its nontoxicity. Evidently, it is important to determine the concentration of the generated singlet oxygen for the practical use of this effect. In this work, we used electron paramagnetic resonance (EPR) spectroscopy for this purpose. EPR spectroscopy allows us to study the interaction of spin centers on the surface of silicon nanocrystals (silicon dangling bonds) with the paramagnetic molecules of triplet oxygen and thus to measure a decrease of their relative amount. The EPR-diagnostics of the ^{1}O_{2} oxygen generation, suggested in the present work, is based on the change in the spin–lattice (T_{1}) and spin–spin (T_{2}) relaxation times of spin centers, i.e., silicon dangling bonds (the so-called P_{b} centers (Bisi et al., 2000)). In this respect, in the present study, these times were measured by the pulsed EPR technique. The results obtained were confirmed by directly measuring the change in the amount of ^{3}O_{2} oxygen upon photoexcitation of micro-PS with the use of EPR technique in the Q band. It should be noted that the photosensitization of O_{2} molecules on the micro-PS surface is one of the possible channels for nonradiative recombination of excitons in silicon nanostructures. Therefore, to achieve the maximum efficiency of ^{1}O_{2} generation in microporous silicon layers, it is important to comprehensively study the transfer and scattering of the excited silicon nanocrystals energy and to determine the characteristic times of these processes. To solve this problem, we analyze the decay kinetics of the photoluminescence (PL) of silicon nanocrystals (nc-Si) in micro-PS layers. The PL intensity of such samples is known to be well described by the so-called stretched exponent (Maly et al., 1996)

where τ is the average PL lifetime and β is the nonexponentiality parameter. The values of τ and β depend on the excitation wavelength λ, temperature (Pavesi & Ceschini, 1993), and intensity and some other parameters (Chen et al., 1992) and are τ = 5–50 μs and β = 0.5–0.7 for PS samples with porosity P=70%. Note that Eq. (1) describes the PL relaxation kinetics in both ensembles of interconnected nc-Si in porous silicon layers (Mihalcescu et al., 1996; Pavesi & Ceschini, 1993) and other solid-state nanostructures, such as closely spaced (1–2 nm) nc-Si in a SiO_{2} matrix (nc-Si/SiO_{2}) (Germanenko et al., 2001; Kanemitsu, 1996), CdSe–ZnSe superlattices (Chen et al., 1992), InGaN layered structures (Pophristic et al., 1998), and nanoporous SiGe solid solutions (Lebib et al., 1999). Numerous attempts were made to explain the causes of the stretched exponent of PL relaxation in the systems given above. For example, the authors of (Chen et al., 1992; Kanemitsu, 1996) relate this character of luminescence drop to the presence of disordered regions in these objects, which can be caused by a shape distribution of nanocrystals or a variation in their spatial arrangement. These factors induce a scatter of the recombination energies and probabilities of electron–hole pairs, which is likely to result in luminescence kinetics of type (1). Other researchers (Germanenko et al., 2001; Linnros et al., 1999) assume that the distortion of a luminescence monoexponential decrease is mainly caused by energy transfer between neighboring nanocrystals. This energy transfer can occur because of exciton migration from smaller to larger nanocrystals owing to a smaller energy gap in the latter nanocrystals. Thus, the exciton lifetime depends on the transfer rate; as a result, the kinetics of PL intensity drop “stretches”. This assumption is supported by the study of the PL time characteristics of nc-Si ensembles separated by sufficiently thick (about 5 nm) spatial SiO_{2} barriers (Vinciguerra, et al., 2000). In this case, nc-Si crystals are supposed to be well isolated; therefore, no energy exchange occurs between them and the lifetime is constant for nc-Si of a given size. Indeed, the PL kinetics detected in such systems turns out to be monoexponential with a characteristic time of 0.2–0.8 ms (Vinciguerra, et al., 2000). However, recent theoretical investigation (Chen, 2003) predicts a stretched exponent for PL kinetics even for individual (isolated) nanocrystals in the case where the rate of trapping of photoexcited charge carriers is higher than the rate of their recombination. Note that Chen (Chen, 2003) assumed that nonradiative carrier capture centers should occur in every nc-Si. However, this assumption was not supported experimentally for luminescent PS samples, since EPR spectroscopy data demonstrate that the number of defects in 1 cm^{3} (about 10^{17}) is much smaller than the number of nc-Si crystals in this volume (about 10^{19}–10^{20} or 10^{21} in the case of a fractal PS structure (Moretti et al., 2007; Nychyporuk et al., 2005)). Thus, most nc-Si do not have trapping sites, which requires an alternative explanation for the nonexponential kinetic curves of PL decay. Therefore the purpose of this work is (i) to experimentally investigate the process of singlet oxygen generation in nc-Si ensembles using electron paramagnetic resonance and photoluminescence spectroscopy and (ii) to theoretically study the PL decrease kinetics and the steady-state PL intensity of ensembles of nc-Si with allowance for exciton migration–assisted energy transfer in them, and (iii) to compare the results obtained in terms of the developed theory with the experimental data on the luminescence and photosensitization properties of nc-Si ensembles.

## 2. Sample preparation and experimental technique

Silicon nanocrystals in the PS layers were formed by a standard method of the electrochemical etching of silicon single crystals in a solution based on hydroﬂuoric acid: HF(48%) : C_{2}H_{5}OH=1 : 1 (e.g., see the review (Bisi et al., 2000)). The *p*-type crystalline silicon with the (100) surface orientation and the resistivity ρ=10…20 Ω cm and 10…20 mΩ cm was used as a substrate for micro-PS and meso-PS. The current density was 50 mA/cm^{2} (micro-PS1 and meso-PS) and 70 mA/cm^{2} (micro-PS2), the etching time was 60 min. After the completion of the pore formation process, PS layer was lifted from the substrate by a short-time increase in the current density to 700 mA/cm^{2}. The sample thickness measured with an optical microscope was 50 μm. Porosity of the samples was determined gravimetrically and was approximately 70 % for micro-PS1 and meso-PS samples, and 85 % for micro-PS2 samples. As determined by transmission electron microscopy, the average size of the nanocrystals was about 2–4 nm (micro-PS) and ~10 nm (meso-PS). A part of micro-PS samples were dried in air (PS1 and PS2 samples). Upon drying in air at room temperature, the PS films were failed by the surface tension force of water and transformed into granules with a lateral size of about 0.5 mm (coarsegrained (CG) PS). Fine-grained (FG) PS was prepared by mechanical milling in air of a CG PS powder placed in an ampule with quartz balls on a vibratory mill at a vibration frequency of 17 Hz and an amplitude of 7 mm for 2 min.

The EPR measurements were performed on a Bruker ELEXSYS-580 EPR spectrometer (operating frequency is 9.5 and 35 GHz, X- and Q-band, recpectively). Pulsed EPR measurements were made at the regime of “spin-echo” (time resolution ~5 ns). The admission of oxygen and the subsequent evacuation to the residual pressure *p* ~ 10^{–5} mbar were performed “in situ” using oil-free vacuum equipment. For Q-band measurements soldered ampoules with PS in oxygen ambient were used because of technical features. The samples were illuminated with light from a halogen lamp at the maximum intensity *I*_{exc} ~2 W/cm^{2} immediately in the resonator of the EPR spectrometer.

The PL of the PS samples were excited by nitrogen laser radiation with a wavelength of 337 nm at pulse duration of 10 ns and pulse energy of 10 μJ. PL spectra were measured on a Solar TII spectrometer equipped with a Hamamatsu MS-101H CCD camera. PL kinetics were recorded with an MDR-12 monochromator, Hamamatsu R237 photomultiplier tube, and an Agilent 54642A digital oscilloscope.

The experiments were carried out at room temperature.

## 3. Results and discussion

### 3.1. Investigation of the EPR spectra of silicon nanocrystalls in the PS1 layers in vacuum, nitrogen and oxygen

Figure 1a shows the EPR spectra of micro-PS samples in oxygen in the dark and under illumination, which were measured at the low power *P*_{b}-centers, which are the dangling bonds of silicon at the Si/SiO_{2} interface (Bisi et al., 2000; Cantin et al., 1995).

As can be seen in Fig. 1a, a noticeable change in the EPR signal amplitude of micro-PS in oxygen did not occur under illumination (curve *2a*), as compared with that in the dark (curve *1a*). At the same time, a considerable variation in the EPR signal amplitude of micro-PS under analogous conditions was detected at *1* and *2*). It is well known that, at a sufﬁciently high intensity of microwave radiation its absorption by *P*_{b} centers was saturated; this manifested itself in a decrease in the EPR signal amplitude (Cantin et al., 1995). This effect was observed in our measurements performed with micro-PS in a vacuum (Fig. 1b, curve *3*). Note that the relaxation of *P*_{b}-centers from an excited state to the ground state occurred by energy transfer to crystal lattice phonons in the case of micro-PS samples in a vacuum. At the same time for micro-PS in an oxygen atmosphere, the magnetic dipoles of ^{3}O_{2} molecules adsorbed on the surface of silicon nanocrystals interacted with the magnetic moments of excited *P*_{b}-centers to induce their rapid relaxation to the ground state. Consequently, the relaxation time of *P*_{b}-centers for micro-PS in an oxygen atmosphere decreased; the saturation effect of microwave absorption weakened; and, as a result, the EPR signal amplitude considerably increased (Fig. 1b, curve *1*). The photosensitization of oxygen molecules occurred under the illumination of micro-PS layers placed in an oxygen atmosphere; consequently, the concentration of ^{3}O_{2} molecules decreased. In turn, this caused an increase in the relaxation time of *P*_{b}-centers; as a result of this, the EPR signal amplitude decreased (Fig. 1b, curve *2*). It should be noted that the change in the EPR signal amplitude upon photoexcitation of PS is almost completely reversible in switching on–switching off cycles if the defect formation under illumination is reduced to a minimum. The samples under investigation satisfy this condition, because they were preliminarily oxidized upon illumination in air for several minutes, which resulted in a considerable decrease in the defect generation rate.

In order to conﬁrm the decisive role of oxygen in the decrease of the EPR signal amplitude upon photoexcitation of nc-Si, we measured the EPR spectra of micro-PS in an atmosphere of nitrogen because of N_{2} molecules are diamagnetic (Fig. 2). As can be seen in Fig.2, a noticeable change in the EPR signal amplitude of micro-PS in nitrogen ambient did not occur under illumination (curve *2*), as compared with that in the dark (curve 1). The amplitude of both spectra is small because of saturation effect. Therefore the relaxation of spins takes place only throw spin-lattice relaxation canal, the spin-spin relaxation is suppressed because of diamagnetic nature of N_{2} molecules. Therefore namely triplet oxygen molecules are responsible for the effective spin relaxation process and an elimination of saturation effect.

Control experiments for meso-PS show that there is no change in EPR spectra amplitude of meso-PS under illumination of the samples in oxygen ambient (Fig.3). This fact points out an absence of singlet oxygen generation effect and confirms the decisive role of excitons in energy transfer process from photoexcited excitons in nc-Si to oxygen molecules adsorbed on their surface. Indeed, the average size of nanocrystals in meso-PS layers is approximately 10 nm. In such systems the exciton binding energy is small in comparison with thermal energy at room temperature (26 meV).

The above data were obtained at high microwave powers_{b} center (dominant in vacuum), its characteristic lifetime in the excited state is shorter than the characteristic time 1/W of absorption of a microwave photon. Consequently, at low microwave powers

Consequently, at low microwave powers

microwave powers _{b} centers and, as a consequence, to a decrease in the absorption of the microwave power. Indeed, the saturation curve for micro-PS in oxygen under illumination (Fig. 4, curve 2) lies lower than that obtained under dark conditions (Fig. 4, curve 1) (effect of generation of ^{1}O_{2} molecules), and the saturation curve for the samples in vacuum (Fig. 4, curve 3) is characterized by a lower amplitude as compared to curve 2, passes through a maximum, and falls off with an increase in the microwave power

where H_{1} is the amplitude of the magnetic ﬁeld strength of the microwave wave, ω_{0} is the frequency of the EPR transition,

because *I(ω; ω*_{0}*)* in the following form:

Taking into account that the EPR spectrum, as a rule, is recorded in the form of the ﬁrst derivative *I'(ω; ω*_{0}*)* of expression (4) with respect to *ω*, we have

The width and the amplitude of the EPR spectrum determined by formula (5) are represented in the form

where the parameters *a* and *b* determine the position of the maximum in the dependence

Relationships (2) and (4)–(6) are valid in the case of a homogeneous broadening of the EPR line due to a ﬁnite lifetime of spin centers in the excited state. However, the EPR line of porous silicon is inhomogeneously broadened and can be described by the Gaussian distribution (Carrington & McLachlan, 1979)

whеre

whеre *a* and *b* and holds true for the homogeneously broadened spectrum. In this case, the calculated curves and the experimental data are in satisfactory agreement (Fig. 4). The ﬁtting parameters were determined to be *a*_{v}=0.09 relative units and *b*_{v}=0.4 mW^{–1} for the sample in vacuum and *a*_{d}=0.10 relative units and *b*_{d} = 1.1 × 10^{–3} mW^{–1} for the sample in the oxygen atmosphere under the dark conditions. According to formulas (7), the saturation curve ^{1}O_{2} molecules that do not contribute to the paramagnetic relaxation. The other part of nanocrystals (with larger sizes), as in the case of meso-PS (see above), do not make a contribution to the photosensitization of oxygen. This is equivalent to the relaxation of P_{b} centers in the atmosphere of ^{3}O_{2} molecules under the dark conditions. It follows from the aforesaid that the quantity

and, hence,

.The quantity ^{1}O_{2} molecules depends on the amount of ^{3}O_{2} molecules that surround a silicon nanocrystal. Figure 5a shows the dependence of EPR signal amplitudes on the pressure of oxygen (*p*) for PS layers in the dark and under illumination. The curve for the sample in oxygen under illumination is also described by*p.* Indeed, the time of energy transfer from an exciton to the ^{3}O_{2} molecule shortened as the concentration of oxygen molecules surrounding a nanocrystal was increased (Gross et al., 2003). In this case, the relaxation time of ^{1}O_{2} remained unchanged (direct singlet–triplet energy exchange between oxygen molecules is spin forbidden). Consequently, the fraction of ^{1}O_{2} molecules adsorbed on the surface of silicon nanocrystals increased. At the same time, this fraction depends on the quantity *p*. Using Eq. (11), we can express the value of *p* = 10^{–5} mbar was chosen as^{1}O_{2} molecules on *p* thus obtained can be directly converted into the concentration of ^{1}O_{2} molecules (*N*_{SO}) taking into account the initial triplet oxygen concentration in silicon pores, which is equal to 2.7 10^{19} cm^{–3} at

An increase in the intensity of illumination of porous silicon nanocrystals caused an increase in amount of ^{1}O_{2} molecules photosensitized on the surface of nc-Si (Gross et al., 2003) As a result of this, the EPR signal amplitude decreased (Fig. 6). In this case, a sharp decrease in the value of *I*_{EPR} was observed up to *I*_{exc}=600 mW/cm^{2}; as *I*_{exc} was further increased, *I*_{EPR} reached an approximately constant value. The latter was likely due to the fact that, at the speciﬁed value, the predominant fraction of oxygen molecules that covered a nanocrystal occurred in a singlet state. Figure 6 also shows the dependence of *N*_{SO} on *I*_{exc}, as calculated using Eq. (11) with consideration for the initial concentration of triplet oxygen.

### 3.2. Investigation of the photosensitization of oxygen molecules by the pulsed EPR technique

The EPR diagnostics of the generation of ^{1}O_{2} oxygen molecules in ensembles of silicon nanocrystals (considered in the preceding section) is based on a change in the relaxation times of spin centers. In this respect, the relaxation times T_{1} and T_{2} of spin centers in the samples under investigation were measured by the pulsed EPR technique based on the spin echo phenomenon (Hahn, 1950). It should be noted that, since the relaxation time T_{1} characterizes the return of the net magnetization component parallel to the constant

magnetic ﬁeld to its equilibrium thermal value M_{0}, this quantity is also referred to as the longitudinal relaxation time. The relaxation time T_{2} characterizes the relaxation of the transverse magnetization component to zero, which does not affect the total Zeeman energy of spins but determines the width of the EPR line. The relaxation times T_{1} and T_{2} can be represented by the following relationships (Carrington & McLachlan, 1979):

where*S* with the times T_{1} and T_{2}. It is assumed that the ﬁeld _{c} (Carrington & McLachlan, 1979). The analysis of the system of equations (12) demonstrates that the relaxation times are related by the expression

In expression (13), _{1} of the spin in the excited state (the contribution*H*_{0}. In our case, the ﬂuctuating magnetic ﬁeld _{b} centers with phonons through the spin–orbit interaction can be represented as the interaction of the spin center with a randomly varying magnetic ﬁeld. This type of relaxation dominates in vacuum. In the presence of triplet oxygen molecules at the nanocrystal surface in the vicinity of the P_{b} center, the mechanism of magnetic dipole–dipole interaction between the spins of the P_{b} center and the ^{3}O_{2} molecule dominates. As a result, the P_{b} centers appear to be in the magnetic ﬁeld induced by the oscillating magnetic moment of the physically sorbed ^{3}O_{2} molecules. It is this type of relaxation that dominates in the oxygen-containing atmosphere and is responsible for the increase in the amplitude of the EPR spectrum in the oxygen atmosphere as compared to vacuum (Fig. 1b). In this case, the lifetime T_{1} of the P_{b} center in the excited state and, hence, the relaxation time T_{2} should decrease in the oxygen atmosphere (see expression (11)). Actually, the analysis of the experimental data obtained by the pulsed EPR technique indicates the presence of this tendency. Figure 7 depicts the relaxation curves for the longitudinal (Fig. 7a) and transverse (Fig. 7b) components of the net magnetization of the micro-PS samples with the characteristic time T_{1} and T_{2}, respectively.

It should be noted that, for PS, the net magnetization of the sample is the sum of the contributions from the spin moments of all P_{b} centers. With the aim of determining the relaxation times, the experimental curves were approximated by the dependences _{1} and T_{2}, respectively. The error in the measurements was equal to 5–7%. The paramagnetic relaxation times thus obtained for micro-PS are listed in Table 1. It can be seen from this table that the illumination of the micro-PS samples in oxygen leads to an increase in the relaxation time T_{1} and, correspondingly, in the relaxation time T_{2}. The observed increase in the relaxation times of spin centers suggests that a number of oxygen molecules transform from the paramagnetic triplet state (participating in the dipole–dipole spin relaxation process) into the diamagnetic singlet state upon photosensitization of oxygen on the surface of nc-Si. Therefore, the pulsed EPR data conﬁrm the generation of ^{1}O_{2} molecules upon photoexcitation of nanocrystals in micro-PS layers. Within the limits of experimental error, the illumination of the meso-PS samples in oxygen is not accompanied by a variation in the relaxation times of spin centers (Table 1). This indicates the absence of generation of singlet oxygen in meso-PS and agrees with the data obtained by the EPR technique under continuous microwave irradiation (see Section 3.1). Apart from the micro-PS and meso-PS samples, the Si samples were also investigated for comparison by the pulsed EPR technique. The relaxation times T_{1} determined for these samples in measurements in vacuum are given in Table 1. It should be noted that the relaxation times T_{1} for c-Si are in agreement with those available in the literature (Lepine, 1972). For the nc-Si, the relaxation times T_{1} were determined for the ﬁrst time. The analysis of the relaxation times T_{1} for all the samples under investigation permitted us to reveal the following tendency: the spin–lattice relaxation process in micro-PS is retarded as compared to that in meso-PS and c-Si. In turn, the relaxation times T_{1} for meso-PS are longer than those for c-Si (Table 1).

The retardation of the spin–lattice relaxation with a decrease in the size of the structure can be associated with the decrease in the efﬁciency of the electron–phonon interaction in silicon nanocrystals as compared to the bulk phase of the material under investigation. Indeed, as the size of the object decreases, the phonon spectrum undergoes substantial changes, including a partial degeneracy of phonon modes (Roodenko et al., 2010). In turn, this affects the character of the interaction between electrons and phonons that are conﬁned in nanostructures and, in particular, leads to a decrease in the efﬁciency of energy transfer from excited spins to the lattice.

### 3.3. EPR spectroscopy of molecular oxygen

The EPR spectra of ^{3}O_{2} molecules in pores of micro-PS are shown in Fig. 8. It should be noted that the investigation of triplet oxygen by the EPR technique was performed in the millimeter (Q) band of microwave radiation in view of a large width (approximately 10 kG) of the EPR spectrum of ^{3}O_{2} molecules (due to the short lifetime in the excited state (Vahtras et al., 2002)). The presence of several lines in the EPR spectrum (Fig. 8) is determined by the strong interaction of the rotational angular momentum K and the spin angular momentum *S* of the oxygen molecule.

The total angular momentum J takes on values K and K ± 1 (because *S* = 1). As a rule, the recorded EPR spectrum contains six lines designated as C, E, F, G, K, and J, which have the

highest intensities (Vahtras et al., 2002). They correspond to the following magnetic dipole transitions J, M_{J} J^{/}, M_{J}^{/}: {1, -1 1, 0} (C); {2, 1 2, 2} (E); {2, 0 2, 1} (F); {6, -2 4, -1} (G); {2, -1 2, 0} (K) and {2, 1 2, 2} (J). It can be seen from Fig. 8 that the illumination of the samples results in a decrease in the amplitude of the EPR spectrum. This suggests a decrease in the concentration of triplet oxygen molecules. The data obtained can be explained by the transition of oxygen molecules to the singlet state and can be considered a direct proof of the generation of ^{1}O_{2} molecules in micro-PS layers. It should be noted that the EPR signal amplitude after the illumination is switched off only partially regains its initial value (before illumination) (Fig. 8). This can be explained by the fact that a considerable number of ^{1}O_{2} molecules pass from pores of the sample into the closed volume of the measuring cell, i.e., into the gaseous medium, in which the lifetime of the singlet state increases to approximately 50 min as compared to approximately 500 µs in pores of silicon (Gross et al., 2003). Since the area under the EPR line is proportional to the number of spin centers, it is easy to estimate the fraction of the newly formed singlet oxygen molecules from the relationship*S*_{light} and *S*_{dark} are the areas under the EPR lines of ^{3}O_{2} molecules under illumination and in the dark, respectively. Therefore, approximately 30% of oxygen molecules upon photoexcitation of nanocrystals in micro-PS layers transform into the singlet state. It should be noted that similar experiments with meso-PS revealed no changes in the amplitude of the EPR spectrum of triplet oxygen molecules in pores of mesoPS. This indicates that ^{1}O_{2} molecules are not formed in this material (see Sections 3.1, 3.2).

### 3.4. Kinetic equations for the relaxation processes in nc-Si ensembles

We now develop kinetic equations for an ensemble of excited nanocrystals in PS samples in vacuum. To his end, we consider a model fragment of the PS microstructure, namely, a chain of spherical nc-Si crystals having a diameter changing in small limits and forming a quantum filament of a variable cross section (Fig. 9a). This model representation agrees with the real structure of micro-PS, whose layers consist of nc-Si crystals no more 5 nm in size. Figure 9 shows micrographs of micro-PS films produced by the anodic oxidation of p-type silicon wafers. They clearly exhibit nc-Si chains forming silicon filaments of a variable cross section (Cullis & Canham, 1991). We number neighboring nanocrystals in a chain beginning from a zeroth crystal with the maximum size as 0, …, m – 1, m, m + 1, …, M, where m = 0, 1, 2, …, M. In the case of a fractal PS model (Moretti et al., 2007; Nychyporuk et al., 2005), m has the meaning of an ordinal number that is repeated when the nanofragment (nanocrystal) scale decreases, d_{m} = d_{0}k^{–m}, where k > 1 is the linear scale reduction coefficient and d_{0} is he diameter of the largest crystal. Let *N*_{m} be the number of nc-Si crystals with ordinal number m; then, *N*_{m} can be written as*N*_{0m}, *N*_{1m}, and *N*_{2m} are the numbers of unexcited nc-Si crystals and nc-Si crystals containing one and two excitons, respectively. The number of nc-Si crystals with more than two excitons is negligibly small as compared to *N*_{2m} because of the rapid Auger recombination processes that occur when the number of excitons in a nanocrystal is larger than one.

Moreover, *N*_{m}). In general, all primed quantities relate to the processes that take place in nc-Si crystals with defects. Then, a set of kinetic equations describing the energy relaxation for *N*_{0m}, *N*_{1m}, and *N*_{2m} after the end of optical excitation is (point above a quantity means differentiation with respect to time t)

(15) |

The meaning of the parameters entering into the set of nonlinear differential equations (12) is as follows: ^{–1}. ^{3} μs^{–1}, it can be formally interpreted as the product of the translational thermal exciton velocity into the transfer cross section. Finally, ^{–1}). In total, all three equations of set (12) yield zero, since

Let us explain the physical meaning of the terms of type^{–8} –10^{–9} s (Proot et al., 1992; Delerue et al., 1993), whereas the characteristic migration and radiative recombination times fall in the microsecond range, as follows from an analysis of the experimental PL relaxation kinetics (see below). Moreover, in the third equation of set (12), we neglected the terms of type^{–9} –10^{–11} s) for the nanocrystals that luminesce in the red spectral region (which have an energy gap 1.3 < Eg < 2.2 eV) is extremely short (Delerue et al., 1993). If we consider the case of nanocrystals that do not luminesce in this spectral region (for which

We substitute

We analyze this expression by analogy with the procedure performed for

Let us analyze the processes that occur in such nc-Si crystals immediately after removing an optical excitation in detail. If the initial values of ^{–6} –10^{–9} s via nonradiative and Auger recombination (Delerue et al., 1993). Beginning from the time when the recombination rates become equal to the rate of filling of these nanocrystals due to the exciton migration from nc-Si crystals with larger ordinal numbers, these processes manifest themselves as negative-feedback systems to some extent. For example, if the inequality

hold true due to the substantial difference in the rates of rapid recombination processes and slow migration processes at any time beginning from approximately 10^{–6} s after removing an optical excitation. With the first equation from set (16), we can express

Based on inequality (17), we neglect

As a result of this simplification, the substitution of

Although set (12) is substantially simplified, it is rather difficult to directly solve Eq. (18) even using numerical methods. First, the total number of equations can be rather large, since m varies from zero to M. The value of parameter M can easily be estimated in a fractal model, since it represents the order of the minimum size nc-Si crystal with a diameter_{0} = 5 nm (see Figs. 9b–1d) and obtain

Thus, M depends on coefficient k of linear scale reduction, which can be only insignificantly higher than unity in the case of crystals having almost the same sizes; as a result, M can be arbitrarily large. For example, at k = 1.02 we have M = 107. Second, the character of the dependence of recombination rate *D* on ordinal number m is unknown. Thus, to solve the set of equations (18), we have to know the dependences of these parameters on the nc-Si crystal size over the entire size range (in our case, from 0.6 to 5 nm). Nevertheless, equations (18) become much simpler if we assume “local quasi-identity” of nc-Si crystals, i.e., if we assume that neighboring nc-Si crystals with numbers m–1, m, and m+1 and several farther neighbors have similar shapes and sizes. The larger the number of almost identical nc-Si crystals in a chain, the more adequate this assumption. In a fractal model of the structure of PS, this assumption means that *k* is close to unity. Then, the number of neighbors of any nc-Si crystal is*D*_{F} is the fractal dimension of nc-Si (which ranges from 2.2 to 2.6) (Nychyporuk et al., 2005). Taking *D*_{F} = 2.4 and *k* = 1.02, we obtain *n* ≈ 1.05. The structure described by these values of parameters *n* and *k* is represented by long filaments made of almost the same nc-Si crystals (the difference in the neighboring crystal sizes is approximately 2%) intersecting with other filaments every 20 nc-Si crystals. As follows from Figs. 9b–9d, this structural model is close to reality. With the assumption of local quasi-identity of nc-Si crystals, we can omit all ordinal indices in Eq. (18) assuming that

and so on. As a result, we can rewrite Eq. (18) in the form

and eventually obtain the following kinetic equation for the relaxation of an optical excitation in an ensemble of nc-Si crystals with allowance for exciton transfer in them:

where we introduced a designation for the sum*D* and *D** should be close. The small difference between them can be caused by an additional interaction of excitons located in neighboring nc-Si crystals (in the case of *D**). In the dipole approximation, we have *D=D**, since the trace of the symmetric dipole–dipole interaction tensor is zero (Carrington & McLachlan, 1979). Therefore, the energy of this interaction is also zero upon averaging over a spherically symmetric exciton charge distribution. The difference in coefficients *D* and *D** can be related to the presence of a quadrupole (or higher moments of) interaction of excitons or an exchange interaction in the case of a substantial overlapping of their wavefunctions. Obtained ordinary differential equation (21) is a Bernoulli equation with constant coefficients and an exponent γ=2 of the highest degree of the unknown function. This equation is solved analytically using the substitution

Obviously, the product _{2} layer (Vinciguerra et al., 2000). The form of the normalized kinetics *D*; see Eq. (21)) to optical excitation relaxation rate

However, a numerical analysis with the Mathcad software package demonstrates that the factor multiplying

### 3.5. Photoluminescence of porous silicon under steady-state photoexcitation conditions

We write the first and second equations from set (12) for a steady-state case (*m* (i.e., nanocrystals of a certain size; see Section 3.4) and I_{ex} is the radiation intensity):

(30) |

Note that an analysis of the last equation in set (24) immediately yields the inequality

is obvious in the steady-state case. It reflects the fact that these relations are determined by the fraction of defect-containing nc-Si crystals of the total number of nc-Si crystals in the same manner. Based on the inequalities, we may neglect

We substitute the latter expression into the equation for nc-Si crystals with defects, which is analogous to the first equation in set (24) in which

Based on the causes described in the previous section, we neglect all quantities in the denominator as compared to

This is the general form of an equation for the number of photoexcited nc-Si crystals of the *m*th order in a fractal chain in the steady-state case. This equation can be solved by an iteration method beginning from *m* = 0. However, this process is related to the difficulties described in the previous section. At the same time, Eq. (26) can be solved in the local nc-Si quasi-identity approximation, which is represented by Eqs. (19). As a result, we obtain the following quadratic equation for *N*_{1}, in which ordinal indices are omitted:

where we introduced a designation for

Note that a more accurate (by several fractions of a percent) solution can be obtained if we do not neglect

in which correcting coefficient

In the limiting case D*=0, Eq. (28) is reduced to the form

that is, it coincides with the solution to set (24) in the absence of the terms describing migration. According to Eq. (29), steady-state PL intensity of nc-Si ensembles^{–3} –10^{–2}). However, according to the experimental data, the PL spectrum amplitude decreases significantly during the photostimulated oxidation of PS samples (see below), whereas the defect concentration increases only by a factor of 1.5–2 (see below). Therefore, as in the case of the kinetic curves of PL decay (Section 3.4), the motion of excitons along a network of coupled nc-Si crystals also substantially affects the steady-state PL intensity. To express diffusion coefficient

as a result, we have

where we neglected *g* and the nc-Si concentration in the ensemble. Although the latter is most difficult, it can be performed in terms of a fractal PS model. Indeed, the number of nc-Si crystals of a given size (in this case, about 3 nm, which corresponds to an energy gap of 1.6 eV (Delerue et al., 1993)) can readily be estimated from the following expression for the number of nc-Si crystals of order *m*:*N*_{0} is the number of nc-Si crystals of the zeroth order, *k*=1.02 is the linear scale reduction coefficient (according to the local quasi-identity approximation; see Section 3.4), *D*_{F}=2.4 is the fractal dimension of PS layers, and *m* ranges from zero to *M* = 107 (see Section 3.4). The total number of nc-Si crystals of all orders is

We assume that *N*_{tot}=10^{21} cm^{–3} and determine *N*_{0}. The order corresponding to an nc-Si diameter *d*_{m}=3 nm is expressed as

as a result, we obtain

Similarly, knowing the total number of nc-Si crystals with defects (about 10^{17} cm^{–3}) or oxygen molecules (at ^{19} cm^{–3}) and taking into account that their specific number per nanocrystal is proportional to the nanocrystal surface area, we can estimate the number of defects (about 10^{15} cm^{–3}) or O_{2} molecules (about 2 10^{17} cm^{–3}) per ensemble of nc-Si crystals of a given size (with an ordinal number *m*=26). At a nitrogen laser radiation intensity of 300 mW/cm^{2} and an quantum energy of 3.7 eV, *I*_{ex} = 5 10^{17} photons/(cm^{2} s) are incident on a sample. Assuming that the absorption cross section of a PS sample is *σ*=10^{–14} cm^{2} (Bisi et al., 2000), we find that the optical photoexcitation rate is ^{-1}. Allowing for ^{–1}, we obtain ^{–1}. The characteristic value of *P*= 70% (table 2). The approximation parameter is ^{–1} (table 2).

Substituting these values of parameters into Eq. (31), we finally obtain the migration coefficient for coarse-grained (*P*=70%) PS samples, ^{3}/s. Using Eq. (28) and the values of all necessary parameters determined above, we can construct a relationship between the number of excited nc-Si crystals

### 3.6. Theoretical analysis of the singlet oxygen generation efficiency

We now consider an ensemble of nc-Si crystals of a certain size under conditions of steady-state photoexcitation in an oxygen-containing medium and still assume that the local nc-Si quasi-identity approximation holds true (see Section 3.4). Then, the condition of dynamic equilibrium between the number of O_{2} molecules in the singlet (*n*_{SO}) and triplet (*n*_{TO}) states is

where ^{3}O_{2} molecules per nc-Si (*n* is the concentration of O_{2} molecules. Here, for simplicity, the number of excited nc-Si crystals (*N*_{1}) is written without superscript (0), which means that a steady-state case is analyzed. Using Eq. (32), we can easily express the stationary concentration of ^{3}O_{2} molecules,

For an oxygen-containing medium, the term *n*_{TO} and some manipulations, we write Eq. (27) in the form

This is a cubic equation for *N*_{1}, which can be solved numerically. We used the Mathcad software package. With Eq. (33) and the solution to Eq. (34), we obtain the following expression for the concentration of photosensitized ^{1}O_{2}:

Figure 11 shows the calculated dependence of the fraction of singlet oxygen on the nc-Si photoexcitation rate at various molecular oxygen concentrations.

Hereafter, by default we use the following parameters for numerical simulation, unless otherwise specified (see Section 3.4): *D*≈*D**=3.9 10^{13} cm^{3}/s, *N*=10^{18} cm^{–3}, *N'*=10^{15} cm^{–3}, _{r}-1=125 μs, _{ET}-1=40 μs (Gross et al., 2003), and _{SO}-1= 500 μs (Gross et al., 2003). The curves in Fig. 11 are similar to the experimental dependence of the concentration of ^{1}O_{2} molecules on the photoexcitation intensity obtained by EPR (see Fig.6). When oxygen partial pressure ^{1}O_{2} increases (Fig. 12a); however, the fraction of O_{2} molecules in the singlet state of the total number of O_{2} molecules decreases beginning from a certain value of _{2} molecules per Si nanocrystal in the ensemble under study is smaller than a certain value, the fraction of ^{1}O_{2}, which is only determined by the relation between the characteristic times of excitation, relaxation, and energy transfer, remains constant with increasing pressure. When the specific number of O_{2} molecules reaches a certain threshold value (more than unity) per nc-Si crystal, this crystal cannot excite all oxygen molecules adsorbed on its surface at the same efficiency. As a result, the fraction of ^{1}O_{2} decreases when the concentration of molecular oxygen increases further (Fig. 12b). The difference between the curves shown in Fig. 12a and the curve obtained experimentally by EPR (see Fig.5b) can be caused by the integral character of EPR: a microwave power signal of absorption by all spin centers (Pb-centers) in a sample is measured in this technique, whereas many of them do not interact with O_{2} molecules at a low oxygen concentration and, thus, are insensitive to a change in the ^{1}O_{2} concentration.

Hence, the EPR diagnostics of singlet oxygen generation is inapplicable in the case of a low concentration of O_{2} molecules. As noted above, the presence of a defect in a luminescent nc-Si crystal fully suppresses its luminescence due to fast (10^{–9} –10^{–11}) nonradiative trapping of nonequilibrium charge carriers. Therefore, this nanocrystal cannot excite an oxygen molecule on its surface, since the energy transfer time is several tens of microseconds (Gross et al., 2003). This conclusion is reflected in a decrease in the fraction of photosensitized ^{1}O_{2} when the number of nc-Si crystals with defects increases (Fig. 13). When concentration N' is equal to the number of nc-Si crystals in the ensemble under study (10^{18} cm^{–3}), the ^{1}O_{2} generation efficiency vanishes (Fig. 13). In the next section, we will show that the calculated dependences have an experimental support.

### 3.7. Effect of the powder granule size of porous silicon on the luminescence properties

Figure 14 shows micrographs of the FG PS powders taken on a scanning electron microscope. Submicronscale granules are clearly visible apart from micronscale granules (Fig. 14a). Moreover, many microgranules consist of a large number of smaller aggregated particles (Fig. 14b), with the number of such agglomerates increasing with the time of sample storage in air. Indeed, nc-Si crystals in PS layers oxidize in time and their surfaces become hydrophilic. Such PS granules coalesce and form coarse particles due to the

hydrogen bonds appearing between them. The image contrast of particles with a diameter of several tens of nanometers in Fig. 14 is low because of their strong static charging. As follows from Fig. 14, simple mechanical milling in a vibratory mill decreases the PS granule size to submicron sizes. This is also indicated by a change in the powder color from dark red–brown (in the case of a CG powder) to light yellow (FG powder), which is likely to be caused by Rayleigh scattering by particles several tens of nanometers in size and structural relaxation (i.e., the relief of microstresses, along which granules predominantly fail). As follows from the EPR spectroscopy data, the concentration of Pb-centers increases almost twofold, from 1.4 10^{17} to 2.8 10^{17} cm^{–3}. We assume that this increase leads to a decrease in the PL decay time and, hence, the PL intensity because of an increase in the probability of nonradiative exciton recombination by defects (Delerue et al., 1993).

An analysis of the relaxation times demonstrates that, for PS1 samples, they increase in the powders subjected to FG milling (Fig. 15a), which is in conflict with the assumption made above.

This experimental fact can be explained in terms of a model of exciton migration along a network of intersecting nc-Si crystals with a preferred motion direction from smaller to larger crystallites (against the gradient of the energy gap). Both radiative and nonradiative recombination is possible in each of the intermediate nc-Si crystals. The experimental fact of an increase in the PL relaxation time for the FG PS powders as compared to the CG powders can be explained by a decrease in the number of possible exciton migration ways when PS films are milled to granules with sizes varying from several tens to several hundreds of nanometers. For every fraction of nc-Si crystals having a certain size, a nonradiative relaxation channel is partly suppressed due to exciton motion from these nanocrystals; as a result, the exciton lifetimes at the PL wavelength corresponding to this nc-Si crystal size increase. It is important that the properties are modified upon crystal refinement only for nc-Si crystals on a fracture surface and that the internal porous structure of PS granules remains unchanged. Then, the detected change in the PL properties of the samples is related to the fact that the number of surface nc-Si crystals becomes comparable with their number in the volume of a PS granule when its size decreases from approximately 50 μm (etched layer thickness) to several tens of nanometers. Therefore, this effect can be detected experimentally. Indeed, the simplest model considerations yield the following estimate of the ratio of the number of surface nc-Si crystals having a spherical shape to their number in the volume of a spherical granule:^{13} and 1.3 10^{13} cm^{3}/s, respectively. Thus, upon fine milling of CG PS samples with a porosity of 70%, diffusion coefficient D* averaged over an ensemble of nc-Si crystals decreases approximately threefold. The weak change in parameter ^{15} cm^{–3} /10^{18} cm^{–3}=10^{–3} for the ensemble of nc-Si crystals under study (see Section 3.4). The developed theory can be used to determine the actual radiative PL lifetime of an ensemble of nc-Si crystals, in contrast to formal approximation parameter τ in Eq. (1) for an stretched exponent. Indeed, when analyzing the experimental kinetics, we obtain the value of approximation parameter_{g}≈1.6 eV. At this energy gap, the characteristic rate of nonradiative recombination at a defect is more than 10^{4} μs^{–1} (Delerue et al., 1993), whereas the exciton migration rate ^{–1}. Therefore, any products of type DN are substantially lower than

### 3.8. Effect of the degree of oxidation on the photosensitization activity of porous silicon

To study the efficiency of ^{1}O_{2} generation on the nc-Si surface, it is important to know its dependence on the photostimulated oxidation of PS samples during the photosensitization of molecular oxygen. Figure 16a shows the EPR spectra of coarse-grained PS1 powders that were recorded in an oxygen atmosphere as a function of the exposure time during irradiation by the light of a galogen lamp. The spectrum amplitude (and P_{b}-center concentration, respectively) is seen to monotonically increase with exposure time t_{exp} (Fig.16b). For PS powders to be applied in photodynamic therapeutic methods, it is

important to study the effect of the number of defects on the efficiency of singlet oxygen generation. Figure 17a shows the PL spectra of CG PS1 powders recorded in vacuum at various times t_{exp} of preliminary holding under the light in an oxygen atmosphere. The PL intensity decreases monotonically according to an increase in the contribution of nonradiative recombination, and the spectral maximum shifts toward short wavelengths due to an increase in the quantum confinement of excitons in Si nanostructures upon an increase in the surface SiO_{2} oxide thickness (Bisi et al., 2000).

Using the dependence of the concentration of P_{b}-centers in an ensemble of nc-Si crystals on the exposure time as a normalized curve (Fig. 16b), we can plot the dependence of the PL intensity at a certain wavelength on the defect concentration (Fig. 18a). The curve in Fig. 18a is seen to agree quantitatively with the calculated curves in Fig. 10a, which indicates that the initial principles of the developed exciton migration theory are consistent.

When the number of defects increases, the ^{1}O_{2} generation efficiency degrades, which is indicated by the monotonic decrease in the quenching function amplitude _{exp} (Fig.17b). The quenching function is the ratio of the PS PL spectrum amplitudes in vacuum and an oxygen-containing medium and characterizes the intensity of exciton recombination followed by energy transfer to oxygen molecules (Gross et al., 2003). Note that the quantity

approximately determines the fraction of oxygen molecules in the singlet state on the assumption that the entire energy of PL quenching of PS samples is consumed for the excitation of molecular oxygen.

Figure 18b shows quantity (36) as a function of the concentration of the defects that form during photostimulated oxidation at a wavelength λ=760 nm. The photosensitized activity of PS layers is seen to monotonically decrease with increasing number of P_{b}-centers, which points to the suppression of the luminescent and, hence, photosensitized properties of nanocrystals with point defects. As in the case of the PL intensity, the calculated curves shown in Fig. 13 agree qualitatively with the experimental curve in Fig. 18b. Thus, the photostimulated oxidation of PS samples leads to a monotonic decrease in the ^{1}O_{2} generation efficiency, which is important for practical application. After photosensitization, ensembles of nc-Si crystals transform into harmless amorphous SiO_{2}, which does not exert a phototoxic effect on biological tissues.

## 4. Conclusions

The generation of singlet oxygen was investigated and its concentration upon photoexcitation of silicon nanocrystals in microporous silicon layers was determined using EPR spectroscopy. It was demonstrated that a considerable part (approximately 40%) of the molecules of triplet oxygen can transform into the singlet state in the course of photosensitization of oxygen molecules in ensembles of silicon nanocrystals. The longitudinal (T_{1}) and transverse (T_{2}) relaxation times of spin centers increase under illumination of the samples in an oxygen atmosphere owing to the decrease in the efﬁciency of the dipole–dipole interaction between silicon dangling bonds and molecules of triplet oxygen as a result of the transformation of a number of the latter molecules into an excited (diamagnetic) state. It was shown that the amplitude of the EPR spectrum of triplet oxygen molecules in pores of silicon decreases upon photoexcitation of the samples, which suggests the formation of singlet oxygen molecules.

Taking into account exciton migration, we developed a quantitative model for the relaxation processes that occur in ensembles of silicon nanocrystals. With this model, we theoretically described the photosensitization of singlet oxygen on the surface of silicon nanocrystals under various external conditions. We experimentally studied the effect of the granule size of a porous silicon powder on its luminescence properties. Upon mechanical fine milling of as-deposited PS films, both the PL intensity and the concentration of point defects increase. The effect of the photostimulated oxidation of PS layers on the photosensitization of molecular oxygen on the surface of the nanocrystals forming the PS was investigated. The photosensitized activity of silicon nanocrystals was found to degrade monotonically during their oxidation. With the developed model, we explained the experimental data by restricted energy transfer between coupled silicon nanocrystals because of breaks in exciton migration trajectories when the granule size of a PS powder decreases and by an increase in the efficiency of nonradiative recombination during the photostimulated oxidation of the nanocrystals, which leads to the suppression of singlet oxygen generation. Moreover, using this model, we can approximate the kinetic curves of PL decay at a high accuracy, determining the actual exciton radiative recombination time in the nanocrystals. Our data indicate that PS is promising for the methods of nontoxic photodynamic therapy of a number of diseases, including oncological ones.