## 3 Electron Spin-Dependent Processes in Si Nanocrystals
### Phenomenological Model of the Photosensitization by Si Nanocrystals  
The photosensized generation of \({}^{(1)}\)O\({}_{2}\) in nc-Si ensembles is quantitatively explained by using a model based on the rate equations for excitons and oxygen molecules [65; 79]. In brief, the number of excitons in the nc-Si ensemble at low temperatures is limited due to the long radiative lifetime of excitons and efficient nonradiative Auger recombination in nc-Si [80]. According to the experimental data the PL intensity of por-Si in vacuum increases linearly with the excitation intensity up to 0.01 W/cm\({}^{2}\) as shown in Figure 25 [65]. At higher excitation intensities an occupation of each nc-Si by two electron-hole pairs can be achieved and the nonradiative Auger recombination results in a saturation behavior of the emission intensity (triangles in Figure 25). The model describes the experimental data both for por-Si in vacuum (curve 1) and after oxygen adsorption, which leads to the superlinear dependence \(I_{\rm PL}\) on \(I_{\rm ex}\) (curves 2,3). The fitting parameters correspond to the product of \((\beta\cdot n_{0})^{-1}=200\) us, which is close to the experimentally determined \(\tau_{\rm ET}\) (see Figure 25). The fitting parameter \(\beta\) can be calculated by considering the microscopic mechanism of the energy transfer [78]. A coupled system, which consists of nc-Si and O\({}_{2}\) molecules, under optical excitation can be described by a simple set of the rate equations [79]:  
\[\frac{dN_{1}}{dt}\ =\ N_{0}\sigma\Phi-\frac{N_{1}}{\tau_{ex}}-\beta N_{1}n_{0},\frac{dn_{1}}{dt}\ =\ \beta N_{1}n_{0}-\frac{n_{1}}{\tau_{\Delta}}\, \tag{16}\]  
where \(N_{0}\) and \(N_{1}\) are the numbers of nc-Si containing 0 and 1 exciton, respectively, \(n_{0}\) and \(n_{1}\) are the numbers of O\({}_{2}\) molecules being in the ground \({}^{3}\Sigma\) and singlet \({}^{1}\Delta\) states, respectively, \(\tau_{\rm ex}\) is the exciton lifetime, \(\tau_{\Delta}\) is the lifetime of the \({}^{1}\Delta\) state, \(\sigma\) is the cross section of optical absorption of nc-Si, \(\Phi\) is the photon flux density of exciting light, \(\beta\) is the coupling coefficient. The latter value describes the energy transfer probability of in nc-Si/O\({}_{2}\) system and the reciprocal product \((\beta n_{0})^{-1}\) corresponds to the experimentally accessible energy-transfer time \(\tau_{\rm ET}\). Figure 25: PL intensity of por-Si in vacuum (triangles, \(h\
u_{\rm PL}\) = 1.47 and 1.5 eV) and with adsorbed O\({}_{2}\) molecules (closed circles, \(h\
u_{\rm PL}\) = 1.47 eV and open circles, \(h\
u_{\rm PL}\) = 1.5 eV) vs excitation intensity. Dashed line is a linear law approximation. The datasets are arbitrarily scaled for clarity. Solid lines are the fits by using Equations (16) and (17) with the following parameters: \(N=10^{19}\) cm\({}^{-3}\), \(\sigma=10^{-15}\) cm\({}^{2}\), \(\tau_{\rm ex}\) = 0.8 ms, \(\tau_{\Delta}\) = 4 ms, \(\beta\) = 5\(\cdot 10^{-16}\) cm\({}^{3}\)/s and \(n\) = 0 (curve 1) and oxygen concentrations are taken as \(n=10^{19}\) sm\({}^{-3}\) (curve 2), \(n\) = 3\(\cdot 10^{19}\) cm\({}^{-3}\) (curve 3). Excitation photon energy \(E_{\rm exc}\) = 2.41 eV, \(T\) = 5 K. (The experimental data are taken from Reference [65]. Reprinted from Reference [79] by permission from Springer Nature Customer Service Centre GmbH. Springer © 2007). To solve Equation (16) the following expressions for the total numbers of interacting species can be used:  
\[N\ =\ N_{0}+N_{1},n\ =\ n_{0}+n_{1}, \tag{17}\]  
where \(N\) and \(n\) are the total numbers of nc-Si and O\({}_{2}\) molecules, respectively. The discussed phenomenological model does not account contributions of the upper excited states of nc-Si and O\({}_{2}\) molecules, the nonradiative recombination of excitons in nc-Si and the reverse energy transfer from \({}^{(1)}\)O\({}_{2}\) to nc-Si. The exciton PL intensity of nc-Si is obtained from the steady state solution of Equations (16) and (17) and from the relation \(I_{\rm PL}=N_{1}/\tau_{\rm ex}\).