## 3 Nonradiative Processes
### Auger Recombination  
In order for the Auger process to occur, at least one negative or positive trion (exciton + one electron or exciton + one hole) has to be initially excited in a nanocrystal, as shown in Figure 2. The trion turns into a high-energy single-particle excitation as indicated by the arrows in the figure.  
The rate of an Auger-type transition can be written as  
\[\tau_{A}^{-1}=\frac{2\Gamma}{\hbar}\sum_{f}\frac{|U_{if}|^{2}}{\Gamma^{2}+(E_ {f}-E_{g})^{2}}, \tag{6}\]  
where \(U_{if}\) is the matrix element of the two-particle Coulomb interaction operator, the two-electron wave functions are approximated by products of the single-electron ones and the \(\delta\)-function expressing the energy conservation in the Fermi's Golden Rule has been broadened with the half-width \(\Gamma=10\) meV. All the possible final states of the Auger electron with energy \(E_{f}\) (shown in Figure 2) are summed up. The initial states were chosen so that electrons and holes are in the LUMO and HOMO states, respectively.  
Usually, the Auger recombination is an efficient process that can \"shunt\" other photon-assisted processes such as light emission or multiple exciton generation. For instance, in Si crystallites, the Auger rates can be sufficiently high, as the measurements [133, 134] and calculations [106, 135, 136, 137, 138] show. Therefore, suppression of the Auger process is an important practical problem.  
It turns out that coating with halogens can slow down the Auger processes in Si NCs. Below we present the calculated Auger rates and demonstrate their reduction due to the surface halogenation of Si crystallites. The calculations take into account both the long  
Figure 2: Auger recombination in nanocrystals via _eeh_ and _ehh_ processes. Arrows indicate Auger transitions between the initial and final states.  
range (\(U_{0}(\mathbf{r_{1}},\mathbf{r_{2}})\)) and the short-range (\(U_{1}(\mathbf{r_{1}},\mathbf{r_{2}})\)) parts of the carrier-carrier interaction. The first one is a macroscopic electrostatic field modified by the nanocrystal boundary,  
\[U_{0}(\mathbf{r_{1}},\mathbf{r_{2}}) = \frac{e^{2}}{\epsilon|\mathbf{r_{1}}-\mathbf{r_{2}}|}+\frac{e^{2} (\varepsilon-1)}{\varepsilon R} \tag{7}\] \[\times\sum_{l=0}^{\infty}\frac{(l+1)P_{l}(\cos\theta)}{l \varepsilon+l+1}\frac{r_{1}^{l}r_{2}^{l}}{R^{2l}}\,,\]  
while the second one describes at a microscopic level the point charges interaction at short distances:  
\[U_{1}(r)=\frac{e^{2}}{r}[Ae^{-\alpha r}+(1-A-1/\varepsilon)e^{-\beta r}]\,. \tag{8}\]  
Here, \(r=|\mathbf{r_{1}}-\mathbf{r_{2}}|\), \(A=1.142\), \(\alpha=0.82/a_{B}\) and \(\beta=5/a_{B}\)[99], where \(a_{B}\) stands for the Bohr radius, and we set \(\epsilon_{1}=1\) as before. The calculated values of \(\tau_{A}^{-1}\) are listed in Table 2 for both _eeh_ and _ehh_ trion annihilation in the Si\({}_{n}\)X\({}_{m}\) crystallites considered here.  
The surface halogenation of Si nanocrystals also slows down the Auger recombination according to the data presented in Table 2. In order to quantitatively analyse the effect of halogenation on the Auger rates, we have calculated the average decimal logarithms of \(\tau_{A}^{-1}\) for the Br-, Cl- and H-passivated nanocrystals. In the case of the _eeh_ Auger recombination one has obtained: \(\langle\lg(\tau_{A(\mathrm{H})}^{-1})\rangle=11.66\), \(\langle\lg(\tau_{A(\mathrm{Cl})}^{-1})\rangle=10.55\) and \(\langle\lg(\tau_{A(\mathrm{Br})}^{-1})\rangle=10.60\). In the case of the _ehh_ Auger recombination the rates also decrease but not so strongly: \(\langle\lg(\tau_{A(\mathrm{H})}^{-1})\rangle=11.86\), \(\langle\lg(\tau_{A(\mathrm{Cl})}^{-1})\rangle=11.44\) and \(\langle\lg(\tau_{A(\mathrm{Br})}^{-1})\rangle=11.20\). It should be noted also that strong reduction of the Auger recombination rates was theoretically shown by [PERSON] [139] for GaSb NCs whose surfaces were passivated with atoms of electronegative elements.