## 2 Light Emission in Nanocrystals
### Silicon Nanocrystals  
Since the 1990s, the optical properties of Si NCs have been widely discussed [25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89], Many methods have been proposed to increase their emissivity, such as doping with shallow impurities [83; 84; 85; 86; 87; 88; 89], growth in different matrices [90; 91], plasma [92; 93], and colloidal solutions [94; 95; 96; 97; 98]. As a result, numerous theoretical predictions and experimental observations of enhanced PL intensity [83; 84; 87; 88], radiative recombination rates [89; 99; 100; 101; 102; 103; 104; 105; 106; 107] and quantum efficiency of photon generation [108] have been reported. It is well known that introduction of shallow impurities is capable of modifying electronic properties of bulk silicon. Similarly, doping with shallow impurities influences the electronic structure of silicon NCs [109; 110; 111; 112; 113; 114; 115; 116; 117], which, in turn, affects the electron-hole radiative recombination. It was revealed that doping of Si nanocrystals with P or Li is (under certain conditions) capable of improving their emittance [84; 87; 89; 118]. In particular, the performed calculations [87; 99; 100; 106] show that doping with P or Li can essentially increase the radiative transition rates. For NCs with \(R\gtrsim 1\) nm, semi-empirical methods are usually employed for computing their electronic structure, such as envelope function approximation [119; 120; 121] or tight binding model [122]. The performed calculations of the phonon-assisted radiative recombination rates in Si crystallites in the SiO\({}_{2}\) matrix yielded the values varying from \(\sim\)\(10^{2}\) to \(\sim\)\(10^{5}\) s\({}^{-1}\) as the crystallite radius decreases from 3 to 1 nm. Within a simple model of an infinitely deep spherical potential well, the dependence of the radiative decay rate on the crystallite radius is \(\tau_{R}^{-1}\propto R^{-3}\)[121]. For the no-phonon radiative transitions, the rates turn out to be much lower, by one to three orders of magnitude with increasing the size within the same range. In this case, \(\tau_{R}^{-1}\propto R^{-8}\)[121], i.e., the rate sharply drops as \(R\) increases. At the same time, doping with phosphorus provides a several orders of magnitude increase of the radiative recombination rates for the no-phonon transitions. The rates become even 1-3 orders greater than those of the phonon-assisted transitions. They slowly decrease with increasing the NC size, especially for high concentration of phosphorus [99]. As calculations show, the radiative transitions including both no-phonon and phonon-assisted ones become faster in small Si crystallites. However, for crystallites with \(R\lesssim 1\) nm, all semi-empirical methods of calculation turn out to be essentially less accurate. In this case, first-principle calculations have to be carried out. In Figure 1, the radiative recombination rates computed at room temperature \(T\) within the [PERSON]'s version [123] of the time-dependent density functional theory (TDDFT) are depicted for H-passivated P-doped (Si\({}_{46}\)H\({}_{60}\)P, Si\({}_{57}\)H\({}_{84}\)P, Si\({}_{86}\)H\({}_{76}\)P, Si\({}_{122}\)H\({}_{100}\)P, Si\({}_{166}\)H\({}_{124}\)P, Si\({}_{238}\)H\({}_{196}\)P), Li-doped (Si\({}_{42}\)H\({}_{64}\)Li, Si\({}_{66}\)H\({}_{64}\)Li, Si\({}_{82}\)H\({}_{72}\)Li, Si\({}_{106}\)H\({}_{120}\)Li, Si\({}_{172}\)H\({}_{120}\)Li, Si\({}_{220}\)H\({}_{144}\)Li), and undoped (Si\({}_{47}\)H\({}_{60}\), Si\({}_{71}\)H\({}_{84}\)SisP\({}_{76}\), Si\({}_{123}\)H\({}_{100}\), Si\({}_{167}\)H\({}_{124}\), Si\({}_{239}\)H\({}_{196}\)) Si nanocrystals. Here, NCs are considered in vacuum, therefore, \(\epsilon_{1}=1\), while for Si \(\epsilon=12\).