## 3 Nonradiative Processes
### Phonon-Assisted Relaxation of Hot Carriers  
It is well known that highly excited (\"hot\") electrons, holes or excitons may relax to the lower states in order to minimize the system energy. In bulk semiconductors and QWS, where the electronic energy spectra are continuous, such a relaxation is accompanied by the emission of phonons caused by the interaction between electrons and lattice vibrations. This interaction also plays a major role in QDs and it is generally accepted that the mechanisms are essentially the same in nanocrystals and bulk materials. The most universal mechanism of coupling between the electrons (or holes) and long-wavelength acoustic phonons is through the volume deformation potential. The bottom of a non-degenerate band (e.g., conduction band in materials with zinc-blend structure) is shifted proportionally to the (local) relative variation of the volume [141],  
\[\hat{H}_{e-AP}=a_{c}\sum_{\
u}\left(\
abla\cdot\hat{\mathbf{u}}_{\
u}(\mathbf{ r})\right)\,,\]  
where \(a_{c}\) is the bulk deformation potential constant, and \(\hat{\mathbf{u}}_{\
u}\) denotes the atomic displacement operator corresponding to an acoustic phonon mode \(\
u\) (it is expressed in terms of phonon creation and annihilation operators in usual way). For optical phonons, there are two mechanisms: (i) a universal optical deformation potential (ODP) coupling and (ii) the Frohlich one characteristic of polar materials. In bulk semiconductors with cubic structure, for long-wavelength optical phonons, the ODP coupling vanishes by symmetry for any non-degenerate band but it is non-zero for holes near the top of the valence band [141]. Its expression can be found in the book [12] (Chapter 8) or in [142]. The Frohlich interaction Hamiltonian for electrons is simply given by  
\[\hat{H}_{e-OP}=-e\sum_{\
u}\hat{\phi}_{\
u}(\mathbf{r})\,\]  
where \(\hat{\phi}_{\
u}\) is the electrostatic potential operator corresponding to an optical phonon mode \(\
u\). With the known electron and hole wavefunctions and phonon displacements obtained from lattice dynamics equations, the calculation of the coupling matrix elements is straightforward. Even though the Frohlich interaction usually dominates in QDs made of polar semiconductors, the ODP mechanism is important for some of them (e.g., InP) as revealed by modelling of QD resonant Raman spectra [142]. The so-called rigid-ion model is more suitable when describing the electron-phonon interaction in multi-valley semiconductors, such as Si and Ge [143], where the processes of inter-valley scattering are often important. Within this model, the electron-phonon interaction is described by the following operator (which describes both acoustic and optical phonons),  
\[\hat{H}_{e-P}=-\sum_{\
u:\mathbf{n},s}\hat{\mathbf{u}}_{\
u}(\mathbf{R}_{ \mathbf{n}s})\cdot\
abla V_{at}(\mathbf{r}-\mathbf{R}_{\mathbf{n}s}).\]  
Figure 3: Calculated rates of carrier trapping on dangling bonds compared to the rates of radiative phonon-assisted transitions and Auger recombination in Si nanocrystals. Here, \(V_{at}(\mathbf{r}-\mathbf{R}_{\mathrm{ns}})\) stands for the electron potential energy in the field of the \(s\)-th atom situated in the \(\mathbf{n}\)-th primitive cell of the lattice and \(\mathbf{R}_{\mathrm{ns}}\) is the position-vector of this atom in equilibrium. In practice, atomic pseudopotentials are used. The discreteness of the energy spectrum in quantum dots implies that the rate of the single-phonon emission should strongly depend on the level spacing [144]; in particular, electronic transitions causing the phonon emission should be impossible if the spacing is larger than the phonon energy. This simple idea gave rise to the \"phonon bottleneck\" concept, which predicts the inefficiency of hot carrier relaxation by emission of phonons in QDs [145]. However, this prediction relies on the assumption that the phonon emission is irreversible, with a probability described by the perturbation theory (Fermi's Golden Rule), which may not be reliable in QDs.