The first prediction of slowed cooling at low light intensities in quantized structures was made by Boudreaux et al . They anticipated that cooling of carriers would require multiphonon processes when the quantized levels are separated in energy by more than phonon energies. They analysed the expected slowed cooling time for hot holes at the surface of highly doped n-type TiO2 semiconductors, where quantized energy levels arise because of the narrow space charge layer (i. e. depletion layer) produced by the high doping level. The carrier confinement in this case is produced by the band bending at the surface; for a doping level of 1 x 1019 cm-3, the potential well can be approximated as a triangular well extending 200 A from the semiconductor bulk to the surface and with a depth of 1 eV at the surface barrier. The multiphonon relaxation time was estimated from
tc ~ «-1exp(AE/kT) (9.4)
where tc is the hot carrier cooling time, rn is the phonon frequency, and A E is the energy separation between quantized levels. For strongly quantized electron levels, with AE > 0.2 eV, tc could be >100 ps according to equation (9.4).
However, carriers in the space charge layer at the surface of a heavily doped semiconductor are only confined in one dimension, as in a quantum film. This quantization regime leads to discrete energy states which have dispersion in k-space . This means the hot carriers can cool by undergoing interstate transitions that require only one emitted phonon followed by a cascade of single-phonon intra-state transitions; the bottom of each quantum state is reached by intra-state relaxation before an inter-state transition occurs. Thus, the simultaneous and slow multiphonon relaxation pathway can be bypassed by single-phonon events and the cooling rate increases correspondingly.
More complete theoretical models for slowed cooling in QDs have been proposed by Bockelmann and co-workers [20,46] and Benisty and co-workers [19,21]. The proposed Benisty mechanism [19,21] for slowed hot carrier cooling and phonon bottleneck in QDs requires that cooling only occurs via LO phonon emission. However, there are several other mechanisms by which hot electrons can cool in QDs. Most prominent among these is the Auger mechanism .
Here, the excess energy of the electron is transferred via an Auger process to the hole, which then cools rapidly because of its larger effective mass and smaller energy level spacing. Thus, an Auger mechanism for hot electron cooling can break the phonon bottleneck . Other possible mechanisms for breaking the phonon bottleneck include electron-hole scattering , deep level trapping  and acoustical-optical phonon interactions [50,51].