For the application of the present hot-carrier model, the evaluation of the acceleration integral (cf. Equation (7.8)) is necessary. To achieve this, the energy distribution of the charge carriers is required. In order to obtain the energy distribution, the Boltzmann Transport equation (BTE) has to be solved, which is challenging in its own right. Unfortunately, HCD is highly sensitive to the high-energy tail of the distribution, and therefore the modeling of the scattering operator requires special attention. In particular, impact ionization scattering as well as electron-electron interactions have to be incorporated. For example, it has been shown that the accuracy of the BTE solution ignoring electron-electron scattering can be seriously impacted [189]. Furthermore, it has been shown that the majority carriers can, in some cases, significantly contribute to the damage, requiring a coupled solution of the BTE for electrons and holes [174]. Finally, since an accurate resolution of the energy distribution at high energies is required, information about the full band structure has to be included into the model. Traditionally, this complicated problem is approached by using the Monte Carlo method [25], which is computationally- and time-intensive, particularly when the high-energy tails of the distribution function have to be resolved in detail [44]. Until a fully functional full-band spherical harmonics expansion (SHE) simulator with electron-electron scattering incorporated has been available the BTE had to be solved using the Monte Carlo approach [25]. The main advantages of the MC method are, that it is a well-known and investigated method, as well suited for the incorporation of the full dispersion relation (cf. Figure 2.1). However, the MC method is capable of resolving the whole phase space only at the cost of a square root dependence on the number of particles per magnitude of resolution in the distribution function. Aside from a limited time-scale, Monte Carlo simulators are not easily extendable to self-consistently solve the BTE, Poisson’s Equation and the HCD model equations. Additionally, it has been shown that the use of strongly simplified models derived from approximations of the acceleration integral in moment based simulators leads to unsatisfactory results [190].
Due to enormous computational burden of the MC method to resolve the high energy tail, a spherical harmonics expansion of the BTE to assess HCD is attractive. Thus, SHE based BTE solvers, such as ViennaSHE [65], have been developed in hopes to overcome the inadequacies of the Monte Carlo method. A SHE based simulator can easily be extended to numerically solve any defect equation self-consistently with the set of SHE-BTE equations. However, the greatest advantage of a SHE based simulator is that there is no numeric noise whatsoever in the energy distribution function. Nevertheless, SHE is still a fairly ‘young’ method. This means that not all possible numeric schemes have been sufficiently explored yet. Additionally, it is quite challenging to incorporate full-band effects into a SHE based simulator [47]. Nevertheless, as a solution technique to the BTE, a spherical harmonic expansion delivers the better noise/performance trade-off compared to Monte Carlo methods [46, 44].