Transport models for semiconductor devices are commonly based on the Boltzmann Transport Equation (BTE) [65]. One method, derived from the BTE, is the drift-diffusion (DD) model. Based on the first two moments of the BTE, the DD model is neglecting non-local effects. This method became very popular for macroscopic devices in the last century. Till today the DD model is used as a trade-off between computational efficiency and accuracy [30, 33, 82, 91–93].
With shrinking device geometry and the need to include hot-carrier effects, this method suffers some severe drawbacks. In particular, the DD model assumes that the average carrier energy is in equilibrium with the local electric field. This assumption has been shown to be invalid as the average carrier energy lags behind the electric field [27, 30]. The DD model is unable to calculate non-local effects and all kinds of hot carrier effects [28, 87, 97]. Despite these limitations, there are some attempts to estimate the high-energy-tail within the DD model [92]. The DD model includes the first two moments of BTE. Including more moments of the BTE leads to the hydrodynamic and energy-transport models. However, for small devices, they often do not deliver the expected improved accuracy. For higher accuracy in smaller devices a six moments model was proposed [29].
In the 1970s a stochastic approach for the solution of the BTE became popular. The Monte Carlo method is based on the calculation of random carrier trajectories in a semiconductor. The trajectory calculation takes into account realistic band-structure models and various kinds of scattering processes in the semiconductor [21, 34, 43, 51, 59, 61].
In the beginning, an analytic model of the semiconductor band structure was sufficient for Monte Carlo simulations, but with the growing significance of hot-carrier effects, a more accurate band structure model was required. This accuracy was achieved by calculating the full-band structure of a semiconductor numerically and embedding it into the Monte Carlo simulation [65].
A problem in hot-carrier modeling is that Monte Carlo is a stochastic method and therefore a huge number of carrier trajectories is needed for obtaining an adequate ensemble of rare states representing the hot-carriers. A solution to this problem was suggested in the late 1980s, known as the backward Monte Carlo method. The theory is based on generating the sates of hot-carriers in a device first and letting the carriers travel back in time to their origin at an injecting contact. With this backward Monte Carlo (BMC) algorithm, the probability of each hot-carrier state can be calculated, and no other carriers than those of interest have to be simulated. This brings a vast decrease in simulation time. The initial version of this algorithm was numerically unstable. Therefore, it was never implemented in a semiconductor device simulator [45, 73].
In 2003, a stable backward Monte Carlo algorithm was proposed [61] and finally implemented in a semiconductor device simulator in 2015 [P4]. The backward Monte Carlo method is able to overcome the statistical drawbacks of the conventional Monte Carlo method. In particular, the capability to include particles with arbitrary energies and the usage of full-bands, is giving this method an advantage over simulators using conventional Monte Carlo algorithms or deterministic solvers [P4, 62].
A deterministic method to solve the BTE is based on spherical harmonics expansion (SHE). This method relies on the spherical symmetry for the band-structure model. Thus, the dispersion relation for higher energies is not well represented [22, 24, 25, 39, 40, 50, 84, 87].
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