# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 2 Fundamentals of Semiclassical Device Modeling
### Kinetic Dissipation and Thermalization Models  
To overcome the well-known limitations of the phenomenological dissipation model in Equation (4), the key step is to replace the RTA scheme with a Boltzmann collision term of the form [2; 18; 20]  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{diss} }=\int d\mathbf{k}^{\prime}\left[P(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime}) f(\mathbf{r},\mathbf{k}^{\prime})-P(\mathbf{r};\mathbf{k}^{\prime},\mathbf{k})f( \mathbf{r},\mathbf{k})\right]\, \tag{10}\]  
where  
\[P(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime})=\left(1-f(\mathbf{r},\mathbf{k}) \right)P_{0}(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime}) \tag{11}\]  
denotes the low-density scattering rate \(P_{0}\) in \(\mathbf{r}\) for the generic transition \(\mathbf{k}^{\prime}\to\mathbf{k}\), weighted by the usual Pauli-blocking factor, and simply reduces to \(P_{0}(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime})\) in the low-density limit (\(f(\mathbf{r},\mathbf{k})\to 0\)). The Boltzmann-type collision operator in Equation (10) provides a kinetic description of energy dissipation in terms of stochastic carrier transitions dictated by the low-density scattering rates \(P_{0}(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime})\). Opposite to the phenomenological nature of the relaxation rate \(\Gamma(\mathbf{r},\mathbf{k})\) in Equation (4), such scattering rates can be microscopically derived in quantum-mechanical terms via the well-known [PERSON]'s golden rule including all relevant interaction mechanisms. Moreover, such a stochastic dynamics may be efficiently and intuitively simulated via a wide variety of Monte Carlo approaches [2]. As anticipated, the Boltzmann-type kinetic treatment in Equation (10) is intrinsically charge conserving, which implies that in this case the possible time variation of the total carrier number inside the device active region \(\Omega\) is due to device-reservoir thermalization processes only, customarily described via the phenomenological IBC scheme in Equation (5). As a result, most of the semiclassical device modeling is currently based on a kinetic treatment of dissipation phenomena inside the device properly combined with a phenomenological treatment of environment-induced carrier thermalization processes. This is typically accomplished via generalized (i.e., particle non-conserving) Monte Carlo simulations able to account for the presence of external reservoirs by means of corresponding injection versus loss terms [15]. As already stressed, the IBC thermalization model in Equation (5) is grounded on an artificial spatial separation between device and external contacts/reservoirs (see Figures 1 and 2); such an abrupt jump is, in general, not well justified, and one may expect that for some device operating conditions the distribution of incoming carriers may display nonequilibrium features as well. To properly describe the (continuous) device-reservoir transition, the key step is to include in the simulated region a significant portion of the electrical contacts as well; to this aim, in addition to the Boltzmann collision operator in Equation (10) (describing energy-dissipation phenomena inside the device), carrier thermalization due to the presence of the electrical contacts can be described again via a corresponding Boltzmann-type operator, whose thermalization rates are different from zero within the contact regions only. As a result, one deals with a kinetic treatment of both dissipation and thermalization phenomena, and the latter may be conveniently simulated via standard (i.e., particle conserving) Monte Carlo simulations [2]. Compared to the phenomenological IBC scheme, such kinetic treatments of the device-reservoir coupling are by far computationally more demanding.