# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 2 Fundamentals of Semiclassical Device Modeling  
The key ingredient of the semiclassical transport theory is the carrier distribution function. More specifically, adopting a simplified model based on a single conduction band, the semiclassical distribution function \(f(\mathbf{r},\mathbf{k})\) describes the probability distribution of the conduction electrons over the single-particle phase-space \(\mathbf{r},\mathbf{k}\), and is thus always positive-definite.  
The primary goal of any semiclassical device simulation is the investigation of the carrier distribution function within the device volume \(\Omega\), often referred to as the device active region. This is the result of a highly non-trivial interplay between the deterministic (i.e., scattering-free) carrier dynamics and the stochastic one induced on the carrier system by dissipation phenomena within the device active region as well as by thermalization processes due to the coupling of the device with external carrier reservoirs (see Figure 1). Regardless of the specific dissipation and thermalization models considered, the time evolution of the carrier distribution function can always be written as the sum of a deterministic (det) and of a stochastic (stoc) contribution,  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{stoc} }=\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{det} }+\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{ stoc}}\,, \tag{1}\]  
where the latter, in turn is the sum of a dissipation (diss) and of a thermalization (th) term,  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{stoc }}=\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{ diss}}+\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{th}}\,. \tag{2}\]In the presence of slowly-varying electromagnetic forces and relatively large device sizes (compared to the typical electron de Broglie wavelength) the deterministic carrier dynamics can be safely described in classical terms via the conventional drift-diffusion model of the Boltzmann transport theory [2; 18; 20]. More specifically, adopting the customary parabolic-band approximation (with effective mass \(m^{*}\)) and in the presence of a static electric field only (i.e., no magnetic fields) described by a force \(\vec{F}\), the deterministic contribution in Equation (1) is given by:  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{det} }=\left(-\vec{v}(\mathbf{k})\cdot\
abla_{\mathbf{r}}-\frac{\vec{F}}{\hbar} \cdot\
abla_{\mathbf{k}}\right)f(\mathbf{r},\mathbf{k})\, \tag{3}\]  
where \(\vec{v}(\mathbf{k})=\frac{\hbar\mathbf{k}}{m^{*}}\) denotes the carrier group velocity.  
As anticipated, in a conventional semiconductor device, the deterministic carrier dynamics in Equation (3) is significantly modified by a variety of stochastic processes described by Equation (2); the latter, in turn, take place both within the device active region due to different scattering mechanisms (carrier-impurity, carrier-phonon, carrier-carrier, etc.) and close to the device spatial boundaries due to carrier exchange between the device and one or more external charge reservoirs. The explicit form of the corresponding dissipation and thermalization contributions in Equation (2) strongly depends on the nature of the transport model adopted. As we now discuss, in any semiclassical simulation scheme, dissipation as well as thermalization processes are always described via spatially local models, and the same applies to the Wigner-function modeling as well (see Section 3). Such local treatments can, however, be subdivided into two different classes: phenomenological models based on relaxation-time schemes where the total electronic charge within the device is not necessarily conserved (open-system paradigm) and kinetic models based on Boltzmann-type schemes where the total number of simulated carriers is always conserved (closed-system paradigm).