# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 2 Fundamentals of Semiclassical Device Modeling
### Phenomenological Dissipation and Thermalization Models  
As far as energy-dissipation within the device active region is concerned, the simplest choice is the adoption of a phenomenological model based on the relaxation-time approximation (RTA) [18; 20], i.e.,  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{diss }}=-\Gamma(\mathbf{r},\mathbf{k})\ \left(f(\mathbf{r},\mathbf{k})-f^{\circ}(\mathbf{r},\mathbf{k})\right)\, \tag{4}\]  
where the relaxation of the carrier distribution in \((\mathbf{r},\mathbf{k})\) toward its equilibrium value \(f^{\circ}(\mathbf{r},\mathbf{k})\) is described in terms of a space- and momentum-dependent relaxation rate \(\Gamma(\mathbf{r},\mathbf{k})\) encoding all relevant scattering mechanisms characterizing the operational conditions of the device under investigation; the space and momentum/energy dependence of \(\Gamma(\mathbf{r},\mathbf{k})\) may be extracted from fully microscopic Monte Carlo simulations [2] or modeled via simplified Fermi's Golden rule treatments. Figure 1: (Color online) Schematic representation of a typical semiconductor-based quantum device as an open system connected to two external charge reservoirs. Here, the distance between the interfaces is \(l\) and \(z\) is the longitudinal transport direction. Reprinted from [67]. As far as the device-reservoir thermalization is concerned, within the semiclassical picture, the latter may be easily described via a net spatial separation between the device active region \(\Omega\) and the external charge reservoirs (given by one or more electrical contacts/device terminals). Within this spirit, the simplest and physically more intuitive thermalization model is the so-called inflow-boundary-condition (IBC) scheme [2], where the carrier distribution function entering the device at its spatial boundary region coincides with the quasiequilibrium distribution in the external charge reservoirs. More specifically, denoting with \(\mathbf{r}^{b}\) the generic point of the device boundary surface \(\sigma^{b}\) and with \(f^{b}(\mathbf{r}^{b},\mathbf{k})\) the carrier distribution function entering the device at the surface point \(\mathbf{r}^{b}\), such a boundary condition can be easily fulfilled/implemented via the following thermalization operator [20]  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{th}}= -\int_{\sigma^{b}}d\sigma^{b}\,\delta(\mathbf{r}-\mathbf{r}^{b})\;v_{\text{in}} (\mathbf{r}^{b},\mathbf{k})\;\left(f(\mathbf{r}^{b},\mathbf{k})-f^{b}(\mathbf{ r}^{b},\mathbf{k})\right)\;, \tag{5}\]  
where  
\[v_{\text{in}}(\mathbf{r}^{b},\mathbf{k})=-\min(\vec{v}(\mathbf{k})\cdot\vec{ n}(\mathbf{r}^{b}),0) \tag{6}\]  
is the incoming component of the carrier group velocity \(\vec{v}(\mathbf{k})\) at the boundary point \(\mathbf{r}^{b}\) (\(\vec{n}(\mathbf{r}^{b})\) being the unit vector normal to the boundary surface \(\sigma^{b}\)). Indeed, according to its definition in Equation (6), \(v_{\text{in}}\) is different from zero for incoming carriers only; moreover, in view of the Dirac delta function, the action of the thermalization operator in Equation (5) is limited to the device boundary surface \(\sigma^{b}\). As a result, the above thermalization scheme is able, as requested, to fix the incoming carrier distribution to that of the external reservoirs, i.e., \(f(\mathbf{r}^{b},\mathbf{k}_{\text{in}})=f^{b}(\mathbf{r}^{b},\mathbf{k}_{ \text{in}})\), and has no influence on the outgoing carrier flux. The simplest physically relevant implementation of the IBC scheme in Equation (5) is that of a one-dimensional system/device (\(\mathbf{r},\mathbf{k}\to z,k\)) of length \(l\) in thermal contact with a left and a right charge reservoir. As schematically shown in Figure 2, in this case, the boundary surface \(\sigma^{b}\) of the three-dimensional case reduces to two points: \(z^{b}=\pm l/2\). This amounts to fixing the value of the incoming distribution function \(f(z,k)\) at these two points, i.e., \(f(-l/2,k>0)\) and \(f(+l/2,k<0)\).