# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 4 Fully Quantum-Mechanical Device Modeling Based on the Density-Matrix Formalism
### Non-Local Relaxation-Time Models
To this aim, our strategy, originally proposed in [61], is to start from the physically sound density-matrix RTA scheme in Equation (29) and apply to it the following Weyl-Wigner transform:  
\[f(\mathbf{r}^{\prime},\mathbf{k}^{\prime})=\sum_{\alpha_{1}^{\prime}\alpha_{2} ^{\prime}}W_{\alpha_{2}^{\prime}\alpha_{1}^{\prime}}(\mathbf{r}^{\prime}, \mathbf{k}^{\prime})\rho_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}. \tag{31}\]The resulting Wigner-function model is given by  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{RTA}}=- \int d\mathbf{r^{\prime}}\,d\mathbf{k^{\prime}}\,\overline{\Gamma}(\mathbf{r}, \mathbf{k};\mathbf{r^{\prime}},\mathbf{k^{\prime}})\left(f(\mathbf{r^{\prime}}, \mathbf{k^{\prime}})-\overline{f}(\mathbf{r^{\prime}},\mathbf{k^{\prime}}) \right)\, \tag{32}\]  
where  
\[\overline{\Gamma}(\mathbf{r},\mathbf{k};\mathbf{r^{\prime}},\mathbf{k^{\prime }})=\sum_{\alpha_{1}\alpha_{2}}W_{\alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k} )\,\frac{\overline{\Gamma}_{\alpha_{1}}+\overline{\Gamma}_{\alpha_{2}}}{16 \pi^{3}}\,W_{\alpha_{1}\alpha_{2}}^{*}(\mathbf{r^{\prime}},\mathbf{k^{\prime }}) \tag{33}\]  
is a fully nonlocal RTA superoperator expressed in terms of the relaxation rates \(\overline{\Gamma}_{\alpha}\) of the density-matrix theory, and \(\overline{f}(\mathbf{r},\mathbf{k})\) is the Weyl-Wigner transform of the equilibrium or quasiequilibrium density matrix \(\overline{\rho}_{\alpha_{1}\alpha_{2}}\). In striking contrast to the standard RTA model in Equation (7), the generalized version in Equation (32) intrinsically ensures the positivity of the spatial charge density as well as of the single-particle state populations; indeed: (i) the density-matrix RTA model in Equation (29) is known to preserve the positive-definite character of the density matrix \(\rho_{\alpha_{1}\alpha_{2}}\); and (ii) the action of the Weyl-Wigner transform in Equation (31) does not alter such property. At this point, it is crucial to recall (see Section 2) that the original local RTA model in Equation (7) may be employed to describe both dissipation and thermalization phenomena, and the same applies to the density-matrix based RTA model in Equation (29) as well. Indeed, as discussed in more detail in [72], the generic relaxation rate \(\overline{\Gamma}_{\alpha}\) may describe energy dissipation inside the device toward an equilibrium-state population (\(\overline{\Gamma}_{\alpha}=\Gamma_{\alpha}^{\text{diss}}\), \(\overline{f}_{\alpha}=f_{\alpha}^{\circ}\)), or device-reservoir thermalization toward the quasiequilibrium state population induced by one or more external charge reservoirs (\(\overline{\Gamma}_{\alpha}=\Gamma_{\alpha}^{\text{res}}\), \(\overline{f}_{\alpha}=f_{\alpha}^{\text{res}}\)). It follows that also the new RTA model in Equation (32) may be employed for the description of dissipation as well as thermalization phenomena, and that in both cases the action of the new RTA superoperator is always non-local. To illustrate the quality and physical soundness of the proposed non-local RTA scheme in Equation (32), we focus first on the study of energy dissipation phenomena. To this end, the new non-local RTA model has been applied once again to the prototypical QW nanosystem in Figure 3 adopting again the very same effective 1D model introduced in Section 3.1. In this case, the in-plane carrier dynamics is neglected and the confined states of our QW system as well as the corresponding dissipation rates are labeled by the integer number n only, i.e., \(\Gamma_{\alpha}^{\text{diss}}\equiv\Gamma_{\text{n}}^{\text{diss}}\). In particular, Figure 9 shows the results obtained by setting \(1/\Gamma_{1}^{\text{diss}}=1\,\text{ps}\) and \(1/\Gamma_{2}^{\text{diss}}=0.7\,\text{ps}\). A comparison with Figures 4 and 5 shows that, here, opposite to the pathological behaviors obtained via the RTA models M2 and M3, both the spatial carrier distributions (Figure 9a) and the corresponding level populations (Figure 9b) are always positive-definite.