# 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  
As discussed in Section 2, both dissipation and thermalization processes are routinely described via the generic RTA scheme in Equation (7). To better clarify the intrinsic limitations of such local model within a Wigner-function simulation, it is useful to analyze its density-matrix version. To this aim, applying to Equation (7) the inverse Weyl-Wigner transform equation (Equation (22)), one gets [61]  
\[\left.\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{RTA }}=-\sum_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\overline{\Gamma}_{\alpha_{1 }\alpha_{2},\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\left(\rho_{\alpha_{1}^{ \prime}\alpha_{2}^{\prime}}-\overline{\varrho}_{\alpha_{1}^{\prime}\alpha_{2}^ {\prime}}\right) \tag{27}\]  
with  
\[\overline{\Gamma}_{\alpha_{1}\alpha_{2},\alpha_{1}^{\prime}\alpha_{2}^{\prime }}=\int\frac{d\mathbf{r}\,d\mathbf{k}}{(2\pi)^{3}}W_{\alpha_{1}\alpha_{2}}( \mathbf{r},\mathbf{k})\overline{\Gamma}(\mathbf{r},\mathbf{k})W_{\alpha_{1}^ {\prime}\alpha_{2}^{\prime}}^{*}(\mathbf{r},\mathbf{k})\, \tag{28}\]  
where \(\overline{\varrho}_{\alpha_{1}\alpha_{2}}\) denotes the inverse Weyl-Wigner transform of the equilibrium or quasiequilibrium distribution function \(\overline{f}(\mathbf{r},\mathbf{k})\) in Equation (7). It is now worth comparing Equation (27) (obtained by applying a semiclassical RTA model to the Wigner function) to the equation obtained applying the RTA scheme straightforwardly to the density-matrix evolution, i.e.,  
\[\left.\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{RTA }}=-\left.\frac{\overline{\Gamma}_{\alpha_{1}}+\overline{\Gamma}_{\alpha_{2}}} {2}\ \left(\rho_{\alpha_{1}\alpha_{2}}-\overline{\rho}_{\alpha_{1}\alpha_{2}} \right)\, \tag{29}\]  
where \(\overline{\rho}_{\alpha_{1}\alpha_{2}}=\overline{f}_{\alpha_{1}}\delta_{ \alpha_{1}\alpha_{2}}\) denotes the (diagonal) equilibrium or quasiequilibrium density matrix (with respect to our single-particle basis \(\alpha\)). Here, \(\overline{\Gamma}_{\alpha}\) can be regarded as a sort of state-dependent effective inverse life-time, involving all relevant dissipation/thermalization mechanisms acting on the carrier in state \(\alpha\); indeed, in the semiclassical limit (\(\rho_{\alpha_{1}\alpha_{2}}=f_{\alpha_{1}}\delta_{\alpha_{1}\alpha_{2}}\)), the density-matrix RTA model in Equation (29) reduces to the well-known injection-loss scheme [15] of conventional device modeling. \[\left.\frac{\partial f_{\alpha}}{\partial t}\right|_{\text{RTA}}=-\overline{ \Gamma}_{\alpha}\left(f_{\alpha}-\overline{f}_{\alpha}\right)=\overline{S}_{ \alpha}-\overline{\Gamma}_{\alpha}f_{\alpha}. \tag{30}\]  
In addition to the population dynamics in (30), the density-matrix RTA model in (29) describes the dissipation/thermalization-induced decay of the interlevel polarization, known as decoherence process. As one can see, while the superoperator in Equation (29) is diagonal in the single-particle basis \(\alpha\), the superoperator \(\overline{\Gamma}\) in Equation (27) is, in general, non-diagonal in such basis. Moreover, while it is possible to show [72] that the density-matrix RTA model in Equation (29) is always positivity preserving, the same does not apply, in general, to the non-diagonal RTA model in Equation (27) [and therefore to its Wigner-function version in Equation (7)], as confirmed by the negative carrier populations in Figures 4 and 5). The only exception is the case of a space- and energy-independent relaxation rate, \(\overline{\Gamma}(\mathbf{r},\mathbf{k})=\overline{\Gamma}_{0}\), as well as a state-independent rate, \(\overline{\Gamma}_{\alpha}=\overline{\Gamma}_{0}\), for which the two RTA models in Equations (27) and (29) coincide. This is indeed confirmed by the convincing results in Figure 4a. The density-matrix analysis presented so far suggests to replace the highly problematic semiclassical-like RTA scheme in Equation (7) with a fully quantum-mechanical Wigner-function model.