# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 4 Fully Quantum-Mechanical Device Modeling Based on the Density-Matrix Formalism  
In view of the intrinsic limitations of local dissipation as well as thermalization models discussed thus far, it is imperative to replace such semiclassical treatments with fully quantum-mechanical schemes. Despite of the level of accuracy/approximation of the latter, intimately related to the nanosystem as well as to the specific phenomenon under investigation, two basic requirements are, however, always mandatory: (i) the positive-definite character of the quantum-mechanical state has to be preserved at any time; and (ii) in the absence of external electro-optical excitations as well as of nonequilibrium carrier injection, the solution of the steady-state Wigner equation (Equation (21)) should coincide with the thermal-equilibrium state. It is in general a highly non-trivial task to check the conformity to the above requirements; to this aim, it is crucial to exploit the close relation between the Wigner function and the corresponding single-particle density matrix [11; 15]. More specifically, adopting the very same notation employed in [67], and by compactly labeling with \(\alpha\) the set of relevant quantum numbers for the single-particle electronic states of our semiconductor nanostructure, the density matrix \(\rho_{\alpha_{1}\alpha_{2}}\) (whose diagonal terms of the density matrix describe the population of the generic single-particle state \(\alpha\), while the off-diagonal terms describe the quantum-mechanical phase coherence (or polarization) between states \(\alpha_{1}\) and \(\alpha_{2}\)) can be written in terms of the Wigner function \(f(\mathbf{r},\mathbf{k})\) as  
\[\rho_{\alpha_{1}\alpha_{2}}=\int\frac{d\mathbf{r}\,d\mathbf{k}}{(2\pi)^{3}}W_{ \alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k})f(\mathbf{r},\mathbf{k})\, \tag{22}\]  
where  
\[W_{\alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k})=\int d\mathbf{r}^{\prime}\phi _{\alpha_{1}}\left(\mathbf{r}+\frac{\mathbf{r}^{\prime}}{2}\right)e^{-i \mathbf{k}\cdot\mathbf{r}^{\prime}}\phi_{\alpha_{2}}^{*}\left(\mathbf{r}- \frac{\mathbf{r}^{\prime}}{2}\right) \tag{23}\]  
denotes the well-known Weyl-Wigner transform [70], and \(\phi_{\alpha}(\mathbf{r})\) the real-space wavefunction of the electronic state \(\alpha\). Applying the inverse Weyl-Wigner transform equation (Equation (22)) to the original Wigner transport equations (Equations (1) and (2)), in the absence of external electromagnetic excitations, the latter can be easily translated into the density-matrix equation  
\[\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}=\left.\frac{\partial \rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{det}}+\left.\frac{ \partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{stoc}} \tag{24}\]  
with  
\[\left.\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{ det}}=\frac{e_{\alpha_{1}}-e_{\alpha_{2}}}{\eta h}\,\rho_{\alpha_{1}\alpha_{2}}\, \tag{25}\]  
where \(\epsilon_{\alpha}\) is the energy of state \(\alpha\), and  
\[\left.\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{ stoc}}=\left.\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{\text{ diss}}+\left.\frac{\partial\rho_{\alpha_{1}\alpha_{2}}}{\partial t}\right|_{ \text{th}}. \tag{26}\]  
As far as the deterministic dynamics is concerned, the density-matrix term in Equation (25) is by far simpler compared to its Wigner-function version in Equation (12): the diagonal terms (\(\alpha_{1}=\alpha_{2}\)) do not change in time, while the non-diagonal terms (\(\alpha_{1}\
eq\alpha_{2}\)) simply rotate with frequency \(\left(\epsilon_{1}-\epsilon_{2}\right)/\hbar\). As far as the dissipation and thermalization terms in Equation (26) are concerned, the latter may be conveniently described via well-established density-matrix models able to fulfill the two basic requirements mentioned above [72].