# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 3 Partially Quantum-Mechanical Device Modeling Based on the Wigner-Function Formalism
### Failure of Local Thermalization Models  
In the presence of a significant device-reservoir thermal coupling, the primary goal of any quantum simulation scheme is the solution of the steady-state Wigner transport equation  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{det} }+\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{diss }}+\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{th }}=0\ ; \tag{21}\]  
in particular, two crucial issues need to be assessed: (i) Is the solution of Equation (21) always unique? (ii) When this is the case, is it always physically acceptable? As we now discuss, the answers to these questions depend crucially on the choice of the thermalization model as well as on the presence/absence of dissipation phenomena within the device active region. In particular, we now briefly review the detailed analysis originally presented in [67]. The latter has shown intrinsic limitations of Wigner-function simulations based on the conventional IBC thermalization scheme in Equation (5), providing the following answers: (i) in the coherent limit, the solution of the steady-state Wigner equation (compatible with given spatial boundary conditions) is definitely not unique; and (ii) when dissipation phenomena are taken into account, the solution, although unique, may be physically unacceptable. It is useful to start with a simplified 1D model, having the notable advantage to allow for an exact analytical treatment; in particular: (i) the deterministic dynamics is described via a delta-like potential profile; (ii) the dissipation dynamics is neglected (so-called coherent limit); and (iii) the device-reservoir thermalization is described via the semiclassical IBC scheme in Equation (5), whose 1D version is schematically depicted in Figure 2. Within such a simplified scheme, it is possible to demonstrate [67] that the solution is not unique. This is a consequence of the double degeneracy of  
Figure 7: **(a,b)** The same as in Figure 6 but considering as initial state the thermal-equilibrium state \(\text{n}=1\). Reprinted from [61]. the 1D electronic scattering states corresponding to our potential profile, which in turn is due to the time-reversal symmetry of the dissipation-free dynamics. In the presence of energy dissipation phenomena, in contrast, the time-reversal symmetry is lost, and the solution of the steady-state Wigner equation is found to always be unique. However, such mathematical solution does not necessarily correspond to a physically acceptable Wigner function, namely to the Weyl-Wigner transform of a genuine single-particle density matrix (see Section 4). To better focus on this intrinsic limitation, let us consider a realistic GaAs-based nanostructure characterized by a rectangular potential barrier and assume a monoenergetic carrier injection from the left spatial boundary only. Moreover, we describe energy dissipation via the RTA scheme in Equation (4), with a space- and energy-independent relaxation rate \(\Gamma_{0}=1/\tau\). The resulting spatial carrier-density profiles corresponding to the coherent limit (solid curve) as well as to two different values of the relaxation time [\(\tau=5\) ps (dashed curve) and \(\tau=50\) fs (dash-dotted curve)] are reported in Figure 8. In the coherent limit, a strongly anomalous feature shows up: the density profile comes out to be rigorously symmetric, despite the assumed left-injection into a significantly high potential barrier. The presence of energy dissipation mechanisms (dashed and dash-dotted curve) induces some (apparently more physical) spatial asymmetry. However, for the parameters considered here, the barrier transmission coefficient is so small (of the order of \(10^{-3}\)) that no charge should be present in the right portion of the device (see also Figure 10 in Section 4.1). Moreover, in addition to this, all three density profiles display negative-value regions.