# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 3 Partially Quantum-Mechanical Device Modeling Based on the Wigner-Function Formalism
### Failure of Local Dissipation Models
The rates entering the Boltzmann collision term in Equation (10) are evaluated via the conventional Fermi's golden rule using as noninteracting states standard 3D plane waves, instead of the nanostructure single-particle wavefunctions, and assuming as main dissipation mechanism the carrier interaction with a bulk GaAs LO-phonon system [2]. The quantities \(P(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime})\) are therefore space-independent. Within such 3D description, the Weyl-Wigner phase-space of the QW device is given by \(\mathbf{r}\equiv\mathbf{r}_{\parallel},z\) and \(\mathbf{k}\equiv\mathbf{k}_{\parallel},k_{z}\). The effective 1D carrier-density profile along the growth direction (\(z\)) in Equation (14) is then now replaced by  
\[n(z)=\frac{1}{(2\pi)^{3}}\,\int d\mathbf{r}_{\parallel}\,d\mathbf{k}\,f( \mathbf{r},\mathbf{k})\, \tag{19}\]  
and the effective 1D level population in Equation (15) by the following subband population  
\[n(z)=\frac{1}{2\pi}\,\int dk\,f(z,k)\, \tag{14}\]0  
Here, \(f_{\mathbf{k}_{\parallel}\mathbf{n}}\) is the population of the generic carrier state in subband n with in-plane wavevector \(\mathbf{k}_{\parallel}\), obtained via the 3D version (\(\text{n}\rightarrow\mathbf{k}_{\parallel}\mathbf{n}\)) of the Weyl-Wigner transform in Equation (15). Figure 6 shows the low-temperature and low-density energy-dissipation dynamics for the GaAs/(Al,Ga)As prototypical QW nanostructure of Figure 3, resulting from the fully 3D Boltzmann  
\[n(z)=\frac{1}{2\pi}\,\int dk\,f(z,k)\, \tag{14}\]1 Bound-state populations [see Equation (15)] as well as the continuum (cont) contribution as a function of time for models of M1 (**a**); M2 (**b**); and M3 (**c**). Reprinted from [61]. bulk model just described. The spatial carrier density profile [Figure 3a] is now only marginally affected by negative-value regions, if compared to the unphysical results obtained via the RTA dissipation model (Figures 4 and 5). However, analogously to the RTA case, a marked slowdown of the excited-level decay is evident: after \(1.6\,\mathrm{ps}\) (dash-dotted curve), the charge distribution is still far from equilibrium (solid curve). Such anomalous scenario is confirmed by an insight into the various subband populations (Figure 3b), showing: (i) a marked slowdown out of the initial subband \(\mathrm{n}=2\); (ii) negative values in subband \(\mathrm{n}=3\); and (iii) despite our zero-temperature analysis, considerable values in the continuum. A further close-up in the analysis may be achieved repeating the simulated experiment reported in Figure 6 with a different initial condition, e.g., the QW low-temperature thermal-equilibrium state \(\mathrm{n}=1\). In this case, a correct scattering superoperator would leave the electronic system unmodified. However, the results, reported in Figure 7, show that the Boltzmann collision term in Equation (10) drives the electronic system out of equilibrium, populates the continuum, and induces negative populations of the excited QW subbands. These outcomes confirm the intrinsic limitations of local treatments, which are known to be intrinsically incompatible with any quantum-mechanical framework. Indeed, both the RTA results in Figures 4 and 5 and the Boltzmann-type ones in Figures 6 and 7 are based on bulk-like scattering models, where the relaxation rate \(\Gamma(\mathbf{r},\mathbf{k})\) in Equation (4) and the semiclassical scattering probability \(P(\mathbf{r};\mathbf{k},\mathbf{k}^{\prime})\) in Equation (10) do not account for the QW electronic subband structure. The two terms in the Wigner transport equation (Equation (1)) are then intrinsically conflicting: the deterministic one accounts for the QW subband structure via its confinement potential \(V(\mathbf{r})\), while the scattering one does not. This contradiction, recently pointed out also within the density-matrix formalism [71], is responsible for the dissipation slowdown as well as for the wrong thermalization dynamics.