# 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 incompatibility between the local dissipation models of the semiclassical theory and the (non-local) deterministic Wigner-function dynamics in Equation (12) is already evident in modeling a basic prototypical quantum-well (QW) device such as the one sketched in Figure 3. The effects of device spatial boundaries as well as the role of device-reservoir thermalization phenomena may be neglected assuming semi-infinite barriers and limiting our investigation to QW bound states only. Due to confinement along the growth (\(z\)) direction, the system three-dimensional (3D) electronic states exhibit the usual subband structure. However, within the prescription of the RTA dissipation model in Equation (4), we neglect the in-plane phase-space coordinates and adopt an effective one-dimensional (1D) description: The set of single-particle quantum numbers then reduces to the discrete index (n) of the confined 1D state only and \((\mathbf{r},\mathbf{k})\rightarrow(z,k)\). Figure 3 shows the three bound energy levels and squared envelope functions for the model case of a GaAs/(Al,Ga)As 12 nm-thick QW, with a conduction band offset of 0.3 eV. In the following, we resume the results of simulated experiments performed on this nanosystem in the low-temperature regime, thus assuming that the QW lowest energy level (n = 1) corresponds to the thermal-equilibrium condition. In particular, we analyze the spatial carrier-density profile, along the growth direction, corresponding to the 1D Wigner function \(f(z,k)\), namely  
\[n(z)=\frac{1}{2\pi}\,\int dk\,f(z,k)\, \tag{14}\]  
and investigate its time evolution, when the system is initially prepared in the first excited level (n = 2), by solving the 1D version of the Wigner transport equation (Equation (1)) with the quantum-mechanical deterministic term in Equation (12) and the conventional RTA scheme in Equation (4). In Figure 4, the dissipation dynamics induced on our prototypical QW system by three different versions of the RTA approach are compared. Within the model labeled M1, we assume a space- and energy-independent rate \(\Gamma_{0}\); this is often called phenomenological or macroscopic RTA approach. Within the model M2, we employ a space-dependent but energy-independent rate \(\Gamma(z)\), assumed to vanish within a given region \(-\frac{\theta}{2}<z<+\frac{\theta}{2}\), and equal to \(\Gamma_{0}\) outside; this may partially mimic the case of material-dependent semiclassical scattering rates, such as in the case of complex multi-layer quantum heterostructures. Finally, within the model M3, we consider a space-independent but momentum/energy-dependent rate \(\Gamma(k)\), vanishing for \(|k|<k_{\text{min}}=\sqrt{2m^{*}\epsilon_{\text{min}}}/\hbar\) and equal to \(\Gamma_{0}\) otherwise; this may describe the typical energy-threshold scenario of electron-optical phonon scattering in a variety of nanomaterials and nanodevices, e.g., new-generation quantum-cascade lasers [15]. The dissipation dynamics resulting from M1 [Figure 4a] agrees with the physically valid scenario in which the charge distribution in the QW ground state n = 1 increases, at the expenses of the one of the initial state n = 2 (thin solid curve); after 1.6 ps (dash-dotted curve) the electronic system is already pretty close to its equilibrium distribution (solid curve). In the case of M2 [Figure 4b] and M3 [Figure 4c], the scenario is substantially different: besides a significant slowdown of the excited-level decay process, negative carrier distributions arise; this crucial difference with respect to M1 is an unambiguous fingerprint of unphysical electronic states. Figure 3: Conduction band profile along the growth direction for the prototypical GaAs/AlGaAs QW nanostructure considered in our simulated experiments. Energy levels of the three 1D bound states are shown, together with the corresponding squared envelope functions. Readapted from [61].