# 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
Moreover, as for the case of model M1, we deal again with a correct relaxation dynamics toward the equilibrium state (solid curve in Figure 9a) without unphysical interlevel relaxation couplings and continuum population. Let us finally apply the proposed non-local RTA scheme in Equation (32) to the investigation of device-reservoir thermalization. To this end, let us consider again the 1D problem of Figure 8, namely the steady-state quantum transport across a rectangular barrier due to a monoenergetic carrier injection from left. In this case, the single-particle states \(\alpha\) corresponding to the 1D rectangular potential profile are conventional scattering states labeled by their 1D carrier wavevector \(k\), and the device-reservoir relaxation rates depend on \(k\) as well, i.e., \(\Gamma_{\alpha}^{\text{res}}\equiv\Gamma_{k}^{\text{res}}\). To mimic the monoenergetic-injection scenario of Figure 8, let us assume a monoenergetic carrier distribution in the left reservoir, i.e., \(f_{k}^{\text{res}}\propto\delta(k-k_{0})\), \(k_{0}\) denoting the wavevector corresponding to the injection energy \(\epsilon_{0}\). As a result, in this case, the device-reservoir coupling is described via a single relaxation rate \(\Gamma_{0}^{\text{res}}\equiv\Gamma_{k_{0}}^{\text{res}}\). The latter can be regarded as the inverse of the so-called device transit time of the semiclassical theory [15], i.e., the ratio between the device length \(l\) and the injection carrier group velocity \(v_{0}=(\hbar k_{0})/m^{*}\), which for our simulation parameters is of the order of \(100\,\text{fs}\). The results obtained via the new non-local RTA model in Equation (32) are reported in Figure 10, where we have assumed again a monoenergetic thermalization with \(\epsilon_{0}=50\,\)meV, and we have employed a device-reservoir relaxation time \(1/\Gamma_{0}^{\text{res}}=100\,\)fs. Opposite to the highly non-physical results of the local IBC scheme in Figure 8, here, the charge distributions are always non-negative, thus confirming the positive-definite character of the proposed non-local RTA model in Equation (32). Moreover, regardless of the dissipation level, the amount of charge in the right portion of the device is now definitely negligible, in total agreement with the fact that for our simulation parameters the transmission coefficient across the rectangular barrier is extremely small (of the order of \(10^{-3}\)). In addition, while in the coherent limit (solid curve) as well as in the weak dissipation regime (dashed curve) one deals with the typical interference scenario between incoming and reflected wavefunctions, in the strong-dissipation regime (dash-dotted curve), such interference process is partially suppressed due to dissipation induced decoherence [20]. Figure 10: The same as in Figure 8 but replacing the semiclassical IBC scheme based on the local RTA model in Equation (7) with the fully non-local model in Equation (32). To this end, the monoenergetic left injection of the IBC scheme employed in Figure 8 is replaced by a thermal coupling of the device with a corresponding monoenergetic distribution in the left reservoir via a relaxation time \(1/\Gamma_{0}^{\text{res}}=100\,\)fs. The coherent-limit result (solid curve) and the weak-dissipation one (dashed curve) are basically coincident; the dash-dotted curve refers to the strong-dissipation regime. Figure 9: The same as in Figure 4 (**a**) and in Figure 5 (**b**) but for the proposed nonlocal RTA model in Equation (32) with \(1/\Gamma_{1}^{\text{diss}}=1\,\)ps and \(1/\Gamma_{2}^{\text{diss}}=0.7\,\)ps. Reprinted from [61]. Comparing once again the present convincing results with the highly problematic ones in Section 3.2, we also notice that, while there, in Figure 8, the curves corresponding to the coherent-limit result (solid line) and the weak-dissipation one (dashed line) differ significantly, here, in Figure 10, they basically coincide (the two profiles are hardly distinguishable).