# 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
A closer inspection of the dissipation dynamics may be provided by looking at the carrier populations \(f_{\rm n}\) of the QW states. The latter are given by the diagonal terms of the single-particle density matrix, which can be obtained from the 1D Wigner function via the inverse of the 1D Weyl-Wigner transform. In Section 4, the link between the density matrix and the Wigner function is more extensively addressed. Here, we just anticipate that the populations \(f_{\rm n}\) can be obtained applying to the 1D Wigner function \(f(z,k)\) the inverse of the 1D Weyl-Wigner transform \(W_{\rm nn}(z,k)\), i.e.,  
\[f_{\rm n}=\int\frac{dz\,dk}{2\pi}W_{\rm nn}(z,k)f(z,k). \tag{15}\]  
Figure 5 shows the time evolution of the populations of the three QW bound states (\(f_{1}\), \(f_{2}\), and \(f_{3}\)) as well as of the continuum, the latter being defined as \(N_{\rm cont}=N_{\rm tot}-\sum_{n=1}^{3}f_{\rm n}\), with \(N_{\rm tot}\) denoting the total number of carriers. As expected, model M1 [Figure 5a] shows a simple exponential depopulation of the initial level \({\rm n}=2\) into the ground state \({\rm n}=1\), while level \({\rm n}=3\) and the continuum are not affected by the dissipation dynamics. Such a plausible scenario is, however, lost in models M2 [Figure 5b] and M3 [Figure 5c], where we do observe: (i) a slowdown of the \({\rm n}=2\) state depopulation; (ii) negative population values for states \({\rm n}=1\) and \({\rm n}=3\); and (iii) particularly for M3, a marked charge transfer into the continuum in spite of our zero-temperature analysis. The unavoidable conclusion is that the conventional RTA dissipation model in Equation (4) does not preserve the positivity of the carrier populations \(f_{\rm n}\), and induces a fictitious interlevel coupling; the latter, in turn, may also artificially generate interlevel phase coherence. The unphysical results of M2 and M3 are not related to the charge non-conserving nature of the RTA model and/or to the effective 1D modeling of our nanodevice. In the following, we attest that analogous problems also arise when: (i) a fully 3D treatment of the prototypical QW nanostructure is adopted; and (ii) the RTA dissipation scheme in Equation (4) is replaced by the (charge-conserving) Boltzmann collision term in Equation (10). Figure 4: Low-temperature dissipation dynamics in the 12 nm thick GaAs/AlGaAs QW of Figure 3. The carrier density \(n(z)\) in Equation (14) is plotted as a function of position at different times (0 ps: thin solid curve; 0.8 ps: dashed curve; and 1.6 ps: dash-dotted curve) for models: M1 (**a**); M2 (**b**); and M3 (**c**). Here, the following parameters are employed: \(1/T_{0}=0.8\) ps for M1, \(1/T_{0}=0.4\) ps and \(a=5\) nm for M2, and \(1/T_{0}=0.5\) ps and \(\epsilon_{\rm min}=40\) meV for M3. The equilibrium carrier density is also shown for comparison (solid curve). Reprinted from [61]. Concerning the latter, it has the well-established in- minus out-scattering structure and may also be written as  
\[\left.\frac{\partial f(\mathbf{r},\mathbf{k})}{\partial t}\right|_{\text{diss}}= \int d\mathbf{r}^{\prime}\,d\mathbf{k}^{\prime}\left[P^{\text{in}}(\mathbf{r}, \mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime})-P^{\text{out}}(\mathbf{r}, \mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime})\right]f(\mathbf{r}^{ \prime},\mathbf{k}^{\prime}) \tag{16}\]  
with  
\[P^{\text{in}}(\mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime})= \delta(\mathbf{r}-\mathbf{r}^{\prime})P(\mathbf{r};\mathbf{k},\mathbf{k}^{ \prime}) \tag{17}\]  
and  
\[P^{\text{out}}(\mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime}) =\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(\mathbf{k}-\mathbf{k}^{\prime} )\int d\mathbf{k}^{\prime\prime}P(\mathbf{r};\mathbf{k}^{\prime\prime}, \mathbf{k}). \tag{18}\]  
Both superoperators, \(P^{\text{in}}\) and \(P^{\text{out}}\), are local in \(\mathbf{r}\); \(P^{\text{out}}\) is local in \(\mathbf{k}\) as well.