# Simulation of Electronic Quantum Devices: Failure of Semiclassical Models
## 4 Fully Quantum-Mechanical Device Modeling Based on the Density-Matrix Formalism
### Non-Local Boltzmann-Type Models
\[P^{\text{in/out}}(\mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime })=\int\frac{d\mathbf{r}^{\prime\prime}\,d\mathbf{k}^{\prime\prime}}{(2\pi)^{ 3}}\;\left(1-f(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime})\right)\, \tilde{P}^{\text{in/out}}(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime}; \mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime}) \tag{37}\]  
with  
\[\tilde{P}^{\text{in}}(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime}; \mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime})=\frac{1}{(2\pi )^{3}}\sum_{\alpha_{1}\alpha_{2}\alpha^{\prime}\alpha_{1}^{\prime}\alpha_{2}^{ \prime}}\Re\left\{W_{\alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k})W_{\alpha_{1} \alpha^{\prime}}^{*}(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime})P_{ \alpha^{\prime}\alpha_{2}\alpha_{2}^{\prime}\alpha_{1}^{\prime}\alpha_{2}^{ \prime}}W_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}^{*}(\mathbf{r}^{\prime}, \mathbf{k}^{\prime})\right\} \tag{38}\]  
and  
\[\tilde{P}^{\text{out}}(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime}; \mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime})=\frac{1}{(2\pi )^{3}}\sum_{\alpha_{1}\alpha_{2}\alpha^{\prime}\alpha_{1}^{\prime}\alpha_{2}^{ \prime}}\Re\left\{W_{\alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k})W_{\alpha^{ \prime}\alpha_{1}^{\prime}}^{*}(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime \prime})P_{\alpha^{\prime}\alpha_{1}^{\prime}\alpha_{1}^{\prime}\alpha_{2}^{ \prime}}^{*}W_{\alpha_{2}^{\prime}\alpha_{2}^{\prime}}^{*}(\mathbf{r}^{ \prime},\mathbf{k}^{\prime})\right\}\;. \tag{39}\]  
It is worth stressing that, despite the strong formal similarity with the conventional Boltzmann transport theory, the generalized Wigner-function scattering rates in Equation (37) are not necessarily positive-definite. The proposed quantum-mechanical generalization of the standard Boltzmann collision term in Equation (10) is thus intrinsically nonlocal. In particular, comparing Equation (37) with its semiclassical counterpart in Equation (11), it is evident that the action of the Pauli exclusion principle within the Wigner phase-space is itself nonlocal: the generalized in and out scattering rates for a given transition \(\mathbf{r},\mathbf{k}\rightarrow\mathbf{r}^{\prime},\mathbf{k}^{\prime}\) depend on the value of the Wigner function in any other phase-space point \(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime}\) via the Pauli factor \(1-f(\mathbf{r}^{\prime\prime},\mathbf{k}^{\prime\prime})\). A detailed investigation of such Pauli-blocking nonlocality is outside the scope of the present work and can be found in [64]. In the low-density limit \(f(\mathbf{r},\mathbf{k})\to 0\), the proposed scattering model in Equation (37) then reduces to:  
\[P^{\text{in}}(\mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime})= \frac{1}{(2\pi)^{3}}\sum_{\alpha_{1}\alpha_{2}^{\prime}\alpha_{2}^{\prime}} \Re\left\{W_{\alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k})P_{\alpha_{1}\alpha_ {2},\alpha_{1}^{\prime}\alpha_{2}^{\prime}}W_{\alpha_{1}^{\prime}\alpha_{2}^{ \prime}}^{*}(\mathbf{r}^{\prime},\mathbf{k}^{\prime})\right\} \tag{40}\]  
and  
\[P^{\text{out}}(\mathbf{r},\mathbf{k};\mathbf{r}^{\prime},\mathbf{k}^{\prime}) =\frac{1}{(2\pi)^{3}}\sum_{\alpha_{1}\alpha_{2}^{\prime}\alpha_{2}^{\prime}} \Re\left\{W_{\alpha_{1}\alpha_{2}}(\mathbf{r},\mathbf{k})P_{\alpha_{1}^{ \prime}\alpha_{1}^{\prime}\alpha_{2}^{\prime},\alpha_{1}\alpha_{2}^{\prime}}^ {*}W_{\alpha_{2}^{\prime}\alpha_{2}}^{*}(\mathbf{r}^{\prime},\mathbf{k}^{ \prime})\right\}\;. \tag{41}\]  
To test the suitability of the proposed nonlocal Boltzmann superoperator, we have applied the latter to the case of the simulated experiment of Figure 6; the resulting energy-dissipation dynamics is presented in Figure 12. Here, opposite to the unphysical behaviors obtained via the semiclassical Boltzmann model (see Figures 6 and 7), both the spatial carrier distributions (Figure 12a) and the corresponding subband populations (Figure 12b) are always positive-definite.