Propagation scheme for nonequilibrium dynamics of electron transport in nanoscale devices

A closed set of coupled equations of motion for the description of time-dependent electron transport is derived. It provides the time evolution of energy-resolved quantities constructed from nonequilibrium Green’s functions. By means of an auxiliary-mode expansion a viable propagation scheme for finite temperatures is obtained, which allows to study arbitrary time dependences and structured reservoirs. Two illustrative examples are presented.

Figure 1

Time-resolved occupation $⟨N⟩$ and net current $⟨J_{\text{net}}⟩$ for different values of the noise amplitude. Noise averages are obtained from $20000$ realizations and $\kappa =0.5\Gamma$. The arrows indicate the time-averaged values $\overline{⟨J_{\text{net}}⟩}$ obtained by sampling the current for times $t>10/\Gamma$.

Figure 2

Time-averaged current vs noise amplitude for different values of the inverse noise correlation time $\kappa$. Noise averages are obtained from $20000$ realizations. Error bars indicate 95% confidence interval for the sample mean. Full lines denote the analytical result given by Eq. (B4).

Figure 3

(Color online) Time-resolved current through left barrier $J_L$ for different values of $W$ in Eq. (42) driven by a bias-voltage pulse according to Eq. (43) with $V_{\text{bias}}=3\Gamma$, $t_{\text{s}}=1/\Gamma$, and $t_{\text{p}}=20/\Gamma$. The WBL corresponds to $W\to \infty$.

Figure 4

(Color online) Energy scheme for $W=\Gamma$ at the plateau of the pulse with $V_{\text{bias}}=V_{\text{max}}=3\Gamma$. Outermost parts show $f_{\alpha }\left(\epsilon \right)\Gamma \left(\epsilon \right)$ for $\alpha =L,R$ (blue/dark-shaded areas). The inner part shows spectral densities of left and right levels. Left panel: $V_{\text{bias}}={\mu }_L-{\mu }_R$ and ${\Delta }_L={\Delta }_R=0$. Right panel: $V_{\text{bias}}={\Delta }_L-{\Delta }_R$ and ${\mu }_L={\mu }_R=0$.

Figure 5

Number of pulse-induced tunneling electrons $N_{\text{p}}$ vs pulse length $t_{\text{p}}$. Symbols indicate numerical results for different values of $W$ in Eq. (42). Straight lines show result of linear fit in the respective range. The dashed line gives the two intercepts $N_{\text{p}}^{\star }$ and $t_{\text{p}}^{\star }$, respectively, for the case of the WBL.

Figure 6

Results of the linear fit to $N_{\text{p}}\left(t_{\text{p}}\right)$ shown in Fig. 5: (a) $N_{\text{p}}$ intercept and (b) $t_{\text{p}}$ intercept.