# Characteristics of Resonant Tunneling in Nanostructures with Spacer Layers
## 3 Results and Discussion  
To verify the correctness of the developed methodology's implementation, the calculations of the current-voltage characteristic for the structure from [23] were carried out. Comparisons of the results from [23] obtained in the framework of solving the stationary Schrodinger-Poisson problem are shown in Figure 3. One can see that the current-voltage characteristic obtained in [23] with a high degree of accuracy coincides with the results obtained using the methodology developed in the present study. For the developed method of calculation based on the solution of the time-dependent Schrodinger-Poisson equations, the electroneutrality condition was fulfilled with a high degree of accuracy (with an error significantly less than 1%), which is demonstrated in Figure 4 (the value \(\Delta\rho_{0}/\rho_{D}=\frac{\int(m(x)-n_{D}(x))dx}{\int n_{D}(x)dx}\) is postp \begin{table}
\begin{tabular}{c c} \hline \hline
**Structure \#1**[30] & **Structure \#2**[26] \\ \hline \hline \multicolumn{3}{c}{Layer structure} \\ \hline n-GaAs, \(n_{D}=6\times 10^{18}\) cm\({}^{-3}\) & n-In\({}_{0.53}\)Ga\({}_{0.47}\)As, \(n_{D}=2\times 10^{19}\) cm\({}^{-3}\) \\ n-GaAs, 40 nm, \(n_{D}=2\times 10^{18}\) cm\({}^{-3}\) & n-In\({}_{0.53}\)Ga\({}_{0.47}\)As, 25 nm, \(n_{D}=3\times 10^{18}\) cm\({}^{-3}\) \\ GaAs, 3 nm (collector spacer) & In\({}_{0.53}\)Ga\({}_{0.47}\)As, 20 nm (collector spacer) \\ Al\({}_{0.3}\)Ga\({}_{0.7}\)As, 3 nm & AlAs, 1.1 nm \\ GaAs, 5 nm (quantum well) & In\({}_{0.8}\)Ga\({}_{0.2}\)As, 45 nm (quantum well) \\ Al\({}_{0.3}\)Ga\({}_{0.7}\)As, 3 nm & AlAs, 1.1 nm \\ GaAs (emitter spacer) & In\({}_{0.53}\)Ga\({}_{0.47}\)As (emitter spacer) \\ n-GaAs, 40 nm, \(n_{D}=2\times 10^{18}\) cm\({}^{-3}\) & n-In\({}_{0.53}\)Ga\({}_{0.47}\)As, 20 nm, \(n_{D}=3\times 10^{18}\) cm\({}^{-3}\) \\ n-GaAs, \(n_{D}=6\times 10^{18}\) cm\({}^{-3}\) & n-In\({}_{0.53}\)Ga\({}_{0.47}\)As, \(n_{D}=2\times 10^{19}\) cm\({}^{-3}\) \\ \hline \hline \multicolumn{3}{c}{Additional parameters} \\ \hline \(m^{*}=0.0667m_{e}\)1 & \(m^{*}_{QW}=0.032m_{e}\); \(m^{*}_{ALAs}=0.08m_{e}\); \\ \(m^{*}_{const}=0.041m_{e}\) & \(m^{*}_{const}=0.041m_{e}\) \\ \(E_{F}=0.1774\) eV & \(E_{F}=0.655\) eV \\ \(\Delta E_{c}=\Delta E^{\text{QW}}_{c}=0.3\) eV & \(\Delta E_{c}=1.2\) eV; \(\Delta E^{\text{QW}}_{c}=1.3785\) eV \\ \(\varepsilon=12.9\) & \(\varepsilon=13.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the investigated resonant tunneling nanostructures. Figure 3: Current-voltage characteristic for the resonant tunneling nanostructure from [23]. The solid blue line is the solution from [23], and the red dots represent the solution obtained as a result of our calculations. Due to the peculiarities of the calculation method proposed, along with the information obtained on the static characteristics of the considered resonant tunneling nanostructures, we also obtained the data for the analysis of transient processes under various conditions. The consideration of spacer layers and heavily doped contacts in the model of resonant tunneling nanostructures boils down to the fact that, at the beginning of the transition process (see Region I in Figure 5), when the applied voltage increases rapidly over 0.5 ps, the nature of the transient process of resonant tunneling nanostructures corresponds to the behavior of the parallel plate capacitor [23]. Then, the transition process associated with the restructuring of the state in the main quantum well begins (see Region II in Figure 5). This transition process occurs over times of the order of the electron lifetime in the quantum well \(\tau=\hbar/\Gamma\) (\(\Gamma\)--resonance-level width), which is consistent with the theoretical studies carried out earlier in the framework of the simplified model [31; 32]. The change in the thickness of the emitter spacer has a weak effect on the transition times for all the structures considered.