Effects of Coulomb interaction on electron dynamics in a double-barrier potential: Decay and trapping

The decay of a resonance and electron trapping in the quantum well region of a double barrier potential are studied. We take into account electron-electron interactions at the Hartree level by coupling the time-dependent Poisson and Schrödinger equations. Since, in this mean-field theory, the effective potential for the electron motion depends on the classical distribution of charge, the Schrödinger equation becomes nonlinear. We have used the electron sheet density in the well as an adjustable parameter that controls the coupling and thus the degree of the nonlinearity. Without nonlinear coupling, the well generally holds resonance states and, if deep enough, possibly bound states. The nonlinear interaction gives rise to shorter decay times for the resonances, and the existence of a critical electron sheet density above which permanent trapping does not occur.

Figure 1

Bare one-dimensional model potential $\left(V_{\mathit{ext}}\right)$ for transport through the DBS. The light and dark shaded contact regions are doped whereas the barriers and quantum well are not. The darker (depletion) regions are charged positively to compensate for the electron charge residing inside the barriers and quantum well.

Figure 2

Time evolution of the probability for being in the well and barriers region for different values of the sheet density $n_{2D}$. The initial state is a Gaussian wavepacket localized in the quantum well with zero average momentum. The width is ${\sigma }_x=12\mathrm{nm}$ and the quantum well depth is $V_{\mathit{\text{well}}}=0$. The chosen values for the two-dimensional electron sheet density, $n_{2D}$, are: 0 (solid line), ${10}^{11}{\mathrm{cm}}^{-2}$ (thick dashed line), $2.5\times {10}^{11}{\mathrm{cm}}^{-2}$ (dashed line), $5\times {10}^{11}{\mathrm{cm}}^{-2}$ (long-dashed line) and ${10}^{12}{\mathrm{cm}}^{-2}$ (dotted-dashed line).

Figure 3

Time evolution of the probability for finding electrons in the well and bariers for different values of the sheet density $n_{2D}$. The initial state is centered in the quantum well and is identical to the one used for Fig. 2. $V_{\mathit{\text{well}}}=-30\mathrm{meV}$. The values for $n_{2D}$ are: 0 (solid line), ${10}^{11}{\mathrm{cm}}^{-2}$ (thick dashed line), $2.5\times {10}^{11}{\mathrm{cm}}^{-2}$ (dashed line), $5\times {10}^{11}{\mathrm{cm}}^{-2}$ (long-dashed line) and ${10}^{12}{\mathrm{cm}}^{-2}$ (dotted-dashed line).

Figure 4

Time evolution of the average energy $E_L\left(t\right)$ for different values of the sheet density $n_{2D}$. The initial state is the one used for Figs. 2, 3. $V_{\mathit{\text{well}}}=-30\mathrm{meV}$. The values for $n_{2D}$ are: 0 (solid line), ${10}^{11}{\mathrm{cm}}^{-2}$ (thick dashed line), $2.5\times {10}^{11}{\mathrm{cm}}^{-2}$ (dashed line), $5\times {10}^{11}{\mathrm{cm}}^{-2}$ (long-dashed line) and ${10}^{12}{\mathrm{cm}}^{-2}$ (dotted-dashed line).

Figure 5

Total initial potential (thick solid line) and potential at $50\mathrm{ps}$ (thin solid line) for a sheet density $n_{2D}=5\times {10}^{11}{\mathrm{cm}}^{-2}$ above the critical value. The initial state (thick dashed line) is the same as that for Fig. 3. The probability density at $t=50\mathrm{ps}$ is also plotted (thin dashed line). In spite of the increase of the density peak right at the center of the well, the total probability in the undoped region (well and barriers) has decreased. $V_{\mathit{\text{well}}}=-30\mathrm{meV}$. The dotted-dashed lines in the well indicate the resonance levels for the potential at $50\mathrm{ps}$.

Figure 6

Total initial potential (thick solid line) and potential at $50\mathrm{ps}$ (thin solid line, barely visible since it essentially overlaps with the initial potential-contrast this behavior with that in the previous figure) for a sheet density $n_{2D}={10}^{11}{\mathrm{cm}}^{-2}$ below the critical value. The initial state (thick dashed line) is the same as that for Fig. 3. The probability density at $t=50\mathrm{ps}$ is also plotted (thin dashed line). As with the previous figure, the density peak at the well center has increased with time, although the total probability has decreased. $V_{\mathit{\text{well}}}=-30\mathrm{meV}$. The dotted-dashed lines in the well represent resonance levels and the dotted line marks the bound, ground state level of the potential at $50\mathrm{ps}$.

Figure 7

Mean energy of the ground state versus $n_{2D}$ for $V_{\mathit{\text{well}}}=-30\mathrm{meV}$. The two lines correspond to a computational box $[-X_L∕2,X_L∕2]$ of two different sizes.