Nonadiabatic Dynamics in Single-Electron Tunneling Devices with Time-Dependent Density-Functional Theory

We simulate the dynamics of a single-electron source, modeled as a quantum dot with on-site Coulomb interaction and tunnel coupling to an adjacent lead in time-dependent density-functional theory. Based on this system, we develop a time-nonlocal exchange-correlation potential by exploiting analogies with quantum-transport theory. The time nonlocality manifests itself in a dynamical potential step. We explicitly link the time evolution of the dynamical step to physical relaxation timescales of the electron dynamics. Finally, we discuss prospects for simulations of larger mesoscopic systems.

Figure 1

(a) Energy diagram of a single-level quantum dot with tunnel coupling $\mathrm{\Gamma }$ to a lead. (b) Adiabatic HXC (XC plus Hartree) potential from [13]. (c) Nonadiabatic HXC potential derived in Eq. (8). Parameters are $U=16\mathrm{\Gamma }$, $T=2\mathrm{\Gamma }$, and $\dot{n}/\mathrm{\Gamma }=1.2,0.8,\dots ,-1.2$ (left to right lines). The black arrow indicates the dynamical step.

Figure 2

Comparison of TDDFT results obtained using ${\epsilon }_{\mathrm{HXC}}^{\mathrm{ad}}$ (dotted lines), ${\epsilon }_{\mathrm{HXC}}^{\mathit{M}}$ (dashed lines), and no HXC potential (dashed-dotted lines) with analytic results from Eq. (3) (solid lines). (a) Densities of the quantum dot subject to a square pulse (inset: HXC potentials) with parameters $U=16\mathrm{\Gamma }$, $T=2\mathrm{\Gamma }$, $\epsilon (t<0)=10\mathrm{\Gamma }$, $\epsilon (t>0)=-6\mathrm{\Gamma }$. (b) Currents in the case of a harmonic gate-voltage drive plotted for driving frequencies in the range $[0.1\mathrm{\Gamma },0.5\mathrm{\Gamma }]$ with $\mathrm{\Delta }\mathrm{\Omega }=0.1\mathrm{\Gamma }$ (note that for visibility lines are shifted by 1). Further parameters are $U=12\mathrm{\Gamma }$, $T=3\mathrm{\Gamma }$, $\epsilon (t)=-6\mathrm{\Gamma }+18\mathrm{\Gamma }\cos (\mathrm{\Omega }t)$.

Figure 3

(a) Setup of 70 quantum dots tunnel coupled to the same electron reservoir. (b) Absolute density change summed over all quantum dots $|\mathrm{\Delta }n(t)|=\sum _{l=1}^{70}|n_l(t)-n_l(t_1)|$ with $t_1\gg t$ for a sudden rise (charging) or drop (discharging) of the gate voltage at $t=0$, calculated using either ${\epsilon }_{\mathrm{HXC}}^{\mathit{M}}$ or ${\epsilon }_{\mathrm{HXC}}^{\mathrm{ad}}$ for all dots. See text for gray area and Ref. [51] for parameters.