Transient regime in nonlinear transport through many-level quantum dots

We investigate nonstationary electronic transport in noninteracting nanostructures driven by a finite bias and time-dependent signals applied at their contacts to the leads. The systems are modeled by a tight-binding Hamiltonian, and the transient currents are computed from the nonequilibrium Green-Keldysh formalism. The numerical implementation is not restricted to weak coupling to the leads and does not imply the wideband limit assumption for the spectral width of the leads. As an application of the method we study in detail the transient behavior and the charge dynamics in single and double quantum dots connected to leads by a steplike potential, but the method allows as well consideration of nonperiodic potentials or short pulses. We show that when the higher-energy levels of the isolated system are located within the bias window of the leads, the transient current approaches the steady state in a nonoscillatory smooth fashion. At moderate coupling to the leads and fixed bias the transient acquires a steplike structure, the length of the steps increasing with system size. The number of levels inside a finite bias window can be tuned by a constant gate potential. We find also that the transient behavior depends on the specific way of coupling the leads to the mesoscopic system.

Figure 1

Schematic picture of the system. The steplike potential is applied between the leads and the central region $S$ at $t=0$.

Figure 2

(Color online) (a) The transient current for one site coupled to single-channel leads. We present curves for different values of the coupling strength $U$ and of the bias $V$. (b) The effect of the coupling amplitude $U$ on the occupation number. The bias is fixed to $V=1.0$ and $kT=0.0001$.

Figure 3

(Color online) The two contributions to the transient current in the left lead. $J_{\alpha }^R\left(t\right)$ is always positive while the lesser contribution $J_{\alpha }^{<}\left(t\right)$ is negative. The chosen values for the coupling strength $U$ are given in the figure. The bias $V=1.0$ and $kT=0.0001$.

Figure 4

(Color online) The imaginary parts of the retarded (a) and lesser (b) two-time Green functions of the coupled single site. The bias $V=1.0$ and the coupling strength $U=1.2$. The oscillations seen along the “diagonal” in (b) corresponding to almost equal times are responsible for the oscillations of the occupation number.

Figure 5

(Color online) The imaginary part of the lesser two-time Green function for the moderate coupling $U=0.75$ (a) and strong coupling $U=1.2$ (b). (c) The real part of the oscillating function $F_2$. The bias $V=1.0$.

Figure 6

(Color online) (a) The transient current in the left lead $J_{\alpha }\left(t\right)$ for different sizes of the 1D central region. The number of sites, $N$, is indicated in the figure. In (a) the coupling to the leads is $U=0.75$ and in (b) $U=0.50$. By decreasing $U$ the shoulders in (a) turn to clear steps in (b). The bias is fixed to $V=2.0$ and $kT=0.0001$.

Figure 7

(Color online) The imaginary part of the left-contact retarded Green function (a) and of the lesser Green function for the four-site system (b). (c) The imaginary part of $G_{11}^R(t_1=10,t_2)$ at different couplings $U$. At small coupling $\mathrm{Im}{G}^R$ resembles the retarded Green function of the isolated system (the curve corresponding to $U=0.0$).

Figure 8

(Color online) (a) The transient current for different values of the gate potential applied on the four-site quantum dot for $U=0.75$. (b) The spectrum of the system as function of the gate potential. It can be checked that the additional shoulders observed in (a) develop when there is at least one state of the QD in the bias window and no state above it. Other parameters: $V=2$ and $kT=0.0001$. Note that the bias is applied asymmetrically—that is, ${\mu }_{\beta }=0.0$.

Figure 9

(Color online) (a) The transient in the left lead $J_{\alpha }\left(t\right)$ for the $4\times 1$ system and for the $2\times 2$ system in the two configurations of the leads as mentioned in the text. (b), (c), (d) The occupation numbers $N_i\left(t\right)$ of the $i$th site for the three cases considered in Fig. 9a. (b) 1D system, (c) symmetric configuration, and (d) asymmetric. Other parameters: $U=0.75$, $V=2.0$, and $kT=0.0001$.

Figure 10

(Color online) The total transient current and the components in the left lead and right lead. Other parameters: $U=0.5$, $V=2$, and $kT=0.0001$.

Figure 11

(Color online) (a) The 3D map of the transient current for a $(2\times 2)$-site QD as a function of time and gate potential $V_g$. The maxima are related to the spectrum of the isolated system given in (c). (b) The 3D map of the transient current for $2\times 2$ the double dot $t_{24}=t_{13}=0.5$. Other parameters: $V=1.0$, $U=0.50$, and $kT=0.0001$. (c) The spectra of the two systems as a function of the gate potential. The lines mark the bias window.