Time-dependent quantum transport with superconducting leads: A discrete-basis Kohn-Sham formulation and propagation scheme

In this work we put forward an exact one-particle framework to study nanoscale Josephson junctions out of equilibrium and propose a propagation scheme to calculate the time-dependent current in response to an external applied bias. Using a discrete basis set and Peierls phases for the electromagnetic field, we prove that the current and pairing densities in a superconducting system of interacting electrons can be reproduced in a noninteracting Kohn-Sham (KS) system under the influence of different Peierls phases and of a pairing field. In the special case of normal systems, our result provides a formulation of time-dependent current-density-functional theory in tight-binding models. An extended Keldysh formalism for the nonequilibrium Nambu-Green’s function (NEGF) is then introduced to calculate the short- and long-time response of the KS system. The equivalence between the NEGF approach and a combination of the static and time-dependent Bogoliubov-de Gennes (BdG) equations is shown. For systems consisting of a finite region coupled to $\mathcal{N}$ superconducting semi-infinite leads, we numerically solve the static BdG equations with a generalized waveguide approach and their time-dependent version with an embedded Crank-Nicholson scheme. To demonstrate the feasibility of the propagation scheme, we study two paradigmatic models, the single-level quantum dot and a tight-binding chain, under dc, ac, and pulse biases. We provide a time-dependent picture of single and multiple Andreev reflections, show that Andreev bound states can be exploited to generate a zero-bias ac current of tunable frequency, and find a long-living resonant effect induced by microwave irradiation of appropriate frequency.

Figure 1

(Color online) The Keldysh contour ${\gamma }_{\text{K}}$ described in the main text. The contour variable $z=t_{-}/t_{+}$ denotes a point on the upper/lower branch at a distance $t$ from the origin while $z=-i\tau$ denotes a point on the imaginary track at a distance $\tau$ from the origin. In the figure, we also illustrate the points $0_{-}$ (earliest point on ${\gamma }_{\text{K}}$), $0_{+}$, and $-i\beta$ (latest point on ${\gamma }_{\text{K}}$).

Figure 2

(Color online) (a) Schematic of the transport setup. A single-level QD with on-site energy ${\epsilon }_C=0.5$ is weakly connected $\left({\Gamma }_L={\Gamma }_R=0.2\right)$ to a left normal lead and a right superconducting lead. In equilibrium, both temperature $T$ and chemical potential $\mu$ are zero. The system is driven out of equilibrium by a steplike voltage bias $U_L=0.3,0.6,0.9,1.2$ in the normal lead. For $U_L<{\Delta }_R$, the dominant scattering mechanism is the AR in which an electron is reflected as a hole and a Cooper pair is formed in lead $R$. (b) Time-dependent current at the left interface (first panel), right interface (second panel), and absolute value of the pairing density on the QD (third panel). The insets show the TD current for the same parameters but ${\Delta }_R=0$, i.e., for a normal $R$ lead. The results are obtained with a time step $\delta =0.05$, cutoff $\Lambda =6$, and a number of scattering states $N_{p,L}=1070$ and $N_{p,R}=1056$.

Figure 3

(Color online) (a) Schematic of the S-QD-S model with ${\Gamma }_L={\Gamma }_R=1.0$, ${\Delta }_L={\Delta }_R=1$, and ${ϵ}_C=0$. This system admits two ABS in the gap. The ABS energy depends on the superconducting phase difference $\chi$ as illustrated in the inset. (b) and (c) Time-dependent current $J_L\left(t\right)$ at the left interface as a function of time for (b) $U_L=3.0,2.0,1.0$ [the curves corresponding to bias $U_L=n.0$ are shifted upward by $0.3\left(n-1\right)$] and (c) $U_L=0.5,0.4,0.3,0.2$ [the curves corresponding to bias $U_L=0.n$ are shifted upward by $0.6\left(n-2\right)$]. The results are obtained with a time step $\delta =0.05$, cutoff $\Lambda =12.1$, and a number of scattering states $N_{p,L}=N_{p,R}=768$ for panel (b) and $\delta =0.05$, $\Lambda =4$, and $N_{p,L}=N_{p,R}=788$ for panel (c).

Figure 4

(Color online) (a) Discrete Fourier transform of $J_L\left(t\right)$ in arbitrary units [the curves corresponding to bias $U_L=0.n$ are shifted upward by $0.7\left(n-3\right)$ while that corresponding to bias $U_L=1.0$ is shifted upward by 2.8. (b) Values of the average current for biases in the subgap region. (c) ABS contribution to the current $J_L\left(t\right)$ for biases $U_L=0.2,0.3,0.4,0.5,0.6$ [the curves corresponding to bias $U_L=0.n$ are shifted upward by $0.8\left(n-2\right)$]. The numerical parameters are the same as in Fig. 3.

Figure 5

(Color online) Time-dependent current at the right interface $J_R$ (first panel) as well as the density $n_0$ (second panel) and pairing density $\left|P_0\right|$ (third panel) on the QD. The curves from bottom to top corresponds to a switch-off time $t_{\text{off}}^{\left(n\right)}=5\pi +n\pi /8$, with $n=0,1,2,3,4$. Since the bias is $U_L=1$, the accumulated phase difference ${\chi }^{\left(n\right)}$ at the end of the pulse is ${\chi }^{\left(n\right)}=2t_{\text{off}}^{\left(n\right)}=n\pi /4$. For the switch-off time $t_{\text{off}}^{\left(n\right)}$, the curves of $J_R$ are shifted upward by $0.3n$, those of $n_0$ by $0.5n$, and those of $\left|P_0\right|$ by $0.2n$. The results are obtained with a time step $\delta =0.05$, cutoff $\Lambda =12.1$, and a number of scattering states $N_{p,L}=N_{p,R}=768$.

Figure 6

(Color online) (a) TD current at the left interface for $U_L=0$ and $U_{\text{r}}=0.05{\omega }_{\text{r}}$ with ${\omega }_{\text{r}}=0.5$, [the curve is shifted upward by 0.4], and 1.5 [the curve is shifted upward by 0.8]. (b) ABS and continuum contribution to the total current in the resonant case ${\omega }_{\text{r}}=1.08$, $U_{\text{r}}=0.05{\omega }_{\text{r}}$, and $U_L=0$. (c) Pairing potential on the QD for the same parameters as in panel (b). The results are obtained with a time step $\delta =0.05$, cutoff $\Lambda =4$, and a number of scattering states $N_{p,L}=N_{p,R}=788$.

Figure 7

(Color online) ABS contribution to the current at the right interface for dc biases with a superimposed microwave radiation described by $U_L\left(t\right)=U_L+U_{\text{r}}\text{sin}\left({\omega }_{\text{r}}t\right)$, with $U_{\text{r}}=0.05{\omega }_{\text{r}}$, ${\omega }_{\text{r}}=1.08$, and $U_L=0.0,0.03,0.1,0.3$. The system is the same as in Fig. 6 with ${\Delta }_L={\Delta }_R=1$, ${\Gamma }_L={\Gamma }_R=1$, and ${ϵ}_C=0$. The time step is $\delta =0.05$.

Figure 8

(Color online) TD picture of MAR. A chain of 21 atomic sites is symmetrically connected with ${\Gamma }_L={\Gamma }_R=2t_C=2$ to two identical superconducting leads with ${\Delta }_L={\Delta }_R=1$. A dc bias $U_L=2\Delta /n$, $n=4,3,2$, is applied to lead $L$ at time $t=0$. The panels show the contour plots of the bond current $J_{n,n+1}\left(t\right)$ across the atomic bonds of region $C$. The results are obtained with a time step $\delta =0.05$, cutoff $\Lambda =4$, and a number of scattering states $N_{p,L}=N_{p,R}=1232$.

Figure 9

(Color online) Photon-assisted MAR in a chain of 21 atomic sites. The equilibrium parameters are the same as in Fig. 8. An ac bias $U_R=U_{\text{r}}\text{sin}\left({\omega }_{\text{r}}t\right)$ in lead $R$ is superimposed to a dc bias $U_L=0.8$ in lead $L$. The panels show the contour plots of the bond current $J_{n,n+1}\left(t\right)$ across the atomic bonds of region $C$ for different values of $U_{\text{r}}=0.0,0.1,0.3,0.5$ and ${\omega }_{\text{r}}=0.4$. The results are obtained with a time step $\delta =0.05$, cutoff $\Lambda =4$, and a number of scattering states $N_{p,L}=N_{p,R}=1232$.