Green's Function Methods for Computing Supercurrents in Josephson Junctions

Abstract

Interest in Josephson junctions (JJs) has increased rapidly in recent years not only because of their use in qubits and other quantum devices but also due to the unique physics supported by the JJs. The advent of various novel quantum materials for both the barrier region and the superconducting leads has led to the possibility of adding new functionalities to the JJs. Thus, there is a growing need for accurate modeling of the JJs and related systems to enable their predictive control and atomistic level understanding. This review presents an in-depth discussion of a Green's function-based formalism for computing supercurrents in JJs. The formulation is tailored for large-scale atomistic simulations and encompasses both dc and ac supercurrents. We hope that this review will provide a timely and comprehensive reference for researchers as well as beginning practitioners interested in Green's-function-based methods to model supercurrents in JJs.

Figures (5)

• Figure 1: Schematic representations of (a) SNS and (b) SS junctions. $u$, $u_L$, and $u_R$ represent various couplings as shown in the figure. $L$ and $R$ refer to left and right, respectively.
• Figure 2: Density of states of the QD in the non-interacting limit when $U=0$.
• Figure 3: Steps involved in supercurrent calculations, shown schematically, are: (i) Green's functions for the non-interacting leads and tunneling barrier are obtained; (ii) Self-energy matrices for the junction are calculated using the lead Green's functions and the interaction matrices; and, (iii) $\hat{G}_{N}$ is used to obtain ABSs and other features such as the real-space mappings of the anomalous matrix elements of the Nambu-Gorkov Green's function.
• Figure 4: Collection of quantities that can be calculated using the Green's function method. (a) The horizontal Brillouin zone (BZ$_{p}$) of MoS$_{2}$ (outer hexagon), and the Brillouin zone (BZ$_{s}$) (inner hexagon) for the $3\times 3$ computational supercell used in Ref. Nieminen2023. Two non-equivalent valley points K and K' are indicated, as well as the four quadrants used to reveal spin-valley coupling in ABSs. (b) and (c) Spin-polarized dispersion of ABSs projected on different quadrants of the Brillouin zone indicated in (a): quadrants Q2 and Q4 (b), and quadrants Q3 and Q1 (c). Note the spin-valley coupling in the quadrants containing the $K$ and $K'$ points. (d) Supercurrent across a monolayer projected in two non-equivalent quadrants (Q1 and Q3). (e) Projection of singlet ($s=0,~m_{s}=0$) and triplet ($s=1,~m_{s}=-1,~0,~1$) components of the real and imaginary parts of the anomalous Green's function at the barrier location. (f) Side view of the atomic configuration of the junction. See Ref. Nieminen2023 for details.
• Figure 5: Josephson current in the ac (voltage biased) regime. (a) dc ($m=0$) component of the current as a function of the bias voltage for increasing tunneling coupling strengths $u/\Delta$ in increments of 0.1 at temperature $k_BT=0.01\Delta$. (b) First four harmonic components of the supercurrent as a function of the bias voltage for $u=0.4\Delta$ and $k_BT=0.01\Delta$. (c) The dc ($m=0$) component of the current as a function of the bias voltage for three different temperatures ($\beta\equiv\Delta/k_B T$) for $u=0.4\Delta$. All computations used an $N=6$ truncation.