Supercurrent diode with high winding vortex

Abstract

Nonreciprocal supercurrent refers to the phenomenon where the maximum dissipationless current in a superconductor depends on its direction of flow. This asymmetry underlies the operation of superconducting diodes and is often associated with the presence of vortices. Here, we investigate supercurrent nonreciprocal effects in a superconducting weak-link hosting distinct types of vortices. We demonstrate how the winding number of the vortex, its spatial configuration, and the shape of the superconducting lead can steer the sign and amplitude of the supercurrent rectification. We identify a general criterion for optimizing the rectification amplitude based on vortex patterns, focusing on configurations where the first harmonic of the supercurrent vanishes. We prove that supercurrent nonreciprocal effects can be used to diagnose high-winding vortex and to distinguish between different types of vorticity. Our results provide a toolkit for controlling supercurrent rectification through vortex phase textures and detecting unconventional vortex states.

Figures (6)

• Figure 1: Image of Josephson junctions with a vortex. (a) Schematic of a Josephson junction in the lattice model with a vortex phase texture. SC-L, N, and SC-R mean the Left-side superconductor (SC), the Normal metal, and the Right-side SC. $N^\mathrm{L}_{x}$, $N^\mathrm{R}_{x}$, $N^\mathrm{N}_{x}$, $N_{y}$, and $a$ denote the number of sites in the left and right-side SCs, and in the normal metal along the $x$-direction, and number of sites along the $y$-direction, and lattice constant. $\bm{r}^\mathrm{L}_{0}=(x^\mathrm{L}_{0},y^\mathrm{L}_{0})$ is the core position. (b) A representative nonreciprocal current phase relation with $\varphi$ being the phase bias across the junction. (c)(d) Sketch of the superconducting weak link with a vortex placed on the left side of the junction. We show two representative vortex positions along the lateral direction. Since the supercurrent pattern is spatially modified by the phase vortex, it can become nonreciprocal, i.e. the forward supercurrent $I_+$ is different from the backward one $I_{-}$. The vortex winding, $V^\mathrm{L}_{0}$, can take any integer number $l$.
• Figure 2: Space dependence of rectification amplitudes with a vortex. (a) Schematic illustration of the Josephson junction with real space coordinates. The analysis is performed for a vortex configuration having winding number $V^\mathrm{L}_{0}=+l$ with $l=1,2,3$. SC-L, N, and SC-R mean the Left-side superconductor (SC), the Normal metal, and the Right-side SC. $N^\mathrm{L}_{x}$, $N^\mathrm{R}_{x}$, $N^\mathrm{N}_{x}$, $N_{y}$, and $a$ denote the number of sites in the left and right-side SCs, and in the normal metal along the $x$-direction, and number of sites along the $y$-direction, and lattice constant. Evolution of the rectification amplitude $\eta$ with regard to the vortex core coordinates for different winding numbers: (b,c) $V^\mathrm{L}_0=1$, (f,g) $V^\mathrm{L}_0=2$, and (j,k) $V^\mathrm{L}_0=3$. Spatially resolved harmonics of the supercurrent: $I_{1}$ (red-solid line), $I_{2}$ (green-dotted), and $J_{1}$ (blue-dashed) indicate the odd-parity first harmonic, the odd-parity second harmonic, and the even-parity first harmonic amplitude, respectively. Longitudinal scan: $I_{1}$, $I_{2}$, and $J_{1}$ at a given $y^\mathrm{L}_{0}$ as a function of $x^\mathrm{L}_{0}$ for (d) $V^\mathrm{L}_0=1$, (h) $V^\mathrm{L}_0=2$, and (l) $V^\mathrm{L}_0=3$. Lateral scan: $I_{1}$, $I_{2}$, and $J_{1}$ at a given $x^\mathrm{L}_{0}$ as a function of $y^\mathrm{L}_{0}$ for (e) $V^\mathrm{L}_0=1$, (i) $V^\mathrm{L}_0=2$, and (m) $V^\mathrm{L}_0=3$. The gray dotted lines refer to the position of the maximal rectification and are a guide to indicate the values $I_{1}$, $I_{2}$, and $J_{1}$. We set $y^\mathrm{L}_{0}$ as (b)(d) $0.5a$ and (f)(h)(j)(l) $5.5a$, and $x^\mathrm{L}_{0}$ as (c)(e)(g)(i) $-23a$ and (k)(m) $-6a$. The aspect ratio is (b)-(i) $\alpha=3/2$ and (j)-(m) $\alpha=1$. The maximal rectification $\eta$ occurs for vortex core positions corresponding to a supercurrent with $I_{1}$, $I_{2}$, and $J_{1}$ components that are comparable in size. The sign change of $\eta$ is related to the vanishing of $J_{1}$ and to the zeros of $I_{1}$ when the amplitude is comparable to $J_{1}$. Multiple sign reversals of $\eta$ are observed for $V^\mathrm{L}_0=3$. Parameters: $|\Delta_0|=0.02t$ (superconducting energy gap amplitude), $t_\mathrm{int}=0.90$ (transparency at the interface), $N^\mathrm{N}_{x}=10$, $N_{y}=30$, and $z^\mathrm{L}_{0}=10a$ (vortex size).
• Figure 3: Evaluation of the first and second harmonics and the rectifications as a function of the space with a vortex. Amplitude of the first harmonics of the Josephson current and the rectification versus the vortex core positions $\bm{r}^\mathrm{L}_{0}=(x^\mathrm{L}_{0},y^\mathrm{L}_{0})$ for different vortex winding $V^\mathrm{L}_0$: (a,b,c) $V^\mathrm{L}_0=1$, (d,e,f) $V^\mathrm{L}_0=2$, (g,h,i) $V^\mathrm{L}_0=3$. $\bar{I}_{1}$, $\bar{J}_{1}$, and $\eta$ indicate the odd- and even-parity first harmonics of the supercurrent evaluated by the direct Cooper pairs tunneling and the amplitude of the rectification. The color bars indicate the amplitude of (a,d,g) $\bar{I}_1(\bm{r}^\mathrm{L}_{0})$, (b,e,h) $\bar{J}_1(\bm{r}^\mathrm{L}_{0})$, and (c,f,i) the rectification amplitude $\eta$. In (a,b,d,e,g,h), gray-dotted and dashed lines stand for the vortex core positions, $y^\mathrm{L}_{0}$ and $x^\mathrm{L}_{0}$, evaluated in Fig. \ref{['fig:2']}. The aspect ratio is (a)-(f) $\alpha=3/2$ and (g)-(i) $\alpha=1$. In (c)(f)(i), the rectification amplitude is evaluated by scanning all the positions of the vortex cores by performing the computation of the supercurrent for the weak link assuming the following parameters: $|\Delta_0|=0.02t$ (superconducting energy gap amplitude), $t_\mathrm{int}=0.90$ (transparency at the interface), $N^\mathrm{N}_{x}=10$, $N_{y}=30$, and $z^\mathrm{L}_{0}=10a$ (vortex size).
• Figure 4: Space dependence of rectifications with two vortices. (a) Schematic illustration of two vortices in the left-side superconductor. SC-L, N, and SC-R mean the Left-side superconductor (SC), the Normal metal, and the Right-side SC. $\bm{r}^\mathrm{L}_{m}=(x^\mathrm{L}_{m},y^\mathrm{L}_{l})$ with $m=1,2$ indicate core position for each vortex. $V^\mathrm{L}_{0,1}$ indicate the winding number of each phase vortex. $N^\mathrm{L}_{x}$, $N^\mathrm{R}_{x}$, $N^\mathrm{N}_{x}$, $N_{y}$, and $a$ denote the number of sites in the left and right-side SCs, and in the normal metal along the $x$-direction, and number of sites along the $y$-direction, and lattice constant. (b,c) Rectification $\eta$ and (d,e) $I_{1}$, $I_{2}$, $J_{1}$ as a function of the literal direction $y^\mathrm{L}_{1}$ for each (b,d) $(V^\mathrm{L}_{0},V^\mathrm{L}_{1})=(+1,+1)$ and (c,e) $(V^\mathrm{L}_{0},V^\mathrm{L}_{1})=(+1,-1)$. We set the core positions as $(x^\mathrm{L}_{0},y^\mathrm{L}_{0})=(-30,0.5)$ and $x^\mathrm{L}_{1}=-15$. $I_{0}=0.122|\Delta_{0}|(2e/\hbar)$ stands for the maximum Josephson current without any phase vortices in superconductors. We select the parameters: $|\Delta_0|=0.02t$ (superconducting energy gap amplitude), $t_\mathrm{int}=0.90$ (transparency at the interface), $N^\mathrm{L}_{x}=N^\mathrm{R}_{x}=45$, $N^\mathrm{N}_{x}=10$, $N_{y}=30$, and $z^\mathrm{L}_{0}=z^\mathrm{L}_{1}=10a$ (each vortex size).
• Figure 5: Evaluation of the first and second harmonics Josephson current with two vortices. (a,b) $\bar{I}_{1}$ and (c,d) $\bar{J}_{1}$ for the $\bm{r}^\mathrm{L}_{1}=(x^\mathrm{L}_{1},y^\mathrm{L}_{1})$ space at (a,c) $(V^\mathrm{L}_{0},V^\mathrm{L}_{1})=(+1,+1)$ and (b,d) $(V^\mathrm{L}_{0},V^\mathrm{L}_{1})=(+1,-1)$. The color bars indicate the amplitude of (a,c) $\bar{I}_1(\bm{r}^\mathrm{L}_{0})$ and (b,d) $\bar{J}_1(\bm{r}^\mathrm{L}_{0})$. We set the core positions as $(x^\mathrm{L}_{0},y^\mathrm{L}_{0})=(-30a,0.5a)$ and the vortex size as $z^\mathrm{L}_{1}=10a$ ($N^\mathrm{L}_{x}=N^\mathrm{R}_{x}=45$ and $N_{y}=30$). The Gray line indicates the scanning site in Fig. \ref{['fig:4']}.