Spin-polarized Andreev molecules and anomalous nonlocal Josephson effects in altermagnetic junctions

Abstract

Altermagnetism has emerged as a promising ingredient for realizing nontrivial Josephson phases, but so far explored in single Josephson junctions. In this work, we consider the coherent coupling of two Josephson junctions with spin-singlet $s$-wave superconductivity and demonstrate that $d$-wave altermagnetism gives rise to spin-polarized Andreev molecules due to the hybridization of Andreev bound states of each junction when the coupling is weak. Interestingly, these spin-polarized Andreev molecules induce an anomalous nonlocal Josephson effect, where the current flow across one Josephson junction due to phase changes across the other junction develops $0-π$ and $φ_{0}$ transitions originating from altermagnetism. Furthermore, the nonlocal Josephson current carried by spin-polarized Andreev molecules exhibits nonreciprocal critical currents, enabling a nonlocal Josephson diode effect whose polarity is tunable by the altermagnetic strength and right phase. Our findings put forward altermagnetism as a promising arena for designing nonlocal spin Josephson phenomena.

Figures (7)

• Figure 1: (a,f) Sketch of the studied Josephson setup, where three spin-singlet $s$-wave superconductors denoted by $S_{\alpha}$ ($\alpha={\rm L,R}$ pink and $\alpha={\rm M}$ gold) are in contact with a $d$-wave altermagnet (blue); the superconductors have lengths $L_{\alpha}$ and pair potentials $\Delta_{\rm \alpha}=\Delta {\rm e}^{i\phi_{\alpha}}$, with $\phi_{\rm M}=0$. (a) and (f) correspond to $S_{\rm M}$ with a long ($L_{\rm M}=100a\gg\xi$) and short ($L_{\rm M}=2a\lesssim\xi$) length, where $\xi$ is the superconducting coherence length and $a$ is the lattice spacing. (b,c) Low-energy spectrum as a function of $\phi_{\rm L}$ at $\phi_{\rm R}=0.7\pi$ and $k_{y}=0.1\pi$ for $L_{\rm M}\gg\xi$, while (g,h) for $L_{\rm M}\lesssim\xi$. The blue (red) color in (b,c,g,h) indicates that such energy levels belong to the Nambu sector formed by spin-$\uparrow$ electrons (holes) with spin-$\downarrow$ holes (electrons), denoted by '$+$' ('$-$'). The light green curves are the energy levels without altermagnetism. The filled yellow and green circles mark the positions where dispersionless levels of the right JJ cross with the dispersive levels of the left JJ in presence of the AM, while the filled gray circles mark the same but in absence of the AM. (d,e) Spin projection along $z$ in the left JJ at $k_{y}=0.1\pi$ for $L_{\rm M}\gg\xi$, while (i,j) for $L_{\rm M}\lesssim\xi$. Parameters: $L_{\rm L} = L_{\rm R} = 100a$, $\mu = 1.5t$, $\Delta = 0.2t$, $\phi_{\rm M} = 0$.
• Figure 2: Josephson current ${\cal I}_{\rm L}$ across the left JJ as a function of $\phi_{\rm L,R}$ in the presence of $d_{x^{2}-y^{2}}$-wave altermagnetism. (a) ${\cal I}_{\rm L}$ for $L_{\rm M}=100a\gg\xi$ and $J_{2}/t=0.1$, while (b,c) for $L_{\rm M}=2a\lesssim\xi$ and $J_{2}/t=\{0.1,0.3\}$. (d,e) Line cuts of ${\cal I}_{\rm L}$ in (b) as a function of $\phi_{\rm L(R)}$ at distinct $\phi_{\rm R(L)}$ in steps of $0.2\pi$ marked by color bars in (b). Rest of parameters same as in Fig. \ref{['Fig1']}
• Figure 3: (a) Critical currents $\mathcal{I}_{\rm cL}^{\pm}$ as a function of $\phi_{\rm R}$ and $J_2$ under $d_{x^{2}-y^{2}}$-wave altermagnetism. (b) Quality factor $\eta_{\rm L}$ as a function of $J_2$ and $\phi_{\rm R}$. (c,d) Line cuts of (b) for distinct values of $J_{2}$ and $\phi_{\rm R}$. The values of $\phi_{\rm R}$ for the line cuts in (c) are marked by color bars in (b). Parameters: $L_{\rm M} = 2a$ and the rest is the same as in Fig. \ref{['Fig1']}.
• Figure 4: Wave function probability density of the first positive state for a JJ with $d_{x^{2}-y^{2}}$-wave altermagnetism as a function of space for $L_M = 100a$ (a) and $L_M=2a$ (b,c), in both cases with $\phi_{\rm R}=0.7\pi$. The green and the red curves indicate two different values of $k_y$. Parameters: $J_1 = 0$, $J_2 = 0.1t$; the rest is the same as in Fig. \ref{['Fig1']}.
• Figure 5: Lowest four positive levels of each $\pm$ sector as a function of $L_{\rm M}$ in the absence (a) and presence of altermagnetism (b,c) at $\phi_{\rm R} = 0.7\pi$, $k_y = 0.1\pi$. The dashed and solid curves correspond to $\phi_{\rm L} = 0.7\pi = \phi_{\rm R}$ and $\phi_{\rm L} = 1.3\pi = -\phi_{\rm R}$ respectively. Parameters: $J_{1,2} = 0$ in (a), $J_1 = 0.1t$ and $J_2 = 0$ in (b), $J_1 = 0$ and $J_2 = 0.1t$ in (c), and the rest as in Fig. \ref{['Fig1']}.