$p$-wave superconductivity and Josephson current in $p$-wave unconventional magnet/$s$-wave superconductor hybrid systems

TL;DR

The paper shows that a noncollinear spin structure in a p-wave unconventional magnet (PUM) can induce p-wave–like superconductivity in p-wave UM–s-wave SC hybrids, producing zero-energy flat bands at the [100] edge and allowing odd-frequency triplet edge pairing. By analyzing surface DOS, edge pair amplitudes, and Josephson transport in high- and low-transparency junctions, the authors reveal how edge states and spin-texture shape current-phase relations, including phi-junction behavior even when s-wave pairing is present on both sides. The work demonstrates that edge-state resonance can enhance higher harmonics and the maximum Josephson current, while the presence or absence of flat bands strongly modulates Ic(T) in low-transparency limits. Overall, the results offer a concrete framework for realizing and probing spin-triplet p-wave superconductivity in PUM–SC hybrids and for designing Josephson devices controlled by edge-state topology.

Abstract

We study the surface density of states in $p$-wave unconventional magnet-spin-singlet $s$-wave superconductor hybrid systems ($p$-wave unconventional magnetic superconductors), by using the effective $p$-wave unconventional magnet model [B. Brekke, et al., Phys. Rev. Lett. 133, 236703 (2024)]. Owing to the noncollinear spin structure along the $x$-direction, the quasiparticle energy dispersion has the spin-triplet $p_x$-wave energy gap structure, and then zero-energy flat bands emerge at the [100] edge. Analyzing the pair amplitude at the [100] edge, odd-frequency spin-triplet even-parity pairing is induced in the presence of zero-energy flat bands, while even-frequency spin-singlet even-parity remains. We also demonstrate the Josephson current in superconducting junctions with $p$-wave unconventional magnet-conventional $s$-wave superconductor hybrid systems. By the cooperation of spin-singlet $s$-wave pair potential and the $p$-wave unconventional magnetic order, the current phase relation shows the $\varphi$-junction in the high-transparency but also the temperature dependence of the Josephson current caused by the coupling of the spin-singlet even-parity pairings in the low-transparency limit, even though $p$-wave superconductivity is mainly realized both in the bulk and at the edge. Our calculations provide the possible superconducting phenomena and transport properties in $p$-wave unconventional magnet-$s$-wave superconductor hybrid systems.

Figures (16)

• Figure 1: (a)(b) Schematic illustration of the two-dimensional $p$-wave unconventional magnet (PUM) at (a) $t_{x}\ne 0$ ($t_y=0$) and (b) $t_y\ne 0$ ($t_x=0$) in Eq. (\ref{['PUM_eq']}). Arrows stand for the noncollinear spin structure in PUM, and blue and red colors indicate the combination of the sublattices, suggested in Ref. BrekkePRL2024. Both $J=t$ and $t_{x,y}$ terms lead to the noncollinear spin structures of $p$-wave unconventional magnetism. (c-f) Fermi surface at (c) $(\mu,t_{x},t_{y})=(-4t,t,0)$, (d) $(-2t,t,0)$, (e) $(-4t,0,t)$, and $(-2t,0,t)$. We set $J=t$ in panels (c)-(f).
• Figure 2: (a) Schematic illustration of PUM-$s$-wave SC hybrid systems ($p_x$-wave UM-SCs) at $(t_{x},t_{y})=(t,0)$. At the [100] edge, (b,c) momentum-resolved surface density of states (SDOS) $D(E)$ at (b) $\mu=-4t$ and (c) $\mu=-2t$. (d,e) SDOS (green solid) and bulk DOS (black dotted lines) normalized by $D_\mathrm{N}$ with the normal state SDOS and bulk DOS at zero energy for (d) $\mu=-4t$ and (e) $\mu=-2t$. Parameters: $(t_{x},t_{y})=(t,0)$, $J=t$, $\Delta=0.01t$, and $\delta=0.01\Delta$.
• Figure 3: (a) Schematic illustration of PUM-$s$-wave SC hybrid systems ($p_y$-wave UM-SCs) at $(t_{x},t_{y})=(0,t)$. At the [100] edge, (b,c) momentum-resolved SDOS $D(E)$ at (b) $\mu=-4t$ and (c) $\mu=-2t$. (d,e) SDOS (green solid) and bulk DOS (black dotted lines) normalized by $D_\mathrm{N}$ with the normal state SDOS and bulk at zero energy for (d) $\mu=-4t$ and (e) $\mu=-2t$. Parameters: $(t_{x},t_{y})=(0,t)$, $J=t$, $\Delta=0.01t$, and $\delta=0.01\Delta$.
• Figure 4: Absolute value of the pair amplitude for (a)(b) even-frequency spin-singlet even-parity (ESE) and (c)(d) odd-frequency spin-triplet even-parity (OTE) pairings at the [100] edge of PUM-SCs at $t_{x}=t$. We plot the pair amplitude at the lowest Matsubara frequency $\omega_{n}=\pi k_\mathrm{B}T$. The function of $|F^{\uparrow\uparrow}_\mathrm{OTE}|$ and $|F^{\downarrow\downarrow}_\mathrm{OTE}|$ for $k_y$ are the same. Parameters: $(t_{x},t_{y})=(t,0)$, $J=t$, $T_\mathrm{c}=0.01t$, and $T=0.025T_\mathrm{c}$.
• Figure 5: Absolute value of the pair amplitude for (a)(b) even-frequency spin-singlet even-parity (ESE), (c)(d) even-frequency spin-triplet odd-parity (ETO), and (e)(f) odd-frequency spin-triplet even-parity (OTE) pairings at the [100] edge of PUM-SCs at $t_{y}=t$. We plot the pair amplitude at the lowest Matsubara frequency $\omega_{n}=\pi k_\mathrm{B}T$. The function of $|F^{\uparrow\uparrow}_\mathrm{ETO}|$ and $|F^{\downarrow\downarrow}_\mathrm{ETO}|$ for $k_y$ are the same. Parameters: $(t_{x},t_{y})=(0,t)$, $J=t$, $T_\mathrm{c}=0.01t$, and $T=0.025T_\mathrm{c}$.