Tunneling conductance in superconducting junctions with $p$-wave unconventional magnets breaking time-reversal symmetry

TL;DR

The paper addresses tunneling transport in superconducting junctions interfaced with $p$-wave unconventional magnets (PUMs) that break time-reversal symmetry. By employing a tight-binding model with an effective TRS-breaking PUM Hamiltonian and calculating tunneling conductance via a recursive Green's function approach and the Lee–Fisher formula, it characterizes momentum- and spin-resolved conductance for spin-singlet and spin-triplet pairings, including helical and chiral states. The main findings reveal an $eV$-asymmetric conductance for helical $p$-wave superconductors due to missing helical edge states, and spin-dependent differences in chiral $d$- and $p$-wave cases, with results aligning qualitatively with prior simplified models. These results illuminate how TRS breaking in PUMs and momentum-dependent spin splitting shape Andreev processes and edge-state contributions, with potential implications for superconducting spintronics. The work validates the simplified odd-function exchange-coupling picture and sets the stage for exploring anisotropic interfaces, Josephson effects, and odd-frequency pairing in PUM-based hybrids.

Abstract

A new type of magnet called $p$-wave unconventional magnet is proposed, stimulated by the discovery of altermagnet. We study the tunneling conductance of $p$-wave unconventional magnet/superconductor junctions by adopting the effective Hamiltonian of $p$-wave unconventional magnets with time-reversal symmetry breaking, suggested in Ref [arXiv: 2309.01607 (2024)]. The tunneling conductance shows an asymmetric behavior with respect to bias voltage in the helical $p$-wave superconductor junctions. It is caused by the missing of helical edge states contributing to the charge conductance owing to the momentum-dependent spin-split feature of the Fermi surface in $p$-wave unconventional magnets. In chiral $d$ and $p$-wave superconductor junctions, the resulting spin-resolved tunneling conductance takes a different value for spin sectors due to the time-reversal symmetry breaking in superconductors. Our results qualitatively reproduce the results based on the simplified Hamiltonian in Ref [J.\ Phys.\ Soc.\ Jpn.\ \textbf{93}, 114703 (2024)], where only the odd function of the exchange coupling of $p$-wave unconventional magnets is taken into account, which gives the shift of the Fermi surface and preserves the time-reversal symmetry similar to the spin-orbit coupling.

Figures (10)

• Figure 1: (a) Image of $p$-wave unconventional magnet (PUM)/superconductor (SC) junctions. We suppose the periodic boundary condition along the $x$-direction and semi-infinite systems along the $y$-direction. We consider two orbital (sublattice) degrees of freedom: orbital A (red) and B (blue) in PUMs and the single-orbital model in SCs (green). To correspond to the $p_y$-wave magnet/SC junction with the interface along the $x$-direction, we consider a junction along the $y$-direction. Black dotted lines indicate the tunneling between A (red) and B (blue) of PUM and SC (green). 0 and 1 indicate the indices in the Green's function of Sect. II. The Fermi surface of PUM and that in the normal state of SC are shown in (b) and (c). (b) $\mu_{p}=-3.85t$ and $t_J=0.25t$, and (c) $\mu_\mathrm{sc}=-2.50t$. The black-dashed lines indicate the overlapping momenta $k_{x1}$ and $k_{x2}$ between PUM and SC.
• Figure 2: Values of $k_{x1}$ (red) and $k_{x2}$ (blue lines) as a function of the strength of the PUM order $t_J$. We choose the chemical potential as $\mu_{p}=-3.85t$.
• Figure 3: (a)(b)(c) Momentum-resolved tunneling conductance $\sigma(eV,k_x)$ for (a) spin-singlet $s$-wave, (b) $d_{x^2-y^2}$-wave, and (c) $d_{xy}$-wave SC junctions. Here, $\sigma_\mathrm{N}$ is the tunneling conductance in the normal state at $eV=0$. (d)(e)(f) Momentum-resolved tunneling conductance with up-spin $\sigma_{\uparrow}(eV,k_x)$ for (d) spin-singlet $s$-wave, (e) $d_{x^2-y^2}$-wave, and (f) $d_{xy}$-wave SC junctions. (g)(h)(i) Momentum-resolved tunneling conductance with down-spin $\sigma_{\downarrow}(eV,k_x)$ for (g) spin-singlet $s$-wave, (h) $d_{x^2-y^2}$-wave, and (i) $d_{xy}$-wave SC junctions. We choose the parameters as $t_J=0.25t$, $U_\mathrm{b}=5t$, $\Delta=0.001t$, and $\delta=0.01\Delta$.
• Figure 4: Tunneling conductance $\bar{\sigma}(eV)$ for (a)(d) spin-singlet $s$-wave, (b)(e) $d_{x^2-y^2}$-wave, and (c)(f) $d_{xy}$-wave SC junctions at (a)(b)(c) $t_J=0$ and (d)(e)(f) $t_J=0.25t$. It is normalized by its value in the normal state at $eV=0$: $\sigma_\mathrm{N}$. We choose the parameters as $U_\mathrm{b}=5t$, $\Delta=0.001t$, and $\delta=0.01\Delta$.
• Figure 5: (a)(b)(c) Momentum-resolved tunneling conductance $\sigma(eV,k_x)$ for spin-triplet SC junctions. (a)helical $p$-wave, (b) $p_{x}$-wave, and (c) $p_{y}$-wave SC junctions. Here, $\sigma_\mathrm{N}$ is the tunneling conductance in the normal state at $eV=0$. (d)(e)(f) Momentum-resolved tunneling conductance with up-spin $\sigma_{\uparrow}(eV,k_x)$ for (d)helical $p$-wave, (e) $p_{x}$-wave, and (f) $p_{y}$-wave SC junctions. (g)(h)(i) Momentum-resolved tunneling conductance with down-spin $\sigma_{\downarrow}(eV,k_x)$ for (d)helical $p$-wave, (e) $p_{x}$-wave, and (f) $p_{y}$-wave SC junctions. We choose the parameters as $t_J=0.25t$, $U_\mathrm{b}=5t$, $\Delta=0.001t$, and $\delta=0.01\Delta$.