Interplay between Superconductivity and Altermagnetism in Disordered Materials and Heterostructures

TL;DR

This work develops a symmetry-based, diffusive framework to study the interplay between superconductivity and altermagnetism. Starting from a nonlinear sigma model, it derives a Ginzburg–Landau free energy containing a novel second-gradient term tied to the altermagnetic tensor $K_{ajk}$, which yields two magnetization mechanisms: a magnetoelectric effect quadratic in the phase gradient and a magnetization generated by spatial variations of $| ext{Delta}|$. The Usadel formalism is then applied to S/AM bilayers and S/AM/S Josephson junctions to quantify proximity-induced magnetization and to reveal $0$-$\pi$ transitions possible in the diffusive regime, with magnetization patterns showing clear imprint of the altermagnet's $d$-wave symmetry and angular dependence via the orientation parameter $\alpha$. A low-energy effective model connects the phenomenological coefficients to microscopic disorder-affected parameters, clarifying how gradients and exchange fields drive the observed phenomena. Overall, the results establish a comprehensive, disorder-tolerant framework for equilibrium and transport properties in superconducting altermagnetic heterostructures, with potential implications for spintronic applications that exploit current- and order-parameter–driven magnetization control.

Abstract

We study the interplay between superconductivity and altermagnetism in disordered systems using recently derived quantum kinetic transport equations. Starting from this framework, we derive the Ginzburg-Landau free energy and identify, in addition to the conventional pair-breaking term, a coupling between the spin and the spatial variation of the superconducting order parameter. Two distinct effects emerge from this coupling. The first is a magnetoelectric effect, in which a supercurrent (i.e., a phase gradient) induces a spin texture; this contribution is quadratic in the phase gradient. The second effect arises when the magnitude, rather than the phase, of the superconducting order parameter varies in space, likewise leading to a finite magnetization. We show that these two contributions compete in the case of an Abrikosov vortex, where both the amplitude and phase of the order parameter vary spatially. The effect associated with amplitude variations also gives rise to a proximity-induced magnetization (PIM) in hybrid structures composed of a superconductor (S) and an altermagnet (AM). Using quasiclassical theory, we analyze the PIM in diffusive S/AM bilayers and S/AM/S Josephson junctions, and determine the induced magnetization profiles. In Josephson junctions, where both the PIM and the magnetoelectric effect coexist, we further predict the occurrence of $0$-$π$ transitions.

Figures (8)

• Figure 1: Magnetization induced in the presence of an Abrikosov vortex in a superconducting altermagnet with vorticity $m=1$. Contributions due to (a) the nonlinear magnetoelectric effect, and (b) due to the variation of $|\Delta|$. (c) Total magnetization $M_a = M_a^{(\varphi)} + M_a^{(|\Delta|)}$. The underlying $d$-wave symmetry of the altermagnet is manifested as four lobes with alternating sign.
• Figure 2: A square altermagnet of dimensions $L\times L$ next to an infinite (bulk) superconductor at $x=0$. The remaining edges of the altermagnet ($x=L$, $y=0$ and $y=L$) are bounded by vacuum. We draw a four-lobed red flower, reminiscent of $d$-wave symmetry, to represent the angle $\alpha$ between the $x$ axis and the crystallographic axis of the altermagnet.
• Figure 3: (a) Magnetization for $K = 0.1$ and different values of $\Gamma$. Increasing $\Gamma$ suppresses the overall magnitude of the magnetization and reduces the characteristic decay length $\xi$. (b) Magnetization for $\Gamma / \Delta_0 = 0.1$ and varying $K$. A larger $K$ enhances the induced magnetization. Around $x\approx\xi \approx 0.4\xi_0$, the magnetization changes sign, reflecting oscillations of the triplet component. The remaining parameters are $L = \xi_0$, $T / \Delta_0 = 0.1$, and $\gamma_B = 1$. The red line in both panels corresponds to the parameters of the low-energy model discussed in Sec. \ref{['Microscopicderiv']}, with $h / \Delta_0 = 1$ and $\tau \Delta_0 = 0.1$.
• Figure 4: Magnetization in a 2D square altermagnet of side $L=\xi_0$ in contact with a superconductor for different values of $\alpha$. For $\alpha=0$, the induced magnetization is uniform along the y-direction and decays exponentially away from the interface. As $\alpha$ increases, a spatial redistribution of the magnetization emerges: negative magnetization localizes near the bottom-left corner, while positive magnetization develops near the top-left corner. At $\alpha=\pi/4$, d-wave symmetry dictates that the magnetization has equal magnitude and opposite sign in these two corners. For the other parameters we have chosen: $T/\Delta_0=0.1$, $\gamma_B=1$, $K=0.1$ and $\Gamma/\Delta_0=0.1$, which corresponds to $h/\Delta_0=1$ and $\tau\Delta_0=0.1$ in the low-energy model of Sec. \ref{['Microscopicderiv']}.
• Figure 5: Circular superconducting island (white circle) of radius $r=\xi_0$ on top of a square altermagnet of side $L=5\xi_0$. The underlying $d$-wave symmetry of the altermagnet becomes particularly evident, giving rise to four distinct lobes in the magnetization with alternating sign. Magnetization oscillations are especially prominent in this geometry. For instance, along the horizontal direction, the magnetization at the interface with the superconductor is initially negative (blue) and then oscillates to a bigger positive (yellow) lobe. For the other parameters we have chosen: $T/\Delta_0=0.1$, $\gamma_B=1$, $K=0.1$ and $\Gamma/\Delta_0=0.1$, which corresponds to $h/\Delta_0=1$ and $\tau\Delta_0=0.1$ in the low-energy model discussed in Sec. \ref{['Microscopicderiv']}.