In conventional superconductors, a Cooper pair is a time-reversed pair of two electrons on the Fermi surface, resulting in a state with zero Cooper pair’s momentum. However, when time-reversal symmetry is broken—for example, by applying a magnetic field—the spin degeneracy of the Fermi surface is lifted. This spin splitting can enable the formation of Cooper pairs with finite center-of-mass momentum, leading to finite-momentum superconductivity. Recently, two-dimensional superconductors with both RSOC and Zeeman coupling (the Rashba–Zeeman model) have been extensively studied, particularly in connection with the superconducting diode effect.

In this work, we propose altermagnets as an alternative mechanism for breaking time-reversal symmetry and theoretically investigate the combined effects of RSOC and altermagnetic splitting on two-dimensional -wave superconductivity. Altermagnets are a class of magnetic materials that exhibit anisotropic spin splitting on the Fermi surface without a net magnetization. By incorporating both the anisotropic spin splitting characteristic of altermagnets and RSOC into a two-dimensional free-electron gas model and assuming spin-singlet d-wave superconductivity, we examine the resulting finite-momentum superconducting states using the Bogoliubov–de Gennes formalism.

Our numerical results demonstrate that finite-momentum superconductivity can be realized across a wide range of parameters in the presence of in-plane altermagnetic order. The interplay between RSOC and altermagnetic spin splitting produces quadrupole deformations of the Fermi surfaces. These deformations enhance the stability of the inter-band pairing through the nesting condition for the formation of Cooper pairs, thereby expanding the superconducting phase. Furthermore, they enable intra-band pairing mechanisms involving contributions from both Fermi surfaces, supporting a single-q finite-momentum superconducting state. This behavior stands in contrast to the conventional Rashba–Zeeman model, where typically only one Fermi surface significantly contributes to finite-momentum pairing.

These findings offer a new perspective on how RSOC and altermagnetism jointly deform Fermi surfaces to stabilize unconventional superconductivity. This deeper understanding could inform the future design of superconducting devices based on altermagnetic materials.

(Written by Kohei Mukasa and Yusuke Masaki).

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