Finite-momentum superconductivity in two-dimensional altermagnets with a Rashba-type spin-orbit coupling

We theoretically study finite-momentum superconductivity in two-dimensional (2D) altermagnets with a Rashba-type spin-orbit coupling (RSOC). We show the phase diagrams obtained by solving a linearized gap equation, considering two directions of the N\'{e}el vector of the 2D altermagnet: parallel to the $xy$ plane (in-plane) and perpendicular to the $xy$ plane (out-of-plane). For the in-plane N\'{e}el vector, we find two different finite-momentum $d_{x^2-y^2}$-wave superconducting states distinguished by a dominant pairing channel: an inter-band pairing or an intra-band pairing. Furthermore, it is shown that an anisotropic deformation of Fermi surfaces caused by spin-splitting effects due to the in-plane N\'{e}el vector and the RSOC can contribute to the stabilization of the finite-momentum superconductivity. We also perform the self-consistent calculations, and verify the existence of these superconducting states in addition to identifying other possible superconducting states and transition lines away from the phase boundary. For the out-of-plane N\'{e}el vector, the finite-momentum superconductivity is realized only in the inter-band pairing mechanism, which is in contrast to the in-plane case.