We investigate the emergence of superconducting phases, both with zero and finite Cooper-pair center-of-mass momenta, in recently proposed unconventional $p$-wave magnets. As a result we find that while these magnetic phases are in principle compatible with a conventional pairing state at zero field, a Fulde-Ferrell phase can generally be promoted as the leading instability under the application of a finite magnetic field. Interestingly, by calculating the efficiency of the superconducting diode effect of this finite-momentum pairing state via a Ginzburg-Landau theory, we uncover that a high efficiency can be obtained in these systems for experimentally relevant spin splittings. Therefore, our prediction reveals that the experimental discovery of these materials represents a promising platform for the construction of superconducting logic circuits that may as a consequence find various technological applications.

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• Landau theory
• Order parameters
• Phase diagrams
• Noncollinear magnets
• Superconductors
• Effective field theory
• Feynman diagrams