Spin susceptibility, upper critical field, and disorder effect in j = 3 2 superconductors with singlet-quintet mixing

Recently, a new pairing state with the mixing between the s-wave singlet channel and the isotropic d-wave quintet channel induced by centrosymmetric spin-orbit coupling has been theoretically proposed in the superconducting materials with j = 3 2 electrons [J. Yu and C.-X. Liu, Phys. Rev. B 98, 104514 (2018)]. In this work, we derive the expressions of the zero-temperature spin susceptibility, the upper critical field close to the zero-field critical temperature T c, and the critical temperature with weak random non-magnetic disorders for the singlet-quintet mixed state based on the Luttinger model. Our study reveals the following features of the singlet-quintet mixing. (1) The zero-temperature spin susceptibility remains zero for the singlet-quintet mixed state if only the centrosymmetric spin-orbit coupling is taken into account and deviates from zero when the non-centrosymmetric spin-orbit coupling is introduced. (2) The singlet-quintet mixing can help enhance the upper critical field roughly because it can increase T c. (3) Although the quintet channel is generally suppressed by the non-magnetic disorder scattering, we find the strong mixing between singlet and quintet channels can help to stabilize the quintet channel. As a result, we still find a sizable quintet component mixed into the singlet channel in the presence of weak random non-magnetic disorders. Our work provides the guidance for future experiments on spin susceptibility and upper critical field of the singlet-quintet mixed superconducting states and illustrates the stability of the singlet-quintet mixing against the weak random non-magnetic disorder.