Angular dependence of superconductivity in CeRh$2$As$2$

CeRh$_2$As$_2$ is one of the few materials where unconventional superconductivity is revealed by the appearance of multiple superconducting phases [1]. When the magnetic field $\mu_{0}H$ is applied parallel to the tetragonal crystallographic axis $c$, this material shows a transition at 4 T from a low-field superconducting state SC1 to a high-field SC2 with a upper critical field $H_{c2} = 14$ T and critical temperature $T_c = 0.26$ K. In contrast, for in-plane fields, only the SC1 with a lower critical field of 2 T is observed. In this particular case, the phase-diagrams and their anisotropy were explained by the influence of anti-symmetric spin-orbit coupling at the Ce sites where the inversion symmetry is broken locally. Above the superconducting state, a phase transition at $T_0 \approx 0.4$ K to a possibly quadrupolar order is present as well, whose influence on the superconducting state remains unknown. Here, we present a comprehensive study of the angular dependence of the upper critical fields and $T_0$ using low temperature magnetic ac susceptibility, specific heat and torque in single crystalline CeRh$_2$As$_2$. The field-induced superconducting state SC2 is strongly suppressed when rotating the magnetic field away from the c-axis and disappears for an angle of $35^\circ$. Furthermore, an analysis of the Pauli and orbital limit as a function of angle reveals that the critical field of SC2 for angles departing from the c axis is attained when the in-plane component of the field reaches the in-plane Pauli limit. This result corroborates the idea of a field-induced odd-parity state with a pseudo-spin d-vector in the plane in CeRh$_2$As$_2$.