Field resilient superconductivity in atomic layer crystalline materials

A recent study [S. Yoshizawa {\it et al}., Nature Communications {\bf 12}, 1462 (2021)] reported the occurrence of field-resilient superconductivity, that is, enhancement of the in-plane critical magnetic field $H^{||}_{\rm c2}$ beyond the paramagnetic limiting field, in atomic-layer crystalline ($\sqrt{7}\times\sqrt{3}$)-In on a Si(111) substrate. The present article elucidates the origin of the observed field-resilient noncentrosymmetric superconductivity in this highly crystalline two-dimensional material. We develop the quasiclassical theory of superconductivity by incorporating the Fermi surface anisotropy together with an anisotropic spin splitting and texture specific to atomic-layer crystalline systems. In Si(111)-($\sqrt{7}\times\sqrt{3}$)-In, a typical material with a large antisymmetric spin-orbit coupling (ASOC), we show an example where the combination of the ASOC and disorder effect suppresses the paramagnetic depairing and can lead to an enhancement of $H^{||}_{\rm c2}$ compared to an isotropic system only when a magnetic field is applied in a particular direction due to an anisotropic spin texture. We also study the parity-mixing effect to demonstrate that the enhancement of $H^{||}_{\rm c2}$ is limited in the moderately clean regime because of the fragile $s$+$p$-wave pairing against nonmagnetic scattering in the case of the dominant odd-parity component of a pair wavefunction. Furthermore, from analysis of the transition line, we identify the field-resilience factor taking account of the scattering and suppression of paramagnetic effects and discuss the origin of the field-resilient superconductivity. Through fitting of the $H^{||}_{\rm c2}$ data, the normal-state electron scattering is discussed with a prime focus on the role of atomic steps on a Si(111) surface.