Magnetic-field induced $s+\mathit{if}$ pairing in Ising superconductors

We show that an in-plane Zeeman field applied to noncentrosymmetric Ising superconductors converts singlet $s$-wave Cooper pairs to equal-spin triplet $if$ pairs. Singlets are converted to triplets as spins originally polarized by spin-orbit interaction are reoriented by the Zeeman field. We demonstrate that the pairing in the triplet $f$ channel(s) decisively affects the critical field. While attraction enhances the critical field, repulsion suppresses it. The singlet to triplet conversion phenomenon has a geometrical origin, and the discussion applies universally to Ising superconductors with generic band structure such as, e.g., monolayer transition metal dichalcogenides.

Figure 1

(a) $B_{\mathrm{c}}\left(T\right)$ obtained from the pair breaking equation ${\alpha }_s(T,B){\alpha }_t(T,B)={\alpha }_{st}^2\left(B\right)$. The brown and the gray dotted curves show repulsion in the $E^{\prime \prime }$ channel, with the dimensionless couplings ${\lambda }_{A_1^{\prime }}=N_0{v}_{s,A_1^{\prime }}$ and ${\lambda }_{E^{\prime \prime }}=N_0{v}_{t,E^{\prime \prime }}$. (b) GL solution for ${\psi }_0$ with $T_{\mathrm{ct}}/T_{\mathrm{cs}}=0.1$. The nodal transition line is determined by the condition $B={\psi }_0$. The black, red, and blue curves are the same as in (a). The blue and purple transitions deviate far from $T_{\mathrm{cs}}$, where the GL solution fails, but still provides a qualitative description. (c),(d) Plots of ${\psi }_0\left(B\right), {\eta }_y\left(B\right)$, and ${\eta }_z\left(B\right)$ at two different temperatures. ${\eta }_z$ is favorable at high magnetic fields below its own critical temperature, in this case $T_{\mathrm{ctz}}=T_{\mathrm{cs}}/2$ for illustrative purposes.

Figure 2

(a) Illustration of the Brillouin zone, showing the sign modulation of SOC and parallelepipeds $\mathbf{B}\times {\mathit{\gamma }}_{\mathbf{k}}·i{\mathbf{d}}_{\mathbf{k}}$. (b) Schematic of the Cooper pairs nested on the spin-split Fermi surface. Cooper pairs are related by the symmetry ${\sigma }_h\mathcal{T}\nearrow ={\sigma }_h\swarrow =\searrow$ and are shown as same color arrows.