Magnetic Field-Induced “Mirage” Gap in an Ising Superconductor

Superconductivity is commonly destroyed by a magnetic field due to orbital or Zeeman-induced pair breaking. Surprisingly, the spin-valley locking in a two-dimensional superconductor with spin-orbit interaction makes the superconducting state resilient to large magnetic fields. We investigate the spectral properties of such an Ising superconductor in a magnetic field taking into account disorder. The interplay of the in-plane magnetic field and the Ising spin-orbit coupling leads to noncollinear effective fields. We find that the emerging singlet and triplet pairing correlations manifest themselves in the occurrence of “mirage” gaps: at (high) energies of the order of the spin-orbit coupling strength, a gaplike structure in the spectrum emerges that mirrors the main superconducting gap. We show that these mirage gaps are signatures of the equal-spin triplet finite-energy pairing correlations and due to their odd parity are sensitive to intervalley scattering.

Figure 1

(a) Schematic band structure in the normal state. The electrons near the $K$ and $K^{\prime }$ valleys are subject to the ISOC ${\beta }_{\mathrm{so}}$, which pins the electron spins to the out-of-plane direction (dashed arrows), and an in-plane magnetic field $B_x$. For finite $B_x$, the spin directions are reoriented (solid arrows). (b) Quasiparticle energy spectrum of Eq. (1) near the Fermi momentum $p_F$ with ${\beta }_{\mathrm{so}}=7\mathrm{\Delta }$, $B_x=2\mathrm{\Delta }$, and $\mu =150\mathrm{\Delta }$. The mirage gaps $\delta$ are shifted images of the main superconducting gap. (c) Density of states $N_s$ for different $B_x$ in the clean limit. All lines for finite $B_x$ have been offset for better visibility. Here, ${\beta }_{\mathrm{so}}=7{\mathrm{\Delta }}_0$ and $T=0.1T_{c0}$, where ${\mathrm{\Delta }}_0$ and $T_{c0}$ are, respectively, the zero-temperature gap and transition temperature in the absence of a magnetic field.

Figure 2

Effects of intervalley scattering. Various components of the Green’s function $\widehat{g}$ at different intervalley scattering strengths (from bottom to top: $\mathrm{\Gamma }=0$, $0.2{\mathrm{\Delta }}_0$, $0.4{\mathrm{\Delta }}_0$, ${\mathrm{\Delta }}_0$, $2{\mathrm{\Delta }}_0$) in valley $K$ with ${\beta }_{\mathrm{so}}=7{\mathrm{\Delta }}_0$, $B_x=2{\mathrm{\Delta }}_0$, and $T=0.1T_{c0}$. All curves for finite $\mathrm{\Gamma }$ have been offset for better visibility. For valley $K^{\prime }$, the sign of $f_y$ reverses, while the signs of $g_0$, $f_0$, and $f_x$ remain unchanged.