Type-II Ising Superconductivity in Two-Dimensional Materials with Spin-Orbit Coupling

Centrosymmetric materials with spin-degenerate bands are generally considered to be trivial for spintronics and related physics. In two-dimensional (2D) materials with multiple degenerate orbitals, we find that the spin-orbit coupling can induce spin-orbital locking, generate out-of-plane Zeeman-like fields displaying opposite signs for opposing orbitals, and create novel electronic states insensitive to the in-plane magnetic field, which thus enables a new type of Ising superconductivity applicable to centrosymmetric materials. Many candidate materials are identified by high-throughput first-principles calculations. Our work enriches the physics and materials of Ising superconductivity, opening new opportunities for future research of 2D materials.

Figure 1

Schematic picture of Ising superconductivity. (a) Monolayer ${\mathrm{MoS}}_2$ with type-I Ising superconductivity sketched in (b). Near $K$ and $K^{\prime }$, spin-up (red) and spin-down (blue) split in energy by a Zeeman-like field, which is opposite for the two valleys. (c) Monolayer SnH with type II-Ising superconductivity sketched in (d). Near $\mathrm{\Gamma }$, spin-up and spin-down are locked to the two opposing orbitals (denoted by solid and dashed lines, respectively). They spit in energy for each orbital by a Zeeman-like field, whose sign is opposite for opposing orbitals. The dashed and solid lines are degenerate in energy.

Figure 2

(a) Band structure of SnH with SOC. Valence bands labeled by a red (blue) dot are mainly contributed by $|+\uparrow ⟩$ and $|-\downarrow ⟩$ ($|+\downarrow ⟩$ and $|-\uparrow ⟩$) orbitals near $\mathrm{\Gamma }$. Inset: Brillouin zone of SnH. (b) Schematic plot of Zeeman splitting ($\delta {\epsilon }_{+}$) between ${\epsilon }_{++}$ and ${\epsilon }_{+-}$ as a function of $k$. (c) The ratio between Zeeman splitting without SOC $2Z$ ($Z=0.1\mathrm{meV}$ in this picture) and the actual Zeeman splitting $\delta \epsilon$ as a function of $k$ in the $\mathrm{\Gamma }-M$ direction for SnH. $\delta \epsilon$ is roughly independent of the polar angle of $\mathit{k}$. Inset: Zeeman splitting $\delta {\epsilon }_{+}$ (red solid line, magnified by 100) and $\delta {\epsilon }_{-}$ (blue dashed line) at $\mathrm{\Gamma }$ ($\mathit{k}=\mathbf{0}$) as a function of Zeeman field $Z$.

Figure 3

(a) The relation between critical field $B_{c2}$, normalized by Pauli limit $B_p$, with respect to temperature $T$, normalized by critical temperature $T_c$ for $k_F=0.02/0.025/0.03/0.035{\AA }^{-1}$. (b) The relation between the critical field with respect to temperature for out-of-plane electric field $E=0.01/0.02/0.03\mathrm{V}/\AA$ at $k_F=0.025{\AA }^{-1}$. Inset: $\delta {\epsilon }_T$ (see main text for definition) for $E=0$ (blue) and $E=0.03\mathrm{V}/\AA$ (red) under 1 meV Zeeman field at $k_F=0.025{\AA }^{-1}$ for varying polar angles of $k_F$. Parameters $\hslash {\omega }_D=0.01\mathrm{eV}$ [39] and $gV\nu =0.205$ are used, giving $T_c\sim 1\mathit{k}$ [27].

Figure 4

Candidate materials of type-II Ising superconductors. Blue markers indicate the relevant bands cross the intrinsic Fermi energy, and green (red) markers indicate the relevant bands are below (above) the intrinsic Fermi energy. Solid (hollow) markers indicate inversion symmetric (asymmetric) materials. Triangular, rectangular, and rhombic markers correspond to $C_3$, $C_4$, $S_4$ symmetric materials. Type-II Ising superconductivity is prominent for $k_F<\overline{k}$. The rightmost dot (${\mathrm{ZrCl}}_2$) features the coexistence of type-I and type-II Ising superconductivity with $2Z/\delta {\epsilon }_T$ always above 10 in our simulations. Therefore, $\overline{k}$ is ill defined for this material. The top axis denotes an estimation of carrier density contributed by this pocket near $\mathrm{\Gamma }$ with $k_F=\overline{k}$.