Probing type-II Ising pairing using the spin-mixing parameter

The immunity of Ising superconductors to external magnetic fields originates from a spin locking of the paired electrons to an intrinsic Zeeman-like field. The spin-momentum locking in non-centrosymmetric crystalline materials leads to type-I Ising pairing in which the direction of the intrinsic field can be deduced from the spin expectation values. Conversely, in centrosymmetric crystals the electron spins locked to the orbitals can form Ising type-II pairs consisting of spin-orbit split doublets. Due to time-reversal symmetry, the doublets are spin degenerate, making it difficult to read the spin polarization of bands and the direction of spin-orbit fields. Here we present an efficient approach to determine the direction of the intrinsic field using the spin-mixing parameter $b^2$. Using first-principles calculations based on the density functional theory, we study monolayer transition metal dichalcogenide superconductors ${\mathrm{PdTe}}_2, {\mathrm{NbTe}}_2$, and ${\mathrm{TiSe}}_2$ with the 1T structure. We calculate $b^2$ for individual Fermi pockets and provide a general picture of possible Ising type-II pairing within the full Brillouin zone. To complement our first-principles results, we use group theory to provide a detailed picture of spin-orbit coupling and spin mixing in the relevant bands forming Fermi pockets. We demonstrate that contrary to the anticipated effects of spin-orbit locking, not every spin-orbit split spin doublet actively participates in Ising pairing. Finally, by connecting the spin-mixing parameter $b^2$ with the intrinsic out-of-plane Zeeman field, we estimate the upper in-plane critical magnetic field.

Figure 1

Calculated orbital and spin-orbital properties of 1T-polytype monolayers of transition metal dichalcogenides. (a) Band pockets in energy window $\pm 50$ meV around the Fermi energy for ${\mathrm{PdTe}}_2$ in the Brillouin zone wedge. The colormap corresponds to $b_{\mathbf{k}}^2$ for SQA $s=z$. (b) orbital resolved band structures of ${\mathrm{PdTe}}_2$ with the indicated bands crossing the Fermi energy. Grey horizontal lines depict the $\pm 50$ meV energy window around the Fermi energy for which the average spin-mixing parameter $b^2$ was calculated. (c)–(e) Fermi contour averaged spin-mixing parameter $b^2$ calculated for the Fermi pockets formed by the bands $B_1$ and $B_2$ depicted in (a) for three spin quantization axes $s=\{x,y,z\}$ aligned with the Cartesian axes. Similarly, the middle and right row for ${\mathrm{NbTe}}_2$ and ${\mathrm{TiSe}}_2$.

Figure 2

Calculated properties for monolayer ${\mathrm{PdTe}}_2$ at 0 K as a function of the Fermi energy. (a) The Fermi contour averaged spin-orbital splitting ${\mathrm{\Delta }}_{\mathrm{so}}$, (b) polarization vector $P$, (c) intrinsic out-of-plane Zeeman field $B_{\perp }$, and (d) in-plane upper critical field $B_{c2,\parallel }$ in units of Pauli field $B_{\mathrm{P}}$.

Figure 3

The upper critical field $B_{c2,\left|\right|}$ versus temperature and the Fermi level for (a) ${\mathrm{PdTe}}_2$ and (b) ${\mathrm{TiSe}}_2$ for band ${\mathrm{B}}_2$ estimated using Eq. (14) from the Supplemental Material [42].