Enhancement of the Upper Critical Field in Disordered Transition Metal Dichalcogenide Monolayers

We calculate the effect of impurities on the superconducting phase diagram of transition metal dichalcogenide monolayers in the presence of an in-plane magnetic field. Because of strong intrinsic spin-orbit coupling, the upper critical field greatly surpasses the Pauli limit at low temperatures. We find that it is insensitive to intravalley scattering and, ultimately, limited by intervalley scattering.

Figure 1

Schematic representation of the conduction band of TMDC monolayers in the vicinity of the corners of the Brillouin zone as described by Eq. (2). The spin splitting in the two valleys is opposite due to the so-called Ising SOC.

Figure 2

Diagrammatic representation of the Dyson equation and the self-energy in the self-consistent Born approximation. The thin arrow represents the “bare”, disorder-free Green’s function ${\mathcal{G}}_{\eta }^{\pm }$, while the thick arrow is the disorder-averaged Green’s function $⟨{\mathcal{G}}_{\eta }^{\pm }⟩$. The dashed and dot-dashed impurity lines represent intra- and intervalley scattering events, respectively.

Figure 3

(a) Diagrammatic representation of the disorder-averaged self-consistency condition given by Eq. (4). (b) Bethe-Salpeter equation for the renormalized vertex functions ${\mathrm{\Pi }}_{\eta }^{s{s}^{\prime }}$. For the definition of diagram elements, see Fig. 2. We use the abbreviation $\overline{s}=-s$.

Figure 4

Upper critical field as a function of temperature in the absence of intervalley scattering for different values of Ising SOC, as described by Eq. (8). The plot on the right shows the same result but with a different scale for the $x$ axis to illustrate the logarithmic divergence at low temperature when ${\mathrm{\Delta }}_{so}\ne 0$.

Figure 5

Upper critical field as a function of the temperature for various strengths of Ising SOC and intervalley scattering: (a) ${\mathrm{\Delta }}_{so}/T_c=0.3$, (b) ${\mathrm{\Delta }}_{so}/T_c=3$, and (c) ${\mathrm{\Delta }}_{so}/T_c=12$. The choice of parameters ${\mathrm{\Delta }}_{so}/T_c=12$ and $1/({\tau }_2{T}_c)=1.5$ [dashed line in (c)] gives a good fit of the experimental data from Ref. [12] taking the $g$ factor to be $g=2$.