Straintronic effect for superconductivity enhancement in Li-intercalated bilayer MoS2

In this study, ab initio calculations were performed to show that the superconductivity in Li-intercalated bilayer MoS2 could be enhanced by applying either compressive or tensile strain. Moreover, the mechanism for superconductivity enhancement for the tensile strain case was found to be different than that of the compressive strain case. Enhanced electron phonon coupling (EPC) under tensile strain could be explained by an increase in the nesting function involved with the change in the Fermi surface topology in a wide range of Brillouin zones. The superconducting transition temperature Tc of 0.46 K at zero strain increased up to 9.12 K under a 6.0% tensile strain. Meanwhile, the enhancement in compressive strain was attributed to the increase in intrinsic electron phonon matrix elements. Furthermore, the contribution from interband scattering was large, which suggested the importance of electron pockets on the Fermi surface. Finally, 80% of the total EPC (λ = 0.98) originated from these pockets and the estimated Tc was 13.50 K.


Evolution of band structure under strain.
Fig. S1: Evolution of band structure at maximum Tc condition, i.e. strain +6.0% (top), 0.0% (middle), -5.5% (bottom). Band structures along high symmetry path (Γ − M − K − Γ) were colored with contribution from Mo orbitals separated into degenerated pair !" + ! ! #" ! (left), "$ + $! (middle) and $ ! (right). The Fermi level was set to zero.    In practical estimation of the superconducting transition temperature based on the phonon linewidth defined in equation (8), the number of k-points, q-points and broadening width ( ), which is introduced in approximating delta functions in equation (8) by gaussians, are important parameters to determine the accuracy.

Evolution of soft mode under compressive strain
Thus, we performed the convergence test on these three parameters. For , a smaller value is better in principle because the gaussian functions are introduced to approximate delta functions. In practice, however, the balance between number of k-points and the value of should be considered in determining the value of . Figure S4 shows the dependence of the density of states (DOS) at the Fermi level, EPC constant (λ) and transition temperature (Tc) on the value of and size of k, q-points. We did the same manner in Quantum espresso (reasonable size of k-mesh) and EPW (dense k-mesh from Wannier interpolation) and here we show only the result from dense k-mesh under -5.5 % strain condition. As we can see in this figure, each value depends on number of k-points and significantly especially when is small. Similarly, when number of k-points is small, especially k = 24 × 24 × 1, the convergence respect to is slow and it does not converge into same value as the other size of k-mesh even at large . Then we concluded that k = 24 × 24 × 1 is not appropriate.
We found that 96 × 96 × 1k-mesh and 12 × 12 × 1 q-mesh with = 0.136 eV (= 0.01 Ry) is sufficient in Quantum espresso balanced with reasonable computational afford. In EPW level, a greater number of k, q points from Wannier interpolation, we use dense 120 × 120 × 1 for both k and q with = 0.05 eV to guarantee the convergence. k120 2 ,q12 2 k120 2 ,q24 2 k120 2 ,q48 2 k120 2 ,q72 2 k120 2 ,q96 2 k120 2 ,q120 2 In order to verify superconducting Tc from McMillan Allen-Dynes formula, we also calculated temperature dependence of superconducting gap from anisotropic Eliashberg function using EPW. The convergence test was done on several combination of number of k-and q-point. We found that k-and q-point dependence is small, and it converged rapidly. In this study we used k = 48 2 and q = 48 2 for superconducting calculation and the Tc was estimated to 15 K for -5.5% compressive strain case.  (9). Top subfigure is zero strain dispersion, middle subfigure is -5.5 % strain dispersion and bottom subfigure is result from equation (9) Phonon frequency, in term of a result from the singularity of Dyson equation, is expressed by

Temperature dependence of superconducting gap from anisotropic Eliashberg function
where ω %& (4) is the phonon frequency of bare ionic system without EP interaction, 6 is the frequency renormalized by including EP interaction and Re[Π( )] is real part of phonon self-energy. The factor 9 %& 9 * Re[Π( ν)] plays important role in determining phonon frequency. Here we visualized the factor into corresponded wave vector and branch in phonon dispersion. We can see from the left figure, the softening due to the large EP matrix element occur at 3 different points in two lowest phonon modes. And the most conspicuous one is the softest mode at = 0.76ΓK .... which contributes largest to superconductivity. We also calculated phonon renormalization from equation (9) by replacing frequency of bare ionic system ( 6 (4) ) with zero strain dispersion. We already confirmed that EP interaction is significantly small at particular modes in zero strain dispersion ( Figure S6: right top). There is discrepancy between renormalized curve and DFPT result for we did not consider strain effect to phonon frequency in phonon renormalization calculation here. The result, however, was in good tendency with DFPT curve.
EP matrix element is labeled with 5 parameters, i.e. phonon wave vector and branch ν, electronic wave vector at band index for initial state and for final state. Decomposed EPC constant on each transition process can be obtained by constraining band index , in summation over all these parameters. Figure S7 shows electron phonon coupling constants on corresponding wave vector colored by the strength of EPC constant λ %& . The same result for all Brillouin Zone was shown in the paper. Band index 2 contributes largely to EPC. EP interaction involved by soft mode = 0.76ΓK .... occurs through only interband scattering process. Thus, the appearance of band index 2 on the Fermi surface is important for EPC enhancement under compressive strain induction.