We investigate Ising superconductivity in two archetypal intrinsic superconductors, monolayer $1H-{\mathrm{TaS}}_2$ and $1H-{\mathrm{NbSe}}_2$, in a bottom-up approach. Using ab initio-based tight-binding parametrizations for the relevant low-energy $d$ bands, the screened interaction is evaluated microscopically in a scheme including Bloch overlaps. In direct space, the screened potential for both systems displays long-range Friedel oscillations alternating in sign. Upon scaling, the oscillation pattern becomes universal, with the location of minima and maxima locked to the lattice. Solving the momentum-resolved gap equations, a chiral ground state with $p$-like symmetry is generically found. Due to the larger Ising spin-orbit coupling, the chiral gap is more anisotropic in ${\mathrm{TaS}}_2$ than in ${\mathrm{NbSe}}_2$. This is reflected in tunneling spectra displaying V-shaped features for the former, in quantitative agreement with low-temperature scanning tunneling experiments on ${\mathrm{TaS}}_2$. At the same time, our results reconcile the apparent discordance with hard gap tunneling spectra reported for the sibling ${\mathrm{NbSe}}_2$.

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• Transition metal dichalcogenides
• Band structure methods
• Many-body techniques
• Methods in superconductivity
• Numerical techniques
• Scanning tunneling spectroscopy