Two-dimensional nature of superconductivity in the intercalated layered systems ${\mathrm{Li}}_x\mathrm{HfNCl}$ and ${\mathrm{Li}}_x\mathrm{ZrNCl}:$ Muon spin relaxation and magnetization measurements

We report muon spin relaxation $\left(\mu \mathrm{SR}\right)$ and magnetization measurements, together with synthesis and characterization, of the Li-intercalated layered superconductors ${\mathrm{Li}}_x\mathrm{HfNCl}$ and ${\mathrm{Li}}_x\mathrm{ZrNCl}$ with/without cointercalation of THF (tetrahydrofuran) or propylene carbonate. The three-dimensional superfluid density $n_s{/m}^{*}$ (superconducting carrier density/effective mass) as well as the two-dimensional superfluid density $n_{s2\mathrm{}\mathrm{D}}{/m}_{\mathrm{ab}}^{*}$ [two-dimensional (2D) area density of superconducting carriers/ab-plane effective mass] have been derived from the $\mu \mathrm{SR}$ results of the magnetic-field penetration depth ${\lambda }_{\mathrm{ab}}$ observed with external magnetic field applied perpendicular to the 2D honeycomb layer of HfN/ZrN. In a plot of $T_c$ versus $n_{s2\mathrm{}\mathrm{D}}{/m}_{\mathrm{ab}}^{*},$ most of the results lie close to the linear relationship found for underdoped high-$T_c$ cuprate (HTSC) and layered organic BEDT (bis(ethylenedithio)) superconductors. In ${\mathrm{Li}}_x\mathrm{ZrNCl}$ without THF intercalation, the superfluid density and $T_c$ for $x=0.17\mathrm{}$ and 0.4 do not show much difference, reminiscent of $\mu \mathrm{SR}$ results for some overdoped HTSC systems. Together with the absence of dependence of $T_c$ on average interlayer distance among ZrN/HfN layers, these results suggest that the 2D superfluid density $n_{s2\mathrm{}\mathrm{D}}{/m}_{\mathrm{ab}}^{*}$ is a dominant determining factor for $T_c$ in the intercalated nitride-chloride systems. We also report $\mu \mathrm{SR}$ and magnetization results on depinning of flux vortices, and the magnetization results for the upper critical field $H_{c2\mathrm{}}$ and the penetration depth $\lambda .$ A reasonable agreement was obtained between $\mu \mathrm{SR}$ and magnetization estimates of $\lambda .$ We discuss the two-dimensional nature of superconductivity in the nitride-chloride systems based on these results.