Two dimensional nature of superconductivity in intercalated layered systems Li$_{x}$HfNCl and Li$_{x}$ZrNCl: muon spin relaxation and magnetization measurements

We report muon spin relaxation ($\mu$SR) and magnetization measurements, together with synthesis and characterization, of the Li-intercalated layered superconductors Li$_{x}$HfNCl and Li$_{x}$ZrNCl with/without co-intercalation of THF (tetrahydrofuran) or PC (propylene carbonate). The 3-dimensional (3-d) superfluid density $n_{s}/m^{*}$ (superconducting carrier density / effective mass), as well as the two dimensional superfluid density $n_{s2d}/m^{*}_{ab}$ (2-dimensional (2-d) area density of superconducting carriers / ab-plane effective mass), have been derived from the $\mu$SR results of the magnetic-field penetration depth $\lambda_{ab}$ observed with external magnetic field applied perpendicular to the 2-d honeycomb layer of HfN / ZrN. In a plot of $T_{c}$ versus $n_{s2d}/m^{*}_{ab}$, most of the results lie close to the linear relationship found for underdoped high-$T_{c}$ cuprate (HTSC) and layered organic BEDT superconductors. In Li$_{x}$ZrNCl without THF intercalation, the superfluid density and $T_{c}$ for $x$ = 0.17 and 0.4 do not show much difference, reminiscent of $\mu$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 2-d superfluid density $n_{s2d}/m^{*}_{ab}$ is a dominant determining factor for $T_{c}$ in the intercalated nitride-chloride systems. We also report $\mu$SR and magnetization results on depinning of flux vortices, and the magnetization results for the upper critical field $H_{c2}$ and the penetration depth $\lambda$. Reasonable agreement was obtained between $\mu$SR and magnetization estimates of $\lambda$. We discuss the two dimensional nature of superconductivity in the nitride-chloride systems based on these results.