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The superconducting properties of the recently discovered double Fe$_2$As$_2$layered high-$T_c$ superconductor RbCa$_2$Fe$_4$As$_4$F$_2$ with $T_c\approx$30~K have been investigated using magnetization, heat capacity,transverse-field (TF) and zero-field (ZF) muon-spin rotation/relaxation($\mu$SR) measurements. Our low field magnetization measurements and heatcapacity (C$_p$) reveal an onset of bulk superconductivity with $T_{\bf c}\sim$30.0(4) K. Furthermore, the heat capacity exhibits a jump at $T_{\bf c}$ of$\Delta$C$_p$/$T_{\bf c}$=94.6 (mJ/mole-K$^2$) and no clear effect of appliedmagnetic fields was observed on C$_p$(T) up to 9 T between 2 K and 5 K. Ouranalysis of the TF-$\mu$SR results shows that the temperature dependence of themagnetic penetration depth is better described by a two-gap model, eitherisotropic $s$+$s$-wave or $s$+$d$-wave than a single gap isotropic $s$-wave or$d$-wave model for the superconducting gap. The presence of two superconductinggaps in RbCa$_2$Fe$_4$As$_4$F$_2$ suggests a multiband nature of thesuperconductivity, which is consistent with the multigap superconductivityobserved in other Fe-based superconductors, including ACa$_2$Fe$_4$As$_4$F$_2$(A=K and Cs). Furthermore, from our TF-$\mu$SR study we have estimated anin-plane penetration depth $\lambda_{\mathrm{ab}}$$(0)$ =231.5(3) nm,superconducting carrier density $n_s = 7.45 \times 10^{26}~ $m$^{-3}$, andcarrier's effective-mass $m^*$ = 2.45\textit{m}$_{e}$. Our ZF $\mu$SRmeasurements do not reveal a clear sign of time reversal symmetry breaking at$T_{\bf c}$, but the temperature dependent relaxation between 150 K and 1.2 Kmight indicate the presence of spin-fluctuations. The results of our presentstudy have been compared with those reported for other Fe pnictidesuperconductors.

Multigap superconductivity in RbCa2Fe4As4F2 investigated using muSR measurements