Oxygen isotope effect on the superfluid density within the $d-$wave and $s-$wave pairing channels of YBa$_2$Cu$_4$O$_8$
We report on measurements of the oxygen isotope ($^{16}$O/$^{18}$O) effect (OIE) on the transition temperature $T_{\rm c}$ and the zero-temperature in-plane magnetic penetration depth $λ_{\rm ab}(0)$ in the stoichiometric cuprate superconductor YBa$_2$Cu$_4$O$_8$ by means of muon-spin rotation/relaxation. An analysis of the temperature evolution of $λ^{-2}_{\rm ab}$ in terms of coexisting $s+d-$wave order parameters reveals that the OIE on the superfluid density $ρ_{\rm s}(0)\proptoλ^{-2}_{\rm ab}(0)$ stems predominantly from the $d-$wave component while the contribution of the $s-$wave one is almost zero. The OIE on the transition temperature $T_{\rm c}$ is found to be rather small: $δT_{\rm c}/T_{\rm c}= -0.32(7)$%, compared to the total OIE on the superfluid density $ρ_{\rm s}(0)$: $δρ_{\rm s}(0)/ρ_{\rm s}(0)= -2.8(1.0)$%. The partial OIE's on the corresponding $d-$wave and $s-$wave components of $ρ_{\rm s}(0)$ are $δρ_{\rm s,d}(0)/ρ_{\rm s}(0)= -3.0(1.2)$%, and $δρ_{\rm s,s}(0)/ρ_{\rm s}(0)= 0.2(1.2)$%, respectively. Our results demonstrate that polaron formation in the CuO$_2$ planes is the origin of the observed OIE in the $d-$wave channel. In the much weaker $s-$wave channel, fermionic quasiparticles are present, which do not contribute to the OIE on $ρ_{\rm s}(0)$. Our results support the original idea of K. Alex Müller on the polaronic nature of the supercarries in high-temperature cuprate superconductors.