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 $T_{\mathrm{c}}$ and the zero-temperature in-plane magnetic penetration depth ${\lambda }_{\mathrm{a}\mathrm{b}}(0)$ 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 ${\lambda }_{\mathrm{a}\mathrm{b}}^{-2}$ in terms of coexisting $s+d-$wave order parameters reveals that the OIE on the superfluid density ${\rho }_{\mathrm{s}}(0)\propto {\lambda }_{\mathrm{a}\mathrm{b}}^{-2}(0)$ stems predominantly from the $d-$wave component while the contribution of the $s-$wave one is almost zero. The OIE on the transition temperature $T_{\mathrm{c}}$ is found to be rather small: $\delta T_{\mathrm{c}}\mathrm{/}{T}_{\mathrm{c}}=-0.32(7)$%, compared to the total OIE on the superfluid density ${\rho }_{\mathrm{s}}(0)$: $\delta {\rho }_{\mathrm{s}}(0)\mathrm{/}{\rho }_{\mathrm{s}}(0)=-2.8(1.0)$%. The partial OIE’s on the corresponding $d-$wave and $s-$wave components of ${\rho }_{\mathrm{s}}(0)$ are $\delta {\rho }_{\mathrm{s},\mathrm{d}}(0)\mathrm{/}{\rho }_{\mathrm{s}}(0)=-3.0(1.2)$%, and $\delta {\rho }_{\mathrm{s},\mathrm{s}}(0)\mathrm{/}{\rho }_{\mathrm{s}}(0)=0.2(1.2)$%, 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 ${\rho }_{\mathrm{s}}(0)$. Our results support the original idea of K. Alex Müller on the polaronic nature of the supercarries in high-temperature cuprate superconductors.