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Self-consistent two-gap approach in studying multi-band superconductivity in NdFeAsO$_{0.65}$F$_{0.35}$

High-quality single crystals of NdFeAsO$_{0.65}$F$_{0.35}$ (the transition temperature $T_{\rm c} \simeq 30.6$~K) were studied in zero-field (ZF) and transverse-field (TF) muon-spin rotation/relaxation ($μ$SR) experiments. An upturn in muon-spin depolarization rate at $T\lesssim 3$~K was observed in ZF-$μ$SR measurements and it was associated with the onset of ordering of Nd electronic moments. Measurements of the magnetic field penetration depth ($λ$) were performed in the TF geometry. By applying the external magnetic field $B_{\rm ex}$ parallel to the crystallographic $c$-axis ($B_{\rm ex}\| c$) and parallel to the $ab$-plane ($B_{\rm ex}\| ab$), the temperature dependencies of the in-plane component ($λ_{ab}^{-2}$) and the combination of the in-plane and the out of plane components ($λ_{ab,c}^{-2}$) of the superfluid density were determined, respectively. The out-of-plane superfluid density component ($λ_{c}^{-2}$) was further obtained by combining the results of $B_{\rm ex} \| c$ and $B_{\rm ex} \| {ab}$ set of experiments. The temperature dependencies of $λ_{ab}^{-2}$, $λ_{ab,c}^{-2}$, and $λ_{c}^{-2}$ were analyzed within the framework of a self-consistent two-gap model despite of using the traditional $α$-model. Interband coupling was taken into account, instead of assuming it to be zero as it stated in the $α$-model. A relatively small value of the interband coupling constant $Λ_{12} \simeq 0.01$ was obtained, thus indicating that the energy bands in NdFeAsO$_{0.65}$F$_{0.35}$ are only weakly coupled. In spite of their small magnitude, the coupling between the bands leads to the single value of the superconducting transition temperature $T_{\rm c}$.