Reflection of two-gap nature in penetration depth measurements of MgB${}_2$ film

The magnetic penetration depth, ${\lambda }^{-2}(T)$, in the basal plane of a magnesium diboride (MgB${}_2$) film was measured using a two-coil mutual inductance technique at 50 kHz. This film has $T_c\simeq 38$ K, $\Delta T_c \leq 1$K,and$\lambda(0)\sim 1500$\mathring{A}\mathrm{.}Atlowtemperatures,$\lambda^{-2}(T)$ shows a clear exponential temperature dependence, indicating s-wave superconducting order parameter symmetry. However, the data are not quantitatively well described by theory assuming a single gap. From the data fit by the full BCS calculation assuming a double gap, the values of the two distinct gaps were obtained: ${\mathrm{\Delta }}_S(0)=2.61\pm 0.41$ meV and ${\mathrm{\Delta }}_L(0)=6.50\pm 0.33$ meV. The contributions of the small and the large gaps to the total superfluid density at T=0 were estimated to be 21% and 79%, respectively. Finally, we consider the effect of gap anisotropy on the penetration depth measurements, and find that the anisotropy does not play a significant role in determining the temperature dependence of the penetration depth.