Exponential Temperature Dependence of the Penetration Depth in Single Crystal ${\mathrm{MgB}}_2$

The temperature dependence of the London penetration depth, $\lambda \left(T\right)$, was measured in both single crystal and polycrystalline ${\mathrm{MgB}}_2$ samples by a high-resolution, radio frequency technique. A clear exponential temperature dependence of $\lambda \left(T\right)$ was observed at low temperature, indicating $s$-wave pairing. A BCS fit to the lowest temperature data gives an in-plane energy gap $\Delta$ of $30\pm 2\mathrm{K}$ $(2\Delta {/T}_c=1.5\mathrm{}\pm 0.1)$, which is significantly smaller than the standard BCS weak coupling value of 3.5. We find that the data are best described by a two-gap model.