Muon spin relaxation study of the magnetic penetration depth in ${\mathrm{MgB}}_2$

The magnetic vortex lattice (VL) of polycrystalline ${\mathrm{MgB}}_2$ has been investigated by transverse-field muon spin relaxation (TF-μSR). The evolution of the TF-μSR depolarization rate σ, which is proportional to the second moment of the field distribution of the VL, has been studied as a function of temperature and applied magnetic field. The low-temperature value σ exhibits a pronounced peak near $H_{\mathrm{ext}}=75\mathrm{}\mathrm{mT}.$ This behavior is characteristic of strong-pinning-induced distortions of the VL which put into question the interpretation of the low-field TF-μSR data in terms of the magnetic penetration depth $\lambda \mathrm{}\left(T\right).$ An approximately constant value of σ, such as expected for an ideal VL in the London limit, is observed at higher fields of $H_{\mathrm{ext}}>0.4\mathrm{}\mathrm{T}.$ The TF-μSR data at $H_{\mathrm{ext}}=0.6\mathrm{}\mathrm{T}$ are analyzed in terms of a two-gap model. We obtain values for the gap size of ${\Delta }_1=6.0\mathrm{}\left(3\right)\mathrm{}\mathrm{meV}$ $[2{\Delta }_1{/k}_B{T}_c=3.6\mathrm{}(2\left)\mathrm{}\right],$ ${\Delta }_2=2.6\mathrm{}\left(2\right)\mathrm{}\mathrm{meV}$ $[2{\Delta }_2{/k}_B{T}_c=1.6\mathrm{}(1\left)\mathrm{}\right],$ a comparable spectral weight of the two bands, and a zero-temperature value for the magnetic penetration depth of ${\lambda }_{\mathrm{ab}}\approx 100\mathrm{nm}.$ In addition, we performed $\mu \mathrm{SR}$ measurements in zero external field. We obtain evidence that the muon site (at low temperature) is located on a ring surrounding the center of the boron hexagon. Muon diffusion sets in already at rather low temperature of $T>\mathrm{}10\mathrm{}\mathrm{K}.$ The nuclear magnetic moments can account for the observed relaxation rate and no evidence for electronic magnetic moments has been obtained.