Study of pressure effect on the magnetic penetration depth in MgB$_2$

A study of the pressure effect on the magnetic penetration depth $\lambda$ in polycrystalline MgB$_{2}$ was performed by measuring the temperature dependence of the magnetization under an applied pressure of 0.15 and 1.13 GPa. We found that $\lambda^{-2}$ at low temperature is only slightly affected by pressure [$\frac{\Delta \lambda^{-2}}{\lambda^{-2}} = 1.5(9)$], in contrast to cuprate superconductors, where, in the same range of pressure, a very large effect on $\lambda^{-2}$ was found. Theoretical estimates indicate that most of the pressure effect on $\lambda^{-2}$ in MgB$_2$ arises from the electron-phonon interaction.