Anisotropies of the lower and upper critical fields in MgB$_2$ single crystals

Abstract

The temperature dependence of the London penetration depth ($\lambda$) and coherence length ($\xi$) has been deduced from Hall probe magnetization measurements in high quality MgB$_2$ single crystals in the two main crystallographic directions. We show that, in contrast to conventional superconductors, MgB$_2$ is characterized by two different anisotropy parameters ($\Gamma_\lambda = \lambda_c/\lambda_{ab}$ and $\Gamma_\xi = \xi_{ab}/\xi_c$) which strongly differ at low temperature and merge at $T_c $. These results are in very good agreement with recent calculations in weakly coupled two bands suprerconductors (Phys. Rev. B, 66, 020509(R) (2002).