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

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 (${\mathrm{\Gamma }}_{\lambda }={\lambda }_c\mathrm{/}{\lambda }_{ab}$ and $\Gamma_\xi = \xi_{ab}/\xi_c$)whichstronglydifferatlowtemperatureandmergeat$T_c $. These results are in very good agreement with recent calculations in weakly coupled two bands suprerconductors (Phys. Rev. B, 66, 020509® (2002).