We report on specific heat, high magnetic field transport and ac−susceptibility measurments on magnesium diboride single crystals. The upper critical field Hc2 for magnetic fields perpendicular and parallel to the Mg and B planes is presented for the first time in the entire temperature range. A very different temperature dependence has been observed in the two directions which yields to a temperature dependent anisotropy with Γ ∼ 5 at low temperatures and about 2 near Tc. A peak effect is observed in the susceptibility measurments for µ0H ∼2 T parallel to the c−axis and the critical current density presents a sharp maximum for H parallel to the ab−plane.

The temperature dependence of the upper (H(c2)) and lower (H(c1)) critical fields has been deduced from Hall probe magnetization measurements of high quality MgB2 single crystals along the two main crystallographic directions. We show that Gamma(H(c2))=H(c2 axially ab)/H(c2 axially c) and Gamma(H(c1))=H(c1 axially c)/H(c1 axially ab) differ significantly at low temperature (being approximately 5 and approximately 1, respectively) and have opposite temperature dependencies. We suggest that MgB2 can be described by a single field dependent anisotropy parameter gamma(H) (=lambda(c)/lambda(ab)=xi(ab)/xi(c)) that increases from Gamma(H(c1)) at low field to Gamma(H(c2)) at high field.

The superconducting phase diagram of MgB 2 was determined from magnetization, magnetotransport, and single-crystal specific-heat measurements. A zero-temperature in-plane coherence length of 9.4 nm is determined. The superconducting anisotropy ␥ increases from a value around 2 near T c to above 4.5 at 22 K. For Hʈc a pronounced peak effect in the critical current occurs at the upper critical field. Evidence for a surface superconducting state is presented for Hʈc that might account for the widespread in reported values of ␥.

The lower (Hc1) and upper (Hc2) critical fields of Mg1−xAlxB2 single crystals (for x = 0, 0.1 and 0.2) have been deduced from specific heat and local magnetization measurements, respectively. We show that Hc1 and Hc2 are both decreasing with increasing doping content. The corresponding anisotropy parameter ΓH c2 (0) = H Very similar values have also been obtained by point contact spectroscopy measurements. The evolution of those gaps with Al concentration suggests that both band filling and interband scattering effects are present.

We demonstrate direct evidence of possible surface superconductivity on small, well-shaped MgB2 single crystals. Transport measurements in the range H < 1.6H c c2 , where H c c2 is the bulk upper critical field for the c axis, show non-Ohmic and strongly angular dependent resistivity. Studies of the alignment of H with selected crystal surfaces, transport and specific heat measurements on the same crystal, and a physical sculpturing of the crystal surfaces using a focused ion beam all support the conclusion. Similar, albeit less pronounced results are obtained for fields in the basal plane.