Nanoscale spatial non-homogeneity of 3D $\Delta _{\pi }$ in Mg$_{1-x}$AlB$_{2}$ single crystals

We have performed local I(V) and dI/dV(V) measurements on high quality Mg$_{1-x}$Al$_{x}$B$_{2}$ single crystals with different Al dopings by means of Scanning Tunneling Spectroscopy (STS). Both temperature and magnetic field dependences of the conductance spectra have been studied both in S-I-N and in S-I-S configurations. The measured spectra revealed only one gap at low temperatures as expected for pure c-axis tunneling. In the case of x=0.1, the measured superconducting gap $\Delta _{\pi {\rm g}}$ appeared highly non-homogeneous in its spatial distribution on nanometer scale, with an amplitude varying between 1.5 meV and 2.3 meV. In pure MgB$_ {2}$ the evolution of the zero bias conductance (ZBC) in magnetic field clearly showed two different regimes with a fast rise up to 1 T, followed by a slower refilling of states for higher fields. This effect results smoothed in doped single crystals. Consistently with the more recent theories, in comparison with the undoped material, we have found that for x=0.1 the $\Delta _{\pi }$ gap value and the H$_{c2}$ critical field increase of about 15{\%} and 30{\%}, respectively.