Two regimes in the magnetic field response of superconducting MgB$_2$

Using Scanning Tunneling Microscope at low temperature we explore the superconducting phase diagram in the $pi$-band of the two-band superconductor MgB$_2$. In this band the peculiar shape of the local tunneling spectra and their dynamics in the magnetic field reveal the complex character of the quasiparticle density of states (DOS). The gap in the DOS is first rapidly filled with states in raising the magnetic field up to 0.5 T and then slowly approaches the normal state value : The gap is observed up to 2 T. Such a change in the DOS dynamics suggests the existence of two terms in the DOS of the $pi$-band: a first one, reflecting an intrinsic superconductivity in the band and a second one, originating from an inter-band coupling to the $sigma$-band. Our findings allow a deeper understanding of the unique phase diagram of MgB$_2$.