Momentum-dependent multiple gaps of MgB$_{2}$ probed by electron tunneling spectroscopy on MgB$_{2}$/native oxide/Pb junctions*

Distinct multi-band superconductivity is a unique feature that distinguishes MgB$_{2}$ from all other phonon-mediated Bardeen-Cooper-Schrieffer (BCS) superconductors. According to a first-principles calculation employing an anisotropic Eliashberg formalism [Choi \textit{et al.} Nature 418, 758 (2002)], there is a distribution of superconducting energy gap values on the Fermi surface of MgB$_{2}$ (two $\sigma $ bands and two $\pi $ bands). However, only two distinct gaps have been observed experimentally, leading to the suggestion that consideration of the fully anisotropic electron-phonon interaction may not be necessary for real MgB$_{2}$ samples. Here, we present an electron tunneling spectroscopy study on MgB$_{2}$/native oxide/Pb junctions that clearly shows the distribution of energy gaps. By deconvoluting the tunneling spectrum based on the density of state of Pb, we derive the momentum-dependent energy gaps of MgB$_{2}$, which are in good agreement with the anisotropic Eliashberg calculation. The result affirms the importance of the anisotropic electron-phonon interaction in MgB$_{2}$ as well as its important impact on device applications.