Extraction of the electron-phonon interaction from tunneling data in the multigap superconductor MgB2

The direct inversion of the Eliashberg equations (EE) in case of a multiband superconductor is a mathematically ill-defined problem, because it is not possible to obtain several band splitted electron-phonon spectral functions alpha2Fij(omega) from a single function of the tunnel current. In the present work we follow another direction and calculate the tunneling density of states (DOS) of MgB2 for different tunneling directions by directly solving the two-band EE in the real-axis formulation. This procedure reveals the fine structures of the DOS due to the optical phonons. Then we show that the numeric inversion of the standard single-band EE, when applied to the two-band DOS of MgB2, underestimates the strength of certain phonon branches (e.g., the E2g) in the extracted alpha2F(omega). The fine structures produced by the two-band interaction at energies between 20 and 100 meV turn out to be clearly observable only for tunneling along the ab planes and at very low temperature. Only in this case it is possible to extract some information on the sigma-band contribution to the spectral functions. For any other tunneling direction, the pi-band contribution is dominant and almost coincides with the whole alpha2F(omega) for tunneling along the c axis. Our results are compared with recent experimental tunneling and point-contact data.