Can one extract the electron-phonon-interaction from tunneling data in case of the multigap superconductor MgB$_2$?

Abstract

In the present work we calculate the tunneling density of states (DOS) of MgB% $_{2}$ for different tunneling directions by directly solving the two-band Eliashberg equations (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 \emph{single-band} EE (the only available method), when applied to the \emph{two-band} DOS of MgB$_{2}$, may lead to wrong estimates of the strength of certain phonon branches (e.g. the $E_{2g}$) in the extracted electron-phonon spectral function $α^{2}F(ω)$. 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, when the extracted $α^{2}F(ω)$ contains the combination $α^{2}F_{σσ}(ω)$\textbf{+}$α^{2}F_{σπ}(ω)$, together with a minor $α^{2}F_{ππ}(ω)$\textbf{+}$α^{2}F_{πσ} (ω)$ component. Only in this case it is possible to extract information on the $σ$-band contribution to the spectral functions. For any other tunneling direction, the $π$-band contribution (which does not determine the superconducting properties of MgB$_{2}$) is dominant and almost coincides with the whole $α^2F(ω)$ for tunneling along the c axis. Our results are compared with recent experimental tunneling and point-contact data.