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

TL;DR

This study calculates the tunneling density of states in MgB₂ using two-band Eliashberg equations, revealing how tunneling direction affects the extraction of electron-phonon spectral functions and emphasizing the importance of considering multiband effects.

Contribution

It demonstrates that standard single-band inversion methods can misestimate phonon contributions in MgB₂ and highlights the significance of tunneling direction in extracting accurate electron-phonon interactions.

Findings

Fine structures in DOS are observable mainly along the ab plane.

Single-band inversion can misrepresent the spectral function in multiband superconductors.

Tunneling direction critically influences the ability to extract band-specific electron-phonon interactions.

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 $\alpha^{2}F(\omega)$. 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, when the extracted $\alpha ^{2}F(\omega)$ contains the combination $\alpha ^{2}F_{\sigma \sigma}(\omega)$\textbf{+}$\alpha ^{2}F_{\sigma \pi }(\omega)$, together with a minor $\alpha ^{2}F_{\pi \pi}(\omega )$\textbf{+}$\alpha ^{2}F_{\pi \sigma} (\omega)$ component. Only in this case it is possible to extract information on the $\sigma$-band contribution to the spectral functions. For any other tunneling direction, the $\pi$-band contribution (which does not determine the superconducting properties of MgB$_{2}$) is dominant and almost coincides with the whole $\alpha^2F(\omega)$ for tunneling along the c axis. Our results are compared with recent experimental tunneling and point-contact data.