Determination of the electron-phonon coupling constants from the experimental temperature dependences of superconducting gaps in MgB$_2$

Experimental temperature dependences $\Delta_{\sigma,\pi}(T)$ of the energy of superconducting gaps for MgB$_2$ samples with the critical temperatures $22$ K${}< T_c < 41$ K have been fitted by selecting the renormalized electron-phonon coupling constants $\lambda_{ij}$ with the use of the Moskalenko-Shul system of equations, the expression for the frequency of collective plasma oscillations obtained by Leggett for two-gap superconductors, and two fitting parameters. We previously obtained the dependences $\Delta_{\sigma,\pi}(T)$ by the multiple Andreev reflection spectroscopy of superconductor-constriction-superconductor junctions based on MgB$_2$ with various degree of disorder of the crystal structure. It has been shown that the intraband pairing constants are decisive for the superconductivity mechanism in MgB$_2$; in this case, $\sqrt{V_{\sigma\sigma}V_{\pi\pi}}/V_{\sigma\pi}=8{-}22$ and the ratio of the interband constants $\alpha$ can range from $3$ to $11$. The set of the Eliashberg coupling constants $\lambda_{ij}^0$ has been qualitatively determined for relatively pure MgB$_2$ with maximum values $T_c\approx40$ K. The leading constant is $0.7<\lambda_{\sigma\sigma}^0\approx \lambda_{\text{eff}}^0<0.9$ and depends on the choice of the upper integration limit in the Bardeen-Cooper–Schrieffer (BCS) model and the effective Coulomb repulsion $\mu_{\text{eff}}^*$. The characteristic ratio for the gap in the $\sigma$ band is $2\Delta_\sigma/k_{\text{B}}T_c=5.0{-}6.5$.