Magnetic impurities in a strongly coupled superconductor

TL;DR

The paper investigates how magnetic impurities affect strong-coupling, phonon-mediated superconductivity by coupling the self-consistent Nagaoka scattering-matrix treatment of impurity electrons with the Migdal-Eliashberg theory. The authors compute the in-gap bound-state energy, the superconducting critical temperature $T_c$, and the tunneling density of states across a range of exchange couplings $J$, impurity concentrations $c_{imp}$, and Kondo scales $T_K$, revealing a single bound state inside the gap and rich Tc behavior. For antiferromagnetic exchange, $T_c(c_{imp})$ exhibits regimes including re-entrant behavior and a crossover from three-$T_c$ to a conventional single $T_c$ as Kondo screening strengthens; for ferromagnetic exchange, superconductivity persists only within a finite temperature window $T_{c1}\le T\le T_{c2}$ up to a critical impurity concentration $c_{imp}^*$. The work highlights dynamical Kondo scattering effects and shows qualitative deviations from weak-coupling predictions, notably in the FM case, with implications for tunneling spectra and potential experimental verification.

Abstract

We revisit certain aspects of a problem concerning the influence of carrier scattering induced by magnetic impurities in metals on their superconducting properties. Superconductivity is assumed to be driven by strong electron-phonon interaction. We use the self-consistent solution of the Nagaoka equations for the scattering matrix together with the Migdal-Eliashberg theory of superconductivity to compute the energy of the in-gap bound states, superconducting critical temperature and tunneling density of states for a wide range of values of the Kondo temperature and impurity concentrations. It is found that similar to the case of the weak coupling (BCS) superconductors there is only one pair of the bound states inside the gap as well as re-entrant superconductivity for the case of antiferromagnetic exchange coupling between the conduction electrons and magnetic impurities. In agreement with the earlier studies we find that the gapless superconductivity can be realized which in the case of antiferromagnetic exchange requires much smaller impurity concentration. Surprisingly, in contrast with the weakly coupled superconductors we find that superconducting transition exhibits two critical temperatures for the ferromagnetic exchange coupling.

Figures (4)

• Figure 1: Dependence of the superconducting critical temperature on the concentration of magnetic impurities for various values of the dimensionless antiferromagnetic exchange coupling computed from equation (\ref{['Eq4PhinTc']}) for $S=1/2$. The calculation of the Matsubara summations have been limited to the range $m\in[-N-1,N]$ with $N=512$.
• Figure 2: Dependence of the superconducting critical temperature on the concentration of magnetic impurities for various values of the dimensionless ferromagnetic exchange coupling computed from equation (\ref{['Eq4PhinTc']}) for $S=1/2$. The calculation of the Matsubara summations have been limited to the range $m\in[-N-1,N]$ with $N=512$.
• Figure 3: Plot of the bound state energy as a function of dimensionless exchange coupling $\nu_FJ$ for different values of the dimensionless electron-phonon coupling $\lambda$. The dependence of the bound state energy on $\gamma$ looks qualitatively similar to the weak coupling case. The only difference arises from the fact that at larger values of $\lambda$ the bound state remains closer to the bottom of the upper Bogoliubov band. All the results shown have been obtained for temperature $T=0.01\Omega$.
• Figure 4: Plot of the tunneling density of states $N(\omega)$, Eq. (\ref{['Nw']}), as a function of frequency for the case of antiferromagnetic exchange (top panel) and ferromagnetic exchange (bottom panel) sign of the spin exchange coupling. Both plots have been obtained for temperature $T=0.05\Omega$.