Suppression of superconducting parameters by correlated quasi-two-dimensional magnetic fluctuations

We consider a clean layered magnetic superconductor in which a continuous magnetic transition takes place inside superconducting state and the exchange interaction between superconducting and magnetic subsystems is weak so that superconductivity is not destroyed at the magnetic transition. An example of such material is RbEuFe$_{4}$As$_{4}$. We investigate the suppression of the superconducting gap and superfluid density by correlated magnetic fluctuations in the vicinity of the magnetic transition. The influence of nonuniform exchange field on superconducting parameters is sensitive to the relation between the magnetic correlation length, $\xi_{h}$, and superconducting coherence length $\xi_{s}$ defining the 'scattering' ($\xi_{h}<\xi_{s}$) and 'smooth' ($\xi_{h}>\xi_{s}$) regimes. As a small uniform exchange field does not affect the superconducting gap and superfluid density at zero temperature, smoothening of the spatial variations of the exchange field reduces its effects on these parameters. We develop a quantitative description of this 'scattering-to-smooth' crossover for the case of quasi-two-dimensional magnetic fluctuations. Since the magnetic-scattering probability varies at the energy scale comparable with the gap, the quasiclassical approximation is not applicable in the crossover region and microscopic treatment is required. We find that the corrections to both the gap and superfluid density grow proportionally to $\xi_{h}$ until it remains much smaller than $\xi_{s}$. When $\xi_{h}$ exceeds $\xi_{s}$, both parameters have much weaker dependence on $\xi_{h}$. Moreover, the gap correction may decrease with increasing of $\xi_{h}$ in the vicinity of the magnetic transition. We also find that the crossover is unexpectedly broad: the standard scattering approximation becomes sufficient only when $\xi_{h}$ is substantially smaller than $\xi_{s}$.