NMR and NQR fluctuation effects in layered superconductors

We study the effect of thermal fluctuations of the $s-\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}$ order parameter of a quasi-two-dimensional superconductor on the nuclear spin relaxation rate near the transition temperature $T_C.$ We consider both the effects of the amplitude fluctuations and the Berezinskii-Kosterlitz-Thouless (BKT) phase fluctuations in weakly coupled layered superconductors. In the treatment of the amplitude fluctuations we employ the Gaussian approximation and evaluate the longitudinal relaxation rate $T_1^{-1}$ for a clean $s-\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}$ superconductor, with and without pair breaking effects, using the static pair fluctuation propagator $\mathcal{D}.$ The increase in $T_1^{-1}$ due to pair breaking in $\mathcal{D}$ is overcompensated by the decrease arising from the single-particle Green’s functions. The result is a strong effect on $T_1^{-1}$ for even a small amount of pair breaking. The phase fluctuations are described in terms of dynamical BKT excitations in the form of pancake vortex-antivortex (VA) pairs. We calculate the effect of the magnetic field fluctuations caused by the translational motion of VA excitations on $T_1^{-1}$ and on the transverse relaxation rate $T_2^{-1}$ on both sides of the BKT transition temperature $T_{\mathrm{BKT}}<T_C.$ The results for the NQR relaxation rates depend strongly on the diffusion constant $D$ that governs the motion of free and bound vortices as well as the annihilation of VA pairs. We discuss the relaxation rates for real multilayer systems where $D$ can be small and thus increase the lifetime of a VA pair, leading to an enhancement of the rates. We also discuss in some detail the experimental feasibility of observing the effects of amplitude fluctuations in layered $s-\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}$ superconductors such as the dichalcogenides and the effects of phase fluctuations in $s-$ or $d-\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}$ superconductors such as the layered cuprates.