Effect of fluctuations on the NMR relaxation beyond the Abrikosov vortex state

The effect of fluctuations on the nuclear magnetic resonance (NMR) relaxation rate $W=T_1^{-1}$ is studied in a complete phase diagram of a two-dimensional superconductor above the upper critical field line $H_{\mathrm{c}2}\left(T\right)$. In the region of relatively high temperatures and low magnetic fields, the relaxation rate $W$ is determined by two competing effects. The first one is its decrease in the result of suppression of the quasiparticle density of states (DOS) due to formation of fluctuation Cooper pairs (FCPs). The second one is a specific, purely quantum relaxation process of the Maki-Thompson (MT) type, which for low field leads to an increase of the relaxation rate. The latter describes particular fluctuation processes involving self-pairing of a single electron on self-intersecting trajectories of a size up to phase-breaking length ${\ell }_{\varphi }$ which becomes possible due to an electron spin-flip scattering event at a nucleus. As a result, different scenarios with either growth or decrease of the NMR relaxation rate are possible upon approaching the normal-metal–type-II superconductor transition. The character of fluctuations changes along the line $H_{\mathrm{c}2}\left(T\right)$ from the thermal long-wavelength type in weak magnetic fields to the clusters of rotating FCPs in fields comparable to $H_{\mathrm{c}2}\left(0\right)$. We find that below the well-defined temperature $T_0^{*}\approx 0.6T_{\mathrm{c}0}$, the MT process becomes ineffective even in the absence of intrinsic pair breaking. The small scale of the FCP rotations ${\xi }_{\mathrm{xy}}$ in such high fields impedes formation of long $(\lesssim {\ell }_{\varphi })$ self-intersecting trajectories, causing the corresponding relaxation mechanism to lose its efficiency. This reduces the effect of superconducting fluctuations in the domain of high fields and low temperatures to just the suppression of quasiparticle DOS, analogous to the Abrikosov vortex phase below the $H_{\mathrm{c}2}\left(T\right)$ line.

Figure 1

Relaxation process of the MT type related to the fluctuation pairing of electrons on self-intersecting trajectories involving their spin-flip scatterings on the investigated nuclei. (a) Initially, an electron moves along trajectory 1 (clockwise, red path); due to several impurity scattering events it returns to the departure point. As a result of the interaction with the nuclei, its spin and momentum flip, and the electron returns along trajectory 2 (counterclockwise, green path), which is the reversed trajectory 1. During this motion it interacts with itself in the past (corresponding superconducting interactions are shaded). Such a process is possible due to the “fast” motion of the electron along the trajectory and retarded character of electron-phonon interaction. (b) Representation of the same process as a Feynman diagram [compare with Fig. 2].

Figure 2

Diagrams for spin susceptibility. The solid lines correspond to free-electron Green's functions, bold wavy lines show the fluctuation propagator, and triangles and rectangles account for electron scatterings on impurities. (a) The DOS correction, (b) the renormalization of the diffusion coefficient (DCR), and (c) the MT process.

Figure 3

Closed integration contour $\mathcal{C}$ in the plane of complex frequencies.

Figure 4

(top) The temperature and magnetic field dependence of the relaxation rate $W^{\mathrm{fl}}$ in the case of a very weak pair breaking ${\gamma }_{\varphi }=0.003$. The thick isoline (red) represents a zero relaxation rate, while the dashed isolines correspond to relaxation rate values of $-1$ and $-2$. The mesh line $t^{*}$ (red) marks the critical temperature for ${\gamma }_{\varphi }\to 0$, while the cyan contour line indicates the value of $W^{\mathrm{fl}}$ at $h_{\mathrm{c}2}\left(t^{*}\right)$ $(-3.04$; see Fig. 6). (bottom) Contour plot of the region close to $h_{\mathrm{c}2}\left(t^{*}\right)$.

Figure 5

The temperature and magnetic field dependence of the relaxation rate $W^{\mathrm{fl}}$ in the case of a strong pair breaking ${\gamma }_{\varphi }=0.3$. The dashed isolines correspond to relaxation rate values of $-1.5$ and $-2$.

Figure 6

(a) Total correction in the 2D case for ${\gamma }_{\varphi }=0.003$ calculated parallel to but at different distances from the line $h_{\mathrm{c}2}\left(t\right)$ in the fluctuation regime. (b) Different colors correspond to different distances from $h_{c2}\left(t\right)$ defined by the scaling parameter $a$, given in the legend. (c) $t^{*}\left({\gamma }_{\varphi }\right)$ shows a linear dependence for all experimentally relevant values of ${\gamma }_{\varphi }$. For fixed temperatures below $t^{*}$ the relaxation rate $W^{\mathrm{fl}}$ monotonically increases with field, while for larger temperatures the relaxation rate is nonmonotonic and grows strongly when approaching $h_{c2}$ from above. The asymptotic value is $t_0^{*}=t^{*}({\gamma }_{\varphi }=0)=0.59$.

Figure 7

Quasi-2D result for different $r$ values very close to $h_{\mathrm{c}2}\left(t\right)$ with $a=1.01$ (see Fig. 6) for ${\gamma }_{\varphi }=0.003$.