4.3.1 T₁(T) in Rb₃C₆₀ and K₃

The average normal state values of (T₁T)^-1 are shown in Fig. 4.15b. At higher temperature there are deviations (Fig. 4.17), but at low temperature, we have no significant evidence for previously reported[83] weakly non-Korringa behaviour. Just below T_c, for $B \approx 1-2$T, we find a slightly enhanced (T₁T)^-1 followed by an strong fall-off below about 0.75T_c, e.g. Fig. 4.22.

  

Figure 4.22: Temperature dependence of the spin relaxation rate of Mu@C₆₀ in K₃C₆₀ at a field of 2.0 T. The fit is to the Hebel-Slichter integral (3.9) with a broadened DOS of the form (3.12).

We find that increasing the disorder in the FLL in Rb₃C₆₀ does not affect the peak height by comparing field-cooled and zero field-cooled (T₁T)^-1 at the peak temperature, T_P, at 1.5 T. However, we find, in Rb₃, that the peak is strongly field-dependent (similar behaviour was independently discovered in NMR T₁ measurements in Rb₂CsC₆₀ by Stenger[174]). It is strongly supressed as the field is increased above about 2T, and is entirely gone by 4.2T (Fig. 4.23). The observed heights of the peaks from fits to a parabola near T_P are shown in Fig. 4.24.

  

Figure 4.23: Magnetic field damping of the Hebel-Slichter peak in Rb₃C₆₀, note the different temperature scales. The longitudinal fields are a) 1.5 T and b) 4.2 T.

In order to discuss analysis of the peaks in terms of the theory outlined in the section IV, we include some brief remarks about the DOS functions used. First we note g_D and g_SC (eqns. (3.12) and (3.13)) do not depend on the sign of the broadening parameter ($\Gamma$ or $\Delta_2$), which we take to be positive. The property of conservation of the total number of electron states, which simply results from the construction of the eigenstates of the superconductor from those of the normal state, can be succinctly written,

The BCS g_S follows this, as does the heuristic form g_D; however, the strong coupling form g_SC (3.13) does not; in fact,

i.e. there are effectively fewer states in g_SC than in the normal state. Of course, in the full strong-coupling theory, $\Delta = \Delta(E,T)$,g_SC will have structure above the gap, and Eq. (4.9) will be obeyed. One consequence of this property of g_SC, is that the integral for (T₁T)^-1 (3.9) will approach a value less than 1 as T approaches T_c from below. The resulting discontinuity scales with $\Delta_2$, and is negligible for small $\Delta_2$. However, if g_SC is used to model the small coherence peaks reported here, $\Delta_2$ is relatively large, and the discontinuity is significant, so that the approximation first introduced by Fibich[122], that the energy dependence of $\Delta_2$ can be neglected in calculating the coherence peak seems untenable in the current context, and in order to use strong-coupling results, one would have to resort to more detailed calculations.[125,118] There is also a significant difference between g_D and g_SC in the $E \rightarrow 0$ behaviour which is important in determining the low temperature behaviour of (T₁T)^-1: $g_D(0) = [1+(\Delta/\Gamma)^2]^{-1/2}$,while g_SC(E) goes to zero with slope $\approx \vert\Delta_2\vert/\Delta_1$ as $E \rightarrow 0$.

  

Figure 4.24: Heights of the coherence peak above the normal state as a function of field. These values were obtained from fits to a parabola near the maximum.

The low field peaks have been fit to Eq.(3.9) using both the DOS functions g_D and g_SC (avoiding the discontinuity mentioned above by only considering the data below T_c) with $\Re\{ {\Delta} \} \propto \Delta_{BCS}(T)$, which is still quite a good assumption in the strong-coupling case.[52] We find that, in order to explain such a small height of the coherence peak, the broadening parameters near T_c must be $\Gamma/\Delta_0 \approx \Delta_2/\Delta_0 \approx 10\%$.If the broadening is due to strong electron-phonon scattering, then it should be significantly temperature dependent. For a Debye phonon spectrum, at low temperature, $\Delta_2 \propto T^{7/2}$.The contribution of the low energy libron peak[194] may modify this somewhat; although, the weak electron coupling to this mode[194] will limit its effect. The results of tunneling experiments could, in principle, provide confirmation of the sharpening of g_S at low temperature, but the published spectra are rather equivocal: low temperature break-junction tunneling spectra show rather broad peaks[195] while point-contact spectra are quite sharp[196] and other STM measurements are fairly broad but show a strong dependence on crystallinity.[197] Recent planar junction tunneling measurements[198] confirm the former behaviour, suggesting an intrinsic nearly temperature-independent broadening mechanism for g_S. The width of the coherence peak is controlled by the balance between the peak contribution to the integral (3.9) and the exponential behaviour due to the opening of the gap; thus, there is significant correlation between the value of $\Delta_0$ and the temperature dependence of the broadening parameter. Generally, a broadening that falls off strongly as the temperature decreases causes the peaks of g_S to sharpen, and the coherence peak to widen; consequently, the value of $\Delta_0$ required to explain the observed peak width will be larger than for a broadening which does not depend as strongly on T.

  

Figure 4.25: The temperature dependence of (T₁T)^-1 at 0.3T at low reduced temperature. a) and b) two samples of Rb₃, and c) K₃.

In high field T₁^-1 becomes too slow to observe at low reduced temperature, so we use the strong field-dependence (Eq. (3.4) and Fig. 4.15) and study this region in a reduced applied field. In this case, over the temperature range 0.5T_c-0.25T_c, (T₁T)^-1 exhibits activated behaviour as shown in Fig. 4.25. Again the values of $\Delta_0$ required to fit this data depend on what broadening is assumed at low temperature. Because of the Fermi-factor in the integral (3.9), the low temperature behaviour of (T₁T)^-1 is determined mainly by the gap and contains no information about the shape of the broadened DOS peaks. However, if there are states within the gap, the temperature dependence may deviate strongly from the activated temperature dependence. We have fit the low temperature data to the same set of models as the peak data at higher field. We have assumed two cases for the temperature dependence of the broadening: strong ($\approx T^{7/2}$) and temperature independent broadening, noting that the real dependence will likely lie somewhere in this range. The results are given in Table 4.6.

  

Figure 4.26: Residual relaxation at low temperature in Rb₃.

In Rb₃C₆₀ at 0.3T we find at lower temperatures a sample dependent residual relaxation that is much more weakly temperature dependent (Fig.4.26). The source of this residual relaxation could be related to crystalline disorder (in alkali site occupation or degree of orientational disorder) and the finite low T zero-bias conductance observed in tunneling (zero applied magnetic field). There is also evidence that the C₆₀ in Fm$\bar{3}$m phases undergo orientational dynamics which freeze out near or below room temperature.[91] Thus, it is possible that the degree of orientational disorder varies with cooling procedure. We have attempted no systematic quench-rate dependences, but there is some evidence that the low T residual relaxation and the large sample dependence of $\lambda$ may be partially due to different cooling procedures. We note that there is no evidence in the Fm$\bar{3}$m materials for a low T polymerized phase, which occurs only for intercalated C₆₀ materials with smaller cubic lattice parameters such as Na₂A. The superconducting transition in K₃C₆₀ is extremely sensitive to radiation damage induced disorder[199]. Such behaviour may[199] be a consequence of the narrowness conduction band. If this is the case, then variation of quenched disorder may also have an unusually large effect. At finite field another source of this relaxation could be the vortex cores (section IV.D). The relaxation rate of the small fast relaxing component is roughly that of the (extrapolated) normal material, but the amplitude is too large for the small ($\xi = 30$ Å) cores. Moreover, it does not appear to change linearly in amplitude with field near 0.3T (see section IV.D). On the other hand, the linear field dependence would be rather difficult to observe because of the intrinsic field dependence of the relaxation rate and the extremely small amplitude. The persistence of some relaxation at low temperature in zero applied field (inset, Fig.4.27) suggests that at least some of this residual relaxation is not due to vortices. The relaxation in zero field, however, can have contributions from both static and dynamic fields.

The small size of the coherence peak is not the result of the perturbing influence of the paramagnetic Mu atom. While it has recently been shown[200] that single paramagnetic atoms locally perturb the surrounding superconductor, the Mu as shown in section V is only very weakly coupled to the conduction electrons. The strong evidence that the Mu perturbation is small is the agreement of the temperature dependence of (T₁T)^-1 with similar NMR experiments.[34] On the other hand, the suppression of the peak can be due to any of the mechanisms discussed in section IV.C or a combination of these mechanisms. Recent tunneling measurements suggest that we should expect some broadening from strong-coupling effects. We expect at most a small amount of anisotropy (large anisotropy is not appropriate to explain either the low temperature behaviour of the TF linewidth $\sigma$ or (T₁T)^-1). Also, because the Fm$\bar{3}$m materials are likely in the extreme dirty limit ($l < \xi$) anisotropy would be eliminated by electron scattering.[104] However, the strong field dependence of the coherence peak is not accounted for explicitly by either of these mechanisms, so we now consider the effect of magnetic field introduced in section IV.D. The observed suppression of the peak by magnetic field occurs in a very different part of the phase diagram than regions (i) and (ii) of Fig. 3.13, where it is certainly expected, so that the gaplessness due to proximity to H_c2 does not account for the observed damping. However, we note that some H_c2 measurements by magnetization exhibit unusual temperature dependence[36] near T_c(0) (Fig. 4.32). Our measurements of T_c(H) via TF $\mu$SR (Table 4.5), though, are consistent with a strongly T dependent H_c2 near T_c(0). We have assumed such a strong T dependence to arrive at the estimate of T_c(4.2T) in Fig. 4.23b. We expect that the effects of Pauli pair-breaking on the coherence peak will also be small because, as the Yosida function behaviour of the NMR Knight shift[34] shows, spin-orbit scattering is very weak in these materials. The crude model used to explain the NMR coherence peak[34] (i.e., Eq.(3.17)) does not satisfactorily explain the damping observed in $\mu$SR or NMR. For example, in Fig. 4.23a at 20K, the value of (T₁T)^-1 is near its value in the normal state, so no weighted average of (T₁T)^-1 with its value in the normal state will give the observed value at roughly the same reduced temperature at 4.2T of $\sim 0.6$ (T₁T)^-1_NOR. Furthermore, this model (which treats the vortices as normal cylinders) is expected to apply[143] only when $B \ll B_{c2}$.The observed strong suppression of the coherence peak occurs in the non-linear region of the phase diagram and may be explained by the theories.[144,147] The detailed mechanism for this suppression could be elucidated by STM measurements which can resolve both the spatial and energy dependences of the superconducting DOS.

  

Figure 4.27: A high statistics zero applied field time spectrum in the Meissner state of Rb₃C₆₀ at 4K. The slow gaussian relaxation is likely due to the diamagnetic interstitial muons relaxing in the distribution of magnetic fields of the randomly oriented static nuclear dipoles, while the small fast relaxing component (inset) is attributed to relaxation of the endohedral Mu@C₆₀. Both dynamic and static random magnetic fields can contribute to this relaxation. Dynamic fields could be due to remnant states within the gap as seen in finite low temperature zero bias conductance in several tunneling experiments. Static fields could be due to nearby nuclear dipoles.