4.2 Transverse Field: The Vortex State Field Distribution

In TF experiments in A₃C₆₀ superconductors, it is found[178,179,180] that a large fraction of the injected muons remain diamagnetic, and their precession signal is broadened below T_c by the inhomogeneous magnetic field distribution of the vortex state.

The lineshape due to the field distribution of a triangular flux line lattice (FLL) with additional effects due to flux-lattice disorder and anisotropy have been studied in detail in the context of high-T_c superconductors.[181,182,77,184] The characteristic features of this lineshape can be related to the spatial distribution of fields of the triangular FLL: there is a sharp low-field cutoff of the lineshape due to the minimum field that occurs in the centre of three vortices; at slightly higher field, there is a sharp peak due to the highly weighted field corresponding to the saddle point midway between two vortices; and there is a sharp high-field cutoff due to the maximum field occurring in the vortex cores. For an ordered triangular FLL, it is found that the second moment of the field distribution is related to the London penetration depth at intermediate fields by[181]

where $\Delta B = \sigma_S(0) / 2 \pi \gamma_\mu$ is the RMS deviation of the field distribution.

To accurately determine $\lambda$ for a perfect triangular FLL, involves more detailed modelling of the field distribution. One such model relies on an approximate low-field solution of the Ginzburg-Landau theory[185], which, more recently, has been extended to higher fields[186] and further simplified.[187] This theory results in a field dependent relationship (for $B \ll B_{c2}$),

where f_v is a universal function of order unity (but sharply field dependent) at low reduced fields. $\lambda$ estimated from (4.6), for $B/B_{c2}\approx 0.02$, will be $\sim 15$% smaller than that of (4.5). Estimates of $\lambda$ can be improved by fitting a model of the asymmetric field distribution, rather than just using the second moment.[188,184] If the fractional volume of the FLL corresponding to the vortex cores is large enough, the high-field cutoff will be observable, and the superconducting coherence length $\xi$ can be measured.[189,190]

It would be surprising to find the flux adopting a perfect triangular FLL in powdered superconductors, such as these, where both the vortex separation and $\lambda$are on the same scale as $\xi_{XTL}$, the coherence length of crystalline order. One would instead expect that the flux lines would exhibit no long-range order.[135] A disordered FLL would possess a field distribution smeared relative to the perfect FLL. Such smearing would make a significant contribution to the second moment of the field distribution, e.g. see[77], making the applicability of the above theories questionable. This is just the situation we find (see, for example, Fig. 4.19a). The line is much broader than in the normal state but exhibits only a slight asymmetry. It is unreasonable to attempt to fit such a smeared lineshape to the full theoretical shape, but Fig. 4.19b shows two simulated lineshapes for comparison. We note that the fluxoid distribution is not melted and only weakly pinned at 3K and 1T, because, for example, shifting the applied field at this temperature causes the line to shift in frequency, but broaden significantly (Fig. 4.30. This is in contrast to crystalline YBa₂Cu₃O_6.95, where[188] shifting the field shifts only the background signal as the FLL is strongly pinned, and to the vortex liquid state, where the (symmetric) line simply shifts without altering shape.

  

Figure 4.19: a) A high statistics $\mu$SR lineshape (FFT of TF asymmetry spectrum) in Rb₃C₆₀ field-cooled to 3K at 1.0T applied transverse field. b) Simulated lineshapes for a perfectly ordered triangular flux-lattice using the Ginzburg-Landau theory.[185,186,187] The simulation parameters ($\lambda$,$\xi$) are (3000Å,30Å) and (1100Å,30Å). The field distributions are convoluted with a gaussian corresponding to the normal state linewidth. The 3000Å simulation also includes a small non-relaxing background signal known to exist in the data.

Clearly, the 1100Å value of the penetration depth deduced from magnetization measurements is inconsistent with the observed lineshape, since no amount of disorder will narrow the line, and estimates of the correlation time for motion of the vortices from NMR[191] strongly suggest that, at low temperatures, there should be no dynamical narrowing of the $\mu$SR line. The high-field cutoff in the FLL field distribution will move down towards the average field as either $\lambda$or $\xi$ increases. Because of this correlation, the absence of a long high-frequency tail in the observed lineshape (fig 4.19a) constrains only the pairs ($\lambda$,$\xi$), i.e. along the line ($\lambda$,30Å) the observed lineshape is consistent with $\lambda$ larger than $\approx 3000$Å with the condition that larger values of $\lambda$ will require a greater degree of disorder to match the observed linewidth. Such an inconsistency between the magnetization and $\mu$SR results is not surprising, since the procedure for obtaining $\lambda$from the magnetization is fraught with difficulties[31,36] of which only some are reduced or eliminated through use of a single crystal instead of powder. We note that other results using NMR[176] and optical[192] methods find $\lambda$ consistent with the above lower limit.

  

Figure 4.20: a) Second moments of the $\mu$SR lineshape as a function of temperature in stars: Na₂CsC₆₀ (0.01T), diamonds K₃C₆₀ (1.0T), Rb₃ (circles 1.0T, triangles 0.5T, nablas 1.5T), and squares a second sample of Rb₃C₆₀ (0.27T). b) Second moments corrected for the normal state values. 4.5

We extract the second moment from the TF data by fitting the time dependent envelope of the precession signal to a gaussian of the form $A\exp(-(\sigma t/\sqrt{2})^2)$. The results are shown in Fig. 4.20a. In the normal state, the lineshape is a narrow gaussian whose width is determined by the distribution of magnetic fields due to the randomly oriented nuclear dipoles (¹³C,²³Na,^39,40,41K,^85,87Rb,¹³³Cs, see Tables ). This normal state width $\sigma_{N}$ is temperature independent in the range between T_c and room temperature (except for Na₂CsC₆₀ discussed below) and adds in quadrature to the $\sigma_{S}$ due to the disordered FLL to determine the overall $\sigma$ below T_c. We use this correction to produce $\sigma_{S}$ shown in in Fig. 4.20b. The temperature dependences are fit to the phenomenological form $\sigma_{S}(0)[1-(T/T_c)^\varpi]$, and the resulting parameters are given in Table 4.5. In spite of the lack of the signature of the FLL in the lineshape, we expect that the overall linewidth is controlled by $\lambda$ (and much more weakly by $\xi$) and that $(\Delta B)^{-1/2}$ will scale with $\lambda$.We have calculated the values $\lambda$ reported in Table 4.5 using the linewidths $\sigma_{S}(0)$ and (4.5). Because of the increase of $\Delta B$ from disorder of the FLL, this conversion will underestimate the actual $\lambda$, so this, or perhaps more conservatively the value from the field dependent theory (which we have not included because of uncertainty in the value of B_c2), should be considered a lower bound for $\lambda$.These results can be compared directly with those of references.[178,179] Note that at low temperature, $\sigma_S(T)$ is quite flat in contrast to the d-wave linear T dependence seen in clean high-T_c[193], suggesting that, in this respect, the A₃C₆₀ superconductors appear conventional. This conclusion is not strong, however, because in d-wave systems, for example, impurity scattering can lead to[105] a weaker low T dependence of $\lambda$.In Rb₃, there is no apparent field dependence to $\sigma_S(T)$ between 0.5 T and 1.5 T, so the applicability of Eq. (4.6) is questionable. Furthermore, there is large sample dependence of $\sigma_{S}$in Rb₃C₆₀ suggesting a strong effect of disorder on $\lambda$.In the dirty limit $\xi \approx l$, where l is the electron mean free path. The penetration depth varies strongly with l via

where n_s is the superelectron density and m^* is the effective mass. Thus if l is on the same order as the coherence length, any sample dependent disorder that contributes to l could alter $\lambda$ strongly, as in the observed sample dependence in Rb₃C₆₀. The low T dependence of $\sigma_{S}$ has been fit to the BCS activated form[107] for the clean-limit[180]; however, the dirty-limit form is probably appropriate to Rb₃C₆₀ and K₃C₆₀ (but possibly not to Na₂Cs). In this case[106], for T<0.5T_c,

where $\Delta_0$ is the low temperature BCS gap parameter. The results of these fits are also given in Table 4.5. We note that some caution must be taken regarding the interpretation of the TF results for Na₂CsC₆₀ because: i) the applied field was very low (not far from H_c1) and ii) $\sigma_{N}$ was rather large and temperature dependent above 50K (Fig. 4.31). The behaviour of $\sigma_N(T)$ may be associated with molecular dynamics but may also be due to muons stopping in unmasked areas of the Al sample cell. However, the consistency of the observed $\sigma_{S}$ with another published measurement[179] at higher field suggests that neither of these effects was appreciable. Furthermore, the Al signal will have no contribution to the signal in high LF discussed in the following section.

The historic evolution of $\mu$SR measurements of $\lambda$in YBa₂Cu₃O_6.95 may provide a guide for the robustness of estimates of $\lambda$ from the second moment of the $\mu$SR lineshape. Despite the more serious consequences of powder averaging in these highly anisotropic systems, the extracted penetration depth from $\mu$SR data (even from symmetric lineshapes, like Fig. 4.19a) has exhibited a variation of only $\sim$30% (including model as well as sample variation). Thus the most serious uncertainty with our determination of $\lambda$ may be the strong sample dependence. The observation of the asymmetric lineshape characteristic of a triangular FLL would certainly make our conclusions much stronger, and such lineshapes are expected in single crystals which can now be made in sufficient size[90] for such an experiment.

We turn now to the temperature dependence of the LF (T₁) relaxation of Mu@C₆₀, noting that the interstitial diamagnetic muons will not contribute to this signal except at (LF) magnetic fields less than a few mT.