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Parameters for the phase boundary between the vortex fluid and vortex glass phases" [15.75].

Materials y

1.41 ±0.10 19.2 ±0.1 12.2 ±0.07 1.64 ± 0.14 29.1 ± 0.1 27.2 ± 1.7

"Values for T and H0 are calculated using Eq. (15.10).

which defines a second-order phase transition. Values of the parameters y, r*(0) and H0 of Eq. (15.10) are given in Table 15.2 for K3C60 and Rb3C60. The values of y found experimentally are consistent with the de Almeida-Thouless relation [15.76] given by Eq. (15.10) with y = 3/2. The vortex pinning mechanism for K3C60 and Rb3C60 is associated with a large density of grain boundaries in these powder samples. While T* coincides with Tc for H = 0, the value of H0 differs from Hcl(0), since H0 relates to the boundary between the vortex glass and the vortex fluid, while Hc2(0) relates to the boundary between the vortex fluid and normal phases.

The granular nature of many of the M3C60 films that have been studied gives rise to much vortex pinning, which often results in an overestimate of HcX. The thermodynamic critical field Hc(0) listed in Table 15.1 is found from the relation

The large values of k (see Table 15.1) indicate that M3C60 compounds are strongly type II superconductors [15.17].

Reports on values for the mean free path I vary from ~2 to 30 Á and I is often reported to be smaller than the superconducting coherence length £0 for single crystals, films, and powder samples. The small value of t is mainly attributed to the merohedral disorder, which is present in all doped fullerene samples, including the best available single crystals (see §7.1.4). In addition, t is smaller for films and powders than for single crystals, since i is highly sensitive to carrier scattering from crystal boundaries. For some single-crystal samples, I is comparable to, or greater than, the diameter of a C60 molecule. But since £ <5C £0 in most cases, or at best í ~ fullerene superconductors are considered to be in the dirty limit defined by i < f0.

where

This conclusion, that the M3C60 compounds are dirty superconductors, is supported by the vortex glass behavior [15.75].

The coherence length which enters the formulae for iicl(0) and Hc2{0), is the Ginzburg-Landau coherence length, and £0 is sensitive to materials properties. The Pippard coherence length £00 in the clean limit is an intrinsic parameter, not sensitive to materials properties, and is related to f0 by

and the BCS theory provides the following estimate for £(X):

yielding in the 120-130 A range for the superconducting fullerenes [15.51,69], consistent with estimates using Eq. (15.13) and mean free path values I in the 5-10 A range (see Table 15.1). Available data thus suggest that a fullerene superconductor may be described as a strongly type II superconductor, which is in the dirty limit and has weak to moderate electron-phonon coupling.

15.4. Temperature Dependence of the Superconducting

Energy Gap

Several different experimental techniques have been applied to measure the superconducting energy gap for M3C60 superconductors, including scanning tunneling microscopy, nuclear magnetic resonance, optical reflectivity, specific heat, and muon spin resonance measurements. The various results that have been obtained are reviewed in this section.

It would appear that scanning tunneling microscopy should provide the most direct method for measuring the superconducting energy gap. Using this technique, the temperature dependence of the energy gap has been investigated for both K3C60 and Rb3C60 [15.53,54], Figure 15.11 shows dl/dV, the derivative of the tip current I with respect to bias voltage V, plotted vs. bias voltage for an Rb3C60 sample with a 60% Meissner fraction at 4.2 K. The sample in this study was characterized by zero-field-cooled and field-cooled magnetization measurements at very low magnetic field (~5 Oe) [15.53]. The experimental data for (dl/dV) are compared in Fig. 15.11 to the expression for (dl/dV)

dl/dV = Re{(eV - iT)/[(eV - iT)2 - A2]1/2}, (15.15)

Fig. 15.11. Plot of dl/dV vs. bias voltage V (meV) for an RbjO,, sample at a temperature of 4.2 K [15.53]. The experimental data for the conductance (solid circles) were calculated numerically from I-V measurements. The data in (a) are fit with the expression dl/dV = eV/[(eV)2 - A2]"2 (solid curve) with A = 6.8 meV and assuming no broadening. The data in (b) are fit with the expression given in Eq. (15.15) which includes a phenomenological broadening parameter T (A = 6.6 meV and T = 0.6 meV).

Fig. 15.11. Plot of dl/dV vs. bias voltage V (meV) for an RbjO,, sample at a temperature of 4.2 K [15.53]. The experimental data for the conductance (solid circles) were calculated numerically from I-V measurements. The data in (a) are fit with the expression dl/dV = eV/[(eV)2 - A2]"2 (solid curve) with A = 6.8 meV and assuming no broadening. The data in (b) are fit with the expression given in Eq. (15.15) which includes a phenomenological broadening parameter T (A = 6.6 meV and T = 0.6 meV).

which is plotted in Fig. 15.11 for the superconducting state normalized to that in the normal state [15.54]

where E = eV is the energy of the tunneling electrons relative to EF, 2 A is the superconducting energy gap, and T is a phenomenological broadening parameter. A good fit between the experimental points and the model is obtained. From data such as in Fig. 15.11, the temperature dependence of the superconducting energy gap A(T) was determined and a good fit for A(T) to BCS theory was obtained (see Fig. 15.12) [15.53] for A(0) = 6.6 meV and T = 0.6 meV, from which (2A(0)/kBTc) was found to be 5.3 for K3C60 and 5.2 for Rb3C60, well above the value 3.52 for the case of a BCS superconductor. These results were interpreted to indicate that the M3C60 compounds are strong coupling superconductors [15.54], The discrepancy between the experimental points and the theoretical curves in Fig. 15.11 has been cited as an indication of systematic uncertainties in the data sets, and a reexamination of the determination of 2A by tunneling spectroscopy seems necessary. Also, other experiments, discussed below, yield values of 2A(0)/fcBrc close to the BCS value.

1 1 , 11

i i

i i

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