Discussion

Several objections may be raised to the type I superconductivity hypothesis; firstly, the value of H_c obtained from the t = 0.57 H_c2(theta) fit in Fig. 4 (43 gauss) is considerably lower than that predicted from the specific heat (75 gauss). The ratio between the value of H_c calculated from the specific heat measurements and the H_c obtained from the H_c2(theta) fits is about 1.8 for two T_c = 1.5 K specimens and the three temperatures where H_c2(theta) measurements were performed. A second problem with the type I explanation is that the H_c2(theta) data are not perfectly flat near theta = 0°, although H_c should be absolutely constant as a function of angle. Both of these discrepancies may be caused by angle-dependent demagnetization effects, which are hard to account for quantitatively.[17]

In order to extract believable values for H_c from the H_c2(theta) fits, demagnetization corrections need to be applied. The angular dependence of the demagnetization factor of an oblate spheroidal superconductor has been worked out by Denhoff and Gygax.[17] However, the application of this formula to the C₄KHg data is not justified since too many parameters which are poorly known (dimensions of the intercalated sample) or not independently known ( and epsilon) are involved. Therefore the demagnetization for the C₄KHg samples has been estimated only for the high-symmetry directions by approximating the sample shape as ellipsoidal.

The samples used in the H_c2(theta) measurements were flat plates with typical dimensions on the order of (2 mm × 2 mm × 0.5mm). Consultation of standard tables[43] shows that a demagnetization correction of about 3 is anticipated for an ellipsoid with radii in the ratios 1:1:0.25 when the field is applied along the perpendicular to the smallest dimension. If this correction is applied to the H_c values determined from the H_c2(theta) fits, one obtains H_c(0) 203 gauss, which is now a factor of 1.8 higher than the H_c(0) of 112 gauss calculated from the specific heat data. Since and , a higher H_c for the pink T_c = 1.5 K specimens implies that they have a higher density of states N(0) than the gold specimens, which is in keeping with their higher T_c.

Now that an estimate for the true value of H_c has been obtained, it is interesting to re-examine Fig. 4. marks the corrected value of the thermodynamic critical field at t = 0.57. The implication of this H_c estimate is that C₄KHg is type I in the approximate range theta < 80 °, and type II for theta > 80 °. Magnetization measurements on C₄KHg are desirable to positively identify type I superconductivity. Specific heat measurements on T_c = 1.5 K pink C₄KHg specimens would also be useful.

The extended linearity of H_c2(T) reported here for T_c = 1.5 K pink C₄KHg specimens is not surprising in light of previous studies of layered superconductors. Positive curvature or extended linearity of H_c2(T) are phenomena common to almost all anisotropic superconducting compounds.[59] Depending on the specific type of superconductor in question, many different explanations might be considered for the anomalous temperature dependence of H_c2. For example, mechanisms ranging from coupling-dimensionality crossover[35] to proximity-effect-induced curvature[7] to magnetoresistive anomalies[16] have been cited as the cause of positive curvature in various superconductors. However, each of these esplanations for its own reasons seems to be inappropriate for GIC's.[10] In order to choose an appropriate model from the many that are available, the best procedure is to consider what is already known about GIC's.

There are two types of models of anisotropic superconductivity that seem to contain the right features for the GIC superconductors: these are the anisotropic Fermi surface models[8,61] and the multiband superconductivity models.[22,2] Anisotropic Fermi surface models are an obvious choice because the quasi-2D band structure of graphite is responsible for many of the characteristic properties of GIC's. The simplest anisotropy-based model available is that developed by Butler[8] to fit H_c2(T) in Nb. This model was adapted by Dalrymple and Prober to fit the critical fields of NbSe₂, which show extended linearity and positive curvature[13] similar to that seen in GIC's. The NbSe₂ Fermi surface has cylindrical pieces at the hexagonal Brillouin zone boundary, and bears a great deal of resemblance to the proposed Fermi surface of many of the GIC's.[28,18] The band structure of C₄KHg has been calculated,[27] but unfortunately not enough quantitative information about the Fermi surface has been reported to allow a detailed comparison between the Butler model and the data for C₄KHg.

Another well-established feature of the GIC's Fermi surface is the presence of multiple bands. Some of the bands at the Fermi surface of graphitic origin are nearly two-dimensional in character. Other bands, which may be either of intercalant or graphitic origin, are more 3D in character. In C₄KHg these 3D bands are derived from hybridized K and Hg levels.[27] Models for H_c2(T) which incorporate the participation of two bands in the superconductivity therefore appear to be a logical choice for GIC's. Entel and Peter have fit H_c2(T) data for Cs_0.1WO_2.9F_0.1, a tungsten fluoroxide bronze, using a two-band Fermi surface model.[22] Al-Jishi[3] has proposed a similar model specifically tailored to fit critical field data on C₈K, although calculations using this model are still preliminary. More theoretical work is needed to interpret the large body of critical field data already available.[30,9]

The idea that both graphitic and intercalant bands participate in GIC superconductivity is sensible for two reasons. One reason is that there are superconducting GIC's (specifically the binary compounds C₈K (Ref. [37]) and C₈Rb (Ref. [36])) which are synthesized from non-superconducting starting materials. Since the KHg-GIC's are synthesized from KHg amalgams which are themselves superconducting, it might be thought that this argument for multiband superconductivity is invalid for them. However, the increase of T_c with stage in the KHg-GIC's is contrary to expectations of the superconducting proximity effect[57,14] if the carbon layers are not also participating in the superconductivity. The large critical field anisotropy seen in GIC's is also difficult to explain if graphitic electrons are not involved.[49,2] If the graphitic electrons are implicated in GIC superconductivity, then it follows that speculation about a coupling-dimensionality crossover in KHg-GIC's[30] due to decoupling of the intercalant layers is not sensible. If the graphitic bounding layers are also superconducting, then it may be their separation which will determine a coupling-dimensionality change. Considerations of this type may be important in discussion of dimensionality change in the high-T_c ceramic superconductors.

It is reasonable to expect a common origin for the extended linearity of H_c2(T) and the deviations from Eq. 1 in the H_c2(theta) fits. However, the deviations in H_c2(T) from the HW theory occur at the lowest obtainable reduced temperatures. The H_c2(theta) fits to Eq. 1, on the other hand, worsen as the temperature is increased toward T_c.[10] Worsening agreement between Eq. 1 and the data as temperature is increased is expected if type I superconductivity is the origin of the discrepancy. This behavior has already been observed for TaN.[56] All the available evidence therefore points to type I superconductivity as the explanation of the poor quality of the fits shown in Fig. .

It was not necessary to make allowance for type I character to fit the H_c2(theta) curves of the gold, mixed-phase samples, a finding which is consistent with the observations of Iye and Tanuma.[30] Type II character for all orientations of low-T_c C₄KHg is due to the fact that the T_c = 0.8 K samples have critical fields almost as high as the T_c = 1.5 K samples. To be more specific, the typical value of for a T_c = 1.5 K C₄KHg specimen is only about 750 gauss, while that for the T_c = 0.8 K specimens is about 650 gauss.[30] The critical field slope at T_c in the HW model is dH_c2/ dT , = , 4 k_B / pi e D, where k_B and e are the usual fundamental constants and D = v_F l/3 is the diffusivity.[42,47] Therefore the higher dH_c2/dT in the mixed-phase samples suggests that they have either a lower mean-free path l or a smaller Fermi velocity v_F. The greater in-plane disorder of the mixed-phase samples would appear to favor the lower mean-free path explanation.