Dimensional Crossover Models

The direct ancestor of the KLB papers, and one of the primary contributions to the AGL model, is the Josephson-coupling theory of Lawrence and Doniach[155]. Lawrence and Doniach started with the idea of a superconductor-insulator (S/I) superlattice whose layers are coupled by Josephson tunnelling of quasiparticles. They modified the GL free-energy expression accordingly, and found an effective mass along the c-direction in terms of the tunnelling parameters. From this, following the work of Kats[127], they showed that the critical field anisotropy was equal to the square root of the ratio of the effective masses.

Klemm, Luther and Beasley[131] studied the GL equations with Josephson-coupling further, and found that they predicted a divergence of H_{c2, _|_ ^c}(T) at a temperature T^* given by xi_{|| ^c}(T^*) = s/sqrt2. In this equation, s is the interlayer spacing corresponding to I_c in GIC's. KLB then did a first-principles calculation of H_c2(theta, T) and discovered that T^* is actually the temperature at which a dimensionality crossover occurs. Dimensionality crossover in the context of layered superconductors means a change in character from bulklike behavior to film-like behavior. This change is caused by a decoupling of the superconducting layers from one another when xi_||^c becomes less than s. The symptoms of 2D behavior are that the H_c2(theta) curves are fit with Tinkham's formula rather than the AGL angular dependence formula, and that H_c2(T) goes as (1 - t)^1/2 rather than the (1 - t) shape found near T_c in 3D superconductors. This type of dimensionality crossover has been experimentally observed both in TMDCIC's[200] and artificially structured superlattices (which are discussed in Chapter ).[208] As an explanation for anomalies in superconducting GIC's, dimensionality crossover has a lot of appeal, since so many of the normal state properties of GIC's are quasi-two-dimensional. For example, the anisotropy of the conductivity in C₄KHg is sigma_a/sigma_c 300.[85]

  
Figure: Theoretical demonstration of dimensional crossover in Josephson-coupled superlattices from the work of Klemm, Luther and Beasley.[131]. r is the parameter which characterizes the dimensionality of coupling. alpha, tau_SO, and H _P == 4 k_B T_c/ pi mu are parameters which characterize the degree of Pauli-limiting (Pauli-limiting is discussed in Section ). The inset shows a plot of T^*/ T_c (where T^* is the dimensionality crossover temperature) versus r.

According to KLB, a layered superconductor will have the possibility of a coupling-dimensionality change when T^* is greater than zero, which occurs when their parameter r is close to 1. The exact condition is

Calculated r values for a number of layered superconductors are shown in Tables and . The r's for the GIC superconductors are shown in Fig. . As can easily be seen, many of the transition metal dichalcogenide superconductors and their intercalation compounds are expected to show the dimensionality crossover effect, and indeed many do, as was discussed in Section .[46,200] On the other hand, the known GIC superconductors are more than an order of magnitude away from the critical value of r. The KLB paper found no anomalies for r >= 10,[131] as Fig. shows. Therefore, dimensionality effects can safely be ruled out as having any impact on the critical fields of the known GIC superconductors. This is particularly interesting in light of the fact that many GIC's have anisotropies 1/epsilon comparable to those of the TMDCIC's. For example, both C₈KHg and TaS_1.6Se_0.4 intercalated with collidine have anisotropies of about 30, yet the TMDCIC is expected to show a dimensionality crossover, and C₈KHg is strongly three-dimensional even at 0 K.

  
Table: Values of the KLB parameter r for the GIC superconductors. means that the value of r was calculated from parameters in the cited references.

At this juncture it is probably wise to discuss the proximity-coupled superlattice models of superconductivity[22,235], which have some similarities to the KLB model. These models are also capable of quantitatively explaining dimensional crossover, but in slightly different systems. Proximity-coupled superlattices are typically superconductor-normal-metal (SN) multilayers, while the Josephson-coupled superlattices described by the KLB theory are typically superconductor-insulator (S/I) multilayers.

The motivation for making this distinction was originally explained by Werthamer, and separately by Saint-James and deGennes.[261,54] At an S/I interface (the insulator may be vacuum), the pairs are confined in the superconductor so that they are almost unaffected by the presence of the boundary. In this case it is appropriate to ignore any variation of the order parameter within the superconducting layers, and to treat the interlayer coupling as due to Josephson tunnelling.[155] On the other hand, at an S/N interface, the pairs can easily diffuse into the normal metal, where they will be destroyed.[54] The destruction of pairs at an S/N contact results in the suppression of the energy gap at the interface, and is one aspect of the proximity effect. Another aspect is that the superconductor can to some extent induce superconductivity in a thin layer of the normal metal.[49] The proximity-coupled theories of Takahashi and Tachiki[235] and Biagi, Kogan, and Clem[22] (BKC) were developed to treat the S/N case.

In these models, the authors parameterize the S and N layers separately, unlike the KLB theory, where the properties of the insulating layers are ignored.[131] Takahashi and Tachiki[235] assign different densities of states N(0), diffusivities D, and BCS interaction energies V to the S and N layers. BKC, on the other hand, attribute to each type of layer a mean-free-path l, superconducting transition temperature T_c, and Fermi velocity v_F.[22] Because v _F and l determine D, while N(0) and V determine (with Omega_D) T_c, it seems that these parameterizations are more or less equivalent.

The full theories both predict positive curvature of H_c2(T). Takahashi and Tachiki found signs of a change in the coupling dimensionality for H_{c2, _|_ ^c}(T).[235] Qualitatively these results are quite similar to those of KLB. The exact criteria for dimensional crossover are different, though: the authors say that the necessary condition for decoupling of the layers (at T > 0) is s/ xi_{|| ^c}(0) >= 0.4. This condition is equivalent, in terms of KLB's r-parameter, to r <= 31.8. Thus the proximity-coupled multilayers decouple at longer coherence lengths (xi_{|| ^c} = 2.5s) than the Josephson-coupled ones (xi_{|| ^c} = 1.7s). Nonetheless, in the proximity-coupled superlattice models, no dimensionality change is anticipated for superconducting GIC's, which all exceed the limit in r by at least a factor of 2. (The possibility of dimensionality crossover in higher-stage GIC's is discussed in Section .) The source of this accelerated decoupling is not discussed in Ref. [235], but one can speculate that it comes from the suppression of the gap near the S/N interface by the proximity effect.

What is even more intriguing for those interested in GIC's is the enhanced critical field found for vecH || ^c. BKC[22] studied only this case, and found that ``the proximity effect alone can produce the positive curvature [PC] in H_c2(T) ... [despite the fact that] no provisions were made for anisotropy or any other effects commonly thought to produce PC in H_c2(T).'' These remarks are a little bit deceptive in that anisotropy has been implicitly introduced into the BKC model through the layer thickness parameters; that is, thin films are known to be anisotropic, and the individual layers have been chosen to be thin films. Nonetheless, it is impressive that positive curvature has been found in a superlattice formalism which is not linked to a dimensionality crossover. Takahashi and Tachiki also found PC for vecH _|_ ^c.[235] These findings would seem to have important implications for GIC's, for whom a possible explanation of PC using dimensionality crossover has already been eliminated.

Unfortunately there are several problems with applying the proximity-coupled superlattice models to GIC's. One objection is that assigning different mean-free-paths, diffusivities, T_c's and BCS interaction energies to the S and N layers seems a bit questionable when the layers are only a few atomic layers thick. These models were intended to be applied to S/N superlattices with layers on the order of a few-hundred Å thick, where each of the constituents individually has transport properties which are little modified from the bulk. For layers on the order of 10 Å thick, the transport properties are so modified from the pristine materials that use of the bulk parameters would be highly erroneous. The basic assumptions of the proximity-effect models are violated in GIC's, where the interaction between the host and intercalant is not a weak perturbation. Perhaps in the limit of very high-stage superconducting GIC's (if they exist!) agreement with the proximity-coupled description might be found.

A second reason for discomfort with the KBC and Takahashi and Tachiki models is the idea that superconducting GIC's can be described as S/N superlattices. While Iye and Tanuma felt that superconductivity in the ternary GIC's was probably due to superconductivity in the intercalant layers,[120] the idea that one component is superconducting and the other not appears somewhat shaky when xi_{|| ^c} 20 I_c, as in C₄KHg. Also, as Al-Jishi pointed out, there is good reason to expect that the graphitic pi-bands contribute to superconductivity in GIC's, as otherwise their large critical field anisotropy is difficult to explain.[4]

All these discussions are actually somewhat academic since the idea that GIC's are S/N superlattices turns out not to be self-consistent. The reason is that all proximity-effect theories[49,235] predict that the T_c's of S/N multilayers must always be less than the bulk transition temperature of the superconducting component. In fact, these models predict that, for small layer thicknesses, T_c must decline monotonically with increasing thickness of the normal layer, all other parameters being held equal. The monotonic decrease of T_c with increasing normal metal thickness is illustrated in Fig. for Pb/Cu bilayers.[261] Apparently no studies of T_c versus normal metal thickness have been performed in metallic superlattices; the thicknesses of both the S and N components are usually varied together. (See Figure .) However, the depression of T_c with increasing normal metal thickness is also expected in the superlattices. If one believes that only the intercalant layer is superconducting in GIC's, and thinks of the carbon (graphene) layer as a normal metal, then one would expect a decreasing T_c with increasing stage. Yet, as is well-known, T_c is higher in stage II than stage I for both KHg- and RbHg-GIC's,[116] in contravention to the expectation of the model.

  
Figure: T_c versus normal-layer thickness for S/N bilayers. Figure taken from Ref. [261] Here D _N and D _S are the thicknesses of the normal and superconducting layers, and T _cS is the bulk T_c of the superconducting component. Approximately T/ T_cS = (1 - t(D_N -> infty))(1 - exp-2D_N/xi), where xi is the dirty-limit Pippard coherence length.

The conclusion is that, while the proximity-coupled models have been extremely successful in fitting data on artificially structured superlattices (for example, those of Nb/Ta[29]), they do not seem to be well-suited to GIC's. Perhaps if high-stage superconducting GIC's are ever discovered, it might be worthwhile to take another look at proximity-coupling. For the known GIC's, though, it is imperative to consider alternative theories.

Even though intuition says that the enhancement of the critical fields in superconducting GIC's is linked to their anisotropy, it is important to consider whether mechanisms not related to anisotropy could also play a role. The major causes of critical field enhancement in isotropic superconductors are spin-orbit scattering and strong-coupling behavior.