Characteristic features of the exotic superconductors

CHARACTERISTIC FEATURES OF THE EXOTIC SUPERCONDUCTORS

B. BRANDOW Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

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Physics Reports 296 (1998) 1-63

Characteristic

features of the exotic superconductors Baird Brandow

Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received

June 1997; editor: D.L. Mills

Contents I. Introduction 2. The exotic superconductors of Uemura 3. Typical features of the exotic superconductors 3.1. Short coherence length 3.2. Large and T* or T normal-state resistivity; resistivity maximum 3.3. Anomalous form of I&z(T) 3.4. Large penetration depth, low chargecarrier density, large effective mass 3.5. Gap magnitude, temperature dependence, and anisotropy 3.6. Absence of the Hebel-Slichter peak, evidence for gap nodes

10 17 22 24

3.7. Gap symmetry of the cuprate superconductors 3.8. Miscellaneous 3.9. Crystal-chemistry features 3.10. Exotic features of the borocarbide superconductors 3.11. Short-chain Chevrel compounds 3.12. Cubic Laves-phase cerium compounds 4. Summary and concluding remarks References Note added in proof

28 31 37 37 39 40 40 45 62

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Abstract The exotic superconductors of this survey are those defined by Uemura and co-workers - the materials which approximately satisfy an empirical relation I”, K I,‘, where 1~ is the London penetration depth. As superconductors these materials are strange in many respects - in their chemistry and crystal structures, in many of their electronic and superconducting properties, and in their often conspicuously high transition temperatures. This category includes all of the presently known high-temperature superconductors. We now examine their unusual features in considerable detail, to sort out the features which are apparently universal (such as strong type-11 behavior and high resistivity), or which are non-universal but common enough to be considered typical for this class (such as gap nodes). Several characteristics of the crystal chemistry are identified. Although fragments of this program have been reported often, this is the first attempt at a comprehensive examination. There appears to be a quasi-continuum of electronic behaviors, ranging from strongly exotic down to barely exotic cases. The location of a material within this continuum may thus depend on the relative strength of some “new” mechanism, as compared to the conventional phonon mechanism. In various places this survey is broadened to include other “strange formula” superconductors, since many other materials share the typical crystal-chemistry features of the exotic materials. @ 1998 Elsevier Science B.V. PACS: 74. Keywords: Superconductivity;

Exotic superconductors

0370-l 573/98/$19.00 0 1998 Elsevier PIZ so370-1573(97)00071-9

Science B.V. All rights reserved

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1. Introduction There are many superconductors which have strange chemical formulas, strange in the sense that they are neither pure metals, alloys, or intermetallic compounds [ 11. The materials of this survey typically include non-metallic or semi-metallic elements, which is counter-intuitive for superconductivity. These strange-formula materials are more than just curiosities, because this category includes all of the presently known high-temperature superconductors. An examination of their unusual features, and a search for common aspects and trends among these features, should therefore be worthwhile. According to the BCS theory [2], based on the conventional phonon mechanism, the various properties of the superconducting state should generally scale with T, in a universal manner. The strong-coupling extension of this theory, due to Eliashberg, allows for deviations from this universality, and this theory is well confirmed by agreement with the properties of many known “strong coupling” superconductors [3,4]. It is therefore striking that, in a number of respects, various examples of the strange-formula materials show prominent departures from the usual BCS or strongcoupling phenomenology. (This is in addition to their often-high T,‘s.) Although the meaning of these anomalies is debatable, as discussed below, they can reasonably be interpreted as evidence that these non-standard superconductors are employing some “new” mechanism(s). This has indeed often been suggested. Arguments of this sort are common for the cuprate superconductors, but there is also a long history of such suggestions for other unusual materials. Examples of these suggestions can be found in most of the published proceedings from superconductivity conferences over at least the last two decades. It has also been argued many times that the electronic properties of various examples of the strange-formula superconductors exhibit significant similarities with each other, and/or with the high T, cuprates [5]. Some of this evidence is well known, but we shall demonstrate here that the similarities are more extensive, and perhaps more dramatic, than has generally been recognized. It is quite impossible to present a “final word” on this subject, because some of the desired data is missing or of questionable validity, and there is certainly more to be learned in evaluating this data. Nevertheless, enough information is now available to arrive at some interesting and hopefully useful conclusions. We focus here on the more extreme examples, the superconductors which have been termed “exotic” by Uemura and coworkers [6]. ’ The evidence suggests a continuum of behavior, with the various strange-formula superconductors ranging from highly exotic to barely exotic. It is, of course, quite possible that some of the strange-formula superconductors are not exotic at all, i.e. they are just conventional phonon-driven superconductors. Considering the present weight of evidence, however, we shall argue (in the concluding section) that such purely phononic cases are unlikely and probably rare. With this background as motivation, the intention in this survey is to pull together the various kinds of evidence that are now available, (a) for significant differences from the conventional superconductors, and (b) for a commonality among these differences, i.e. for the existence of some general trends. We, of course, do not catalogue every anomaly of every exotic superconductor; the features to be discussed must have been observed in several of the material families, so that a trend or tendency can be argued. The choice of features discussed here is representative and extensive, but probably not exhaustive. It is inevitably somewhat subjective. Although the literature already ’ In Ref. [6], see in particular Uemura et al. (1989, 1991, 1993) and Uemura and Luke (1993). See also [16-19,361.

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contains a number of pieces of this program [5], their scope has been so limited that the overall picture has remained unclear. This is the first attempt at a comprehensive examination. We turn now to the question of whether these exotic features mean that some “new” mechanism is operating in the exotic superconductors, presumably together with the conventional phonon mechanism. There is now a widespread agreement that some non-phononic mechanism must be operating in the cuprates, although this opinion is not universal. The most commonly quoted evidence is the extraordinary high T,‘s (now ranging up to 164 IS [7]), the evidence for gap nodes, and the very small isotope effect. (Nevertheless, even for these features a phonon source can be argued [8,9].) For most of the other exotic superconductors, however, there is no such agreement. This is because it is usually quite hard to obtain clear evidence against the adequacy of the conventional phonon mechanism. For example, an Eliashberg analysis can be ambiguous even if there is good tunnelling data. The problem is that if there is a non-phononic contribution to the gap A, the Eliashberg tunnelling analysis will increase the overall normalization of the spectral function a2F, or otherwise distort this function, in order to fit this A. Then, since A has been used as input, a rather good calculated value of T, can often be expected, simply because the gap ratio 2A/kaTc often lies in the usual “strong coupling” range. Another example of the difficulty is in calculation of the electronphonon coupling parameter A by band-theoretic methods. In a number of cases this has led to large 1 values which are claimed to sucessfully explain the observed T,‘s. The problem here is that these “ab initio” calculations ignore the effect of the correlations resulting from a Hubbard interaction U. Theoretical studies have shown that these “strong correlations” can significantly reduce the value of 2 [lo]. In spite of these problems, however, a clear inadequacy of the phonon mechanism has been claimed in at least one case (Nb3Ge) [ 111. The problem of reconciling high T,‘s and small isotope-shift c1values has also led to evidence against a pure phonon mechanism for some higher-T, cuprates (YBCO and Bi-2212), but the result of this analysis was ambiguous for La2_,SrXCu0,, and Bal_,KXBi03 [ 121. There is also a significant recent development using the Eliashberg formalism to analyze the imaginary part of infrared conductivity. Application to B~J&Bi03 (“BKBO”) data has provided an electron-phonon ;1 of only about 0.2, in contrast to the value 1% 1 which is needed to explain the high T, [ 131. This is strong evidence that the conventional mechanism is inadequate for this material, and this is also consistent with the argument [lo] that the correlations produced by a Hubbard U can strongly reduce A. It was once believed that the Eliashberg theory predicts an upper limit for T,, as a function of 1, but since the work of Allen and Dynes [3] it has been recognized that T, continues to increase with increasing il. There is undoubtedly an upper limit to A in real materials, set by lattice instability, but this limit is unfortunately still unknown [4]. A study of cuprates within the Eliashberg framework has suggested that the phonon mechanism is inadequate, on the basis of apparent numerical inconsistencies, but the experimental uncertainties at that time prevented a firm conclusion [4]. In view of the frequent ambiguity, the question of some new mechanism as a source of the “exoticity” cannot be settled here. The case for another mechanism operating (in addition to the phonons) is necessarily of a circumstantial nature, at least for most of these materials. This case is based on the wide range of observed anomalies, and on the rather systematic occurrence of these anomalies among many materials, including some materials which are clearly problematic for the phonon mechanism. The “universal relation” of Uemura (Section 2) is important evidence here, as are also the typical shared features in the crystal chemistry of these materials. (This is especially

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so for the large U’s; see an argument in Section 4.) This evidence is presented here for readers to weigh. It is only fair to reveal our bias in this matter. We believe that there is indeed “method in the madness”. It is of course unreasonable to suppose that any “new” mechanism should manifest itself only in the cuprate family of superconductors. We find the observed trends to be consistent with, and suggestive of, a particular one of the many new mechanisms proposed for the cuprate superconductors - a valence-fluctuation mechanism [ 14,151. The overlap between the predicted and the observed properties is really quite extensive. Even the non-universality that is found for a number of the exotic features can easily be rationalized. Furthermore, the perspective of this mechanism, and the apparent requirements for realizing this mechanism in nature, direct attention to some common features that might otherwise be unappreciated. Specifically, this mechanism is able to rationalize the typical crystal-chemistry features of the exotic and other strange-formula superconductors, in addition to rationalizing the typical anomalous electronic properties. Although a preliminary explanation of exotic features in terms of this mechanism has been published [ 151, the mass of data presented here now calls for a more extensive discussion. We intend to publish this elsewhere, as a theoretical sequel to this report. An outline of the resulting theoretical picture is presented in the concluding section of this report. We mention only one aspect here, as a forewarning: This mechanism is generally expected to produce an anisotropic s-like gap, possibly with gap nodes, the node issue depending on details of the band-structure. (The heavy-fermion materials have extra complexity, however, and therefore may have more complicated gap forms.) Now, a comment for readers who are mainly interested in the high-T, cuprate superconductors: We must emphasize that the experiments discussed in this report are those which demonstrate similarities among the various exotic superconductor families; apart from a discussion of the gap symmetry, there is no attempt here to cover the great mass of other cuprate experimental results. We recognize that some readers may find this objectionable. Our response is that we consider the similarities to be just as important as the differences, and in any event the similarities need to be explained. This is a different perspective from much of the cuprate literature, which has often claimed a uniqueness for the cuprate family of superconductors. Some revision of this common perspective seems warranted because it is now clear that the similarities are indeed quite extensive. This is another aspect for readers to weigh. It is tempting to conclude this report with a tidy phenomenological definition of the exotic superconductors, but we have not done so. We feel that the nature of the boundary between exotic and conventional still involves too many questions. We merely list the anomalous features which appear to be universal for the exotic materials, meaning here the materials denoted exotic by Uemura et al. In Section 2 we briefly describe the work of Uemura and collaborators, which led them to categorize a number of superconductors as “exotic”. Section 3 lists the typical or characteristic features of the exotic superconductors, and these features are then examined in some detail. A summary and concluding remarks are in Section 4.

2. The exotic superconductors of Uemura A convenient focus and starting point for this discussion has been provided by Uemura and his many coworkers [6], who discovered that a number of the strange-formula superconductors share a

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curious property. They have used I&R (muon spin relaxation) as a tool to determine the London penetration depth AL, for a number of these unusual superconductors. The remarkable discovery was that these materials approximately obey a new “universal” relation, T, cc Ai2, even though they represent quite different crystal structures and chemical families. (This &_ is the extrapolated T = 0 value). The main results to date are illustrated in Fig. 1, and perhaps more clearly via the logarithmic scale in Fig. 2, which is derived (with a minor modification) from the data in Fig. 1. The examples plotted in these figures include superconducting cuprates, a bismuthate (Bal_,KXBi03, or “BKBO”), A3CG0’s (alkali fullerenes), an A- 15, Chevrels, organics, and also heavy-fermion superconductors. The more recent additions are the “almost heavy-fermion” materials &Fe [16], UPd2A13 and U2PtC2 [17], the layered dichalcogenide NbSe2 [18], and the spine1 compound LiTi [19] which was actually the original high-T, oxide. (Their most complete version of Fig. 2 is in Ref. [19].) The Tc’s of these examples span more than two decades, from N 100 to < 1 K. Uemura et al. have categorized these materials as “exotic” superconductors, hence the title of this survey. Although “exotic” is a term that could easily be abused, their extensive work has made this acceptable and reasonably unambiguous. (This should not be confused with the term “unconventional superconductivity”, which is usually intended to mean superconductivity with novel quantum numbers, i.e. non-s-wave and/or non-singlet Cooper pairs. In a less specific sense, of course, any departure from the phononic BCS or strong-coupling systematics is “unconventional”.) It is clear from these figures that the exotic superconductors do not all fall on a single line, but instead occupy a band of considerable width. The “universal line” of Uemura et al. is the upper edge of this band. Within this band the cuprates show some interesting further structure, as a function of the amount of hole-doping, but we shall not consider this structure here. (Similar structure can be seen in the Chevrel examples.) It is tempting to conclude that the position of a material within this band denotes its degree of exoticity. Thus, for example, some of the Chevrel compounds are

B. Brandow I Physics Reports 296 (1998) or n,“‘/(m’/m,) 1o12

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apparently quite exotic, whereas V3Si, the only A-15 example, appears only modestly exotic. For comparison Fig. 2 also shows some conventional superconductors, which are seen to lie far away from this band. It is interesting that niobium appears in an intermediate position between the exotic and the ordinary superconductors. This is intriguing because niobium is the element with the highest T,, and it is also unusual in having intrinsic type-II behavior, and furthermore it is an element which enters prominently in a number of the exotic and other strange-formula superconductors. (We note in passing that vanadium [20], technetium [21] and lanthanum [22] are also elements with intrinsic type-II behavior. Technetium also has an exceptionally high T, of 7.8 K [21], second only to niobium among the elements.) In theory, AL2 is proportional to oz, the square of the plasma frequency. It is therefore noteworthy that the correlation T, oc 0; has also been directly observed, for several cuprates, where or is derived from room-temperature optical reflectivity data [23]. Noting that ~0; 0: n/m*, and comparing with the Hall-effect determination of the charge-carrier density ~1, the authors concluded that the correlation is at least partially due to a strong hole-doping dependence of the effective mass m* [24]. Uemura and coworkers have suggested that their T, c( AL2 relation may indicate a boson condensation, possibly of pairs which persist in a considerable temperature range above T,. (The plot of Fig. 2 was designed to illustrate this boson condensation aspect.) A number of other studies have also pursued this theme, focussing mainly on the boson limit of BCS pairing, i.e. the limit of very small Cooper pairs (t/a 4 1, where { = coherence length, a = lattice parameter) [25]. This is an interesting and possibly useful idea, but this falls far short of being a proper dynamical theory. (The smallness of t/a is generally assumed rather than being properly explained and calculated. Also, the empirical l’s are generally at least several times a, so the relevance of the boson limit is

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problematic [26].) We consider it likely that the boson limit does have some relevance here, but the reason for this and the limitations of this concept remain to be clarified. We shall take up this issue in the theoretical sequel to this survey. Another attempt to understand the Uemura relation is based on the concepts of strong disorder and incipient localization [27]. This work has focussed particularly on the relatively quite large normalstate resistivities of these materials just above their z’s, and on the effects of induced disorder such as from radiation damage. It is certainly true that the “strange formula” materials (including all of the exotic superconductors) generally have considerably higher defect concentrations than the simple metals, and there are cases where a high degree of disorder is apparently unavoidable - e.g. in the A&‘s [28]. Nevertheless, many experimental studies of exotic materials have concluded that the samples are relatively “clean”, in the sense that “dirty” phenomenology (for mean-freepath
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exotic. The main concern here is to understand

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3. Typical features of the exotic superconductors We begin here by listing a number of electronic properties which are anomalous from the standpoint of BCS behavior, but which occur fairly often in the exotic materials - often enough to be considered typical or characteristic for these materials. We also list some crystal-chemistry features which are characteristic for the exotic superconductors. These latter features are also shared by most of the other strange-formula superconductors, and this fact provides a strong motivation for examining this broader class of materials. The anomalous electronic properties include: (a) extremely short coherence length, (b) extremely high normal-state resistivity, with low-power-law (T2 or T) behavior suggesting strong electronelectron or quasiparticle scattering, (c) a highly anomalous form of Hc2( T), with upwards curvature over a broad range of T and a strong linear variation as T --+0, (d) large penetration depth [which together with (a) causes type-II behavior, typically “extreme” type-II with K $11, (e) low chargecarrier density, of order one per formula unit (or per “active site”, see below), (f) an enhanced electron effective mass, i.e. a reduced dispersion for the normal-state quasiparticle spectrum, as compared to the prediction from band theory together with the (1 + A) state-density enhancement factor from the electron-phonon coupling, (g) in some cases coexistence of superconductivity with a lattice of rare-earth-ion magnetic moments, (h) sometimes very large values for the dimensionless ratios 2A/kBT,, AC/yK, ( i ) sometimes a “more square” form of A(T), i.e. a scaled form which lies above the familiar BCS form, (j) sometimes evidence for gap nodes, (k) sometimes absence of a Hebel-Slichter peak in NMR (nuclear magnetic resonance), (1) a small isotope effect, and (m) in some of the “heaviest” and relatively low-T, materials, a peak in resistivity at a T of order ten times T,. Obviously, not all of these features apply to every material. Some of these have been confirmed in only a few examples, and for some features there are also counterexamples where conventional behavior is found. Nevertheless, the evidence does seem to show that these are genuine trends which deserve some quasi-universal explanation. This list is representative but probably not exhaustive. In the crystal-chemistry category, the typical features are: (i) nearly universal [31] existence of an apparent large-U unit (an ion or cluster of ions), thus suggesting a Hubbard or Anderson-lattice form of model Hamiltonian, (ii) existence of non-large-U ions, usually non-metal or semi-metal p-electron ions, and/or sometimes highly electropositive elements, so that the various constituents involve quite different electronegativities, and (iii) frequent observation of a low-dimensional aspect, and/or clustering, in the crystal structure. We shall refer to the large-U unit as the “active site”. We now examine these items. 3.1. Short coherence length Extremely short in-plane coherence lengths c have been determined for some of superconductors: about 16A for YBa2Cu307-,, [32], 13.6A for Tl-2223 [33], 9.7 A and 9fl A for Bi-2212 [34]. Since the in-plane lattice parameter a is about 4A, these
the cuprate for Bi-2223 results give determined.

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Use of fluctuation theory in the data analysis can give a smaller 5 [34, 351.) These results are enormously different from the typical t/a ratios of - 1O3 for ordinary phononic superconductors (see below). It is quite remarkable that several other exotic superconductors are found to have nearly the same l/a ratios. The comparison seems more appropriate for c/a rather than 5 alone, since in many of these exotics the appropriate lattice parameter is large. Also, t/a relates more directly to microscopic theories. In cases with several “active” (U-bearing) units in a unit cell, for example in the A3Ch0 materials, the nearest-neighbor distance d,,n (e.g. between the C6,, centers) is probably still more appropriate. (Uemura et al. [36] have expressed essentially this idea in the form t/d M constant, where d is the mean spacing between the charge carriers - the in-plane spacing for the two-dimensional materials. The connection is that the charge carrier density in an exotic material is typically around one per active site.) It is therefore undoubtedly significant that the minimum t/a or t/d,, ratios found in several of the other exotic superconductor families are actually around 2.5-3.5: PbMo& (l= 23 A [37], a = 6.54A), Rb&o (c = 20-248, [38] or 27-30 8, [39], afcc= 14.3 A, d,, = afC,/2‘I2= 10 A), K-(ET)2Cu[N(CN)2]Br [lac = 24 8, [40], a = 12.9 A, c = 8.5 A, d,,, = (a2 + c2)‘12/2 = 7.7 A]. (It has been argued that the short l’s of the A3Cs0’s are merely due to the very short mean-free-paths resulting from the C 60 orientational disorder; see the following section on resistivity. But it is also quite possible that the intrinsic values &, are very short in these materials. This issue has not been settled experimentally, since there is a wide range of estimates for the transport mean free path.) It is not at all clear why these t/d,, ratios are so similar, nor why the apparent limiting values are around 2.5-3.5. Nevertheless, this is a dramatic demonstration that these very different materials have some profound similarity. For comparison, some
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for an electron-doped Ndz_,Ce,CuOb single crystal the 300K resistivity was 600 @cm [44]. For the A-l 5’s this is 70-90 ~0 cm [45,46]. For the Chevrels, although a single-crystal measurement on PbMo& gave about 600 u0 cm, this was considered excessive and was attributed to the presence of microfractures in this very brittle material [47]. Indirect determinations for thin-film samples gave about 300 uR cm for PbMo&&, and 200 uQ cm for Cu2.rMo6Sx [48], while later measurement of a zone-refined single crystal of CU,.~MO&(~ = 11.3 K) showed 61 uR cm [49], similar to the A-15 values. The resistivity of BKBO samples is highly variable, ranging from 240 to several thousand u0cm [50, 511. A single-crystal B~,61(0.4Bi03 sample with T, = 30K showed 800 l&cm at 300K [52]. The strongly varying magnitudes and T-dependences of many samples could be reconciled with a model consisting of a metal and a semiconductor in parallel [51]. This strange behavior is apparently due to semiconducting material at grain boundaries [53], or to some other intergrowth of metallic and semiconducting material on a microscopic scale. The measured resistivities of K3Ce0 and Rb3C6,, are also problematical. Both single-crystal [54] and thin-film [55] measurements of the T + 0 extrapolated normal value gave p. > 1000 pfi cm. The Rb3Cb0 value at 300 K is about 3000 @cm, and this continues rising steeply through 5000 uR cm at 500 K with only a beginning of downwards curvature [55]. This high-T behavior is clearly incompatible with the Mott limit, as discussed below. The thin-film samples have the potential problem of insulating material (C 60, AICeO, A4CG0, or A6CG0) at the grain boundaries [53], and the alkali-doping of Ceo single crystals may also be inhomogeneous, with well-doped material only near the surface. It therefore appears that the several indirect determinations of p. [56], of 200-600 uR cm for Rb3Ceo (120-500 ~0 cm for K3CG0) are more representative for the intrinsic value. The higher-T results should presumably be scaled down proportionally. For the organic superconductors, the direct single-crystal measurements of room-temperature resistivity (in-plane) are in the range 20-50mR cm [57]. These values are certainly larger than any reasonable estimate of the relevant Mott limit, and this gives credence to the claim [58] that the data are suffering from microcracks and/or from an artifact of the extreme anisotropy, which can confine the current to only a fraction of the sample cross-sectional area. On the other hand, the observed temperature dependence may still be meaningful, and this shows a very remarkable behavior. The higher-T, examples typically show a broad resistivity maximum at T N 100 K, preceeded by a very steep rise and followed by a gentle fall, with the peak resistivity being as much as three orders of magnitude larger than the value just above c. The most extreme example of this is shown in Fig. 3. This behavior has been found in the T, > 10 K materials IC-(ET)~X for X=CU(SCN)~ [59], Cu[N(CN)JBr [60], Cu[N(CN)JCl [61], and also in the lower-T, materials (DMET)*AuBr2( T, = 1.9 K) [62], Ic-(ET)2Ag(CN)2H20 (T, = 5K) [63], L-(BETS)2GaC14 (T, = 8K) [64], and two JC-(ET)~ materials containing Cu(CF3), (T, = 4 and 9 K) [65]. (It is therefore somewhat surprising that this was not found in the T, = 11.2 K material rc-(ET),Cu(CN)[N(CN),] [66]. In K-(ET)&&(CN) 3 with T, = 3.8 K there is also no peak, but here there is a distinct shoulder around 100K [67], and in a-(ET)2(NH4)Hg(SCN) 4 with T, = 1.15 K there is a shoulder at 30 K [68].) A recent addition to this family, cr-(EDT-TTF)[Ni(dmit)J with T, = 1.3 K, shows a similar broad resistivity peak around 14 K [69]. There has not been a satisfactory explanation for this striking behavior. Corresponding thermal anomalies have been found in in-plane lattice parameters, in the X= Cu(SCN)z and Cu[N(CN)2]Br materials [70], and it has been proposed that the resulting changes in electronic structure lead to the observed resistivity peaks [71]. This proposal replaces one problem by another: if the lattice anomalies are really the sources of the resistivity peaks,

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Fig. 3. Resistivity of K-(ET)~CU(SCN)Z. Note the initial quadratic behavior, the broad maximum enormous increase between TC= 10.4 K and this maximum. From Saito et al., Ref. [59].

around 85K, and the

then an explanation is required for these lattice anomalies. As an alternative, we note that similar resistivity behavior is sometimes found in the heavy-fermion compounds (both superconducting and non-superconducting) [72], of course, at a lower temperature scale, as illustrated in Fig. 4. (Lattice-parameter anomalies are also typical for the heavy-fermion materials [72].) This similarity suggests that the resistivity maximum in the organics is a valence-fluctuation or Kondo-related effect. This form of resistivity maximum is easily explained in terms of the Kondo-temperature or thermal moment-unbinding aspect of valence-fluctuation theory [73], and it is therefore quite reasonable to regard this maximum as a signature that the normal (above T,) state is a valence-fluctuation state. (This will be discussed in the theoretical sequel.) This form of resistivity maximum has also been found in a short-chain Chevrel compound (Section 3.11), and in the low-T, (= 0.93 K) superconductor Sr2Ru04 [74], which has the same crystal structure as the prototype cuprate La2Cu04. In Sr2Ru04, however, this is seen only in the c-axis resistivity. In the heavy-fermion materials, the resistivities at the p maxima are mostly in the range 120-250 uR cm [72]. Their 300 K resistivities are generally smaller than these local-maximum values. They are usually not much smaller, but can be as low as 75 ~0 cm (for single-crystal CeCu2Si2). For the “almost heavy” material I&Fe the 300 K resistivity is about 130 ~0 cm [75]. For NbSez this is about 160 uR cm [76]. An anomalous temperature dependence somewhat like that of the organic superconductors has been found in the spine1 oxide LiTi [77]. There is a broad peak centered around 200K, but with the difference that the residual or background resistivity is now very large - larger than the increase between T, and the maximum. The overall scale of this resistivity is far too large to be intrinsic, and is undoubtedly due to the grain boundaries and porous geometry of the sintered ceramic sample. (The intrinsic resistivity of LiTi has apparently not been determined.) The overall form of this temperature dependence closely resembles some of the data [50, 511 for the problematic material BKBO, where there is clear evidence that this behavior results from some microscopic intergrowth

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T(K) Fig. 4. Resistivities of several heavy-fermion compounds. Although form is always similar to this data. From Steglich, Ref. [72].

CeCuz:Si2 data is quite sample-dependent,

its general

of metallic and semiconducting material [51]. Thus, although somewhat similar to the organic data, a valence-fluctuation interpretation does not seem appropriate for the available LiTi204 data. As already mentioned, the Mot&-Ioffe-Regel limit for metallic resistivity [78] is quite relevant here. A typical estimate for this is PMOtt= 3ha/e*, which is for a simple cubic lattice with one valence electron per site, lattice constant a, and a mean-free-path 1 = a. This result is independent of the scattering mechanism which provides the short mean-free-path [79]. For a = 4 A (about right for the cuprates) this estimate gives &ott = 480 ufl cm, for a = 6.5 8, (for the Chevrels) this gives P&n = 780 pfl cm, and for a = 10 A (the C& separation in the alkali fullerenes) this gives 1200 uR cm. It is clear that the 300K resistivities of the exotic superconductors are typically a considerable fraction of these PMOttestimates. To describe the temperature dependence of the approach to the Mott limit, a “parallel conductor” where the subscripts denote the total, the model is often used, [p&T)]= [P&T)]-’ + [P&‘, corresponding non-saturating expression, and the saturation or Mott-limit value [80]. There is actually a simple and quite general microscopic justification for this parallel model; this follows from the rather obvious requirement that an electron (or quasiparticle) cannot be scattered until it has travelled a distance of at least one bond length or atomic separation from the previous scattering [81]. One can therefore reasonably conclude that the present high-resistivity materials should generally exhibit some downwards curvature, in the high-temperature range ( 2 300 K). The A-15 compounds are well known for their saturation tendency [46, 80, 821, i.e. downwards curvature and apparent asymptotic approach to a constant (pSat) at high temperature. This tendency is also prominent in the Chevrel compounds [48]. It therefore seems strange that the cuprates have generally not shown any sign of such saturation [83], except for underdoped materials [84]. After correcting for thermal expansion, however, some downwards curvature can be seen even in the cuprates [85]. It is also interesting here to note an older observation that superconductivity in d-electron (transition metal) compounds is typically associated with a saturation tendency [86]. With the reasonable inference

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of large intrinsic magnitudes for these compounds, this can now be seen as further evidence for the common association of very large resistivity with (possibly) exotic superconductivity. This latter work also found a correlation between superconductivity and low-power-law resistivity near T, (quoted there as T3 behavior); this aspect is examined below. The Mott limit helps to explain why the residual resistivities p. (T + 0 extrapolation values) of K3CG0 and Rb3Ceo are so enormous - already a considerable fraction of p&R % 1200 pR cm. This is apparently due to the known frozen-in orientational disorder of the Ceo molecules, and the sensitivity of the relevant transfer integrals to this disorder [28]. On the other hand, at high T ( 2500 K) the directly-measured Rb3CGo resistivity almost certainly exceeds the above estimate for &,n. This is likely to be true even if the p M 5000 yR cm value at 500K is scaled down as indicated above, because p continues to rise steeply above this temperature. We note however that the a factor in the above expression for PM,,tt can be replaced by a2/Z. (This follows from Ref. [78] when one assumes 1 #a.) This can make quite a difference, since the appropriate a = 10 A (distance between the Cbo clusters) is large. The a2 factor comes from the Fermi surface area, and the often-large a2 value for the exotic materials is a reflection of the low charge-carrier density, typically around one per active lattice site (Section 3.4). Presumably, a strong electron-electron scattering (see below) should lead to a minimum mean-free-path 1 z a, since the quasiparticle interaction arises mainly from the large U on the active sites. But if there is also a strong electron-phonon contribution (which musl be the case at sufficiently high temperature), its effect should reduce the minimum I to around an interatomic distance (Z 1.4 A in CGo). The Mott limit for the metallic Ceo compounds should therefore be considerably larger than the above estimate. The fact that such very large resistivities pose a serious problem was emphasized by Anderson and Yu [29]. They estimated transport electron-phonon coupling constants lit, of order 100 for some of the exotics (mainly A-15’s). This result is strongly inconsistent with the L’s (-1.5) estimated from the corresponding T,‘s, and also inconsistent with conventional notions of electron-phonon coupling. They therefore described these problematic superconductors as “bad actors”. The AndersonYu arguments assumed that electron-phonon coupling must be the only source for both the resistivity and the superconductivity. Although Gurvitch [87] has resolved this quantitative paradox, Anderson and Yu were certainly correct in emphasizing the problem of why these relatively high-T, materials typically have such enormous resistivities, far greater than the strong-coupling example of lead. The typical low charge-carrier density of these materials is surely an important part of the explanation, as mentioned in connection with the Mott limit, but it now seems clear that this is not the whole story here. Another important aspect is electron-electron or quasiparticle scattering. The rather high state densities of these typically-narrow-band materials should enhance the relative importance of the quasiparticle scattering contribution, and likewise the reduced metallic screening which results from the low carrier density, together with the “lumpiness” (tight-binding aspect) of the relevant Bloch orbitals, so it is not unreasonable that this contribution may be dominating or at least contributing significantly in the exotic materials. [Arguments for the weakness of electron-electron scattering have usually been based on electron-gas (jellium) estimates.] An important question is whether one finds the expected power-law signature for quasiparticle scattering. Usually, the proper signature is p = p. + AT2 at the lowest temperatures. The issue here is not whether there is any Tz component in the resistivity, but whether this is sufficiently prominent so that there is some temperature region above T, where this is clearly observable. In the case of the superconducting cuprates, however, the proximity of &r to the van Hove saddle-point

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energy leads to an unusual band geometry, with the consequence that quasiparticle scattering is predicted here to give p - p. IX T, provided that I.+ - cvn/
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0.6 P 0.4

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0

1.v

2.u

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T Fig. 5. The dimensionless and scaled saturating resistivity, p = r*/(T’ + 1) (solid line), which results from the assumption of a pure T2 form for the non-saturating resistivity expression pns (dashed line), according to the parallel-conductivity model. The Mott-limit value here is 1.0. This form resembles the observed saturating resistivities of A-15, Chevrel, and other d-electron superconductors, as can be seen in Refs. [45,46,48,80,82,86].

An obvious problem for a genuine T2 resistivity mechanism is that the T2 behavior cannot continue up to high temperature without conflict with the experimental data, and also conflict with the relevant Mott limit. We therefore recall here that the above-mentioned parallel conductivity model is well justified [81], and note that this provides at least a conceptual resolution of this problem. In fact, we have found that a plot of the parallel-conductivity formula, based on a pure T2 assumption for the non-saturating expression &T), does indeed look qualitatively like the typical resistivity data for A-15 and Chevrel materials quoted above. This plot, in the dimensionless form T2/(T2 + I), is shown in Fig. 5. It is significant that at high temperatures (above the Debye temperature) the T2 resistivity of quasiparticle scattering rises faster than the linear contribution from electron-phonon scattering, so that the quasiparticle scattering must eventually dominate. The resulting more rapid approach to the Mott limit provides a rationalization for the very large resistivity magnitudes (say at 300K), and also for the frequent observed saturation tendency in the exotic materials. A further subtlety is that a high resistivity magnitude does not necessarily prove dominance by a purely electronic contribution_ The high-resistivity transition metals mentioned at the beginning of this section may be examples, where phonon-induced s-d scattering is a reasonable alternative source for their high resistivities [42]. There are exotic or at least strange-formula cases (BKBO and VN) for which a quantitative study, based on tunnelling-derived spectral functions a2F, has apparently demonstrated the possible adequacy of the electron-phonon mechanism [ 1 lo]. (This was also shown for V and Nb, where this conclusion may seem more natural.) But this work actually shows only that electron-phonon scattering could account for the entire resistivity; this does not prove that the phonon

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mechanism is totally responsible here. This is because, as noted in the Introduction, the derived a2F may be artificially incorporating a non-phonon contribution to the pairing. Consequently, the use of this CX*Fas the transport spectral function may also be including a non-phonon contribution in the resistivity. On the one hand, this argument for a purely conventional electron-phonon resistivity seems at first sight to be strengthened by reasonable fits to the temperature dependences as well as the general magnitudes of the resistivities, but on the other hand the question of uniqueness (e.g. a bound on any additional T* contribution) was not addressed. There are basically two fitting parameters in this work - a plasma frequency (which sets the scale for p-p0 below 100-200 K), and a saturation (Mot&limit) value which determines any downwards curvature at higher temperatures. It was however acknowledged that this analysis did not adequately treat the initial (lowest T) powerlaw behavior, which is a major and crucial weakness in the present context. It was also noted that the deduced saturation value is sometimes poorly determined unless the resistivity data is available for several hundred degrees above room temperature. But in spite of these problems, there was one case for which the resulting apparent electron-phonon contribution was found to be clearly inadequate. This was for (Nd,Ce)&u04, the only cuprate examined in this study. It is significant here that the missing (non-phonon) part of the resistivity was found to be quadratic. This agrees with the expectation of a T* dependence for the electron-doped cuprate, because of its large value [89] of /sF - svn], according to the calculations of Ref. [88]. 3.3. Anomalous form of H,*(T) A highly anomalous form of H,,(T) has been observed in four cuprate materials - T12BazCu0, (Tl-2201) [ 11 l] (with T, lowered by overdoping), Bi2Sr2Cu0, (Bi-2201) [112], YBa2(Cuo.97Zno.03) 0,_a [ 1131, and in the electron-doped and rare-earth local moment material Sm,&eo.&uO~_, [ 1141. These materials were chosen for their relatively low T,‘s ( <20K), so that Hc2 could be accessed down to T E 0 with available magnets. (Throughout this section we are focussing on the case with field perpendicular to the planes.) Their HE2’s showed strong upwards curvature over essentially the entire range of T below T,, and no region of downwards curvature. In contrast to nearly all previous HCz’s for other materials, the region of steepest slope extended down to T = 0 (T <<1 K), with no sign of levelling off. An example is shown in Fig. 6. A major consequence of this upwards curvature and extended slope is that the TM 0 values of Hc2 are an order of magnitude larger than the conventional estimates based on the slope near T,. If this feature of anomalously large Hc2(0) is common for the exotics, as we suspect, then the quoted r values (often based only on low-field measurements) may need to be revised downwards for a number of these materials. Readers are reminded that in the standard theory [ 1151, valid for the more conventional superconductors, HC2 is proportional to (T, - T) just below T,, then at lower T it exhibits downwards curvature and approaches its T = 0 value with zero slope. It is quite striking that both of these anomalous HC2(T) features (broad region of upwards curvature, and steep slope as T + 0) have now also been observed in Bai_,K,Bi03 (BKBO) [116] which is three-dimensional, and in the Chevrel-related chain compound TlMo3Se3 [ 1171 which is quasi-one-dimensional. These features have also been often reported in the quasi-two-dimensional organic superconductors: /%(ET)213 [ 1181, and IC-(ET)*X for X= Cu(SCN)* [ 1193, Cu[N(CN)JBr 1120,=11, WCWN(CNM [661,and 13 [122]. In the lower-T, material K--(MDT-TTF)~AuI~ [123], the curvature becomes normal (downwards) for T/T, 5 3. It has been argued, however, that since

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0

2

4

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IO

I?%+%

TEMPERATURE (K) Fig. 6. &(I”) data for T12BazCuO6, with T, depressed below 20 K by overdoping with excess oxygen. Note the prominent upwards curvature, and the steep slope as T -+ 0. From Mackenzie et al. (1993), Ref. [ 1111.

most of this organic data was obtained from resistance measurements, the results may have been falsified by flux flow [ 120, 1221. (In most of these organic studies H,, has been identified with the point of 50% resistivity, even though the resistive transitions broaden greatly with increasing field.) In the case of K-(ET&u(SCN)~ the same anomalous features were found by DC magnetization [ 1241, but the criterion used there to identify Hc2 was actually instead the criterion for the irreversability line. Other magnetic [ 1201 and specific heat [ 122, 1251 data have given much larger slopes dHcJdT near T,, and a later magnetization study has confirmed that this large slope continues down to T/T, M 0.4, the lowest T in that study [ 1261. It therefore seems certain now that the common 50%-resistivity recipe has falsified the determination of H&T) for the organics. Nevertheless, a close inspection of the more sound data (see Fig. 3 of Ref. [ 1261 and Fig. 5 of Ref. [ 1221) does suggest some upwards curvature for T/T, 2 0.7. These organic materials clearly illustrate the subtlety and danger of mis-identifying Hc2, a subject we return to below. Further study would be worthwhile, to look more carefully for a genuine upwards curvature in the quasi-two-dimensional organic superconductors. Upwards curvature has also been found in many other materials, although the older data often does not extend to small T/T, values. This feature of upwards curvature was first recognized in the layered transition-metal dichalcogenides (NbSe 2, etc. including many intercalated materials) [127]. Besides these and the other materials already mentioned, this feature has been found in A- 15’s [128], Chevrels [ 1291, A3CG0’s [130], Ba&, [131], LiTi204 [132], in the almost-heavy materials &Fe [133] and URu2SiZ [134], as well as in elemental niobium and vanadium [135], tungsten and molybdenum bronze materials [ 1361, polymeric sulphur nitride (SN), [ 1371, intercalated graphite [ 1381, and in artificial multilayer films [ 139, 1401. It is striking that this list includes examples from nearly all of the exotic families (an exception being heavy fermions), and furthermore that the remaining examples are mostly “strange formula” materials which are reasonable candidates for exotic status. This adds to the suspicion that many more of the strange-formula superconductors may be exotic. It should therefore be interesting to see more Hc2 measurements on strange-formula materials, as well as measurements to lower temperatures for some of the preceeding materials, which

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should be feasible with the higher-field magnets now available. (We are not discussing the “true” heavy-fermion superconductors here because they show a variety of behaviors [72, 1411. In this respect as well as others, these materials are more complicated than the other exotic superconductors.) An upwards curvature is visible in some samples of NbN, while absent in other samples, but this material is known to have serious problems of stoichiometry and sample morphology [ 1421. The character of this H,,(T) data can be roughly divided into three types: (i) There are “totally anomalous” cases, with no downwards curvature and with a steep slope as T + 0, as shown in Fig. 6. (ii) There are “barely anomalous” cases, such as the A-15’s and Chevrels, where the upwards curvature is quite mild and is restricted to low fields and T near T,, with most of the H&T) curve looking ordinary except for a large overall magnitude and perhaps an extended linear region. Apparently in these cases the upwards-curving tendency is largely dominated by the conventional downwards tendency, so that over most of the temperature range the former is revealed in only a relative sense, as an upwards deviation from the conventional form. [An explanation for this upwards deviation could perhaps also resolve the well-known problem of the too-large values typically required for the “impurity” spin-orbit parameter A,,,, or the corresponding scattering rate l/rsO, when fitting this Hc2 data by the conventional theory [ 143,144]. Renormalization of the Pauli-limiting term HP by the state-density enhancement factor (1 + 1) has been shown to help resolve this problem, but it is not clear whether this provides the complete answer [ 1441.1 (iii) There are also “medium anomalous” cases, with prominent upwards curvature at higher T, followed by the usual downwards curvature at smaller T/TC. Such cases may or may not exhibit a large slope at small T/T,. (Although a steep slope at rather small T/T, suggests a finite slope at T = 0, there is at least one material where saturation to an apparently vanishing slope was found to occur quite abruptly, at T/TCm 0.1 [ 1451.) There is a cuprate case which falls in this latter or medium-anomalous category: YBa2Cu307-6 (YBCO) film samples with T, = 91 K. The data was obtained by means of explosive flux compression, leading to H&O) z 138 T and thus cab - 15 A [ 1461. The T, is, of course, much higher than in the preceeding cuprates of the totally anomalous category. In the present case, the resistive transitions broaden considerably with increasing H, which makes the proper identification of HC2 a rather subtle problem. There is some “medium anomalous” data for the electron-doped and local-momentcontaining Nd1.84Ce0.1&u04_y with T, = 22.5 K [44], but there is other data for this material which shows a more strongly anomalous form [ 1471. The interest in H,,(T) for layered materials (NbSe2, etc. their intercalates, graphite intercalates, and also artificial multilayer materials) was initially focussed on the case of H parallel to the layers, where H&O) is typically found to be extremely large, often corresponding to a perpendicular coherence length smaller than the interlayer separation. This is partially a geometrical effect arising from the layering, however, which is not our concern here. For H applied perpendicular to the layers (the present interest), the case of a normal metallic interlayer material (e.g. Cu or Ag), has shown some upwards curvature [140], consistent with theories including the proximity effect [148]. In contrast, the theory for the case of insulating interlayer material, with Josephson tunnelling between the superconducting layers, did not lead to upwards curvature [149], although later treatments have obtained this feature by appealing to a proximity effect within the insulating layers [ 1501. A very different approach has been to consider the expected in-plane anistropy (azimuthal variation of Fermi velocity and gap magnitude). This has provided a satisfactory amount of upwards curvature [ 15 11. Careful study of such anisotropy effects for the three-dimensional case of niobium has led to good agreement for both the anomalous T-dependence and the directional anisotropy of HC2 [ 1521. Because

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of the importance of the Fermi-velocity and gap anisotropies, we digress now for some elementary remarks about these anisotropies. Consider that the external field is applied along a direction which we label i. The theoretical expression for the corresponding & will naturally involve an integral over the corresponding parallel momentum k,. In the case of a three-dimensional material this integration can be expected to average over some of the band anisotropy, and it should therefore reduce the effects of such anisotropy. In contrast, for a layered crystal structure and 2 perpendicular to the layers, the associated weak k, dependence of the band structure should lead to a more coherent and therefore more prominent effect from any in-plane band anisotropy. This argument would seem to explain, or at least help to explain, why so many of the materials with prominent upwards curvature have a two-dimensional (planar) aspect. Furthermore, the “strange formula” materials, which typically involve d or f or cluster molecular orbitals, are natural candidates for a tight-binding form of band structure. They are thus a priori more likely to have strong band anisotropy than the more conventional s-p-band metals. There is also a significant corollary of this argument. The corresponding gap anisotropy is determined by an integration (within the gap equation) which effectively averages somewhat over the band structure anisotropy. It follows that strong gap anisotropy is i priori more likely in quasi-twodimensional materials (where this should be mainly in-plane anisotropy) than in materials with a more three-dimensional structure. (This conclusion might, of course, be either weakened or strengthened if the pair interaction also has strong anisotropy, as is the case in Ref. [ 141.) We return to this point in Sections 3.5, 3.6. It would appear from the foregoing that the “barely anomalous” and perhaps some of the “medium anomalous” cases might be adequately explained by the conventional electron-phonon theory, when this is worked out with sufficient detail. At least this is plausible, since this has been demonstrated for niobium. But the “totally anomalous” cases are clearly a serious problem and seem to require a different explanation. The large slope at T M 0 is also puzzling, as this is quite contrary to conventional theory and to most previous experience. The exceptions are the above-mentioned data for organic superconductors (which may be wrong in this regard), data for UBei3 (explained by a strong magnetic field effect on the normal heavy-fermion state [141]), and in the “bipolaron” theoretical studies [153]. The latter work has clearly predicted upwards curvature over the entire range of T < T,, although this theory has the problem of a divergence as T + 0. This divergence has been removed by appealing to some “dirt” or disorder [153]. This assumption makes Hc2 finite at T = 0, while still leaving a steep slope in H&T) as T + 0. It is significant that this so-called bipolaron theory of Hc2 is actually based on the boson limit t/a ---t0, and thus on the model of a charged boson gas; there is no assumption here which is specific to the bipolaron scenario. The reasonable qualitative agreement with experiment [ 11 l-l 14,116,117] is therefore an indication that the boson limit does have some relevance for the exotic superconductors. (We shall comment on this in the theoretical sequel.) It should also be noted that the steep slope at T -+ 0 cannot be explained by simply appealing to nodes in the gap function, such as in a d-wave gap, because a “conventional” calculation of Hc2 for this case has produced the usual zero slope at T = 0 [ 1541. Readers should be aware that a number of other possible sources for upwards curvature in Hc2 have been identified: (1) Sample inhomogeneity can produce a distribution of local G values. The resulting upwards curvature should be confined to the vicinity of T,, within a width comparable to

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that of the zero-field resistive transition, in contrast to the typical exotic behavior. (2) There can be upwards curvature in granular superconductors when the grain size is comparable to or less than the coherence length, but apparently only at temperatures below an initial downturn [155]. (3) An upturn can result from the Eliashberg theory in the regime of extremely strong electron-phonon interaction, i, 2 4 [4, 1561. (4) Upwards curvature can result from a large amount of disorder or nonmagnetic “dirt”, with this behavior possibly extending even into the regime of Anderson localization [157]. (5) Partial nesting of the Fermi surface, due to a charge-density-wave or spin-density-wave state, is claimed to provide upwards curvature [ 1581. The “totally anomalous” H&T) data has led to several more recent theoretical proposals: (6) The feature identified as HC2 may instead be the irreversability line. The latter is expected to exhibit steady upwards curvature and a steep slope at T = 0, so this possibility offers a simple explanation for the anomalous form of HC2(T). As already mentioned this misidentification has been made for the K-(ET), organics, and some investigators [ 1591 have claimed that this is the case also for the low-T, cuprate data. But this seems unlikely for the low-T, cuprates, since the slopes of their resistivity curves typically do not change with increasing H. (7) Excellent fits to the highly anomalous data for Tl-2201 [ 11 l] and Bi-2201 [ 1121 have been obtained by appealing to a high concentration of magnetic impurities [ 1601. The materials are assumed to be “clean” with respect to non-magnetic scattering, whereas the spin-flip scattering is assumed to be very large, z,f’ BnT,, and also temperature dependent. Good agreement is obtained by assigning a suitable form of temperature dependence. However, the assumptions seem ad hoc and unrealistic. (8) It has been argued that the applied magnetic field can alter the form of the gap tinction, specifically by admixing a d(xy) component into a zero-field d(x2-y’) gap state, provided that the pair interaction l&, has a suitable k-dependence (a k. k’ component). With these assumptions an excellent fit to the Tl-2201 data [ 11 I] was obtained [ 1611. (9) The replacement of plane-wave orbitals by Landau-level orbitals for an Abrikosov vortex lattice has been shown to provide upwards curvature near T, [ 1621. But this study also obtained a reduction of HC2 at T = 0, as compared to the conventional theory, which is contrary to the general trend for the exotic superconductors. (See for example Ref. [3 121.) (10) A recent proposal involves a first-order transition at HC2, where this transition consists of a quantum-renormalized Lindemann criterion melting of the vortex lattice [163]. A good fit is obtained for the Tl-2201 data [ 11 I] below 1 K, by using a [l - (T/T*)0-4] form, where the power law is fixed by the theory and T* is a fitting parameter. This melting picture is claimed to also explain why the cases of continuous upwards curvature (the above “highly anomalous” cases) are associated with a lack of broadening in the resistive transitions. However, it is unclear why a melted vortex lattice state should be identified with the normal state. The applicability of this theory is argued to require strong anisotropy, which seems to be contradicted by the “highly anomalous” case of the cubic material BKBO [116]. (11) The rapid variation in H&T) at low T has been suggested to result from enhancement of thermal fluctuations by the applied magnetic field [164]. A problem for this proposal is that the fluctuations always lower HC2, and they therefore cannot explain the sometimes exceedingly large values of HC2 as T -+ 0. ( 12) The existence of a saddle-point feature (“van Hove singularity”) in the band structure, located near the Fermi energy, has been shown to provide a broad range of upwards curvature [ 1651. This upwards curvature begins at T, but does not extend all the way down to T = 0 (where the slope of HC2 is found to vanish). But in the special case where the Fermi energy coincides with the saddle-point energy, the upwards curvature does continue all the way to T = 0, giving an HC2 form like that of Fig. 6. It is noteworthy that the data of Fig. 6 was

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obtained from overdoped Tl-2201, where this energy coincidence may conceivably be realized. But this special band-structure feature is unlikely to occur throughout all of the materials where upwards curvature has been observed. Although most of these proposals seem unlikely candidates to explain the present data, especially in view of the consistency of this data across many material families, most of the latter proposals do involve aspects that should be considered in a general theory. The proper identification of Hc2 must be recognized as a potentially serious problem. At temperatures somewhat below the apparent onset of the superconductivity, and at lower temperatures, there are effects of flux flow that are still controversial [ 1661. Such controversy is outside the scope of this survey. In contrast, the region around the onset is apparently dominated by fluctuations of the order parameter [34,35,167], although for AC susceptibility measurements there is evidence that even data is this region can be affected by flux flow [ 1681. If however the totally anomalous & results are essentially correct, as we are assuming, then two of the above-mentioned ideas stand out with considerable appeal: the effect of a nearby van Hove singularity, and the boson-limit model. The boson limit cannot be literally correct, of course, but it does have additional support in terms of exceptionally small t/a ratios (Section 3.1) and prominent critical-fluctuation behavior in the specific heat near T, (Ref. [169]). The small s/a’s show that at least in some cases the pairing strength must be distributed over much of the Brillouin zone. The latter feature may well be the main requirement for the boson-like properties. Since a broad k-space distribution of pairing is not incompatible with a BCS-like gap equation (with a suitable pair interaction), this does not necessarily require “pre-formed pairs” in a temperature range above T, - a range greater than the critical fluctuation region. We shall return to this idea in the theoretical sequel. In closing this discussion, we must mention a recent magnetoresistive study of HC2in Bi-2212 (TC= 92-95 K) which has shown strong and extensive upwards curvature [170], in good agreement with the boson-limit theory. This work claims to have reduced the ambiguity in identifying HC2 by measuring the resistivity perpendicular to the planes (and thus parallel to the field). 3.4. Large penetration depth, low charge-carrier density, large efSective mass All of the exotic materials to date are intrinsic type-II superconductors, with IC= 3LL/l$1 and typically IC> 10. The exotic superconductors are therefore strong or extreme type-II materials. It is not clear yet whether the converse is true, but this conclusion is very appealing. Any strongly type-II material should therefore be examined for other signs of exotic behavior. In addition to 5 being anomalously small, the penetration depth AL is anomalously large. This is quite evident in the Uemura plot of Fig. 1, where the horizontal axis is proportional to ;li2. (The latter is almost true also in Fig. 2.) In simplified theory (ignoring band-structure details) one finds Ai2 0: n/m*, where n is the charge-carrier density and m* is an effective mass. The &‘s are anomalously large because n is anomalously small, and also because m* is usually anomalously large. The charge-carrier density n is typically of order one per formula unit, or more accurately, per active (U-bearing) site or cluster. This makes IZ conspicuously small compared to ordinary metals, because the exotic materials typically have a large volume (and many atoms) per formula unit. (The case of A3CG0’s may seem exceptional, with three charge carriers per “active” C60 unit, but here the active molecular orbital tl, has a spatial degeneracy of three, so in a sense even this exception seems to confirm the rule.) In many cases the carrier density is obvious on chemical grounds, because of a closely related insulating material which clearly has no carriers, together with the obvious effect

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of ionic charge transfer (as in A3&, LiTi204, NbSe2, and the organics). This chemical reasoning must sometimes be supplemented by cluster calculations which delineate the molecular-orbital level structure of the cluster units, e.g. of CeO in A 3C 60, and MO&& in the Chevrels. Also, or alternatively, band calculations may indicate just one (or a few) partially filled band(s), as in the cuprates and BKBO. But these clues are insufficient in the case of the A-15’s because of their complex band structures. (Nevertheless, the position of V3Si on the Uemura plot confirms a small effective n/m*.) The heavy-fermion (and almost-heavy) materials are also problematic in this regard because of their often complex band structures, and because of their extremely large m* values. In such general cases, as illustrated in a study of Chevrels [48], one can either appeal to a Hall-effect measurement (usually assuming a single band in the analysis, and a simple band dispersion), or one can use a combination of several measured properties (e.g., o$ and y) in order to deduce n. One should be skeptical of the resulting numbers, however, because the formulas typically used are likely to be oversimplified. In spite of these difficult cases, it must be emphasized that the carrier density is low in all of the exotic cases where this density has been determined in a reasonably direct manner. The effective mass m+ tends to be large for two reasons: On the one hand, conventional band calculations usually show that these are narrow-band or high-state-density materials. (Some exceptions are the bismuthate BKBO, and NbN.) On the other hand, in cases where an m* (or a state density) has been determined empirically, this m* tends to be even larger than in the band calculation. In the superconducting cuprates the mass enhancement factor due to many-body effects is 22. This is based on the band-theoretic width for the antibonding pda band, typically about 3.5 eV, and an estimate of about 1.4 eV for the corresponding empirical bandwidth, obtained from tight-binding band structure fits to angle-resolved photoemission data for a number of cuprates [ 1711. Analysis of several kinds of data has also led to a factor of about 2 for the corresponding enhancement in K3Ch0 [ 1721. In SrzRu04 the total mass enhancement beyond band theory has been reported as factors of two or more [ 1731, and as 3 -4 [ 1741. In the organics the mass enhancement, beyond band theory and the (1 + 3L) factor from electron-phonon coupling, has been estimated to be a factor of 3-5 [ 1751. Estimates of the total mass enhancement factor (beyond band theory) are in the range of 5-10 [ 175,176]. For LiTi an enhancement factor of 1.6 has been obtained, beyond band theory and the (1 + A) factor [77]. In the heavy-fermion materials the band-theoretic effective mass is already quite large (due to the smallness of transfer integrals involving f orbitals), but the many-body enhancement factors are also large, typically 2 10 [ 1771. In some of the A-l 5 materials with higher c’s (Nb3Sn, V3Si, V,Ga) there is also an indication of mass enhancement (larger state density) beyond band theory and the (1 + 1) factor. The factors for the additional enhancement are in the range 1.25-1.8, according to analysis of Eliashberg calculations based on tunnelling-derived spectral functions [178]. On the other hand, this study found no such indication for Nb3A1 and Nb3Ge. Paramagnon-theoretic comparisons of susceptibility, specific heat, T,, and band-theoretic state density have found substantial electronic mass enhancements for V,Si, V3Ga, V, and VN, but not for Nb3Sn [ 1791. (The various conclusions for niobium are inconsistent; two of these paramagnon studies finding an enhancement and one of them not. The V3Si and V3Ga mass enhancements are consistent with those of the Eliashberg analysis.) Nevertheless, for all of these materials (A-15’s, V, Nb, VN) the paramagnon analysis showed evidence for substantial electron-electron interaction. In most of these cases, it is clear that a substantial part of the mass enhancement cannot be due to electron-phonon coupling. In the case of cuprates, the reduction of the quasiparticle dispersion has been observed by angle-resolved photoemission and inverse photoemission over an energy range of

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(Q -.+I 5 0.3 eV, b eyond which the quasiparticle peaks become too broad to resolve [ 17 1,180]. This energy range is far greater than the range of phonon energies for these materials (fiw 5 0.08 eV). Such mass enhancement over an extended energy range is a typical consequence of “strong correlations”, as expected for a large U interaction. 3.5. Gap magnitude, temperature dependence, and anisotropy The gap ratio 2A/kBc sometimes has the ordinary BCS magnitude, ~3.5 (e.g. in K3Ch0 and Rb$& [l81,182],in BKBO [183], and in LiTi204 [184]), or somewhat larger (e.g. 3.7 in NbSe* [185], 4.0-4.3 in other studies of BKBO [186], and 4.0 in another study of LiTi204 [187]), but it is also sometimes much larger, being -5-8 for cuprates [43,188]. A recent break-junction tunnelling study of single-crystal La l.ssSr0.&u04 has found a gap ratio of 8.9 f 0.2 [189]. An enormous value of 18 has been reported for an organic, /?-(ET)ZA~12 [190], although a later study of this material found the ratio to be in the range 4 to 10 [ 1911. The latter study also provided a value of 8 for /?L-(ET)213 (Tc = 1.35 K). F or another organic, k--(ET)2C~(SCN)2, the point-contact tunnelling results were bimodal, most of the data giving an ordinary gap ratio, but some of the data giving a ratio about five times larger [ 1921. A plausible scenario for these strange organic results has been proposed, in terms of a series of superconducting-normal junctions resulting from high pressure in the vicinity of the point contacts [193]. A break-junction tunnelling study of the heavy-fermion material UBe13 gave a gap ratio of 4.2 [ 1941, and a point-contact Andreev scattering study of UPt, gave 4.0 [195]. In contrast, a point-contact study of the almost-heavy material URu2Si2 gave a gap ratio at least 20% below the BCS value [196]. Larger gap ratios of 5.0 to 10.8 have been estimated for the heavy-fermion superconductors by fitting the decrease of l/T, below T,, in NMR, assuming a gap form with line nodes [ 1971. But this procedure assumes a standard BCS temperature dependence for the gap magnitude, which may well be wrong and which can thus invalidate the results. Enlarged gap ratios up to 4.46 (Nb,Al) [178] and 4.9 (Nb,Sn) [198] have been found in A-15 materials. These latter ratios are reasonably consistent with the strong-coupling Eliashberg theory, although it is worth noting that the A-15 materials exhibit relatively large deviations from the general trend of the more conventional strong-coupling materials [ 1991. We expect that the larger gap ratios are associated with strong gap anisotropy, since (a) in an anisotropic case, by far the most prominent feature in the tunnelling state density is found at the gap maximum (in contrast to the gap average) [200], and (b) as argued in Section 3.3, a “planar” band structure (with weak k, dependence) is a priori more favorable for a strong gap anisotropy, in particular, for a strong in-plane gap anisotropy. This is consistent with the available evidence. Further evidence for a tendency to stronger gap anisotropy in two-dimensional materials is presented in Section 3.6, although there is also a caveat discussed below. Another anomalous feature is that “the gap” (whatever this may represent) is sometimes found to have a temperature dependence that is “more square” (flatter, and then more abruptly falling) than in the familiar BCS form for A(T). This feature is well known for the cuprates [201], and it has also been found for A&,‘s [ 1821. There is also an indication of this behavior in V3Si, from a uSR (muon spin relaxation) experiment [ 191, although an infrared surface impedance study for V3Si found the normal BCS form [202]. Also for the organic IC-(ET)~I) a more-square form can be deduced, from the form of the specific heat below T, [203], and likewise for rc-(ET)zCU(SCN)Z [122]. In contrast, A(T) has the conventional BCS form for BKBO [186], LiTizOd [184], and for UBe13 [ 1941.

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There is an interesting paradox for the A3Ce0 materials, where infrared reflectivity data has shown a BCS gap ratio but a non-BCS (“more square”) d(T) form, for both the K3 and Rb3 materials [ 1821, in contrast to point-contact tunnelling experiments which provide much larger gap ratios (5.3 for K3Cb0, 5.2-5.4 for Rb3Ch0) but with a conventional A(T) form [204]. A resolution for this conflict has been found [205] by using the theory of pair-breaking due to paramagnetic impurities, wherein the excitation gap (onset of absorption in infrared reflectivity) can be smaller than the order parameter (peak of state density in tunnelling). This modelling was empirically rather successful, but without any clear magnetic aspect in the actual samples this seems artificial. A more reasonable explanation, we believe, is simply that the gap has a substantial anisotropy, while still retaining the full point-group symmetry of the crystal (in this sense it remains “s-like”), and it also remains nodeless. Then the infrared reflectivity identifies the minimum gap value (2A,i,/ksTc RZ3.5 for both materials), while the peak in tunnelling conductance identifies the gap maximum. These identifications are quite straightforward and elementary, as can be seen in a model state-density plot for a case of nodeless anisotropy [206]. This scenario is supported by other infrared experiments which suggest a distribution of gap values [207]. This leaves the more-square A(T) form unexplained, but the similarity to cuprates suggests that this may be due to a rather abrupt onset of pair-breaking, due to a rapid increase of quasiparticle damping as T approaches T,. The above-mentioned discrepancies for BKBO and LiTiz04 do not fit this anisotropy scenario, and are more likely due to inhomogeneous sample stoichiometry and a resulting surface proximity effect, and also to a difference in the data analyses for LiTi204. Nevertheless, there is evidence for strong gap anisotropy in some other threedimensional materials. The A-15 materials Nb3Sn [208] and V3Si [202] have surprisingly strong gap anisotropy, with their minimum values of 2A/kBz being only about 1.0. Broad distributions of gap values in these materials have been observed via infrared surface impedance [202], and by point-contact tunnelling into different planes of a single crystal [208]. Further support for strong gap anisotropy in these materials can be found in their specific heat data [209]. In the usual C/T vs. T* plots one sees nearly linear behavior (i.e. C c( T3) at T well below T,; only a hint of the expected exponential behavior is found, and this is limited to a small region near T = 0. It appears that a small exponential region is being partially obscured by dirt or disorder effects. (Similar behavior has been found for LiTi204 in Ref. [77].) Large gap anisotropies have also been deduced for several of the A-l 5’s by studying their T, reductions due to radiation damage [210]. More extreme cases of gap anisotropy - cases with gap nodes - are discussed in Sections 3.6 and 3.7.

3.6. Absence of the Hebel-Slichter peak, evidence ,for gap nodes The observation of the Hebel-Slichter peak (coherence peak) in NMR l/T, data just below T,, for the conventional superconductors, was one of the early triumphs for the BCS theory. In contrast, the absence of this peak has been established in cuprates [211], in most heavy-fermion materials [212,213], in the quasi-two-dimensional organic compound ~-(ET)2Cu~(CN)z]Br [214], and in the quasi-one-dimensional organic ( TMTSF)$104 [2 151. In all of these cases the l/T, exhibits a very steep drop immediately below T,. The 2D organic data [214] is compared with some cuprate (YBCO) data in Fig. 7, which demonstrates that these materials have essentially the same l/T, behavior below T,. On the other hand, a Hebel-Slichter peak is seen in A3Ce0’s [ 181,216], although its observation requires that the applied field (needed for NMR) be kept rather low. This low-field requirement had earlier been studied in detail for the A-l 5 material V3Sn, where the Hebel-Slichter feature can

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I

I

0

1:

A

:

I

A

d

I

I

K-ET,Cu[N(CN),]Br

13C

NMR Ho = 6kG l

YBa,Cu,O,

63Cu(2) NQR

0.1

0.01

A 0 0.001

Fig. 7. Scaled comparison of l/T1 data for T < T,, for Ic-(ET)zCu[N(CN)zBr (13C NMR) and YBazCu307 (63Cu NQR). Note the similar steep fall of l/T] commencing at T,, which is opposite to the conventional Hebel-Slichter (coherence peak) behavior, and also the non-exponential (actually T3) behavior at T< TC. From de Soto et al., Ref. [214].

be clearly seen [217], and this requirement has been further examined in a recent review [218]. The Hebel-Slichter feature is also found in the Chevrel compounds T1M06Se7.5 and Sni.1M06Se7.5, although in the former this is largely obscured by strong quasiparticle damping [219]. This is also found in the nonmagnetic ternary rhodium borides [220], and in U6Mn and U&o [221]. It has apparently not been possible to obtain any satisfactory NMR data for a bismuthate superconductor (BKBO). Strong gap anisotropy (with or without nodes) obviously reduces the strength of the state-density peak at the gap maximum, and this weakening must reduce or possibly even eliminate the HebelSlichter peak in l/T,. Several studies have found that highly anisotropic gap forms with nodes suffice to strongly reduce [222] or even totally eliminate [223], the Hebel-Slichter peak. The latter studies [223] claim that such anisotropy by itself (i.e. without help from quasiparticle damping) is able to explain the very steep drop at i’& including even the slight upwards curvature in l/T which can often be seen just below T,. (There appears to be an inconsistency in this regard between the conclusions of Refs. [222, 2231.) Of course, strong quasiparticle damping just below T, can also help here, but by itself this damping must still leave a residual hint of a Hebel-Slichter feature (a downwards curvature near T,) [219, 2241. Thus, damping alone cannot explain the very steep drop with upwards curvature just below T,. Such an abrupt drop, commencing immediately at T,, therefore must be a sign of strong gap anisotropy, perhaps assisted by a rapid gap opening (from enlarged 2A/kBTc and/or a more square form of A(T)). The examples just quoted therefore seem to support the idea that the three-dimensional materials tend to have more isotropic gaps, while the

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quasi-two-dimensional materials tend to be strongly anisotropic, as argued in Section 3.3. However, this contention involves several subtleties. In the materials with planar crystal structure, the cylindrical character of the band structure (the consequence of weak k, dependence) suggests that this should be mainly in-plane gap anisotropy. (This was argued in Section 3.3.) On the other hand, one of the apparently node-containing (and thus very anisotropic gap) heavy-fermion superconductors (UBel3 ) has a cubic crystal structure. This is an example of how extraordinarily anomalous the heavy-fermion superconductors can be. It was also noted in Section 3.5 that there is evidence for considerable gap anisotropy (without nodes) in the cubic A3Ch0 and A-15 materials. In these latter cases the anisotropy is evidently not sufficiently strong to eliminate the above-mentioned Hebel-Slichter peaks. In the more-anisotropic case of V,Si, it would thus appear that the small A values are confined to rather small regions of the Fermi surface, such that the distribution of A values remains largely concentrated near the maximum value. It is interesting that a Hebel-Slichter peak has been found in the quasi-two-dimensional organic ( MDT-TTF)2A~12 (T, = 4.1 K) [225], which has a K-type crystal structure [226]. We suggest that this difference from the other organic superconductors mentioned above is due to the unsymmetrical nature of the MDT-TTF molecule [226], which should cause some disorder and thus a reduction of the gap anisotropy. This argument presumes that the highly anisotropic case of K--(ET),Cu[N(CN),]Br has an anisotropic-s type of gap form (Section 3.7) so that the degree of gap anisotropy can be reduced by disorder. (Without this assumption it is hard to reconcile the data for these two k--structure materials.) This NMR data therefore provides some evidence for an anisotropic-s gap form in the 2D organics. There is clear evidence for gap nodes in some of the exotic materials - in hole-doped cuprates and in heavy-fermion superconductors. This is revealed mainly by power-law rather than exponential behavior of various properties at T < T,, although in the cuprate case there is further evidence. The gap-node evidence for heavy-fermion materials is discussed in Refs. [72,212]. In the planar organic superconductors the evidence has been conflicting. Several experiments have shown powerlaw behavior of the penetration depth [227], while other penetration-depth experiments have shown conventional behavior [228]. In some of the latter papers it was argued that the apparent (power-law) evidence for gap nodes in the organic materials is an artifact of flux flow. This controversy has now been resolved in favor of gap nodes, by the recent observation of T3 behavior in l/T, data [214], and by the more recent observation of T3 specific heat in Ic-(ET),Cu[N(CN)JBr [229]. For the hole-doped cuprates there is much evidence that the gap is strongly anisotropic and perhaps always has nodes. The gap-node evidence is of several kinds: (a) An apparent direct observation of gap nodes via angle-resolved photoemission [230,23 11. (b) The low-temperature (T < T,) behavior of the penetration depth, as a function of T [232], of sample quality [233], and of applied field H [234]. (c) Power-law ( T3) behavior of l/T, in NMR at T 4 T, [211,235]. (d) Power-law (T2) behavior in the specific heat for La2_,SrXCu04 [236] and possibly for YBCO [237], and a clear H’12T term in the specific heat in a magnetic field [237]. (e) Magnetothermal conductivity in untwinned singlecrystal YBCO, a thermal analog of the Hall effect [238]. (f) In conventional tunnelling data, structure “within the gap” has been observed in several cuprates [239,240]. In some cases the apparent subgap state density is even non-monotonic (hook-like) [240], which is the result expected for a highly anisotropic s-like gap with nodes [200]. On the other hand, there is some recent evidence against gap nodes in La1.85Sr0.,&u04, from very flat-bottomed tunnelling data in a single-crystal breakjunction experiment [189], although this conflicts with the evidence in (d). (The NMR l/T, data for

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this material is also unusual, see Ohsugi et al. in Ref. [211].) It must be recognized however that inconsistent tunnelling results (sometimes simple-s-like and sometimes d-like) have been obtained by both the break-junction and the point-contact techniques, for several cuprates [241], so caution is still necessary in interpreting the tunnelling data. Many other experiments are at least consistent with gap nodes, but the ones just mentioned seem to provide the most direct evidence. In contrast, there is considerable evidence that the electron-doped (Nd,Ce)$u04 does not have gap nodes [242]. This evidence is from an exponential temperature dependence of the penetration depth, and from essentially symmetry-independent Raman spectra (similar spectra in the Alg, Big, and BZg channels). (The penetration-depth evidence has, however, recently been challenged [243].) Nevertheless another electron-doped material, the infinite-layer material SrO.sL~.lCuOZ, shows a sudden drop of l/T, just below Z [244]. This is probably evidence for strong gap anisotropy, but this does not necessarily imply that there are gap nodes. These two electron-doped cuprate superconductors may therefore be similar with regard to gap anisotropy and the existence (or non-existence) of gap nodes, although a qualitative difference has not been ruled out. 3.7. Gap symmetry of the cuprate superconductors The topic of the cuprate gap symmetry is a departure from our rule that only features found in several of the exotic material families should be discussed here. We are commenting on this issue not only because this is such a famous and much-discussed problem, but because this is relevant for the overall picture of the exotic superconductors. This issue certainly cannot be settled here. The status of the available evidence has been reviewed many times [245], usually with the conclusion that the symmetry is probably d-like, i.e., changing sign when the tetragonal CuOZ plane is rotated 90”. We believe that this conclusion is premature, because of some experimental problems and also because of considerable evidence to the contrary. We must also admit to an ulterior motive here, as mentioned in the Introduction and explained further in the concluding section. Although the following is focussed on the experimental evidence, the remarks are obviously biased. There is indeed some apparently strong evidence for the d-wave case, due to a number of phasesensitive experiments using Josephson tunnelling in a variety of geometries [245], and there is consequently now a widespread opinion that this evidence has settled the issue. But there is also some contrary evidence which indicates that this issue is not settled yet. At the outset it should be recognized that most and perhaps all of these “d-wave” experiments are subject to problems which could falsify their conclusions. These problems have recently been summarized in Ref. [246]. Furthermore, much of the other (non-quantum-interference) evidence which is claimed to support a d-wave gap is actually only evidence for gap nodes. It is important to recognize that gap nodes are also allowed in the highly anisotropic s-like case [200] - s-like in the sense of having no sign change under rotation by 90”. (In general, by “s-like” we mean a gap having the full point-group symmetry of the lattice.) In the cuprate case the tetragonal symmetry of the CUOZ planes strongly suggests that an s-like anisotropic gap should have either no nodes or eight nodes [200]. Both of these cases can be qualitatively described as “s+g” gap forms, where “g” refers to a real Z= 4 component (a cos 44 component); the difference between these cases arises from the relative strength of the “g” component. Probably the most direct evidence for an s-like gap form is the observation of substantial c-axis Josephson tunnelling into a conventional s-wave superconductor (Pb), in spite of twinning in the

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YBCO sample (which should average away any orthorhombic effects) [247]. This experiment has now been reproduced several times with a variety of refinements, thus ruling out several of the initial objections [247]. Other s-wave evidence has come from tunnelling through thin-film grain boundaries [248]. There is also a more detailed grain-boundary tunnelling experiment (with YBCO) which supports an 8-node anisotropic-s gap form [249]. Further evidence comes from the abovementioned conventional tunnelling experiments (Section 3.6) which show “within the gap” structure which is non-monotonic, i.e. hook-like [240]. This result is inconsistent with a simple d-wave gap form, but this agrees with the prediction for an anisotropic-s form with nodes (an 8-node form) [200]. Although the angle-resolved photoemission data for the gap in optimally doped Bi-2212 is consistent with a simple d-wave (x2 - y2) gap form [230,23 I] we note that this is also consistent with an anisotropic-s form where the minimum gap value is close to zero [231]. It is also significant that the angle-resolved data for Bi-22 12 has been found to depend rather strongly on the sample stoichiometry - the doping 6 in BiZSr2CaCu 20 8+S. Overdoped samples have shown a substantial gap magnitude at a place where the simple d-wave (x2 - y2) form would vanish [250]. This is clearly inconsistent with the usual d-wave form, but is consistent with an anisotropic-s form. To date however the overdoped gap magnitude has been determined only along the CuO bond direction, and at 45” to this direction, so it is unclear whether there is a sign change (and thus a node) in between these directions. Thus, depending on the sign of the gap at 45” there could be either a moderate or a very strong s-like gap anisotropy here, although more complex possibilities are not ruled out. On the other hand, Raman scattering in two overdoped materials (Bi-2212 and Tl-220 1) has provided evidence for an apparently quite isotropic (and thus nodeless) s-like gap [251]. Below T,, the angle-resolved photoemission data for underdoped Bi-2212 is similar to that of the optimally-doped (highest-T,) case; this is consistent with the x2 - y2 d-wave gap form, and also consistent with an anisotropic-s form in which the gap minimum is nearly vanishing. But there is also a surprising new feature here. In the underdoped case the gap is found to persist, with the same form, up to temperatures far above T, [252,253]. Some would argue that this is evidence for a persistence of pairing above T,, so that the superconducting transition occurs by a condensation of pre-formed pairs, but there are problems for this interpretation: (a) If the electrons are paired above c, how can the Fermi surface survive, as it evidently does? (b) If pairing above T, is a vital ingredient for cuprate superconductivity, why is this apparent only in the case of underdoping? More photoemission experiments are currently scheduled in order to hopefully clarify the interpretation of this data [254]. There is another problem here, related to the fact that the above-T, data for a number of other properties have also been described in terms of a “pseudogap” (earlier thought to be a “spin gap”), which likewise is apparent only in the underdoped regime [255] (The properties which indicate a pseudogap include l/TIT and the Knight shift in NMR, static spin susceptibility, specific heat, Hall effect, thermopower, and infrared spectroscopy.) This problem involves the van Hove (saddle point) singularity in the state density, below but near to the Fermi level, which is predicted by band theory and which is now experimentally well verified for several cuprate materials [89]. This state-density feature might account for the pseudogap evidence, at least qualitatively (see however the Note added in proof). There are several calculations to date which embody essentially this idea, and which suggest that this is a viable explanation [256]. (Note that this differs from the earlier “van Hove scenario”, where at optimum doping the Fermi level is assumed to coincide with the saddle-point energy [257].) This suggests that the above-T, gap from photoemission might likewise have a more prosaic interpretation. One of the proposed interpretations [253] is that the Fermi

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surface is switching over to the form expected for a lightly doped Mott insulator material, in which there is an antiferromagnetic spin order (spin density wave), and where there is thus no longer a Fermi surface crossing along the (rc, 0)-to-(rr, X) line in the k,, k,, plane. Since the pseudogap features extend essentially to the point of optimum doping, however, it seems to us more reasonable that this may arise from a microscopic inhomogeneity, possibly due to inhomogeneity in the oxygen doping (or other relevant doping); this would involve microscopic regions with such a local antiferromagnetic band structure, coexisting with other microscopic regions having the optimal-doping type of Fermi surface. (This coexistence would explain why, in the underdoped region, there is typically no discontinuity or other conspicuous feature in the evolution of properties as a function of doping.) A realization of such a microscopic coexistence is found in the recent observation of stripe correlations [258], but the data presently available is insufficient to establish a connection with the underdoped photoemission data. A suggestion which could neatly resolve some of these problems is that in the underdoped cases the superconducting transition may be due to thermal disordering between weakly coupled (Josephson coupled) stripe regions, with the pairing persisting to some higher T within each stripe region [259]. Much more data could be discussed and critiqued, but this would not settle the d vs. s issue. The main fact is that the present evidence is inconsistent. Perhaps the gap form is more subtle than either the pure d-wave form or the 8-node anisotropic-s form, as has often been suggested, or perhaps there are important experimental subtleties which are not yet sufficiently understood, such an argued in Ref. [246]. We think the latter alternative is more likely. There are several further indications or plausibility arguments for an s-like gap which are based on considerations of overall consistency: (1) This is suggested by the usually small but quite variable and sometimes large isotope effect in cuprates [30]. An s-like gap form means that a presumed “new” or unconventional mechanism can cooperate with the conventional phonon mechanism, in contrast to the case of a d-wave gap, and the relative importance of these two mechanisms can therefore vary considerably for different materials, and even for different samples of nearly the same material (e.g. with different doping or other alloying). [For a d-wave gap function an extremely small isotope effect has been calculated, far smaller even than what is typically observed in cuprates, although it was argued that this inconsistency can be removed by appealing to strongly anharmonic phonons [260]. Another d-wave study has claimed to resolve this problem via pair-breaking from the quasiparticle damping due to electron-phonon scattering [261], but the resulting variation of the isotope effect with T, is unsatisfactory.] (2) The reported absence of gap nodes in the electron-doped material (Nd, Ce)zCuOa supports the anisotropic-s form, since for this gap symmetry a modest change in the relevant parameters might create or remove the nodes, by changing the degree of anisotropy. (3) In the same vein, the nodes in an anisotropic-s case could possibly be made to vanish via effective angle-averaging due to various “dirt” effects. This scenario is suggested by the reported inconsistency for (La, Sr)2Cu04 (Section 3.6), and by the occurrence of both simple-s-like and d-like tunnelling data for several cuprates [241]. (This perspective has led us to regard the tunnelling data with the sharpest inner-gap structure [239,240] as being probably the most intrinsic.) The recent Raman evidence for a rather isotropic gap in overdoped Bi-22 12 and Tl-2201 [25 l] might be explained either as a parameter effect or as the result of angle-averaging due to the higher concentration of doping centers [262]. (4) An s-like form is also suggested by the striking similarities between the cuprates and the other exotic superconductors, over a broad range of exotic features, as shown throughout this survey. The evidence that there are some three-dimensional (cubic) materials with

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substantial gap anisotropy (A3CG0’s and A-15’s, Section 3.5, and probably also CeCo2, Section 3.12) likewise can be seen to increase the plausibility of an s-like form for the cuprate gap. In this view, the cuprate gap form is merely a more extreme case of the anisotropic-s gap forms in these other exotic materials. (A reason for expecting stronger gap anisotropy in quasi-two-dimensional materials was argued in Section 3.3.) The contrast in NMR data for Ic-structure organic materials also supports this general picture - the non-existence and existence of a Hebel-Slichter peak for an ordered and a disordered material, respectively (Section 3.6). In Section 3.10 a similar difference is found between YNi2B2C (with gap nodes) and ThPt,B,C (no nodes but still a strong gap anisotropy). The complex gap behaviors of some heavy-fermion materials could of course be used as a counter-argument, but their uranium ions involve an extra degree of complexity (Section 2) which may allow these cases to differ from the other exotics. Much of the appeal of the d-wave gap form for the cuprates comes from the fact that this form follows from a well-known theoretical picture involving the exchange of virtual spin fluctuations or paramagnons [263]. There are however some serious problems for this picture, which we shall discuss in the theoretical sequel to this survey. There is, of course, strong evidence for antiferromagnetic spin fluctuations, in the NMR and inelastic neutron scattering data, but it may well be important here to distinguish between real and virtual spin fluctuations. 3.8. Miscellaneous A small oxygen isotope effect is well known in cuprates, but this is actually quite variable and not always small [30]. In samples optimized for highest T, the exponent LYis typically < 0.1, and in YBCO it is about 0.05. This typically increases as T, decreases, and in de-tuned materials with very low z’s the c( may be close to the BCS value of 0.5. This behavior is suggestive of a combination of phonon and non-phonon contributions to the pairing. (Some quantitative analysis of this twomechanism idea is presented in Refs. [4,12].) But this is by no means the only possibility. Suggestions for the cuprates which are consistent with a purely phononic mechanism include strong phonon anharmonicity [264], strong coupling with a highly repulsive Coulomb pseudopotential (p* 9 0.1, a feature which was considered unlikely) [ 121, and the strongly energy-dependent state density arising from a van Hove singularity near the Fermi level [8]. (This latter suggestion has however been shown to be inadequate [ 121.) The organic superconductors have also been found to have a small isotope effect. Upon replacing the double-bonded carbon atoms by 13C, between and within the inner rings of the ET molecules, the shift in T, was undetectably small in IC-(ET)~X for X = 13, (SCN)*, and Cu[N(CN)*]Br [265]. Replacing the sulphur atoms in the ET molecule gave a small isotope exponent, a = 0.17 f 0.15 [266]. There is also a strange inverse (opposite sign) isotope effect from hydrogen -+ deuterium substitution. Recent work with higher precision has found an isotope effect of a = 0.26 f 0.11 from simultaneous carbon and sulphur substitution in K-(ET)&u(SCN)~ [267]. For the other exotic materials the results are more ambiguous. In the bismuthates, one study [268] of oxygen substitution provided an exponent CI of 0.2-0.25 for both Ba0.61(0.4Bi03 (BKBO) and BaPb0.75Bi0.2503. But other studies of BKBO have found oxygen exponents of 0.35 f 0.05 [269] and 0.41 i 0.03 [270]. Although these larger values would appear to suggest a strong-coupling phonon model [271], it has also been demonstrated that the latter (and highest) value is the one to be expected in a weak-coupling phonon model [272]. The apparent inconsistency of these CI values has apparently now been resolved by a recent demonstration that 01 varies considerably with the

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potassium content in BKBO [273]. This has confirmed the existence of a genuine small-a case. However, the fact that the smaller a’s are associated with lower z’s is a sign of further complexity here, and it has been suggested that the electron-phonon coupling A may also be sensitive to the potassium content [273]. For both Rb3Ce0 and I&& the carbon isotope exponent has been found to be 0.30 (& 0.05 for Rb and f 0.06 for K) [274]. This value is consistent with a combination of phonon and non-phonon mechanisms, but it is also consistent with a strong electron-phonon coupling together with a rather large effective Coulomb coupling p*. In contrast to this ambiguity, in LiTiLOd where one would expect the Li to merely donate an electron, the Li isotope shift was indeed found to be essentially zero [275]. To summarize, the typically small but highly variable oxygen isotope effect in the cuprates strongly suggests a non-phonon mechanism [ 123, but does not rule out an unconventional phonon mechanism. In other exotics the isotope effects usually seem a bit small, but not enough so to reach any conclusions about the mechanism. In several cases the stoichiometry can be varied to adjust the carrier density (cuprates, Bal_,KXBi03 =BKBO,Pd,_,AgXH [276], NbN,_,C, [277], Lil+xTi2_-x04 [278] or Lil-XTiZOa [279]), and usually this strongly affects T,. For example, with appropriate silver doping the T, of PdH rises from around 9 to 17 K [276]. An unusual case is NbN,_,C,, which shows a uery broad maximum of about 18 K at intermediate x [277]. For the cuprates, BKBO, and LiTi204 the superconductivity exists only in a rather narrow range of doping X, and for the first two of these the superconducting regime lies close to an SDW or CDW (spin or charge density wave) instability. For BKBO and for BaBil-XPbXOs the shape of T,(X) [280] is very similar to that for the cuprates. The absence of superconductivity outside of this narrow doping range suggests that this T,(x) behavior is not due merely to sharp structure in the electronic state density, in contrast to what one would expect from the conventional electron-phonon picture [ 151. (The cutoffs at large and small x do not seem to be due entirely to instabilities.) In LiTi204 the maximum T, (just over 13 K) is found in Li-deficient material Lil--xTi204, for x M 0.25. This is at the boundary for a material instability - transfer of Ti ions into Li (tetrahedral) sites [279]. A periodic and rather dense array of rare-earth local moments has been found to be compatible with superconductivity in cuprates, Chevrels, and rhodium borides, and there are several lesserknown ternary superconductor families which also exhibit this feature [281]. This apparently has a straightforward explanation: the orbital wave functions of the charge-carrier Bloch states are highly nonuniform in space, and are quite small at the locations of the rare-earth ions. This has been confirmed in several cases by band calculations. Nevertheless, this feature is certainly exotic from the perspective of conventional metals. The ratio of the specific heat jump, AC/yT,, has been determined for a number of materials. For the “pure” heavy-fermion materials this ratio is highly variable, ranging from about 0.6 to 2.6 [72], although the smallest values (well below the BCS value of 1.43) are suspected to be artifacts due to poor samples. In U(Bel_xBx)13, however, this ratio reaches the far higher value of 3.76 [282], which is apparently the largest value ever obtained from direct measurement. In the almost-heavy material &Fe this is 2.1 [283]. In IC-(ET)~CU(SCN)~ the ratios of 1.5 [125] and 2.8 [284] have been reported, while about 1.4 has been found for /?-(ET)ZA~IZ [285]. In A-15 materials this ratio ranges up to about 2.3 [ 1781. The A-15 values are consistent with the Eliashberg theory, but just as for the gap ratio there are relatively large deviations from the general trend of the more conventional strongcoupling materials [ 1991. In the other exotic materials with higher T,‘s, the large phonon contribution makes it difficult to reliably extract the electronic component of the specific heat. Nevertheless, the

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remarkably high value of about 4.8 has been reported for the cuprate YBa#_r307-s (YBCO) [286]. This value seems too large because, although the value chosen for y was estimated in several ways, this turned out to be essentially the band-theoretic value, which thus does not include the mass enhancement observed in angle-resolved photoemission (Section 3.4). (It therefore appears that the quoted value of 4.8 should be divided by about a factor of two.) From the preceeding data it is clear that the specific-heat ratio in the exotics is sometimes very much larger than the BCS value, and there is also a possibility that this can be somewhat smaller than the BCS value. (There is one material, CeCoz, for which a conspicuously small ratio of 0.85 seems to be well established [287]. This form of its specific heat has suggested a strong gap anisotropy, possibly with gap nodes.) It is important here to recognize that a “more square” form of d(T), with more rapid variation just below T,, can help to enlarge this ratio [288], and of course a less rapid variation should have the opposite effect. The suitably scaled ratio of the Pauli susceptibility to the linear specific heat coefficient, R = dk$ 3&)(xIy), is unity for a free-electron system. A deviation of R from unity may therefore provide information about the Stoner enhancement factor for the susceptibility, S = (1 + Ft)-‘, where F,” is a Landau parameter describing the quasiparticle interaction averaged over the Fermi surface. Unfortunately it is quite difficult to determine S accurately, because S differs from R due to (a) enhancement of y by (1 + 13), where il is the electron-phonon coupling parameter, (b) core diamagnetic and orbital paramagnetic contributions which need to be subtracted from the measured x, and (c) a possible paramagnon effect. (The orbital contribution is difficult to obtain accurately, although a reasonable estimate based on the presumably dominant Van Vleck contribution is quite feasible [289]. Furthermore, a possibly significant renormalization of x by the electron-phonon coupling has been argued [290].) An apparently naive but nevertheless interesting approach has been to simply ignore these difficulties, and examine the R’s obtained directly from the measured x and y (at the lowest temperatures for which these are available). There is a plot of y vs. x for uranium-containing heavy-fermion materials [291] and also a plot of R vs. y for a considerable variety of materials, including heavy-fermion materials [292]. [A curious consequence of this R vs. y plot was to reveal an unexpected distinction between the “officially heavy” materials with y > 400 mJ/mol K2 (Stewart, in Ref. [72]), and the several “almost heavy” superconductors U6X, URu2Siz, U,PtC,, and UPd2A13. These two classes were found to be separated by a region of y without any known examples of superconductors. This non-superconducting region was attributed to a crossover between the energy scales for the electronic excitations (Fermi energy) and for the phonons (Debye temperature).] Tabulations of R’s may be found in Refs. [293, 2941. It is expected that as y increases, the relative importance of the core and orbital magnetic terms should decrease in comparison to the desired Pauli paramagnetic susceptibility contribution, so the R’s are presumably most meaningful for the “heavier” (larger y) materials. (In the heaviest materials, however, the weak crystal-field splitting of the f electrons can play a role in reducing the susceptibility.) Among the heavy-fermion superconductors, the “heaviest” examples (UBe 13, UPt3) have R’s close to unity, while the “almost-heavy” examples &Fe and U2PtC2 have R’s of 1.4 and 2.1, respectively. Among the narrow-band transitionmetal compounds tabulated in Ref. [293] (largely A-15’s), most of the superconducting materials have R’s of 2.0 f 0.5. (It was also noted there that among closely related materials a larger R value is generally correlated with a smaller T,, as one would expect.) This simple approach is evidently invalid for the superconducting nitrides (NbN, VN, TIN, ZrN), which have small R’s ranging from 0.4 to 1.1. These nitrides have y’s an order of magnitude smaller than the high-T, examples of the

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A-l 5’s, whose y’s are in turn an order of magnitude smaller than for the heavy-fermion materials, so the necessary corrections are naturally expected to play a more prominent role here. These nitrides have however been studied in the context of paramagnon theory, where for VN there was clear evidence for a large Stoner enhancement (S z 3) [295], which thus also implies some additional mass (state-density) enhancement. Paramagnon analyses have also been done for Nb and V, and for several of the higher-T, A-15’s, with the conclusion that these materials generally have substantial Stoner enhancements (S- 1.3-2 for Nb, -1.6-2.5 for V, -2-3 for A-15’s) [179]. (These determinations for Nb and V are also consistent with calculated band-theoretic Stoner enhancements [296].) For a series of quasi-one-dimensional organic superconductors, Stoner enhancements of 1.6-3.7 have been estimated by comparing measured susceptibilities with optically-determined bandwidths [297]. Overall, therefore, we are led to conclude that the exotic superconductors typically have a strongly repulsive quasiparticle interaction at the Fermi surface, an interaction which acts like an effective or renormalized Hubbard U. This conclusion is of course consistent with the ubiquitous T2 or T dependence of resistivity in these materials, but this association is not watertight. The Ft which causes the Stoner enhancement (1 + Ft)-’ is composed of ftt as well as ftl, with only the latter corresponding to an effective Hubbard U. The “exchange” (as contrasted to “direct”) part of f,t provides a part of the Stoner enhancement, via the imperfectly screened long-range Coulomb interaction. This exchange part of f,, can of course also contribute to the resistivity. This concludes our main discussion of the exotic electronic features. It would clearly be interesting to have measurements of the preceeding properties for more of the exotic and other strange-formula superconductors, although there is undoubtedly more relevant data than we are aware of. The more prominent of these anomalous properties are summarized in Table 1. Here “+” and “-” mean that there is reasonably good evidence for the feature occurring or not occurring, respectively. In many cases however a “+” refers to only some members or even only one member of the material family. A blank entry can mean insufficient evidence, or “not conspicuous for this property”, or that this does not apply (e.g. concerning a lattice of rare-earth local moments). (Also, we have not searched as diligently for negative evidence such as the consistent absence of gap nodes.) These entries often involve a value judgment, and we have tended to be conservative here about whether the evidence is adequate. For the gap and/or specific-heat ratios we have indicated the more striking cases, without regard to whether these seem consistent with strong-coupling (Eliashberg) theory. Readers will surely have noticed that some of the features discussed above can reasonably be interpreted as strong-coupling (Eliashberg) effects [3,4]. Probably the most often quoted examples are elevated values for the gap ratio 2A/kBTC and the specific-heat ratio AC/yT, [ 178, 1991. Thus, for several features the interpretation is admittedly ambiguous. This circumstance, sometimes accompanied by a large electron-phonon coupling i obtained from a band-theoretic calculation, has often been used to argue that in this or that exotic material the phonon mechanism seems to be adequate. On the other hand, it was noted in the Introduction that a band-theoretic calculation may considerably overestimate I, because of the reduction due to strong correlations resulting from the U interaction [lo]. It is also important to recognize that such discussions ignore the problem of how there can be any superconductivity at all, or how there can even be a nonmagnetic state, in the presence of a large U interaction and typically rather narrow bands. (This problem was pointed out long ago by Anderson [5].) The sometimes extremely small values of (/a are also a serious problem for the conventional picture, at least in the cuprate and Chevrel cases. These

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remarks demonstrate the need to examine a wide variety of properties before deciding whether or not a strong-coupling phonon interpretation is adequate. We have focussed on what we consider to be the main exotic electronic properties. There are undoubtedly more features which could be shown to be typical or characteristic, and we shall now mention where some other reasonable candidates are discussed. There is a review by Levin and coworkers [298] which shows that the cuprates and the heavy-fermion materials have a number of similarities in their normal-state behaviors. After allowing for the very different energy scales, they found similarities in transport data (Hall effect and thermopower), in nuclear magnetic resonance data, and in inelastic neutron scattering data. The transport aspect brings to mind the sign change in magnetoresistance which is typical for heavy-fermion materials (positive and thus band-like far below the Kondo temperature, and then negative and Kondo-like at higher temperatures), with the crossover occurring well below the Kondo temperature [299]. It would be interesting to see whether this occurs in the IC-(ET)~ organic superconductors, in view of their typically prominent resistivity maximum. Analogs of the broad “mid-range” absorption continua of the cuprate infra-red and Raman data have been found in infrared data for a number of other materials: A3C&, and organic superconductors, URu2Siz, BKBO, Sr2Ru04, various perovskite titanates MTi03, and a variety of other conducting transition-metal oxides [300]. Acoustic attenuation also deserves more study, although sample quality may often prohibit this. Another arena to explore is that of low-temperature lattice anomalies, either as observed directly (thermal expansion, lattice stiffness) or as inferred from the specific heat. A number of further analogies between the heavy-fermion materials (and &Fe), and the A-15 and Chevrel materials, have been pointed out by DeLong [301], based in part on earlier remarks by Anderson and Yu [29]. These include unusual effects of impurities (both nonmagnetic and magnetic), strong T-dependences of x, C/T, elastic constants, thermal expansion, and the apparent Debye temperature, and also the frequent non-BCS temperature dependence of properties below T, (Hc2, l/T,, specific heat, ultrasonic attenuation, and electronic thermal conductivity). Among the “heavier” (larger r) materials, DeLong and coworkers also noted a correlation between upwards curvature in Hcz, the existence of T2 resistivity, and the detailed low-temperature (T, H) behavior of the magnetoresistance, and they argued that this suggests some sort of (unspecified) novel pairing mechanism [302]. At this stage of experimental and theoretical understanding, it is difficult to say whether any of the present features are essential for a material to be considered exotic, i.e. if there are any other reliable or “universal” criteria besides that of falling within the exotic band in the Uemura plot. One of the above features does seem to clearly stand out as being universal here, for the materials with varying degree of exoticity - this is intrinsic type-11 behavior. It is an interesting question whether all intrinsic type-II materials deserve to be considered somewhat exotic, or perhaps only the more strongly type-II examples, with IC greater than some number of order 10. (Or perhaps all compounds, excluding niobium? We return to the niobium problem in Section 4, and conclude there that niobium is purely conventional after all, in spite of its several indications of mild exoticity.) It is not certain that even the more strongly type-II materials are all exotic, although this seems very likely. A quadratic (or linear) low-temperature resistivity has turned out to be surprisingly common, although this may be due in part to some required minimum of disorder within the samples [87,107]. Another apparently universal feature is a room temperature resistivity much larger than that of lead. (This is not so for niobium, nor does niobium show T2 behavior.) A low charge-carrier density may also be universal, although there are a number of exotic materials for which this has not been directly

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verified. Also found often enough and without counterexample, thereby suggesting universality, are a region of upwards curvature in H&T), and an effective mass enhancement beyond the prediction from the combination of band theory and the (1 + 2) factor from the electron-phonon coupling. 3.9. Crystal-chemistry

features

Several of the typical crystal-chemistry features are obvious. (a) Each of the exotic materials contains disparate types of atoms or ions (sometimes polyatomic ions), with large electronegativity differences. The d-electron materials typically contain non-metallic or semi-metallic elements, and sometimes also extremely electronegative elements. For the f-electron materials the electronegativity differences can be smaller, with the other elements being also metallic. The f-electron materials thus seem to involve a smaller energy scale, apparently due to the smaller f-to-non-f hybridization matrix elements. (In the f-electron intermetallic compounds, the metallic screening of Us may leave this U still relatively large, from the standpoint of this reduced energy scale.) (b) Effective low-dimensionality in the crystal structure is quite common - the quasi-two-dimensionality in cuprates, K-(ET)~ organics, transition-metal dichalcogenides, and now in the borocarbides. Quasione-dimensionality is also found, in the exotic TMTSF organics and in some other strange-formula superconductors, including (SN),, TINb3Se3, NbSe3, and Nb3Sq. (Whether the A- 15 materials belong in this category, due to their chains of transition-metal ions, is less clear. We have argued in Ref. [ 151 that the “active” d orbitals in these chains can have relatively weak overlaps with their counterparts in the neighboring d ions.) (c) Crystal structures containing obvious cluster units are rather common, as in A3C6,-,‘s (CeO molecules), Chevrels (M6Xs units), organics (donor and acceptor molecules), and also the Rh.+B4 units of the rhodium borides. In the organic superconductors the features of low-dimensional&y and clustering are seen to coexist. The strong tendency for clustering in the strange-formula superconductors was recognized long ago [303]. The existence of a large-U unit in each of the cuprates, borocarbides, LiTi204, NbSe2, and the heavy-fermion superconductors is easy to accept in view of their relatively isolated d-electron or f-electron ions. In the case of BKBO it has been demonstrated that the Bi 6s orbitals have a substantial U [304]. Beyond this, however, such an identification involves more subjectivity or theoretical bias. There is some bias in our view [ 151 that an effective Hubbard U should often be associated with a cluster or polyatomic-ion unit: a C 60 ion, M6 subunit of the Chevrel M6XR ion, and the (ET)2 bimolecular unit of the two-dimensional organics. Among the least clear cases are &Fe (are the “active” orbitals the d’s, or the f’s, or both, or neither?), and the A-15’s (individual d ions, or chains of these ions?). For the present, therefore, we merely assert that a large-U unit can be plausibly argued for every one of the officially exotic superconductors, and likewise for the majority of the other strange-formula superconductors [31]. Further discussion of these issues does not seem possible without a more specific theoretical framework, such as that of Ref. [ 151 (for example, the comment about A-15’s in the preceeding paragraph). We shall discuss the crystal-chemistry features further in the theoretical sequel to this survey. 3. IO. Exotic features

of the borocarbide

superconductors

As examples of some of the foregoing properties, and thus as candidates for exoticity, it is interesting to examine the recent borocarbide superconductors. Measurement of the penetration depth

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has put YNizB& at the far edge of the Uemura band (actually a bit beyond this), leading to the description of this material as “marginally exotic” [305]. Consistent with this, the coherence lengths are considerably larger than the extreme cases of Section 3.1. Published < values for the Y-Pd (mixed-phase) material are 50-60 A [306,307], and for YNi2B2C and LuNi2B2C they are 60-lOO A [305, 306, 308, 3091. The resulting rc= AL/r for YNi2B2C is around 13 [305], so this is strongly type-II but not “extremely” so. (The most extreme cases, e.g. Tl-2223 [33], PbMo& [310], Rb3Cb0 [38], and URu2Si2 [72] have ICN 100.) It will be interesting to see whether the higher-T, Pd material is less marginal, if and when this can be made more stoichiometric. In spite of this “marginality”, the borocarbides exhibit an impressive number of the other characteristic features described above. They include large-U ions (Ni, Pd, Pt), and more electronegative counterions (boron and carbon). (Also, as in many ternary superconductors, there are extremely electropositive rare-earth ions which donate electrons.) There is reduced dimensionality (planar character), and insensitivity to the presence of magnetic rare-earth ions (e.g. in ErNi2B2C). In singlecrystal YNi2B2C there is an extended region of upwards curvature in &, from T, down to about 0.5T,, although downwards curvature was found below this temperature [31 I]. In the most recent data, however, there is no sign of downwards curvature, down to the lowest reported temperature of 0.3T, [3 121. (In view of the planar crystal structure, it is quite remarkable that the magnetic properties are essentially isotropic [31 l-3 131.) There is a modest enlargement of the specific heat ratio AC/yT, (1.77, compared to the BCS value of 1.43) in single-crystal YNi2B2C [314]. In single-crystal LuNi2B2C, however, the extremely large value of 3.4 has been found for this ratio [3 151. The 300 K resistivity is also large (as compared to Pb), about 80 ulR cm (in-plane) for single-crystal YNi2B2C and about 50 usl cm for single-crystal LuNi2B2C [3 121. The same study [3 121 found power-law ( Tn) resistivity for both of these materials, with IZ= 2.0 for LuNi2B2C and n =2.2 for YN&B&. (An earlier resistivity measurement of polycrystalline YNi2B2C found good agreement with the BlochGruneisen expression for electron-phonon scattering, but in which the exponent in the power-law prefactor T” was adjusted and was found to be anomalously small, n = 2.5 *0.3 [3 161.) The T, of the Y-Pd material has the remarkably high value of 23 K, equal to the highest value known (for Nb3Ge) before the cuprates. There is evidence for some enhancement of the effective mass beyond band theory, by a factor of 1.2-1.4, and over an extended range, via photoemission data [3 171. A boron isotope-effect exponent of 0.25~tO.04 has been found for YNi2B2C [318]. Also, in the NMR data for YNi2B2C there is no Hebel-Slichter peak [319]. As explained above, this probably indicates strong gap anisotropy. This extensive list leaves no reasonable doubt that the borocarbides are genuinely exotic. (Note also a comparison with the cuprates [5].) We must also mention, however, that break-junction tunnelling data has shown a conventional weak-coupling BCS gap ratio (2d/kBT, = 3.5), and a BCS temperature dependence for A(T), for both YNi2B2C and LuNi2B2C [320]. The differential conductance displays rather sharp inner-gap features, which the investigators attributed to multiple Andreev reflections. On the other hand, a recent optical reflectivity study has found larger gap ratios (3.9-5.2) in the Lu- and Y-nickel compounds, but still with a BCS form for A(T) for both compounds [321]. In this summary we have favored single-crystal data, because the properties for polycrystal samples have often been found to differ significantly. There is a power-law (T3) behavior in the specific heat of YNi2B2C at T 4 T, [322], which appears to be due to gap nodes. (This is a much cleaner power-law behavior than in the specificheat data discussed at the end of Section 3.5.) On the other hand, the NMR l/T1 has been found to level off in this temperature region [3 191. This was attributed to normal-state regions due to local

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inhomogeneity, possibly due to vacancies, so this data does not bear on the issue of gap nodes. In contrast, in NMR data for ThPt,B,C a weak Hebel-Slichter feature has been found, and below T, the l/T1 was found to decrease in an activated manner, demonstrating an absence of gap nodes and a minimum gap value such that the minimum gap ratio 2A/kBTc is only about i of the BCS value [323]. This indicates an s-like gap with strong anisotropy, although not enough anisotropy to produce gap nodes. The properties of a closely related superconductor, La3Ni2B2N3_6 with T, = 12.25 K, have recently been reported [324]. This has the same N&B2 layers, but they are now separated by (LaN)3 layers. This material has strong type-II behavior (rc = 33) but relatively low resistivity (32 uR cm at 300 K). Its gap and specific-heat ratios have essentially the BCS values, and its low-temperature specific heat is exponentially damped, indicating absence of gap nodes. Nevertheless there is a conspicuously anomalous feature - upwards curvature in Hc2( T) between T, and about 0.6T,. Still another related material with some exotic properties is the recently discovered superconductor LaNiC* with T, = 2.7 K [325]. This is reported to have a 300 K resistivity of 190 uR cm, a specificheat ratio ACjyr, of 1.20, and a T3 specific heat well below T,. 3. II.

Short-chain

Chevrel compounds

The materials Cs2Moi2Sei4 and Rb4M01sSeZ0 are short-chain relatives of the Chevrel compounds, whose Mo,2Se14 and Mo18Sezo clusters can be viewed as polymers of the basic MO&Sex Chevrel cluster. Although their Tc’s are low, ~4.5 K, a recent study [326] has revealed that these materials have a number of exotic features: (a) They are extreme type-II materials, with Ai_- lo4 A and a 5 as small as 17A. (b) The charge-carrier densities are very low. For the cesium compound this density was estimated to be around 2 per MO i2 cluster. (c) The reported 300K resistivities are enormous, about 6 mR cm. The experience with ordinary Chevrel compounds (Section 3.2) suggests that this huge magnitude is extrinsic, and probably due to microcracks. The intrinsic resistivities may nevertheless be large, but they are presently unknown. (d) For the cesium compound the specificheat ratio AC/yT, is quite large, 2.5 (as extrapolated to an idealized sharp transition). The estimated gap ratio 2A/kBT, is also large, 4.35. These large values are consistent with strong electron-phonon coupling [4, 178,199] but of course they are not a proof of such coupling. (e) Although the H&T) forms are fairly normal, there is a hint of some upwards curvature near T,. There are not enough data points to be sure of this. The coherence lengths are shortest (-20 A) parallel to the c axis, and larger (-50 A) perpendicular to c, as in the case of a planar electronic structure. This is consistent with the crystal structure. Although the MolzSei4 and Moi8Sezo clusters are arranged in chains (along the c-axis), within these chains the cluster ends are separated by several (2 or 4) alkali ions. These chains are staggered so that the top of one cluster is adjacent to the bottom of a cluster in a neighboring chain. A molecularorbital study [327] has shown that the main electronic couplings (transfer integrals) are between the ends of these staggered clusters, thus imparting some transverse character to the band structure. (The inter-cluster transfers involve Se 4p orbitals as bridges.) The result is a distorted Chevrel-type band structure [327], with an anisotropic three-dimensional character. These short-chain compounds are therefore rather different from the infinite-chain compound TIMo3 Se3. There are also features which suggest a valence-fluctuation type of normal state. There is a broad resistivity maximum around 130 K (for the cesium compound only), similar in form to those of

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Figs. 3 and 4. (Unlike the LiTi204 and BKBO cases the resistivity increases here by a large factor, >>1, between T, and the maximum.) There is also a strong increase of the susceptibility with decreasing T, which is reminiscent of the behavior in heavy-fermion materials. These features can easily be understood in terms of valence-fluctuation theory [73] as will be discussed in the theoretical sequel. 3.12.

Cubic Laves-phase cerium compounds

The cubic Laves-phase (C-15) superconductors CeCoz and CeRu2 have long been known [ 11, but it is only recently that single-crystal samples of these materials have been produced (Refs. [328] and [329], respectively). Although their T,‘s are not high (1.4 and 6.2 K, respectively), their other properties reported to date show a number of exotic features. Their 300 K resistivities are quite high, 95 and 80 uR cm respectively. At low temperatures both materials show quadratic resistivity, with magnitudes consistent with the Kadowaki-Woods plot [96] for heavy-fermion and other valence-fluctuation materials. The resistivities of both materials also show a prominent saturation tendency; their overall resistivity shapes both resemble that of Fig. 5 here. Both materials have strong type-II superconductivity, with Ginzberg-Landau IC values of 8.7-13 and 16-21, respectively. The l&(T) curve has been determined for CeRuz and is what we have called “medium anomalous”, with upwards curvature between T, and about 0.65 T, and then downwards curvature for lower T. For CeCoz, there is de Hass-van Alphen data showing mass enhancement factors (experiment/band-theory) of typically 2-3 for a number of Fermi-surface sheets [330], consistent with the enhancement of the specific-heat coefficient y. The specific heat of CeCo2 [287] is relatively low just below T,, giving a AC/yTc ratio of 0.85 which is far below the BCS value of 1.43. At lower T the specific heat shows approximately a T2 behavior, down to the lowest data at O.l2T,. The combination of these features has suggested a strongly anisotropic gap form [287] and gap nodes are a possibility here.

4. Summary and concluding remarks In a broad sense the focus of this survey is on what we have termed the strange-formula superconductors, compounds which typically include non-metallic or semi-metallic elements, or at least elements with considerably different electronegativities, and in which large-U units (ions or clusters) are plausibly identifiable. The work of Uemura and coworkers [6] on a number of these compounds has demonstrated that their examples share a remarkable property, namely T, 0; Ac2. This indicates some profound similarity, and it also makes these materials quite different from the conventional phonon superconductors. This led Uemura and coworkers to categorize these superconductors as “exotic”. Their examples have become fairly numerous and now cover a wide range of materials, including high-T, cuprates and also bismuthate, C6,,, A-15, Chevrel, organic, and heavy-fermion superconductors, together with the more recent additions of &Fe, NbSe2, and LiTi204. The extent to which the Uemura relation holds throughout the other strange-formula superconductors is unknown, but one can reasonably expect that a number of the others will also be found to fall within the Uemura band, or to at least lie close to this. Earlier, Anderson and Yu [29] had emphasized that some of the then-known strange-formula superconductors (as of 1984) have astonishingly high

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resistivities, which led them to label these as “bad actors”. Their main examples (A-15’s, Chevrels) and their suggestion of NbSe2 are also on the exotic list of Uemura, and their other examples and suspected cases (rhodium borides, NbN, NbC) are reasonable candidates. Since the underlying significance of the Uemura relation was unclear, we have used this as a guide to look for other anomalous features shared by these materials, features which hopefully might be easier to understand. This survey has demonstrated that there are indeed a number of other anomalous features which characterize the exotic superconductors, besides the very large resistivities. Some of the important differences are the often extremely short coherence lengths (only a few lattice spacings), the typical T2 or T resistivity above T,, and the highly anomalous H,,(T) form, with extensive upwards curvature and sometimes a large slope as T + 0. Another feature emphasized here is the remarkable occurrence of a prominent resistivity maximum, in several of the lower-T, material families. (The temperature of this maximum is typically about ten times T,.) A number of other unusual features which are typical (though not necessarily universal) for the exotic or other strange-formula superconductors have been pointed out by many investigators [5]. We mention here such other well-known features as strong or extreme type-II behavior, low charge-carrier density, and sometimes also a small (and variable) isotope effect, a large gap ratio 2A/kBTc, absence of a Hebel-Slichter peak, and evidence for gap nodes. Taken all together, these features confirm a profound difference from the conventional phonon-driven superconductors. Based on this broad range of features, evidence has been presented here to argue that the borocarbides, the short-chain Chevrel compounds, and the cubic Laves-phase cerium compounds are also exotic. The element niobium provides an interesting test case. It has several features suggesting a mild degree of exotic superconductivity: a position in the Uemura plot intermediate between the exotic and the conventional superconductors, intrinsic type-II behavior, the highest T, among the elements, a low-power-law resistivity, and upwards curvature in Hc2. On the other hand there are strong arguments uguinst exoticity here: ( 1) The Hc2( T) and its anisotropy have been explained quantitatively in terms of the Fermi velocity and gap anisotropies [152]. (2) The relatively high T, is well explained in terms of a large state density, consistent with the systematics of the other transition metals [331,332]. (3) The power-law behavior of the low-temperature resistivity has been shown to be mainly T3 instead of T2, indicating dominance of phonon-mediated s-d scattering, with only a weak electron-electron scattering component [42]. (4) The room-temperature resistivity is lower than that of lead, in contrast to all of the more-established exotic superconductors. (5) The high metallic magnitude of the charge-carrier density is expected to produce a strong screening of the U (as well as of the longer-range part of the Coulomb interaction), according to the conventional estimates [332,333]. This strong screening of U is apparently confirmed by a reasonable agreement between the observed [ 1791 and the band-theoretic (local-density approximation) [296] Stoner enhancement for niobium (S 5 2). The high state density corresponds to a relatively small Fermi velocity and a large effective mass, which apparently reduce 4 and enlarge & sufficiently to produce the type-II behavior. Thus, the weight of the evidence for niobium is strongly against this having any degree of exotic character. This example is instructive since it represents a conventional superconductor masquerading as slightly exotic, in contrast to the more frequent case of exotic materials that have been misinterpreted as conventional. We emphasize that for this present non-exotic conclusion the most crucial feature is the high metallic charge-carrier density. Having argued the importance of a low carrier density, so that the Coulomb (or Hubbard) interaction is not too strongly screened, it is appropriate to mention the case of extreme low-carrier-

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density superconductors. These are doped semiconductors (hole-doped GeTe and SnTe, electrondoped SrTi03), where the carrier density is only of order 10-4-10-’ per formula unit [334-3371. The Tc’s of these materials are all quite low, < 0.55 K as expected in view of the low state densities. An interesting feature of these materials, both experimentally and theoretically, is that they are all rather strongly type-II, with K 5 10. This is attributed to low-carrier density, large effective mass, and small Fermi velocity, but it is worth noting that the penetration depth of SrTi03 was found to be anomalously large even compared to the theoretical estimates [334]. These materials were originally thought to be reasonably well explained within the conventional framework of a screened Coulomb and electron-phonon interaction [334]. The state density was argued to be relatively enhanced in these materials (i.e. not as low as might be expected) due to (a) a large band-structure effective mass, and (b) a multi-valley form of band structure, which enlarges the Fermi-surface area. It was argued that the most important feature for these materials is that a majority of the pair-pair interactions within the gap equation are inter-valley, i.e. between the separated valleys of the conduction-band minima (or valence-band maxima). The corresponding matrix elements of the Coulomb interaction thus involve large momentum transfers q, and are therefore sufficiently small compared to the phonon contribution. However, later study has led to a much different picture for SrTi03. Evidence now favors a single-valley conduction band minimum for this material [338]. The plasmon-exchange mechanism has been argued to be important here [339], but this has also been criticized on the grounds that this mechanism is greatly weakened by vertex and other beyond-Migdal corrections [340]. Other theoretical efforts, which stay within the conventional electron-phonon framework, are described in Ref. [336]. We are concerned that there should be a rather large Hubbard U interaction in this material, just as is expected in the case of LiTi204, and it is therefore unclear whether doped SrTi03 can differ in a major way from the exotic material LiTi204. Other examples of extreme low-carrier-density superconductors, probably less well characterized, are listed in Table 4 of Roberts in Ref. [ 11; see also a recent study of Pb(T1 )Te [341]. In view of the frequent ambiguity about whether the conventional phonon mechanism is adequate, when considering a limited number of properties, we want to emphasize some aspects which are particularly troublesome for the conventional theory - the generally large U’s, and the exceedingly short coherence length in some of these materials. In the smallest-t cases, one can see that the pairing must be distributed over much and perhaps all of the Brillouin zone. The doubters of any “new” mechanism have argued that such a very small 4: is merely the consequence of a very small Fermi velocity and a large T,, presuming of course that the latter has a conventional explanation. However, it is essential here to recognize the large-U and narrow-band nature of these materials, the narrow-band aspect being equivalent to the small Fermi velocity. In addition to the problem of understanding how such materials can superconduct at all, there is the problem noted long ago by Anderson [5] that such materials are expected to be unstable towards a magnetic (spin density wave) state or to a lattice-distortion (charge density wave) state. This constitutes a serious problem which the advocates of the purely conventional picture generally ignore. Indeed, the exotic superconductors are often found experimentally to be close to such an instability. There are many precedents for the suggestion that a new mechanism may be operating here. This possibility was raised long before the discovery of the cuprate superconductors. There is no need here to summarize these many suspicions, the more specific suggestions, or the now large number of serious theoretical efforts. However, one of the earliest of these expressed suspicions deserves mention here, because from the present perspective this seems insightful. Even in the early years of

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the BCS theory, when this theory seemed all-powerful, Matthias suggested a “magnetic” mechanism [342]. His examples were of course superconducting and (mostly) not magnetic, but they were often closely related to magnetic materials. One might now argue that the unifying theme was large-U materials. With this view, and thanks to the recent intense study of the cuprate superconductors, his suggestion seems more sensible now than it probably did during his era. On the other hand, it must be noted that much of the evidence claimed by Matthias was based on the transition metals and on alloys between these metals. This metal and alloy evidence has been discredited by calculations explaining the observed properties within the conventional BCS framework of electron-phonon and screened Coulomb interactions [33 1,332]. Matthias apparently did not realize that the “exotic” and other “strange formula” materials are the more natural and proper arena for his proposal (as least according to the present view). This is understandable because much of the systematics described in this survey, and indeed a majority of the exotic superconductors, were unavailable in his era. We came serendipitously to this interest in the exotic superconductors, through a valence-fluctuation (VF) theory we have developed for the cuprate superconductors [ 14, 151. This VF mechanism explains many anomalous features of the cuprates - some even semi-quantitatively - and it does so in a straightforward and fairly ab initio manner. Most of those unconventional cuprate features are also typical, although often not universal, for the other exotics. Furthermore, from the theoretical standpoint there is a tendency for the main qualitative features of this theory to be exhibited throughout the strange-formula materials. (This is not to say that all strange-formula materials should be superconductors, however, see the major caveats below.) It is therefore a reasonable possibility that this mechanism is operating throughout the exotic materials, together with the conventional phonon mechanism, and that this is the source of their exotic&y. For present purposes, the main results of this theory can be summarized as follows: (1) In the paramagnetic VF state (the presumed normal or above-T, state) the Fermi surface should typically be quite close to that of the conventional band theory, but with a reduced dispersion, i.e. a mass enhancement or “heaviness” beyond that due to band theory and to the electron-phonon coupling. The correlations produced by U are by no means weak, but they nevertheless lead to this rather ordinary quasiparticle band structure, similar in form to the conventional band-theoretic result. (2) The gap is s-wave-like, in the sense that the gap function exhibits the full point-group symmetry of the crystal lattice. The VF mechanism can, therefore, cooperate with the phonon-mechanism, thus among the various materials there can be a considerable range of values for the coherence length and for the isotope effect, depending on the relative strengths of the VF and the phonon mechanisms. (3) The pair interaction Vkkt is active throughout the entire Brillouin zone. It exhibits a very strong k dependence, which is the source of the net pair attraction and the often-high T,. This k-dependence arises mainly from the k-dependence of the “hybridization” matrix elements - the analogs of the 3d-2p coupling in the cuprates. Since the hybridization k-dependence is an obvious consequence of the tight-binding treatment of the relevant orbitals, this latter feature should be generic throughout the strange-formula materials. To obtain superconductivity, however, there are some strong caveats. The trick is to find such materials which are adequately represented by an Anderson-lattice form of model Hamiltonian, and which furthermore exhibit an appropriate parameter regime. (4) Away from the Fermi surface the magnitude of the resulting gap function, 1Ak 1, grows in rough proportion to IE~--EF],and it thus tends to be largest at the zone center and the zone corners. (The sign of Ak is opposite in these two regions; there is a node line or surface close to the Fermi surface.) The pairing is therefore distributed throughout the entire Brillouin zone. This novel Ak form can

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explain the sometimes extremely short coherence length, and quite possibly also the strong upwards curvature and/or extended linearity in H,,(T). (5) “The gap” (essentially, dk on the Fermi surface) can be highly anisotropic. It can even exhibit nodes (eight nodes in the case of cuprates), in cases where the node line or surface intersects the Fermi surface. The degree of anisotropy, and thus the question of gap nodes, is decided by band-structure details. Empirically, the likelihood of finding strong gap anisotropy (and thus possibly gap nodes) is increased by reduced dimensionality, e.g. in this materials with planar character. We have shown that it is reasonable to expect these features, and that in planar materials the anisotropy should be mainly of in-plane character (Section 3.3). This strong gap anisotropy can explain the absence of a Hebel-Slichter peak in NMR, as well as large values of the gap ratio 2A/k,T,, and also “within the gap” structures in tunnelling. (In the gap ratio, the effect of a strong pair-breaking reduction of T, is largely compensated by a very small “bare” gap ratio for the VF mechanism, at least for the cuprates, so the large gap ratio is due mainly to the gap anisotropy.) (6) At the Fermi surface, where the pair interaction V(kF, kF) is essentially equal to the Landau interaction f(kF T kF J), this interaction can be strongly repulsive. This provides a source for Stoner enhancement, for short quasiparticle lifetime in the normal state, and for high resistivity with low-power-law ( T2 or T) behavior. (The typically low charge-carrier density also contributes to the high resistivity.) Furthermore, the tendency towards strong quasiparticle scattering can produce a strong pair-breaking effect, lowering T, considerably below the prediction of the simple BCS gap equation. The onset of the strong pair-breaking is rather abrupt, and it can thus produce a “more square” form of A(T) and a large value of the specific-heat ratio fX/yT,. (7) The Kondo temperature or thermal moment-unbinding aspect of valence-fluctuation theory tends to produce a resistivity maximum, with the form observed in several of the lower-T, material families, and at a temperature of about ten times T,. Furthermore, we have found that this aspect can also explain the “universal relation” of Uemura. (8) Finally, we want to emphasize that the obvious non-universality of many of the exotic features (see Table 1) can easily be rationalized within this theoretical picture. We view the typical crystal-chemistry features as aspects which are necessary to enable the VF mechanism, by means of helping to justify an Anderson-lattice form of model Hamiltonian. A largeU unit (ion or cluster) is often obvious in the exotic materials, and this can at least be plausibly argued for the remaining exotic cases. The same can be said for most of the other strange-formula superconductors, although there are some exceptions [31]. Likewise, in these materials one generally finds reasonable candidates for the “conduction” band(s) of an Anderson-lattice form of model Hamiltonian. A less obvious aspect which is probably also essential is an uncommonly small value for the direct hopping between the “localized” (active U-bearing) orbitals on neighboring sites. This requirement serves to rationalize the typical features of reduced dimensionality and/or clustering in the crystal structure, together with a strong crystal field. (The covalent interaction with the counterions generally leads to a strong crystal field splitting for the “localized” orbitals.) The electronegativity difference between the large-U units and the counterions (usually non-metallic or semimetallic elements) can further help, together with the strong crystal field and the large U. In favorable cases these features can combine to keep the extraneous orbital or band states away from the Fermi level (as in covalent bonding), thereby making the compound more suitable for description by an Anderson lattice model. A consequence of this isolation of the desired active ingredients is that the charge-carrier density becomes quite low. It thus appears that a low-chargecarrier density should be a typical characteristic of the VF mechanism. This feature also appears to be essential, from the standpoint that a higher carrier density may well provide enough screening of

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the U interaction to disable the VF mechanism. With sufficient reduction of U the VF mechanism becomes disabled, and at some point the purely conventional phonon mechanism should be enabled. In between these regimes there may be a magnetic phase. We must also point out that a large electronegativity difference is consistent with the finding for cuprates that the charge-transfer energy parameter must be large in order to enable the VF mechanism [ 141. This bare-bones theoretical outline obviously needs much more discussion and elaboration. Although results for the cuprates have been published [14, 151, and also a preliminary discussion for other exotic superconductors [ 151, we intend to present a more detailed exposition elsewhere as a sequel to this survey. We simply want to add here one further remark which follows from this theoretical perspective: Because of the typically large U’s in the strange-formula superconductors, the fact that a given strange-formula compound manages to superconduct at all is already evidence suggesting that it probably has some exotic character [31]. In closing this survey, we would like to list the features which can or should be taken as the phenomenological requirements for exotic status. Reasonable agreement with the Uemura relation c cx L;2 must guarantee admission to the club, by definition. Recognizing that our knowledge is incomplete, however, we can only list the other features which seem to be universal for the present exotic examples. Intrinsic and typically very strong type-II behavior is universal for these materials, and likewise the reasonable inference of a large U interaction. The observation of quadratic or linear resistivity just above T, has turned out to be surprisingly common for these materials, and this also appears to be a universal feature (with the caveats of a possible required minimum of sample disorder [87, 1071 and of a somewhat higher power-law exponent for quasi-one-dimensional materials, according to theory [ 1091). A very large room-temperature resistivity, as compared to the value for lead (the strong-coupling prototype), also seems to be universal. On the other hand, although we have emphasized the importance of a low charge-carrier density, which we expect to be universal (there are no known exceptions), there are a number of exotic materials for which this has not been directly verified. Other features which have been observed sufficiently often and without counter example, suggesting universality, are a region of upwards curvature in HC2(T), and an effective mass enhancement beyond the prediction from the combination of band theory and the (1 +L) enhancement factor from the electron-phonon coupling. There is certainly more to be learned about this exotic phenomenology, and more features can probably be added to this list in the future.

Acknowledgements This survey has been greatly helped by many scientists who have guided me to or otherwise advised me about many of the present references, and who in some cases have shared their data before publication. These people are too numerous to list individually, but their help is very much appreciated. Undoubtedly some significant papers have been overlooked, and for these I apologize. This work was partially supported by the US Department of Energy. References [I]

The number of such “strange formula” superconductors is now quite large. Many examples can be seen in the old compilation of B.W. Roberts, J. Phys. Chem. Ref. Data 5 (1976) 581. Many ternary examples are listed by

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[2] [3]

[4] [5]

[6]

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D.C. Johnston and H.F. Braun, in: 0. Fischer, M.B. Maple (Eds.), Superconductivity in Ternary Compounds II, Springer, Berlin 1982, p. 11, and by R.N. Shelton, in: W. Biickel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kemforschungszentrum Karlsruhe, 1982, p. 123. A more recent compilation (for compounds with T, > 1 K) is available in Appendix C of J.C. Phillips, Physics of High-T, Superconductors, Academic Press, San Diego, 1989. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108 (1957) 1175; J.R. Schrieffer, Theory of Superconductivity, Benjamin, New York, 1964. P.B. Allen, R.C. Dynes, Phys. Rev. B 12 (1975) 905, J. Phys. C 8 (1975) L158. There are a number of other studies of Eliashberg theory in the regime of extremely strong coupling, 1% 2: S.G. Louie, M.L. Cohen, Sol. State Commun. 22 (1977) 1; V.Z. Kresin, H. Gutfreund, W.A. Little, ibid. 51 (1984) 339; J.P. Carbotte, F. Marsiglio, B. Mitrovic, Phys. Rev. B 33 (1986) 6135; L.N. Bulaevskii, O.V. Dolgov, JETP Lett. 45 (1987) 526; L.N. Bulaevskii, O.V. Dolgov, M.O. Ptitsyn, Phys. Rev. B 38 (1988) 11290; F. Marsiglio, P.J. Williams, P.J. Carbotte, ibid. 39 (1989) 9595; Physica C 162-164 (1989) 1493. J.P. Carbotte, Rev. Mod. Phys. 62 (1990) 1027, and references therein. The idea of a combination of phonon and non-phonon mechanisms is also examined in Ref. [12]. Such discussion is usually found in conference proceedings, especially in overview or summary papers. Several of these commentaries are noted in Ref. [ 151. One of these is a listing of anomalous features of ternary superconductors: P.W. Anderson, in: G.K. Shenoy, B.D. Dunlap, F.Y. Fradin (Eds.), Ternary Superconductors, Elsevier, NorthHolland, 1981, p. 309. Although this was long before the discovery of the cuprate superconductors, it is now apparent that there is a striking overlap with the anomalous cuprate properties (see discussion in Ref. [15]). Other notable examples are Ref. [29] and also a case where a large number of similarities between cuprate and organic superconductors are pointed out; R.L. Greene, in: V.Z. Kresin, W.A. Little (Eds.), Organic Superconductivity, Plenum, New York, 1990, p. 7. Further examples are mentioned throughout the present report, especially near the end of Section 3.8. Similarities between cuprate and heavy-fermion superconductors were shown by R. Toumier, A. Sulpice, P. Lejay, 0. Laborde, J. Beille, J. Magn. Magn. Mat. 76-77 (1988) 552. Similarities between nickel borocarbide and cuprate superconductors have been discussed by G. Baskaran, J. Phys. Chem. Sol. 56 (1994) 1957. Similarities among a number of the exotic and other strange-formula superconductors have also been argued by advocates of bipolaronic superconductivity: C.S. Ting, K.L. Ngai, C.T. White, Phys. Rev. B 22 (1980) 2318; B.Ya. Moizhes, LA. Drabkin, Sov. Phys. Solid State 25 (1984) 1139; A.S. Alexandrov, J. Ramringer, S. Robaszkiewicz, Phys. Rev. B 33 (1986) 4526; S. Robaszkiewicz, R. Micnas, J. Ranninger, ibid. 36 (1987) 180; R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. Another survey motivated by the bipolaron concept has emphasized the doped-semiconductor and mixed-valent aspects of all the oxide superconductors: J.L. de Jongh, Physica C 152 (1988) 171. Y.J. Uemura, V.J. Emery, A.R. Moodenbaugh, M. Suenaga, D.C. Johnston, A.J. Jacobson, J.T. Lewandowski, J.H. Brewer, R.F. Kiefl, S.R. Kreitzman, G.M. Luke, T. Riseman, C.E. Stronach, W.J. Kossler, J.R. Kempton, X.H. Yu, D. Opie, H.E. Schone, Phys. Rev. B 38 (1988) 909. Y.J. Uemura, G.M. Luke, B.J. Stemlieb, J.H. Brewer, J.F. Carolan, W.N. Hardy, R. Kadono, J.R. Kempton, R.F. Kiefl, S.R. Kreitzman, P. Mulhem, T.M. Riseman, D.L. Williams, B.X. Yang, S. Uchida, H. Takagi, J. Gopalakrishnan, A.W. Sleight, M.A. Subramanian, CL. Chien, M.Z. Cieplak, G. Xiao, V.Y. Lee, B.W. Statt, C.E. Stronach, W.J. Kossler, X.H. Yu, Phys. Rev. Lett. 62 (1989) 2317; Y.J. Uemura, L.P. Le, G.M. Luke, B.J. Stemlieb, W.D. Wu, J.H. Brewer, T.M. Riseinan, C.L. Seamon, M.B. Maple, M. Ishikawa, D.G. Hinks, J.D. Jorgensen, G. Saito, H. Yamochi, ibid. 66 (1991) 2665; Y.J. Uemura, A. Keren, L.P. Le, G.M. Luke, B.J. Stemlieb, W.D. Wu, J.H. Brewer, R.L. Whetten, S.M. Huang, S. Lin, R.B. Kaner, F. Diederich, S. Donovan, G. Gruner, K. Holczer, Nature 352 (1991) 605; Y.J. Uemura, Physica C 185-189 (1991) 733; Y.J. Uemura, A. Keren, L.P. Le, G.M. Luke, W.D. Wu, Y. Kubo, T. Manako, Y. Shimakawa, M. Subramanian, J.L. Cobb, J.T. Markert, Nature 364 (1993) 605; Y.J. Uemura, G.M. Luke, Physica B 186-188 (1993) 223; Y.J. Uemura, L.P. Le, G.M. Luke, Synth. Met. 5557 (1993) 2845; Y.J. Uemura, A. Keren, L.P. Le, G.M. Luke, W.D. Wu, J.S. Tsai, K. Tanigaki, K. Holczer, S. Donovan, R.L. Whetten, Physica C 235-240 (1994) 2501. For independent data, see P. Birrer, D. Cattani, J. Cors, M. Decroux, 0. Fischer, F.N. Gygax, B. Him, E. Lippelt, A. Schenck, M. Weber, Hype&e Int. 63 (1990) 103; P. Birrer, F.N. Gygax, B. Him, E. Lippelt, A. Schenck, M. Weber, D. Cattani, J. Cors, M. Decroux, 0. Fischer, Phys. Rev. B 48 (1993) 16589; C. Niedermayer, C. Bernhard, U. Binninger, H. Gltickler, J.L. Tallon, E.J. Ansaldo, J.I. Budnick, Phys. Rev. Lett. 71 (1993) 1764; 72 (1994) 2502;

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E.J. Ansaldo, J.J. Boyle, C. Niedermayer, H. Gliickler, J.L. Tallon, A. Mawdsley, D. Pooke, C.E. Stronach, D.R. Noakes, R.S. Cary, M.R. Davis, G.D. Morris, Hyperfine Int. 86 (1994) 505; C. Niedennayer, C. Bernhard, J.I. Budnick, J. Magn. Magn. Mat. 140-144 (1995) 1287; C. Bernhard, C. Niedermayer, U. Binninger, A. Hofer, C. Wenger, J.L. Tallon, G.V.M. Williams, E.J. Ansaldo, J.I. Budnick, C.E. Stronach, D.R. Noakes, M.A. Blat&ton-Mills, Phys. Rev. B 52 (1995) 10488. [71 L. Gao, Y.Y. Xue, F. Chen, Q. Xiong, R.L. Meng, D. Ramirez, C.W. Chu, J.H. Eggert, H.K. Mao, Phys. Rev. B 50 (1994) 4260. PI C.C. Tsuei, D.M. Newns, C.C. Chi, P.C. Pattnaik, Phys. Rev. Lett. 65 ( 1990) 2724; C.C. Tsuei, Physica A I68 (1990) 238. [91 A.A. Abrikosov, Physica C 233 (1994) 102; 244 (1995) 243; Phys. Rev. B 52 (1995) R 15 738. theory, see J.-H. Kim, K. Levin, R. Wentzcovitch, [lOI For the three-band Hubbard model in valence-fluctuation A. Auerbach, Phys. Rev. B 40 (1989) 11378; 44 (1991) 5148; J.-H. Kim, K. Levin, in: Y. Bar-Yam, T. Egami, J. Mustre-de-Leon, A.R. Bishop (Eds.), Lattice Effects in High-T, Superconductors, World Scientific, Singapore, 1992, p. 472; J.-H. Kim, Z. Tesanovic, ibid., p. 460. A similar result has been found for the one-band Hubbard model: M.L. Kulic, R. Zeyher, Phys. Rev. B 49 (1994) 4395. [I 11 K.E. Kihlstrom, P.D. Hovda, V.Z. Kresin, S.A. Wolf, Phys. Rev. B 38 (1988) 4588. [ITI T.W. Barbee III, M.L. Cohen, D.R. Penn, Phys. Rev. B 44 (1991) 4473. The conventional phonon mechanism has been found to be inadequate for YBCO even with the inclusion of a van Hove singularity in the state density; see J.P. Carbotte, E.J. Nicol, in: Y. Bar-Yam, T. Egami, J. Mustre-de-Leon, A.R. Bishop (Eds.), Lattice Effects in High-T, Superconductors, World Scientific, Singapore, 1992, p. 189. [I31 F. Marsiglio, J.P. Carbotte, A. Puchkov, T. Timusk, Phys. Rev. B 53 (1996) 9433; see also A.V. Puchkov, T. Timusk, W.D. Mosley, R.N. Shelton, ibid. 50 (1994) 4144; F. Marsiglio, J.P. Carbotte, ibid. 52 (1995) 16 192: C. Jiang, E. Schachinger, J.P. Carbotte, D. Basov, T. Timusk, ibid. 54 (1996) 1264. P41 B.H. Brandow, Int. J. Mod. Phys. B 8 (1994) 2667. There is also a preliminary report: B.H. Brandow. J. Phys. Chem. Sol. 54 (1993) 1137. u51 B.H. Brandow, Int. J. Mod. Phys. B 8 (1994) 3859. [161 G.M. Luke, L.P. Le, B.J. Stemlieb, Y.J. Uemura, J.H. Brewer, R. Kadono, R.F. Kiefl, S.R. Kreitzman, T.M. Riseman, CL. Seaman, Y. Dalichaouch, M.B. Maple, J.D. Garrett, Hypefine Int. 64 (1990) 517. [I71 W.D. Wu, A. Keren, L.P. Le, G.M. Luke, B.J. Stemlieb, Y .J. Uemura, N. Sato, T. Komatsubara, G.P. Meisner. Hyperfine Int. 85 (1994) 425. [I81 L.P. Le, G.M. Luke, B.J. Stemlieb, W.D. Wu, Y.J. Uemura, J.W. Brill, H. Drulis, Physica C 185-189 (1991) 2715. [I91 W.D. Wu, A. Keren, L.P. Le, G.M. Luke, B.J. Stemlieb, Y.J. Uemura, D.C. Johnston, B.K. Cho, P. Gehring, Hyperfine Int. 86 (1994) 615. WI S.J. Williamson, Phys. Rev. B 2 (1970) 3545. PII ST. Sekula, R.H. Kemohan, G.R. Love, Phys. Rev. 155 (1967) 364; G. Kostorz, L.L. Isaacs, R.L. Panosh. C.C. Koch, Phys. Rev. Lett. 27 (1971) 304. P21 P.H. Pan, D.K. Finnemore, A.J. Bevolo, H.R. Shanks, B.J. Beaudry, F.A. Schmidt, G.C. Danielson, Phys. Rev. B 21 (1980) 2809; A.S. Alexandrov et al., Phys. Rev. Lett. 79 (1997) 1551. v31 S. Tajima, T. Nakahashi, S. Uchida, S. Tanaka, Physica C 156 (1988) 90. v41 We note that such a strong dependence of I$ on the hole doping “x”, in fact a dependence of the correct sign and general magnitude, was found in the calculations of Ref. [ 141; see Fig. 11. I251 A.S. Alexandrov, J. Ranninger, S. Robaskiewicz, Phys. Rev. B 33 (1986) 4526; R. Friedberg, T.D. Lee, ibid. 40 (1989) 6745; R. Friedberg, T.D. Lee, H.C. Ren, Phys. Lett. A 152 (1991) 417 and 423; R. Micnas, J. Ranninger, S. Robaskiewicz, Rev. Mod. Phys. 62 (1990) 113; M. Randeria, J.-M. Duan, L.-Y. Shieh, Phys. Rev. Lett. 62 (1989) 98 I ; Phys. Rev. B 41 (1990) 327; C.A.R. Si de Melo, M. Randeria, J.R. Englebrecht, Phys. Rev. Lett. 7 I (1993) 3202; F. Pistolesi, G.C. Strinati, Phys. Rev. B 49 (1994) 6356; Physica C 2355240 (1994) 2359; R. Hausmann, ibid. 49 (1994) 12975. The boson-condensation aspect is also supported by studies of critical fluctuations: AS. Alexandrov, J. Ranninger, Sol. State Commun. 81 (1992) 403; T. Schneider, H. Keller, Int. J. Mod. Phys. B 8 (1993) 487. WI See, for example, the discussion of Pistolesi and Strinati in Ref. [25], which argues that the exotic superconductors all remain in the not-too-small
48

[27]

[28] [29] [30]

[31]

[32] [33]

[34] [35] [36]

[37] [38]

[39]

[40]

[41]

B. Brandow / Physics Reports 296 (1998)

1-63

potential (the shift between the normal and superconducting states) remains valid. Some important implications are that (a) the Pauli exclusion aspect, which is inherent in the BCS gap equation, is still significant for these materials, and (b) the superconducting transition does not involve a condensation of pre-formed (above-T,) pairs. V.J. Emery, S.A. Kivelson, Phys. Rev. Lett. 74 (1995) 3253. Much of the previous work on superconductivity in strongly disordered materials has been summarized by P.A. Lee, T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. M.P. Gelfand, J.P. Lu, Phys. Rev. Lett. 68 (1992) 1050; Phys. Rev. B 46 (1992) 4367; M.P. Gelfand, Superconductivity Rev. 1 ( I 994) 103. P.W. Anderson, C.C. Yu, in: F. Bassani, F. Fumi, M.P. Tosi (Eds.), Highlights of Condensed Matter Theory, Proc. Int. School Phys. “Enrico Fermi”, Course 89, North-Holland, Italia, 1985, p. 767. M.K. Crawford, M.N. Kunchur, W.E. Farneth, E.M. McCarron III, S.J. Poon, Phys. Rev. B 41 (1990) 282; J.P. Franck, J. Jung, A.A.-K. Mohamed, S. Gygax, G.I. Sproule, ibid. 44 (1991) 5318; H.J. Bornemann, D.E. Morris, ibid. 44 (1991) 5322; J.P. Franck, S. Harker, J.H. Brewer, Phys. Rev. Lett. 71 (1993) 283; J.P. Franck, S. Harker, M. Yu, Physica B 194-196 (1994) 2087; J.P. Franck, in: D.M. Ginsberg (Ed.), Physical Properties of HighTemperature Superconductors, vol. IV, World Scientific, Singapore, 1994, p. 189; G. Zhao, V. Kirtikar, K.K. Singh, A.P.B. Sinha, D.E. Morris, A.V. Inyushkin, Phys. Rev. B 54 (1996) 14 956; G. Zhao, J.W. Ager III, D.E. Morris, ibid. 54 (1996) 14982. See also Refs. [8,12]; D. Zech et al., Nature 371 (1994) 681. There are admittedly a few problematic cases and possible counterexamples among the strange-formula superconductors, where a suitable large-U unit is unclear or missing. Examples include intercalated graphite (KC*), polymeric sulphur nitride (SN),, and the recent silicon clathrate (Na, Ba),.!&: H. Kawaji, H. Horie, S. Yamanaka, M. Ishikawa, Phys. Rev. Lett. 74 (1995) 1427. For the latter material, a band-structure calculation has shown considerable barium Sd-orbital admixture at the Fermi level [S. Saito, A. Oshiyama, Phys. Rev. B 51 (1995) 26281, and this is supported by an NMR study [F. Shimizu, Y. Maniwa, K. Kume, H. Kawaji, S. Yamanaka, M. Ishikawa, Phys. Rev. B 54 (1996) 13 2421. There could be a significant U interaction associated with these 5d orbitals. U. Welp, W.K. Kwok, G. W. Crabtree, K.G. Vandervoort, J.Z. Liu, Phys. Rev. Lett. 62 (1989) 1908; Z. Hao, J.R. Clem, M.W. McElfresh, L. Civale, A.P. Malozemoff, F. Holtzberg, Phys. Rev. B 43 (1991) 2844. G. Brandstatter, F.M. Sauerzopf, H.W. Weber, F. Ladenberger, E. Schwarzmann, Physica C 235-240 (1994) 1845. See also J.R. Thompson, J.G. Ossandon, D.K. Christen, B.C. Chakoumakos, Y.R. Sun, M. Paranthaman, J. Brynestad, Phys. Rev. B 48 (1993) 14 031. Q. Li, M. Suenaga, T. Hikata, K. Sato, Phys. Rev. B 46 (1992) 5857; A. Pomar, M.V. Ramallo, J. Mosqueira, C. Torron, F. Vidal, ibid. 54 (1996) 7470. R. Ikeda, T. Tsuneto, J. Phys. Sot. Japan. 60 ( 1991) 1337. Y.J. Uemura, Physica B 169 (1991) 99; Y.J. Uemura, L.P. Le, G.M. Luke, B.J. Stemlieb, J.H. Brewer, T.M. Riseman, G. Saito, H. Yamochi, in: V.Z. Kresin and W.A. Little (Eds.), Organic Superconductivity, Plenum, New York, 1990, p. 23. S. Foner, E.J. McNiff Jr., E.J. Alexander, Phys. Lett. 49A (1974) 269; K. Okuda, M. Kitagawa, T. Sakakibara, M. Date, J. Phys. Sot. Japan 48 (1980) 2157. S. Foner, E.J. McNiff Jr., D. Heiman, S.-M. Huang, R.B. Kaner, Phys. Rev. B 46 (1992); 14936; G. Spam, J.D. Thompson, R.L. Whetten, S.-M. Huang, R.B. Kaner, F. Diederich, G. Gruner, K. Holczer, Phys. Rev. Lett. 68 (1992) 1228; J.G. Hou, X.-D. Xiang, V.H. Crespi, M.L. Cohen, A. Zettl, Physica C 228 (1994) 175. See also A. Otsuka, T. Ban, G. Saito, H. Ito, T. Ishiguro, N. Hosoito, T. Shinjo, Synth. Met. 55-57 (1993) 3148. This latter work obtained a much smaller 5 of 13 A for Rb 3C 60, by use of fluctuation theory in the analysis of magnetization data. C.E. Johnson, H.W. Jiang, K. Holczer, R.B. Kaner, R.L. Whetton, F. Diederich, Phys. Rev. B 46 (1992) 5880; K. Holczer, R.L. Whetton, Carbon 30 (1992) 1261; C. Politis, V. Buntar, V.P. Seminozhenko, Int. J. Mod. Phys. B 7 (1993) 2163; A.P. Ramirez, Supercond. Rev. 1 (1994) 1; S. Chu, M.E. McHenry, Phys. Rev. B 55 (1997) 11722. W.K. Kwok, U. Welp, K.D. Carlson, G.W. Crabtree, K.G. Vandervoort, H.H. Wang, A.M. Kini, J.M. Williams, D.L. Stupka, L.K. Montgomery, J.E. Thompson, Phys. Rev. B 42 (1990) 8686; H. Ito, M. Watanabe, Y. Nogami, T. Ishiguro, T. Komatsu, G. Saito, N. Hosoito, Physica C 185-189 (1991) 2659; J. Phys. Sot. Japan 60 (1991) 3230; M. Lang, F. Steglich, N. Toyota, T. Sasaki, Phys. Rev. B 49 (1994) 15 227. J.P. Moore, R.S. Graves, J. Appl. Phys. 44 (1973) 1174.

B. Brandow I Physics Reports 296 (I 998) I- 63

49

[42] G.K. White, S.B. Woods, Phil. Trans. Roy. Sot. (London) A 251 (1959) 273. This data is plotted more completely and conveniently by G. Gladstone, M.A. Jensen, J.R. Schrieffer, in: R.D. Parks (Ed.), Superconductivity, vol. 2, Marcel Dekker, New York, 1969, p. 665. A later study of niobium has demonstrated a dominant T3 behavior in this metal: G.W. Webb, Phys. Rev. 181 (1969) 1127. [43] B. Batlogg, Physica B 169 (1991) 7. [44] Y. Hidaka, M. Suzuki, Nature 338 (1989) 635. [45] V.A. Marchenko, Sov. Phys. Solid State 15 (1973) 1261; H. Devantay, J.L. Jorda, M. Decroux, J. Muller, R. Fhikiger, J. Mater. Sci. 16 (1981) 2145. [46] G.W. Webb, Z. Fisk, J.J. Engelhardt, S.D. Bader, Phys. Rev. B 15 (1977) 2624. [47] R. Fhikiger, R. Baillif, E. Walker, Mat. Res. Bull. 13 (1978) 743. [48] J.A. Woollam, S.A. Alterovitz, Phys. Rev. B 19 (1979) 749. We have used their data plots to scale from the quoted indirect po values up to p (300K). [49] K. Kitazawa, T. Matsuura, S. Tanaka, in: G.K. Shenoy, B.D. Dunlap, F.Y. Fradin (Eds.), Ternary Superconductors, Elsevier, North-Holland, New York, 1981, p. 83. [50] H. Sato, T. Ido, S. Tajima, M. Yosida, K. Tanabe, S. Uchida, Physica C 185-189 (1991) 1343. [51] E.S. Hellmann, E.H. Hartford Jr., Phys. Rev. B 47 (1993) 11346. [52] AI. Golovashkin, A.V. Gudenko, A.M. Tskhovrebov, L.N. Zherikhina, M.L. Norton, Physica C 231 (1994) 319. [53] F. Stepniak, P.J. Benning, D.M. Poirier, J.H. Weaver, Phys. Rev. B 48 (1993) 1899. [54] X.-D. Xiang, J.G. Hou, G. Briceiio, W.A. Vareka, M. Mostovoy, A. Zettl, V.H. Crespi, M.L. Cohen, Science 256 (1992) 1190. [55] T.T.M. Palstra, A.F. Hebard, R.C. Haddon, P.B. Littlewood, Phys. Rev. B 50 (1994) 3462. [56] L.D. Rotter, Z. Schlesinger, J.P. McCauley Jr., N. Coustel, J.E. Fischer, A.B. Smith III, Nature 355 (1992) 532; 0. Klein, G. Gruner, S.-M. Huang, J.B. Wiley, R.B. Kaner, Phys. Rev. Lett. 46 (1992) 11247; X.-D. Xiang, J.G. Hou, V.H. Crespi, A. Zettl, M.L. Cohen, Nature 361 (1993) 54; J.G. Hou, V.H. Crespi, X.-D. Xiang, W.A. Vareka, G. Briceiio, A. Zettl, M.L. Cohen, Sol. St. Commun. 86 (1993) 643; J.G. Hou et al., in Ref. [38]. [57] A. Ugawa, G. Ojima, K. Yakushi, H. Kuroda, Synth. Met. 27 (1988) A445; A.M. Kini, U. Geiser, H.H. Wang, K.D. Carlson, J.M. Williams, W.K. Kwok, K.G. Vandervoort, J.E. Thompson, D.L. Stupka, D. Jung, M.H. Whangbo, Inorg. Chem. 29 (1990) 2555; L.I. Buravov, A.V. Zvarykina, N.D. Kushch, V.N. Laukhin, A.A. Merzhanov, A.G. Khomenko, E.B. Yagubskii, Sov. Phys. JETP 68 (1989) 182; S. Kahlich, D. Schweitzer, H.J. Keller, Sol. State Commun. 76 (1990) 933; P. Auban-Senzier, C. Bourbonnais, D. Jerome, C. Lenoir, P. Batail, E. Canadell, Synth. Met. 55-57 (1993) 2542. [58] See P. Auban-Senzier et al., in Ref. [57]. [59] K. Oshima, H. Urayama, H. Yamochi, G. Saito, J. Phys. Sot. Japan 57 (1988) 730; Physica C 153-155 (1988) 1148; S. Glrtner, E. Gogu, I. Heinen, H.J. Keller, T. Klutz, D. Schweitzer, Sol. State Commun. 65 (1988) 153 1; A. Ugawa et al., in Ref. [57]; L.I. Buravov et al., in Ref. [57]; 0. Klein, K. Holczer, G. Griiner, J.J. Chang, F. Wudl, Phys. Rev. Lett. 66 (1991) 655; G. Saito, H. Yamochi, T. Nakamura, T. Komatsu, M. Nakashima, H. Mori, K. Oshima, Physica B 169 (1991) 372. [60] J.M. Williams, A.M. Kini, U. Geiser, H.H. Wang, K.D. Carlson, W.K. Kwok, K.G. Vandervoort, J.E. Thompson, D.L. Stupka, D. Jung, M.-H. Whangbo, in: V.Z. Kresin, W.A. Little (Eds.), Organic Superconductivity, Plenum, New York, 1990, p. 39; H. Sato, T. Sasaki, N. Toyota, Physica C 185-189 (1991) 2679; Y.V. Sushko, V.A. Bondarenko, R.A. Petrosov, N.D. Kushch, E.B. Yagubskii, M.A. Tanatar, V.S. Yefanov, ibid. 1855189 (1991) 2681; G. Saito et al., in Ref. [59]. [61] J.E. Schirber, D.L. Overmyer, K.D. Carlson, J.M. Williams, A.M. Kini, H.H. Wang, H.A. Charlier, B.J. Love, D.M. Watkins, G.A. Yaconi, Phys. Rev. B 44 (1991) 4666. [62] K. Kikuchi, K. Murata, Y. Honda, T. Namiki, K. Saito, K. Kobayashi, T. Ishiguro, I. Ikemoto, J. Phys. Sot. Japan 56 (1987) 2627; K. Kikuchi, Y. Honda, Y. Ishikawa, K. Saito, I. Ikemoto, K. Murata, H. Anzai, T. Ishiguro, K. Kobayashi, Solid State Commun. 66 (1988) 405. [63] H. Mori, I. Hirabayashi, S. Tanaka, T. Mori, Y. Maruyama, H. Inokuchi, Synth. Met. 41-43 (1991) 2255. [64] H. Kobayashi, H. Tomita, T. Udagawa, T. Naito, A. Kobayashi, Synth. Met. 70 (1995) 867. [65] U. Geiser, J.A. Schlueter, K.D. Carlson, J.M. Williams, H.H. Wang, W.-K. Kwok, U. Welp, J.A. Fendrich, J.D. Dudek, C.A. Achenbach, A.S. Komosa, P.M. Keane, D. Naumann, T. Roy, J.E. Schirber, W.R. Bayless, J. Ren, M.-H. Whangbo, Synth. Met. 70 (1995) 1105.

50

B. Brandowl Physics Reports 296 (1998)

I-63

[66] T. Nakamura, T. Komatsu, G. Saito, K. Kato, Y. Maruyama, T. Mori, K. Oshima, T. Osada, S. Kagoshima, N. Miura, T. Takahashi, Synth. Met. 55-57 (1993) 2905. [67] T. Komatsu, T. Nakamura, N. Matsukawa, H. Yamochi, G. Saito, H. Ito, T. Ishiguro, M. Kusunoki, K. Sakaguchi, Solid State Commun. 80 (1991) 843; G. Saito, H. Yamochi, T. Nakamura, T. Komatsu, N. Matsukawa, T. moue, H. Ito, T. Ishiguro, M. Kusunoki, K. Sakaguchi, T. Mori, Synth. Met. 55-57 (1993) 2883. [68] H.H. Wang, K.D. Carlson, U. Geiser, W.K. Kwok, M.D. Vashon, J.E. Thompson, N.F. Larson, G.D. McCabe, R.S. Hulscher, J.M. Williams, Physica C 166 (1990) 57. [69] M. Inokuchi, H. Tajima, T. Ohta, H. Kuroda, A. Kobayashi, A. Sato, T. Naito, H. Kobayashi, J. Phys. Sot. Japan 65 (1996) 538. [70] Y. Watanabe, T. Sasaki, H. Sato, N. Toyota, J. Phys. Sot. Japan 60 (1991) 2118 and 3608. [71] N. Toyota, T. Sasaki, Solid State Commun. 74 (1990) 361. [72] The properties of heavy-fermion materials are reviewed by G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755; W. Schlabitz, J. Baumann, B. Pollit, U. Rauchschwalbe, H.M. Mayer, U. Ahlheim, C.D. Bredl, Z. Phys. B 62 (1986) 171; U. Rauchschwalbe, Physica B 147 (1987) 1; H.R. Ott, Prog. Low Temp. Phys. 11 (1987) 215; Z. Fisk, D.W. Hess, C.J. Pethick, D. Pines, J.L. Smith, J.D. Thompson, J.O. Willis, Science 239 (1988) 33; F. Steglich, J. Phys. Chem. Sol. 50 (1989) 225; N. Grewe, F. Steglich, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 14, North-Holland, Amsterdam, 1991, p. 343. The more recent “almost heavy” superconductors UPQAls and UNizAls are described by C. Geibel, S. Thies, D. Kaczorowski, A. Mehner, A. Grauel, B. Seidl, U. Ahlheim, R. Helfiich, K. Petersen, C.D. Bredl, F. Steglich, Z. Phys. B 83 (1991) 305; C. Geibel, C. Schank, S. Thies, H. Kitazawa, C.D. Bredl, A. Biihm, M. Rau, A. Grauel, R. Caspary, R. Helfrich, U. Ahlheim, G. Weber, F. Steglich, ibid. 84 (1991) 1; C. Geibel, U. Ahlheim, C.D. Bredl, J. Diehl, A. Grauel, R. Helfrich, H. Kitazawa, R. Kiihler, R. Modler, M. Lang, C. Schank, S. Thies, F. Steglich, N. Sato, T. Komatsubara, Physica C 185-189 (1991) 2651; F. Steglich, U. Ahlheim, A. Biihm, C.D. Bredl, R. Caspary, C. Geibel, A. Grauel, R. Helfrich, R. Kiihler, M. Lang, A. Mehner, R. Modler, C. Schank, C. Wassilew, G. Weber, W. Assmus, N. Sato, T. Komatsubara, ibid. 185-189 (1991) 379. [73] B.H. Brandow, Phys. Rev. B 37 (1988) 250. [74] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz, F. Lichtenberg, Nature 372 (1994) 532; F. Lichtenberg, A. Catana, J. Mannhart, D.G. Schlom, Appl. Phys. Lett. 60 (1992) 1138. [75] L.E. DeLong, G.W. Crabtree, L.N. Hall, H. Kierstead, H. Aoki, S.K. Dhar, K.H. Gschneidner Jr., A. Junod, Physica B 135 (1985) 81. [76] H.N.S. Lee, H. McKinzie, D.S. Tannhauser, A. Wold, J. Appl. Phys. 40 (1969) 602; H.N.S. Lee, M. Garcia, H. McKinzie, A. Wold, J. Sol. St. Chem. 1 (1970) 190. [77] J.M. Heintz, M. Drillon, R. Kuentzler, Y. Dossmann, J.P. Kappler, 0. Durmeyer, F. Gauthier, Z. Phys. B 76 (1989) 303. [78] N.F. Mott, Metal-Insulator Transitions, Taylor & Francis, London, 1974, pp. 28 et seq. [79] This is the limit for the more conventional electron scattering mechanisms: electron-phonon, electron-electron, and lattice defects. The limit is considerably larger for metallic resistivity in highly disordered systems with Anderson localization (see Ref. [78]). [80] H. Wiesmann, M. Gurvitch, H. Lutz, A. Ghosh, B. Schwarz, M. Strongin, P.B. Allen, J.W. Halley, Phys. Rev. Lett. 38 (1977) 782. [81] M. Gurvitch, Phys. Rev. B 24 (1981) 7404. [82] Z. Fisk, G.W. Webb, Phys. Rev. Lett. 36 (1976) 1084. [83] M. Gurvitch, A.T. Fiory, Phys. Rev. Lett. 59 (1987) 1337; M. Gurvitch, A.T. Fiory, L.S. Schneemeyer, R.T. Cava, G.P. Espinosa, J.V. Waszczak, Physica C 153-155 (1988) 1369; S. Martin, A.T. Fiori, R.M. Fleming, L.F. Schneemeyer, J. Waszczak, Phys. Rev. B 41 (1990) 846. [84] H. Takagi, B. Batlogg, H.L. Kao, J. Kwo, R.J. Cava, J.J. Krajewski, W.F. Peck Jr., Phys. Rev. Lett. 69 (1992) 2975; Y. Nakamura, S. Uchida, Phys. Rev. B 47 (1993) 8369. [85] B. Sundqvist, B.M. Andersson, Sol. St. Commun. 76 (1990) 1019; B.M. Andersson, B. Sundqvist, J. Niska, B. Loberg, K. Easterling, Physica C 170 (1990) 521; B. Sundqvist, E.M.C. Nilsson, Phys. Rev. B 51 (1995) 6111. [86] Z. Fisk, A.C. Lawson, Solid State Commun. 13 (1973) 277. [87] M. Gurvitch, Physica B 135 (1985) 276.

B. Brandowl Physics Reports 296 (1998)

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51

[88] P.C. Pattnaik, C.L. Kane, D.M. Newns, C.C. Tsuei, Phys. Rev. B 45 ( 1992) 57 14; S. Gopalan, 0. Gunnarsson, O.K. Andersen, ibid. 46 (1992) 11 798; see also R. Hlubina, T.M. Rice, ibid. 51 (1995) 9253. [89] The empirical correlation between linear resistivity and small values of ]sr - EVH~has recently been summarized by D.H. Lu, M. Schmidt, T.R. Cummins, S. Schuppler, F. Lichtenberg, J.G. Bednorz, Phys. Rev. Lett. 76 (1996) 4845. [90] The quasiparticle-scattering contribution to resistivity necessarily involves umklapp, as discussed for example by A. Oshiyama, K. Nakao, H. Kamimura, J. Phys. Sot. Japan 45 (1978) 1136; M. Kaveh, N. Wiser, Adv. Phys. 33 (1984) 257; L.N. Bulaevskii, ibid. 37 (1988) 443; L.N. Bulaevskii, V.B. Ginodman, A.V. Gudenko, M.V. Kartsovnik, P.A. Kononovich, V.N. Laukhin, I.F. Shchegolev, Sot. Phys. JETP 67 (1988) 810; K. Miyake, T. Matsuura, C.M. Varma, Solid State Commun. 71 (1989) 1149; E.S. Hellman, Phys. Rev. B 39 (1989) 9604; S. Fujimoto, H. Kohno, K. Yamada, J. Phys. Sot. Japan 60 (1991) 2724; M.J. Lercher, J.M. Wheatley, Phys. Rev. B 52 (1995) R7038. Umklapp was omitted in the calculations of Ref. [88], which dealt only with the quasiparticle lifetime; there has not been any umklapp study with a sufficiently realistic band structure and Fermi surface for cuprates. [91] Y. Kubo, Y. Shimakawa, T. Mar&o, H. Igarashi, Phys. Rev. B 43 (1991) 7875. [92] C.C. Tsuei, A. Gupta, G. Koren, Physica C 161 (1989) 415. [93] 0. Klein et al., in Ref. [56]; X.-D. Xiang, J.G. Hou, B. Briceiio, W.A. Vareka, R. Mostovoy, A. Zettl, V.H. Crespi, M.L. Cohen, Science 256 (1992) 1190; W.B. Zhao, X.D. Zhang, Z.Y. Ye, J.L. Zhang, C.Y. Li, D.L. Yin, Sol. St. Commun. 85 (1993) 945; T.T.M. Palstra, A.F. Hebard, R.C. Haddon, P.B. Littlewood, Phys. Rev. B 50 (1994) 3462; T.T.M. Palstra, R.C. Haddon Sol. St. Commun. 92 (1994) 71; See also V.H. Crespi, J.G. Hou, X.-D. Xiang, M.L. Cohen, A. Zettl, Phys. Rev. B 46 (1992) 12 064. [94] M. Gurvitch, A.K. Ghosh, H. Lutz, M. Strongin, Phys. Rev. B 22 (1980) 128; A.M. Gurvitch, in: H. Suhl and M.B. Maple (Eds.), Superconductivity in d- and f-Band Metals, Academic Press, New York, 1980, p. 317. [95] V.B. Ginodman, A.V. Gudenko, P.A. Kononovich, V.N. Laukhin, I. Shchegolev, JETP Lett. 44 (1986) 673; L.N. Bulaevskii in Ref. [90]; L.N. Bulaevskii et al., in Ref. [90]; L.I. Buravov et al. in Ref. [57]; S. Kahlich, D. Schweitzer, H.J. Keller, Solid State Commun. 76 (1990) 933; P. Auban-Senzier et al., in Ref. [57]. The quasi-one-dimensional organic superconductors have more complicated resistivities, due apparently to temperaturedependent disorder and to phase transitions. See for example K. Bechgaard, K. Cameiro, M. Olsen, F.B. Rasmussen, C.S. Jacobsen, Phys. Rev. Lett. 46 (1981) 852; S.S.P. Parkin, D. Jerome, K. Bechgaard, Mol. Cryst. Liq. Cryst. 79 (1982) 213; D.U. Gubser, W.W. Fuller, T.O. Poehler, J. Stokes, D.O. Cowan, M. Lee, A.N. Bloch, ibid. 79 (1982) 225. [96] K. Kadowaki, S.B. Woods, Sol. St. Commun. 58 (1986) 507, and references therein; K. Kadowaki, Physica B 186-188 (1993) 727; Y. Maeno et al., J. Phys. Sot. Japan 66 (1997) 1405; K. Miyake et al., Solid State Commun. 7 (1989) 1149. [97] L.E. DeLong, G.W. Crabtree, L.N. Hall, D.G. Hinks, W.K. Kwok, R.B. Mattingly, Jpn. J. Appl. Phys. 26 (1987) Suppl. 26-3, CL19; L.E. DeLong, G.W. Crabtree, L.N. Hall, D.G. Hinks, W.K. Kwok, S.K. Malik, Phys. Rev. B 36 (1987) 7155. [98] P. Garoche, J.J. Veyssie, P. Manuel, P. Molinie, Sol. St. Commun. 19 (1976) 455. [99] J.A. Benda, Phys. Rev. B 10 (1974) 1409; A.H. Thompson, Phys. Rev. Lett. 35 (1975) 1786. [ 1001 C.K. Chang, M.L. Cohen, A.F. Garito, A.J. Heeger, C.M. Mikulski, A.G. MacDiarmid, Sol. St. Commun. 18 (1976) 1451; R.H. Friend, D. Jerome, S. Rehmatullah, A.D. Yoffe, J. Phys. C: Sol. St. Phys. 10 (1977) 1001. [loll Y. Maeno et al., J. Phys. Sot. Japan 66 (1997) 1405. [102] S.D. Bader, S.K. Sinha, Phys. Rev. B 18 (1978) 3082. [103] V.H. Crespi, M.L. Cohen, Phys. Rev. B 52 (1995) 3619. [ 1041 W.A. Vareka, A. Zettl, Phys. Rev. Lett. 72 (1994) 4121. [105] P.C. Klipstein, A.G. Bagnall, W.Y. Liang, E.A. Marseglia, R.H. Friend, J. Phys. C 14 (1981) 4067. [106] C. Jullien, J. Ruvalds, A. Virosztek, 0. Gorochov, Solid State Commun. 79 (1991) 875. [107] M. Gurvitch, Phys. Rev. Lett. 56 (1986) 647. [ 1081 See the publications in Ref. [90], and also T.T.M. Palstra, R.C. Haddon, Solid State Commun. 92 (1994) 71. In the latter, the plotted A coefficients (for p w.4T2) of organic materials lie far above the Kadowaki-Woods line of Ref. [96]. These large A’s are probably erroneous, due to experimental problems in determining the true resistivity

B. BrandowlPhysics

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magnitudes. There is evidence for this in the case of /I-(ET)&, where the plotted A value is about 200 times larger than the experimental value reported by Bulaevskii et al., in Ref. [90]. [109] A. Oshiyama, K. Nakao, H. Kamimura, J. Phys. Sot. Japan 45 (1978) 1136. [ 1 lo] N. Tralshawala, J.F. Zasadzinski, L. Coffey, W. Gai, M. Romanlis, Q. Huang, R. Vaglio, K.E. Gray, Phys. Rev. B 51 (1995) 3812.

[ 1111 A.P. Mackenzie,

S.R. Julian, G.G. Lonzarich, A. Carrington, S.D. Hughes, R.S. Liu, DC. Sinclair, Phys. Rev. Lett. 71 (1993) 1238; A.P. Mackenzie, S.R. Julian, A. Carrington, G.G. Lonzarich, D.J.C. Walker, J.R. Cooper, DC. Sinclair, Physica C 235-240 (1994) 233.

[ 1121 MS. Osofsky, R.J. Soulen Jr., S.A. Wolf, J.M. Broto, H. Rakoto, J.C. Ousset, G. Coffe, S. Askenazy, P. Pari, I. Bozovic, J.N. Eckstein, G.F. Virshup, Phys. Rev. Lett. 71 (1993) 2315; 72 (1994) 3292 (E). [ 1131 D.J.C. Walker, 0. Laborde, A.P. Mackenzie, S.R. Julian, A. Carrington, J.W. Loram, J.R. Cooper, Phys. Rev. B 51 (1995) 9375. [ 1141 Y. Dalichaouch, B.W. Lee, C.L. Seaman, J.T. Markert, M.B. Maple, Phys. Rev. Lett. 64 (1990) 599. [115] N.R. Werthamer, E. Helfand, P.C. Hohenberg, Phys. Rev. 147 (1966) 295; K. Maki, ibid. 148 (1966) 362. [ 1161 U. Welp, W.K. Kwok, G.W. Crabtree, H. Claus, K.G. Vandervoort, B. Dabrowski, A.W. Mitchell, D.R. Richards, D.T. Marx, D.G. Hinks, Physica C 156 (1988) 27; G.W. Crabtree, W.K. Kwok, U. Welp, R. Burriel, H. Claus, K.G. Vandervoort, J.Z. Liu, J. Magn. Magn. Mat. 76-77 (1988) 547; W.K. Kwok, U. Welp, G.W. Crabtree, K.G. Vandervoort, R. Hulscher, Y. Zheng, B. Dabrowski, D.G. Hinks, Phys. Rev. B 40 (1989) 9400; M. Affronte, J. Marcus, C. Es&be-Filippini, A. Sulpice, H. Rakoto, J.M. Broto, J.C. Ousset, S. Askenazy, A.G.M. Janser, ibid. 49 (1994) 3502; V.F. Gantmakher, L.A. Klinkova, N.V. Barkovskii, G.E. Tsydynzhapov, S. Wiegers, A.K. Geim, ibid. 54 (1996) 6133. [117] R. Brusetti, A. Briggs, 0. Laborde, M. Potel, P. Gougeon, Phys. Rev. B 49 (1994) 8931. [ 1181 K. Murata, M. Tokumoto, H. Bando, H. Tanino, H. Anzai, N. Kinoshita, K. Kajimura, G. Saito, T. Ishiguro, Physica B 135 (1985) 515; F. Creuzet, G. Creuzet, D. Jerome, D. Schweitzer, H.J. Keller, J. Physique Lett. 46 (1985) L-1079; M. Tokumoto, K. Murata, H. Bando, H. Anzai, G. Saito, K. Kajimura, T. Ishiguro, Sol. St. Commun. 54 (1985) 1031, Physica B 143 (1986) 338; L.N. Bulaevskii, V.B. Ginodman, A.V. Gudenko, JETP Lett. 45 (1987) 451; V.N. Laukhin, S.H. Pesotskii, E.B. Yagubskii, ibid. 45 (1987) 501; D. Schweitzer, K. Polychroniadis, T. Klutz, H.J. Keller, I. Henning, I. Heinen, U. Haeberlen, E. Gogu, S. Gartner, Synth. Met. 27 (1988) A 465; E. Gogu, D. Schweitzer, H.J. Keller, Physica C 153-155 (1988) 491. [119] K. Oshima et al., in Ref. [59]; K. Murata, Y. Honda, H. Anzai, M. Tokumoto, K. Takahashi, N. Kinoshita, T. Ishiguro, Synth. Met. 27 (1988) A 341; A.G. Swanson, J.S. Brooks, H. Anzai, N. Konoshita, M. Tokumoto, K. Murata, Sol. St. Commun. 73 (1990) 353. [120] W.K. Kwok et al., in Ref. [40]. [121] G. Saito et al., in Ref. [59]. [122] J. Wosnitza, X. Liu, D. Schweitzer, H.J. Keller, Phys. Rev. B 50 (1994) 12747. [123] K. Kanoda, K. Kato, Y. Kobayashi, M. Kato, T. Takahashi, K. Oshima, B. Hilti, J. Zambounis, Synth. Met. 55-57 (1993) 2871. [124] See A.G. Swanson et al., in Ref. [119]. [125] J.E. Graebner, R.C. Haddon, S.V. Chichester, S.H. Glarum, Phys. Rev. B 41 (1990) 4808. [126] H. Mori, K. Nakao, S. Nagaya, I. Hirabayashi, S. Tanaka, K. Oshima, G. Saito, Synth. Met. 41-43 (1991) 2159. See also M. Lang et al., in Ref. [40]. [127] J.A. Woollam, R.B. Somoano, P. O’Connor, Phys. Rev. Lett. 32 (1974) 712; J.A. Woollam, R.B. Somoano, Phys. Rev. B 13 (1976) 3843; N. Toyota, H. Nakatsuji, K. Noto, A. Hoshi, N. Kobayashi, Y. Muto, Y. Onodera, J. Low Temp. Phys. 25 (1976) 485; D.E. Prober, R.E. Schwall, M.R. Beasley, Phys. Rev. B 21 (1980) 2717; J.L. Vincent, S.J. Hillenius, R.V. Coleman, Phys. Rev. Lett. 44 (1980) 892; M. Ikebe, N. Kobayashi, K. Katagiri, Y. Muto, Physica B 105 (1981) 435; M. Ikebe, Y. Muto, Synth. Met. 5 (1983) 229; R.V. Coleman, G.K. Eiserman, S.J. Hillenius, A.T. Mitchell, J.L. Vicent, Phys. Rev. B 27 (1983) 125; Y. Onuki, S. Yamanaka, R. Inada, M. Kido, S. Tanuma, Synth. Met. 5 (1983) 245; B.J. Dalrymple, D.E. Prober, J. Low Temp. Phys. 56 (1984) 545; V.A. Kulbachinskii, Synth. Met. 41-43 (1991) 1925. [128] T.P. Orlando, E.J. McNiff Jr., S. Foner, M.R. Beasley, Phys. Rev. B 19 (1979) 4545; H. Devantay et al., in Ref. [45]; S. Foner, E.J. McNiff Jr., Solid State Commun. 39 (1981) 959.

B. Brandow I Physics Reports 296 (1998) [129] [ 1301 [ 13 11 [132] [133]

[I341

[135] [136]

[ 1371 [138] [139]

[ 1401

[141]

[142] [143]

[ 1441 [145]

[ 1461

[ 1471 [148]

I-63

53

S.A. Alterovitz, J.A. Woollam, Phil. Mag. B 40 (1979) 497; W.M. Miller, D.M. Ginsberg, Phys. Rev. B 28 (1983) 3765. G.S. Boebinger, T.T.M. Palstra, A. Passner, M.J. Rosseinsky, D.W. Murphy, 1.1. Mazin, Phys. Rev. B 46 ( 1992) 5876; C.E. Johnson et al., in Ref. [39]. M. Baenitz, M. Heinze, K. Liiders, H. Werner, R. Schliigl, M. Weiden, G. Spam, F. Steglich, Sol. St. Commun. 96 (1995) 539. Y. Ueda, T. Tanaka, K. Kosuge, M. Ishikawa, H. Yasuoka, J. Sol. St. Chem. 77 (1988) 401. L.E. DeLong, G.W. Crabtree, L.N. Hall, H. Kierstead, H. Aoki, S.K. Dhar, K.A. Gschneidner Jr., A. Junod, Physica B 135 (1985) 81; L.E. DeLong, G.W. Crabtree, L.N. Hall, D.G. Hinks, W.K. Kwok, SK. Malik, Phys. Rev. B 36 (1987) 7155; L.E. DeLong, L.N. Hall, S.K. Malik, G.W. Crabtree, W. Kwok, K.A. Gschneidner Jr., J. Magn. Magn. Mat. 63-64 (1987) 478. The upwards curvature is, however, barely visible in a recent single-crystal study; E. Yamamoto, M. Hedo, Y. Inada, T. Ishida, Y. Haga, Y. Onuki, J. Phys. Sot. Japan 65 (1996) 1034. W. Schlabitz et al., in Ref. [72]. In single-crystal studies this upwards curvature has been shown to arise only when the field is applied perpendicular to the tetragonal c axis: T.T.M. Pal&a, A.A. Menovsky, J. van den Berg, A.J. Dirkmat, P.H. Kes, G.J. Nieuwenhuys, J.A. Mydosh, Phys. Rev. Lett. 55 (1985) 2727; L.E. DeLong, D. Togliatti, G.W. Lehman, W.-K. Kwok, G.W. Crabtree, K.G. Vandervoort, D.G. Hinks, Physica B 163 (1990) 499. S.J. Williamson, Phys. Rev. B 2 (1970) 3545; E. Seidl, H.W. Weber, H. Teichler, J. Low Temp. Phys. 30 (1978) 273. M.R. Skokan, R.C. Morris, W.G. Moulton, Phys. Rev. B 13 (1976) 1077; R.K. Stanley, R.C. Morris, W.G. Moulton, Phys. Rev. B 20 (1979) 1903; C. Schlenker, H. Schwenk, C. Escribe-Filippini, J. Marcus, Physica B 135 (1985) 51 I; M. Boujida, C. Escribe-Filippini, J. Marcus, C. Schlenker, Physica C 153-155 (1988) 465. L.J. Azevedo, W.G. Clark, G. Deutscher, R.L. Greene, G.B. Street, L.J. Suter, Sol. St. Commun. 19 ( 1976) 197; P. Barrett, G.M. Brimlow, W.G. Herrenden-Harker, M.G. Priestley, J. Phys. F 11 (1981) 859. Y. Iye, S-I. Tanuma, Synth. Met. 5 (1983) 257; L.A. Pendrys, R.A. Wachnik, F.L. Vogel, P. Lagrange, ibid. 5 (1983) 277. 1. Banerjee, Q.S. Yang, C.M. Falco, I.K. Schuller, Sol. St. Commun. 41 (1982) 805; 1. Banerjee, I.K. Schuller, J. Low Temp. Phys. 54 (1984) 501; C.S.L. Chun, G.-G. Zheng, J.L. Vicent, I.K. Schuller, Phys. Rev. B 29 (I 984) 4915; K. Kanoda, H. Mazaki, N. Hosoito, T. Shinjo, ibid. 35 (1987) 6736. T. W. Haywood, D.G. Ast, Phys. Rev. B 18 (1978) 2225; S.T. Ruggiero, T.W. Barbee Jr., M.R. Beasley, Phys. Rev. Lett. 45 (1980) 1299, Phys. Rev. B 26 (1982) 4894; H. Nakajima, M. Ikebe, Y. Muto, H. Fujimori, J. Appl. Phys. 65 (1989) 1637. The peculiar form of&(T) in UBei3 has been explained in terms of a strong field dependence of the normal-state properties: M. Tachiki, T. Koyama, S. Takahashi, Physica B 135 (1985) 57; J.P. Brison, J. Flouquet, G. Deutscher, J. Low Temp. Phys. 76 (1989) 453. J.Y. Juang, D.A. Rudman, R.B. van Dover, W.R. Sinclair, D.D. Bacon, Adv. Cryo. Eng. 32 (1986) 651; J.S. Moodera, P.M. Tedrow, R. Meservey, ibid. 32 (1986) 679. See, for example, 0. Fischer, Appl. Phys. 16 (1978) 1; S.A. Alterovitz, J.A. Woollam, Solid State Commun. 25 (1978) 141; M. Decroux, 0. Fischer, in: 0. Fischer, M.B. Maple (Eds.), Superconductivity in Ternary Compounds II, Springer, Berlin, 1982, p. 57; P.M. Tedrow, R. Meservey, Phys. Rev. Lett. 43 (1979) 384; Phys. Rev. B 25 (1982) 171; P.M. Tedrow, J.T. Kucera, D. Rainer, T.P. Orlando, Phys. Rev. Lett. 52 (1984) 1637. T.P. Orlando, E.J. McNiff Jr., S. Foner, M.R. Beasley, Phys. Rev. B 19 ( 1979) 4545; T.P. Orlando, M.R. Beasley, Phys. Rev. Lett. 46 (1981) 1598. See Fig. 4 of R.V. Coleman et al., in Ref. [127]. J.L. Smith, J.S. Brooks, C.M. Fowler, B.L. Freeman, J.D. Goettee, W.L. Hults, J.C. King, P.M. Mankiewich, E.I. De Obaldia, M.L. O’Malley, D.G. Rickel, W.J. Skocpol, J. Low Temp. Phys. 95 (1994) 75; J. Supercond. 7 (1994) 269; J.D. Goettee, J.S. Brooks, W.J. Skocpol, J.L. Smith, D.G. Rickel, B.L. Freeman, C.M. Fowler, P.M. Mankiewich, E.I. DeObaldia, M.L. O’Malley, Physica B 194-196 (1994) 1805. C.L. Seaman, N.Y. Ayoub, T. Bjornholm, E.A. Early, S. Ghamaty, B.W. Lee, J.T. Markert, J.J. Neumeier, P.K. Tsai, M.B. Maple, Physica C 159 (1989) 391. K.R. Biagi, V.G. Kogan, J.R. Clem, Phys. Rev. B 32 (1985) 7165; S. Takahashi, M. Tachiki, ibid. 33 (1986) 4620; D. Feinberg, S. Theodorakis, A.M. Ettouhami, ibid. 49 (1994) 6285.

54

B. Brandow I Physics Reports 296 (1998)

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[149] L.N. Bulaevskii, Sov. Phys.-JETP 38 (1974) 634. [150] S. Theodorakis, Z. Tesanovic, Phys. Lett. A 132 (1988) 372; Phys. Rev. B 40 (1989) 6659; T. Koyama, N. Takezawa, Y. Naruse, M. Tachiki, Physica C 194 (1992) 20. [ 1511 D.W. Youngner, R.A. Klemm, Phys. Rev. B 21 (1980) 3890. Earlier studies indicating upwards curvature due to Fermi-velocity anisotropy include PC. Hohenberg, N.R. Werthamer, Phys. Rev. 153 (1967) 493; P. Entel, M. Peter, J. Low Temp. Phys. 22 (1976) 613. [152] E. Seidl et al., in Ref. [135]; W.H. Butler, Phys. Rev. Lett. 44 (1980) 1516. [ 1531 In the “bipolaron theory” studies the boson limit t/a + 0 has been assumed, and thus i&(T) has been derived for a charged boson gas. This provides upwards curvature for all T < T,, but this model has the problem that Hc2 + DC) as T + 0: A.S. Alexandrov, J. Ranninger, S. Robaskiewicz, Phys. Rev. B 33 (1986) 4526; A.S. Alexandrov, D.A. Samarchenko, S.V. Traven, Sov. Phys. JETP 66 (1987) 567. To remove this divergence, appeal has been made to some “dirt” in the system: A.S. Alexandrov, Phys. Rev. B 48 (1993) 10 571. See also Ref. [170]. [ 1541 C.T. Rieck, Ph.D. Thesis, Universitat Hamburg, 1991, unpublished. [155] G. Deutscher, 0. Entin-Wohlman, Y. Shapira, Phys. Rev. B 22 (1980) 4264; G. Deutscher, Physica B 109-l 10 (1982) 1629. [156] L.N. Bulaevskii, O.V. Dolgov, M.O. Ptitsyn, Phys. Rev. B 38 (1988) 11290. [157] S. Maekawa, H. Ebisawa, H. Fukuyama, J. Phys. Sot. Japan 52 (1983) 1352; L. Coffey, K. Levin, K.A. Muttalib, Phys. Rev. B 32 (1985) 4382; L.N. Bulaevskii, M.V. Sadovskii, J. Low Temp. Phys. 59 (1985) 89; G. Kotliar, A. Kapitulnik, Phys. Rev. B 33 (1986) 3146. [158] A.M. Gabovich, A.S. Shpigel, Phys. Rev. B 38 (1988) 297. [ 1591 S.H. Han, C.C. Almasan, M.C. de Andrade, Y. Dalichaouch, M.B. Maple, Phys. Rev. B 46 (1992) 14 290. [160] YuN. Ovchinnikov, V.Z. Kresin, Phys. Rev. B 52 (1995) 3075; 54 (1996) 1251. [161] T. Koyama, M. Tachiki, Physica C 263 (1996) 25. [162] S.R. Bahcall, Phys. Rev. Lett. 75 (1995) 1376. [163] G. Kotliar, C.M. Varma, Phys. Rev. Lett. 77 (1996) 2296. [164] J.R. Cooper, J.W. Loram, J.M. Wade, Phys. Rev. B 51 (1995) 6179. [165] R.G. Dias, J.M. Wheatley, Solid State Commun. 98 (1996) 859. [166] See for example, M. Suzuki, M. Hikita, Jpn. J. Appl. Phys. 28 (1989) L 1368; K. Kadowaki, J.N. Li, J.J.M. Franse, Physica C 170 (1990) 298; T.K. Worthington, F.N. Holtzberg, C.A. Feild, Cryogenics 30 (1990) 417; H. Raffy, S. Labdi, 0. Laborde, P. Monceau, Physica C 184 (1991) 159. [167] R. Ikeda, T. Ohmi, T. Tsuneto, J. Phys. Sot. Japan 58 (1989) 1377; 59 (1990) 1397; 60 (1991) 1051, Phys. Rev. Lett. 67 (1991) 3874; S. Ullah, A.T. Dorsey, Phys. Rev. Lett. 65 (1990) 2066; Phys. Rev. B 44 (1991) 262; Z. Tesanovic, L. Xing, L. Bulaevskii, Q. Li, M. Suenaga, Phys. Rev. Lett. 69 (1992) 3563; R. Ikeda, T. Tsuneto, Physica B 194-196 (1994) 1503. [168] A.P. Malozemoff, T.K. Worthington, Y. Yeshurun, F. Holtzberg, P.H. Kes, Phys. Rev. B 38 (1988) 7203; G.S. Boebinger et al., in Ref. [130]. [169] See A.S. Alexandrov, J. Ranninger, and also T. Schneider, H. Keller, in Ref. [25]. [170] AS. Alexandrov, V.N. Zavaritsky, W.Y. Liang, P.L. Nevsky, Phys. Rev. Lett. 76 (1996) 983. [ 1711 D.S. Dessau, Z.-X. Shen, D.M. King, D.S. Marshall, L.W. Lombardo, P.H. Dickinson, A.G. Loeser, J. Di Carlo, C.-H. Park, A. Kapitulnik, W.E. Spicer, Phys. Rev. Lett. 71 (1993) 2781; Z.X. Shen, D.S. Dessau, Phys. Rep. 253 (1995) 1. [ 1721 A.P. Ramirez, M.J. Rosseinsky, D.W. Murphy, R.C. Haddon, Phys. Rev. Lett. 69 (1992) 1687. [173] M. Schmidt, T.R. Cummins, M. B&k, D.H. Lu, N. Niicker, S. Schuppler, F. Lichtenberg, Phys. Rev. B 53 (1996) R14761; T. Yokoya, A. Chainani, T. Takahashi, H. Ding, J.C. Campuzano, H. Katayama-Yoshida, M. Kasai, T. Tokura, ibid. 54 (1996) 13311; T. Katsufuji, M. Kasai, Y. Tokura, Phys. Rev. Lett. 76 (1996) 126. [ 1741 A.P. Mackenzie, S.R. Julian, A.J. Diver, G.J. McMullan, M.P. Ray, G.G. Lonzarich, Y. Maeno, S. Nishizaki, T. Fujita, Phys. Rev. Lett. 76 (1996) 3786. [175] J. Singleton, J.F. Pratt, M. Doporto, T.J.B.M. Janssen, M. Kurmoo, J.A.A.J. Perenboom, W. Hayes, P. Day, Phys. Rev. Lett. 68 (1992) 2500; J. Caulfield, W. Lubczynski, F.L. Pratt, J. Singleton, D.Y.K. Ko, W. Hayes, M. Kurmoo, P. Day, J. Phys.: Condens. Matter 6 (1994) 2911; J. Caulfield, W. Lubczynski, W. Lee, J. Singleton, F.L. Pratt, W. Hayes, M. Kurmoo, P. Day, Synth. Met. 70 (1995) 8 15; see also J. Wosnitza, G.W. Crabtree, H.H. Wang, K.D. Carlson, M.D. Vashon, J.M. Williams, Phys. Rev. Lett. 67 (1991) 263.

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1-63

55

[ 1761 K. Komelsen, J.E. Eldridge, C.C. Homes, H.H. Wang, J.M. Williams, Sol. St. Commun. 72 (1989) 475; N. Toyota, E.W. Fenton, T. Sasaki, M. Tachiki, ibid. 72 (1989) 859; T. Sasaki, H. Sato, N. Toyota, ibid. 76 (1990) 507. [177] See a series of conference reports: the first nine articles in J. Magn. Magn. Mat. 76-77 (1988). [178] B. Mitrovic, E. Schachinger, J.P. Carbotte, Phys. Rev. B 29 (1984) 6187. [ 1791 H. Rietschel, H. Winter, Phys. Rev. Lett. 43 (1979) 1256; T.P. Orlando, M.R. Beasley, ibid. 46 (1981) 1598; T.P. Orlando, in: W. Biickel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kemforschungszentrum Karlsruhe, 1982, p. 75. A similar study of V3Ga was done by A. Jut-rod, ibid. p. 89. [ 1801 G. Mante, R. Claessen, T. Buslaps, S. Harm, R. Manzke, M. Skibowski, J. Fink, Z. Phys. B 80 (1990) 181; R. Manzke, T. Buslaps, R. Claessen, M. Skibowski, J. Fink, Phys. Scripta 41 (1990) 579; T. Takahashi, T. Watanabe, T. Kusunoki, H. Katayama-Yoshida, J. Phys. Chem. Solids 52 (1991) 1427; C.G. Olson, R. Liu, D. W. Lynch, R.S. List, A.J. Arko, B.W. Veal, Y.C. Chang, P.Z. Jiang, A.P. Paulikas, Phys. Rev. B 42 ( 1990) 38 I. [ 1811 R.L. Kiefl, W.A. MacFarlane, K.H. Chow, S. Dunsiger, T.L. Duty, T.M.S. Johnston, J.W. Schneider, J. Sonier, L. Brard, R.M. Strongin, J.E. Fischer, A.B. Smith III, Phys. Rev. Lett. 70 (1993) 3987. [182] L. Degiorgi, G. Briceno, M.S. Fuhrer, A. Zettl, P. Wachter, Nature 369 (1994) 541. [183] H. Sato, H. Takagi, S. Uchida, Physica C 169 (1990) 391; F. Sharifi, A. Pargellis, R.C. Dynes, B. Miller, E.S. Hellman, J. Rosamilia, E.H. Hartford Jr., Phys. Rev. B 44 (1991) 12521; M. Kosugi, J. Akimitsu, T. Uchida, M. Furuya, Y. Nagata, T. Ekino, Physica C 229 (1994) 389. A gap ratio of 3.8-3.9 was found by Q. Huang, J.F. Zasadzinski, N. Tralshawala, K.E. Gray, D.G. Hinks, J.L. Peng, R.L. Greene, Nature 347 (1990) 369. [184] T. Ekino, J. Akimitsu, Physica B 165-166 (1990) 1599. [185] B.P. Clayman, R.F. Frindt, Solid State Commun. 9 (1971) 1881. [I861 P. Samuely, N.L. Bobrov. A.G.M. Jansen, P. Wyder, S.N. Barilo, S.V. Shiryaev, Phys. Rev. B 48 (1993) 13904; P. Szabo, P. Samuely, L.N. Bobrov, J. Marcus, C. Escribe-Filippini, M. Affronte, Physica C 235-240 (1994) 1873. A similar gap ratio (4.2) and BCS temperature dependence was also found in a single-crystal infra-red study: A.V. Puchkov et al., in Ref. [13]. A BCS temperature dependence was also found by H. Sato et al. in Ref. [183] and by M. Kosugi et al., in Ref. [183]. [187] K.W. Ng, Z.G. Khim, D.P. Shum, E.L. Wolf, Surf Sci. 181 (1987) 37. [188] Z. Schlesinger, R.T. Collins, F. Holtzberg, C. Feild, G. Koren, A. Gupta, Phys. Rev. B 41 (1990) 11237; B.N.J. Persson, J.E. Demuth, ibid. 42 (1990) 8057; B. Fried& C. Thomsen, M. Cardona, Phys. Rev. Lett. 65 (1990) 915; L.D. Rotter, Z. Schlesinger, R.T. Collins, F. Holtzberg, C. Field, U.W. Welp, G.W. Crabtree, J.Z. Liu, Y. Fang, K.G. Vandervoort, S. Fleshler, ibid. 67 (1991) 2741. [I891 T. Ekino, T. Doukan, H. Fujii, F. Nakamura, S. Sakita, M. Kodama, T. Fujita, Physica C 263 (1996) 249. Compare also with J. Chen, J.F. Zasadzinski, K.E. Gray, J.L. Wagner, D.G. Hinks, Phys. Rev. B 49 (1994) 3683, where a less sharp BCS tunnelling spectrum is found for HgBa~CuO~,~. [ 1901 ME. Hawley, K.E. Gray, B.D. Terris, H.H. Wang, K.D. Carlson, J.M. Williams, Phys. Rev. Lett. 57 (1986) 629. [191] A. Nowak, U. Poppe, M. Weger, D. Schweitzer, H. Schwenk, Z. Phys. B 68 (1987) 41. [192] H. Bando, S. Kashiwaya, H. Tokumoto, H. Anzai, N. Kinoshita, H. Kajimura, J. Vat. Sci. Technol. A 8 (1990) 479. [193] M. Weger, A. Nowak, D. Schweitzer, J. Kubler, J.M. Van Bentum, C.S. Sommers, in: Organic Superconductivity eds. V.Z. Kresin, W.A. Little, Plenum, New York, 1990, p. 211. [194] J. Moreland, A.F. Clark, R.J. Soulen Jr., J.L. Smith, Physica B 194-196 (1994) 1727. [195] Y. DeWilde, J. Heil, A.G.M. Jansen, P. Wyder, R. Deltour, W. Assmus, A. Menovsky, W. Sun, L. Taillefer, Phys. Rev. Lett. 72 (1994) 2278. See also G. Gall, C. Bruder, H.v. Lohneysen, Phys. Rev. B 52 (1995) 6801. [196] K. Hasselbach, J.R. Kirtley, P. Lejay, Phys. Rev. B 46 (1992) 5826. [197] M. Kyogaku, Y. Kitaoka, K. Asayama, C. Geibel, C. Schank, F. Steglich, J. Phys. Sot. Japan 61 (1992) 2660; Y. Kitaoka, M. Kyogaku, K. Asayama, H. Nakamura, C. Geibel, C. Schank, F. Steglich, Physica B 186-188 (1993) 229. [198] A. Junod, in: W. Biickel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kemforschungszentrum Karlsruhe, 1982, p. 89. [ 1991 F. Marsiglio, J.P. Carbotte, Phys. Rev. B 33 (1986) 6141. [200] G.D. Mahan, Phys. Rev. B 40 (1989) 11317; A.J. Fedro, D.D. Koelling, ibid. 47 (1993) 14342. [201] R. Hackl, in: H. Kuzmany, M. Mehring, J. Fink (Eds.), Electronic Properties of High-T, Superconductors and Related Compounds, Springer, Berlin, 1990, p. 312; L.C. Brunel, S.G. Louie, G. Martinez, S. Labdi, H. Raffy,

56

[202] [203] [204]

[205] [206] [207] [208] [209] [210] [211]

[212]

[213]

[214]

[215] [216] [217] [218] [219] [220] [221] [222] [223]

B. Brandowl Physics Reports 296 (1998)

1-63

Phys. Rev. Lett. 66 (1991) 1346; T. Stauffer, R. Nemetschek, R. Hackl, P. Miiller, H. Veith, ibid. 68 (1992) 1069; Y. Li, J.L. Huang, CM. Lieber, ibid. 68 (1992) 3240; J. Ma, C. Quitmann, R.J. Kelley, H. Berger, G. Margaritondo, M. Onellion, Science 267 (1995) 862; L.V. Gasparov, P. Lemmens, M. Brinkmann, N.N. Kolesnikov, G. Giintherodt, Phys. Rev. B 55 (1997) 1223. Many earlier references are described in C. Bandte, P. Hertel, J. Appel, Phys. Rev. B 45 (1992) 8026. D.B. Tanner, A.J. Sievers, Phys. Rev. B 8 (1973) 1978. See Fig. 4 of J. Wosnitza et al., Ref. [122]. Z. Zhang, C.-C. Chen, S.P. Kelty, H. Dai, C.M. Lieber, Nature 353 (1991) 333; Z. Zhang, C.-C. Chen, C.M. Lieber, Science 254 (1991) 1619; P. Jess, S. Behler, M. Bemasconi, V. Thommen-Geiser, H.P. Lang, M. Baenitz, K. Liiders, H.-J. Guntherodt, Physica C 235-240 (1994) 2499. C.M. Lieber, Z. Zhang, in: H. Ehrenreich, F. Spaepen (Eds.), Solid State Physics, vol 48, Academic Press, Boston, 1994, p. 349; see p. 377. G.D. Mahan, in Ref. [200]. L.D. Rotter et al., in Ref. [56]; S.A. Fitzgerald, S.G. Kaplan, A. Rosenberg, A.J. Sievers, R.A.S. McMordie, Phys. Rev. B 45 (1992) 10 165. V. Hoffstein, R.W. Cohen, Phys. Lett. A 29 (1969) 603. G.R. Stewart, B.L. Brandt, Phys. Rev. B 29 (1984) 3908. D.E. Farrell, B.S. Chandrasekhar, Phys. Rev. Lett. 38 (1977) 788. See for example T. Imai, T. Shimizu, H. Yasuoka, Y. Ueda, K. Kosuge, J. Phys. Sot. Japan. 57 (1988) 2280; T. Imai, H. Yasuoka, T. Shimizu, Y. Ueda, K. Yoshimura, K. Kosuge, Physica C 162-164 (1989) 169; K. Asayama, G.-Q. Zheng, Y. Kitaoka, K. Ishida, K. Fujiwara, Physica C 178 (1991) 281; Y. Kitaoka, K. Ishida, S. Ohsugi, K. Fujiwara, G.-Q. Zheng, K. Asayama, Appl. Magn. Reson. 3 (1992) 549; S. Ohsugi, Y. Kitaoka, K. Ishida, G. Zheng, K. Asayama, J. Phys. Sot. Japan 63 (1994) 700. D.E. MacLaughlin, C. Tien, W.G. Clark, M.D. Lan, Z. Fisk, J.L. Smith, H.R. Ott, Phys. Rev. Lett. 53 (1984) 1833; C. Tien, D.E. MacLaughlin, M.D. Lan, W.G. Clark, Z. Fisk, J.L. Smith, H.R. Ott, Physica 135B (1985) 14; K. Asayama, Y. Kitaoka, Y. Kohori, J. Magn. Magn. Mat. 7677 (1988) 449; Y. Kitaoka et al., in Ref. [197]; H. Tou, Y. Kitaoka, K. Asayama, C. Geibel, C. Schank, F. Steglich, J. Phys. Sot. Japan 64 (1995) 725; Y. Kohori, K. Matsuda, T. Kohara, J. Phys. Sot. Japan 65 (1996) 1083; Solid State Commun. 95 (1995) 121; K. Matsuda et al., Phys. Rev. B 55 (1997) 15223. An exception is UNizA13, where no feature was observed in l/T, at the bulk Tc temperature, suggesting that the powdering of the sample has disrupted the superconductivity: M. Kyogaku et al., in Ref. [197]. It should also be noted that a mild Hebel-Slichter peak was found in 63Cu NQR data for CeCu2Si2 [D.E. MacLaughlin, C. Tien, L.C. Gupta, J. Aarts, F.R. de Boer, Z. Fisk, Phys. Rev. B 30 (1984) 15771, in contrast to the NMR data for this material (D.E. MacLaughlin et al., in Ref. [212]; C. Tien et al., in Ref. [212]). S.M. DeSoto, C.P. Slichter, A.M. Kini, H.H. Wang, U. Geiser, J.M. Williams, Phys. Rev. B 52 (1995) 10364; H. Mayaffre, P. Wzietek, D. Jerome, C. Lenoir, P. Batail, Phys. Rev. Lett. 75 (1995) 4122; K. Kanoda, K. Miyagawa, A. Kawamoto, Y. Nakazawa, Phys. Rev. B 54 (1996) 76. M. Takigawa, H. Yasuoka, G. Saito, J. Phys. Sot. Japan 56 (1987) 873. S. Sasaki, A. Matsuda, C.W. Chu, J. Phys. Sot. Japan 63 (1994) 1670; V.A. Stenger, C.H. Pennington, D.R. Buffinger, R.P. Ziebarth, Phys. Rev. Lett. 74 (1995) 1649. Y. Masuda, N. Okubo, J. Phys. Sot. Japan 26 (1969) 309. See also S. Wada and K. Asayama, ibid. 34 (1973) 1168. C.H. Pennington, V.A. Stenger, Rev. Mod. Phys. 68 (1996) 855. Y. Kitaoka, S. Ohsugi, K. Asayama, T. Ohtani, Physica C 192 (1992) 272; S. Ohsugi, Y. Kitaoka, M. Kyogaku, K. Ishida, K. Asayama, T. Ohtani, J. Phys. Sot. Japan 61 (1992) 3054. K. Kumagai, F.Y. Fradin, in: W. Biickel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kemforschungszentrum Karlsruhe, 1982, p. 227. Y. Kohori, T. Kohara, Y. Kitaoka, K. Asayama, M.B. Maple, M.W. McEltiesh, J. Phys. Sot. Japan 56 (1987) 1645. Y. Hasegawa, H. Fukuyama, J. Phys. Sot. Japan 56 (1987) 877 and 2619. K. Asayama et al., in Refs. [211, 2121; K. Ishida, Y. Kitaoka, N. Ogata, T. Kamino, K. Asayama, J.R. Cooper, N. Athanassopoulou, J. Phys. Sot. Japan 62 (1993) 2803; K. Ishida, Y. Kitaoka, K. Asayama, K. Kadowaki,

B. Brandow I Physics Reports 296 (1998)

T. Mochiku, ibid. 63 (1994) 1104. For studies including quasiparticle (1990) 6805; J.P. Lu, Mod. Phys. Lett. B 6 (1992) 547.

57

l-63

damping,

see B.W. Statt, Phys. Rev. B 42

[224] P.B. Allen, D. Rainer, Nature 349 (1991) 396. [225] Y. Kobayashi,

T. Nakamura,

T. Takahashi,

K. Kanoda, B. Hilti, J.S. Zambounis,

[226] G.C. Papavassiliou, G.A. Mousdis, J.S. Zambounis, Met. 27 (1988) B 379.

Synth. Met. 70 (1995) 871.

A. Terzis, A. Hountas, B. Hilti, C.W. Meyer, J. Pfeiffer, Synth.

[227]

K. Kanoda, K. Akiba, K. Suzuki, T. Takahashi, Phys. Rev. Lett. 65 (1990) 1271; K. Kanoda, Y. Tsubokura, K. Ikeda, T. Takahashi, N. Matsukawa, G. Saito, H. Mori, T. Mori, B. Hilti, J.S. Zambounis, Synth. Met. 55-57 (1993) 2865; T. Takahashi, K. Kanoda, G. Saito, Physica C 185-189 (1991) 366; L.P. Le, G.M. Luke, B.J. Stemlieb, W.D. Wu, Y.J. Uemura, J.H. Brewer, T.R. Riseman, C.E. Stronach, G. Saito, H. Yamochi, H.H. Wang, A.M. Kini, K.D. Carlson, J.M. Williams, Phys. Rev. Lett. 68 (1992) 1923; D. Achkir, M. Poirier, C. Bourbonnais, G. Quirion, C. Lenoir, P. Batail, D. Jerome, Phys. Rev. B 47 (1993) 11 595; T. Tsubokura, M. Miyamoto, K. Ikeda, T. Nakamura, T. Takahashi, Synth. Met. 70 (1995) 913.

[228]

D.R. Harshman, R.N. Kleiman, R.C. Haddon, S.V. Chichester-Hicks, M.L. Kaplan, L.W. Rupp Jr., T. Pfiz, D.L. Williams, D.B. Mitzi, Phys. Rev. Lett. 64 (1990) 1293; M. Lang, N. Toyota, T. Sasaki, H. Sato, ibid. 69 (1992) 1443, Phys. Rev. B 46 (1992) 5822; M. Lang, N. Toyota, T. Sasaki, Synth. Met. 55-57 (1993) 2401; M. Dressel, S. Bruder, G. Grilner, K.D. Carlson, H.H. Wang, J.M. Williams, Phys. Rev. B 48 (1993) 9906.

[229] Y. Nakazawa,

K. Kanoda, Phys. Rev. B 55 (1997) R 8670.

[230] Z.-X. Shen, D.S. Dessau, B.O. Wells, D.M. King, W.E. Spicer, A.J. Arko, D. Marshall, L.W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, A.G. Loeser, C.H. Park, Phys. Rev. Lett. 70 (1993) 1553; Z.-X. Shen, W.E. Spicer, D.M. King, D.S. Dessau, B.O. Wells, Science 267 (1995) 343. There is also recent photoemission evidence for a similar gap form in YBa2CU306.95 : M.C. Schabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D.A. Bonn, R. Liang, W.N. Hardy, Phys. Rev. B 55 (1997) 2796. [231] The initial papers of the Argonne photoemission group found evidence suggesting an eight-node anisotropic-s gap form: H. Ding, J.C. Campuzano, A.F. Bellman, Y. Yokoya, M.R. Norman, M. Randeria, T. Takahashi, H. KitayamaYoshida, T. Mochiku, K. Kadowaki, G. Jennings, Phys. Rev. Lett. 74 (1995) 2784; M.R. Norman, M. Randeria, H. Ding, J.C. Campuzano, Phys. Rev. B 52 (1995) 615. In later work this conclusion was found to be an artifact due to the superlattice structure of Bi-2212: M.R. Norman, M. Randeria, H. Ding, J.C. Campuzano, A.F. Bellman, Phys. Rev. B 52 (1995) 15 107; H. Ding, M.R. Norman, J.C. Campuzano, M. Randeria, A.F. Bellman, Y. Tokoya, T. Takahashi, T. Mochiku, K. Kadowaki, ibid. 54 (1996) R 9678. Their data is now consistent with a simple d-wave (x2 - y2) gap form, but we must point out that this is also consistent with an anisotropic-s gap where the minimum gap value is close to zero. [232] W. Hardy, D.A. Bonn, D.C. Morgan, R. Liang, K. Zhang, Phys. Rev. Lett. 70 (1993) 3999; J.E. Sonier, R.F. Kiefl, J.H. Brewer, D.A. Bonn, J.F. Carolan, K.H. Chow, P. Dosanjh, W.N. Hardy, R. Liang, W.A. MacFarlane, P. Mendels, G.D. Morris, T.M. Riseman, J.W. Schneider, ibid. 72 (1994) 744; T. Jacobs, S. Sridhar, Q. Li, G.D. Gu, N. Koshizuka, ibid. 75 (1995) 4516. [233]

D. Achkir, M. Poirier, D.A. Bonn, R. Liang, W.N. Hardy, Phys. Rev. B 48 (1993) 13 184.

[234] A. Maeda, Y. line, T. Haniguri, N. Motohira, K. Kishio, T. Fukase, Phys. Rev. Lett. 74 (1995) T. Hanaguri, Y. Iino, A. Maeda, T. Fukase, Physica C 246 (1995) 223.

1202. See also

[235] Y. Kitaoka, S. Hiramatsu, Y. Kohori, K. Ishida, T. Kondo, H. Shibai, K. Asayama, H. Takagi, S. Uchida, H. Iwabuchi, S. Tanaka, Physica C 153-155 (1988) 83; J.A. Martindale, S.E. Barrett, K.E. O’Hara, C.P. Slichter, W.C. Lee, D.M. Ginsberg, Phys. Rev. B 47 (1993) 9155. See also Ref. [219]. [236] N. Momono, N. Momono, [237]

M. Ido, T. Nakano, M. Oda, Y. Okajima, K. Yamaya, Physica C 233 (1994) 395; M. Ido, M. Oda, C. Manabe, T. Nakano, ibid. 263 (1996) 225.

K.A. Moler, D.J. Baar, J.S. Urbach, R. Liang, W.N. Hardy, A. Kapitulnik, Phys. Rev. Lett. 73 (1994) 2744; R.A. Fischer, J.E. Gordon, S.F. Reklis, D.A. Wright, J.P. Emerson, B.F. Woodfield, E.M. McCarron III, N.E. Phillips, Physica C 252 (1995) 237; K.A. Moler, D.L. Sisson, J.S. Urbach, M.R. Beasley, A. Kapitulnik, D.J. Baar, R. Liang, W.N. Hardy, Phys. Rev. B 55 (1997) 3954; 77 (1996) 3059; H. Aubin et al., Phys. Rev. B 78 ( 1997) 2624.

[238] F. Yu, M.B. Salamon, A.J. Leggett, WC. Lee, D.M. Ginsberg,

Phys. Rev. Lett. 74 (1995) 5136.

58

B. Brandow I Physics Reports 296 (1998)

1-63

[239] J. Geerk, G. Linker, 0. Meyer, Q. Li, R.-L. Wang, X.X. Xi, Physica C 162-164 (1989) 837; M. Gurvitch, J.M. Valles Jr., A.M. Cucolo, R.C. Dynes, J.P. Garno, L.F. Schneemeyer, J.V. Waszczak, Phys. Rev. Lett. 63 (1989) 1008; I. Takeuchi, J.S. Tsai, Y. Shimakawa, T. Manako, Y. Kubo, Physica C 158 (1989) 83. [240] T. Ekino, J. Akimitsu, Phys. Rev. B 40 (1989) 6902; G. Briceno, A. Zettl, Sol. St. Commun. 70 (1989) 1055; J.M. Valles Jr., R.C. Dynes, A.M. Cucolo, M. Gurvitch, L.F. Schneemeyer, J.P. Garno, J.V. Waszczak, Phys. Rev. B 44 (1991) 11986 (see Fig. 10); S.I. Vedeneev, A.G.M. Jansen, P. Samuely, V.A. Stepanov, A.A. Tsvetkov, P. Wyder, ibid. 49 (1994) 9823; A.I. Akimenko et al., J. Low Temp. Phys. 107 (1997) 511. [241] T. Ekino, T. Dot&an, H. Fujii, J. Low Temp. Phys. 105 (1996) 563; K. Kitazawa, H. Sugawara, T. Hasegawa, Physica C 263 (1996) 214. [242] D.H. Wu, J. Mao, S.N. Mao, J.L. Peng, X.X. Xi, T. Venkatesan, R.L. Greene, SM. Anlage, Phys. Rev. Lett. 70 (1993) 85; A. Andreone, A. Cassinese, A. DiChiara, R. Vaglio, A. Gupta, E. Samelli, Phys. Rev. B 49 (1994) 6392; S.M. Anlage, D.-H. Wu, J. Mao, S.N. Mao, X.X. Xi, T. Venkatesan, J.L. Peng, R.L. Greene, ibid. 50 (1994) 523; C.W. Schneider, Z.H. Barber, J.E. Evetts, S.N. Mao, X.X. Xi, T. Venkatesan, Physica C 233 (1994) 77; B. Stadtlober, G. Krug, R. Nemetschek, R. Hackl, J.L. Cobb, J.T. Markert, Phys. Rev. Lett. 74 ( 1995) 4911. [243] J.R. Cooper, Phys. Rev. B 54 (1996) R 3753. [244] T. Imai, C.P. Slichter, J.L. Cobb, J.T. Markert, J. Phys. Chem. Solids 56 (1995) 1921. [245] R.C. Dynes, Solid State Commun. 92 (1994) 53; J.R. Schrieffer, ibid. 92 (1994) 129; D. Pines, Physica C 235-240 (1994) 113; C. Kallin, A.J. Berlinsky, Hyperhne Int. 86 (1994) 489; C.M. Varma, J. Phys. Chem. Sol. 56 (1995) 1685; D.J. Scalapino, Phys. Repts. 250 (1995) 331; D.J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515; J. Anne& N. Goldenfeld, A.J. Leggett, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors, vol. 5, World Scientific, Singapore, 1996, p. 375; Y. Gim, A. Mathai, R.C. Black, A. Amar, F.C. Wellstood, J. Phys. I (France) 6 (1996) 2299. [246] R.A. Klemm, Phys. Rev. B 55 (1997) 3249; see also the survey by Y. Gim et al., in Ref. [245]. [247] A.G. Sun, D.A. Gajewski, M.B. Maple, R.C. Dynes, Phys. Rev. Lett. 72 (1994) 2267; AS. Katz, A.G. Sun, R.C. Dynes, K. Char, Appl. Phys. Lett. 66 (1995) 105; R. Kleiner, A.S. Katz, A.G. Sun, R. Summer, D.A. Gajewski, S.H. Han, S.I. Woods, E. Dantsker, B. Chen, K. Char, M.B. Maple, R.C. Dynes, J. Clarke, Phys. Rev. Lett. 76 (1996) 2161; A.G. Sun, A. Truscott, A.S. Katz, R.C. Dynes, B.W. Veal, C. Gu, Phys. Rev. B 54 (1996) 6734; R. Gross, P. Chaudhari, A. Gupta, G. Koren, Physica C 166 (1990) 277. [248] P. Chaudhari, S.-Y. Lin, Phys. Rev. Lett. 72 (1994) 1084. [249] S.-Y. Lin, P. Chaudhari, N. Khare, T. Shaw, L.S. Yu-Jahnes, A. Gupta, E. Stepantsov, Z. Ivanov, unpublished. [250] R.J. Kelley, C. Quitmann, M. Onellion, H. Berger, P. Almeras, G. Margaritondo, Science 271 (1996) 1255. [251] C. Kendziora, A. Rosenberg, Phys. Rev. B 52 (1995) R 9867; C. Kendziora, R.J. Kelley, M. Onellion, Phys. Rev. Lett. 77 (1996) 727. [252] H. Ding, T. Yokoya, J.C. Campuzano, T. Takahashi, M. Randeria, M.R. Norman, T. Mochiku, K. Kadowaki, J. Giapintzakis, Nature 382 (1996) 51. [253] A.G. Loeser, D.S. Dessau, Z.-X. Shen, Physica C 263 (1996) 208; A.G. Loeser, Z.-X. Shen, D.S. Dessau, D.S. Marshall, C.H. Park, P. Foumier, A. Kapitulnik, Science 273 (1996) 325; D.S. Marshall, D.S. Dessau, A.G. Loeser, C.-H. Park, A.Y. Matsuura, J.N. Eckstein, I. Bozovic, P. Foumier, A. Kapitulnik, W.E. Spicer, Z.-X. Shen, Phys. Rev. Lett. 76 (1996) 484 1; J.M. Harris, Z.-X. Shen, P.J. White, D.S. Marshall, MC. Schabel, J.N. Eckstein, I. Bozovic, Phys. Rev. B 54 (1996) R 15665. [254] M. Norman, M. Onellion, private communications. [255] Entre to the pseudogap literature may be found in Refs. [252, 253, 2561. [256] See, for example, D.M. Newns, C.C. Tsuei, R.P. Huebener, P.J.M. van Bentum, P.C. Pattnaik, CC. Chi, Phys. Rev. Lett. 73 (1994) 1695; M. Houssa, M. Ausloos, K. Durczewski, Phys. Rev. B 54 (1996) 6126; G.C. McIntosh, A.B. Kaiser, ibid. 54 (1996) 12569; S. Sarkar, A.N. Das, ibid. 54 (1996) 14974; J.R. Cooper, J.W. Loram, J. Phys. I (France) 6 (1996) 2237; J.-X. Li, C.-D. Gong, Physica C 274 (1997) 149. [257] See, for example, CC. Tsuei, C.C. Chi, D.M. Newns, P.C. Pattnaik, M. Daumling, Phys. Rev. Lett. 69 (1992) 2134. [258] See, for example, J.M. Tranquada, B.J. Stemlieb, J.D. Axe, Y. Nakamura, S. Uchida, Nature 375 (1995) 561; J.M. Tranquada, J.D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, B. Nachumi, Phys. Rev. B 54 (1996) 7489; MI. Salkola, V.J. Emery, S.A. Kivelson, Phys. Rev. Lett. 77 (1996) 155; J.M. Tranquada, J.D. Axe, N. Ichikawa, A.R. Moodenbaugh, Y. Nakamura, S. Uchida, ibid. 78 (1997) 338. Extensive discussion of stripe and

B. Brandow I Physics Reports 296 (I 998) l-63

[259] [260] [261] [262]

[263] [264] [265]

[266] [267] [268] [269] [270] [271] [272] [273] [274] [275] [276] [277] [278]

[279] [280] [281] [282] [283] [284] [285] [286]

[287] [288]

59

pseudogap concepts and evidence is provided in R.S. Markiewicz, Phys. Rev. B, in press and in a recent review: R.S. Markiewicz, J. Phys. Chem. Sol. 58 (1997) 1179. S.A. Kivelson, private communication. H.-B. Schiittler, C.-H. Pao, Phys. Rev. Lett. 75 (1995) 4504. K. Hanzawa, J. Phys. Sot. Japan 64 (1995) 4856. This is consistent with the view of the Tc suppression from overdoping presented in Ref. [15], namely, by a local suppression of superconductivity around the doping centers. There is some evidence supporting this view: G.Q. Zheng, T. Kuse, Y. Kitaoka, K. Ishida, S. Ohsugi, K. Asayama, Y. Yamada, Physica C 208 (1993) 339; I.F. Schegolev, N.N. Kolesnikov, V.N. Kopylov, T.G. Togonidze, O.M. Vyaselev, J. Phys. I (France) 6 (1996) 2265. See the reviews by D.J. Scalapino and D. Pines in Ref. [245]. V.H. Crespi, M.L. Cohen, Phys. Rev. B 44 (1991) 4712. A.M. Kini, J.D. Dudek, K.D. Carlson, U. Geiser, R.A. Klemm, J.M. Williams, K.R. Lykke, J.A. Schlueter, H.H. Wang, P. Wurz, J.R. Ferraro, G.A. Yaconi, Physica C 204 (1993) 399; K.D. Carlson, A.M. Kini, J.A. Schlueter, H.H. Wang, J.B.D. Sutin, J.M. Williams, J.E. Schirber, E.L. Venturini, W.R. Bayless, ibid. 227 ( 1994) 10. K.D. Carlson, A.M. Kini, J.A. Schlueter, U. Geiser, R.A. Klemm, J.M. Williams, J.D. Dudek, M.A. Caleca, K.R. Lykke, H.H. Wang, J.R. Ferraro, Physica C 215 (1993) 195. A.M. Kini, K.D. Carlson, H.H. Wang, J.A. Schlueter, J.D. Dudek, S.A. Sirchio, U. Geiser, K.R. Lykke, J.M. Williams, Physica C 264 (1996) 81. B. Batlogg, R.J. Cava, L.W. Rupp Jr., A.M. Mujsce, J.J. Krajewski, J.P. Remeika, W.F. Peck Jr., AS. Cooper, G.P. Espinosa, Phys. Rev. Lett. 61 (1988) 1670. S. Kondoh, M. Sera, Y. Ando, M. Sato, Physica C 157 (1989) 469. D.G. Hi&, D.R. Richards, B. Dabrowski, D.T. Marx, A.W. Mitchell, Nature 335 (1988) 419. W. Jin, M.H. Degani, R.K. Kalia, P. Vashishta, Phys. Rev. B 45 (1992) 5535. C.-K. Loong, D.G. Hinks, P. Vashishta, W. Jin, R.K. Kalia, M.H. Degani, D.L. Price, J.D. Jorgensen, B. Dabrowski, A.W. Mitchell, D.R. Richards, Y. Zheng, Phys. Rev. Lett. 66 ( 1991) 3217. G.-M. Zhao, D.E. Morris, Phys. Rev. B 51 (1995) 12848. C.C. Chen, CM. Lieber, J. Am. Chem. Sot. 114 (1992) 3141; Science 259 (1993) 655. R.G. Goodrich, Phys. Rev. B 44 (1991) 5326. W. Biickel, B. Stritzker, Phys. Lett. A 43 (1973) 403; B. Stritzker, Z. Phys. 268 (1974) 261. N. Pessall, R.E. Gold, H.A. Johansen, J. Phys. Chem. Solids 29 (1968) 19; E.L. Haase, J. Ruzicka, in: S.A. Wolf, V.Z. Kresin (Eds.), Novel Superconductivity, Plenum, New York, 1987, p. 39. D.C. Johnston, H. Prakash, W.H. Zachariasen, R. Viswanathan, Mater. Res. Bull. 8 (1973) 777; D.C. Johnston, J. Low Temp. Phys. 25 (1976) 145; R.W. McCallum, D.C. Johnston, CA. Luengo, M.B. Maple, ibid. 25 (1976) 177. E. Moshopoulou, P. Bordet, J.J. Capponi, C. Chaillout, B. Souletie, A. Sulpice, J. Alloys and Compounds 195 (1993) 81; E. Moshopoulou, P. Bordet, A. Sulpice, J.J. Capponi, Physica C 2355240 (1994) 747. See, for example, M. Shirai, N. Suzuki, K. Motizuki, J. Phys.: Condens. Matter 2 (1990) 3553; C.C. Chantey, R. Ramakumar, K.P. Jain, Phys. Rev. B 48 (1993) 13098. See R.N. Shelton in Ref. [I]; M. Ishikawa, J.-L. Jorda, A. Junod, in: W. Biickel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kemforschungszentrum Karlsruhe, 1982, p. 141. W.P. Beyermann, R.H. Heffner, J.L. Smith, M.F. Hundley, P.C. Canfield, J.D. Thompson, Phys. Rev. B 51 (1995) 404. K.N. Yang, M.B. Maple, L.E. DeLong, J.G. Huber, A. Junod, Phys. Rev. B 39 (1989) 151. B. Andraka, J.S. Kim, G.R. Stewart, K.D. Carlson, H.H. Wang, J.M. Williams, Phys. Rev. B 40 (1989) 11345. K. Andres, H. Schwenk, H. Veith, Physica B 143 (1986) 334. N.E. Phillips, R.A. Fisher, J.E. Gordon, S. Kim, A.M. Stacy, M.K. Crawford, E.M. McCarron 111, Phys. Rev. Lett. 65 (1990) 357; N.E. Phillips, R.A. Fisher, J.E. Gordon, Prog. Low Temp. Phys. 8 (1992) 267, see p. 314. See also J.W. Loram, K.A. Mirza, Physica C 153-155 (1988) 1020, where the specific-heat ratio of 4.1 was obtained for YBCO. Y. Aoki, T. Nishigaki, H. Sugawara, H. Sato, Phys. Rev. B 55 (1997) 2768; Physica B 237-238 (1997) 302. M.L. Cohen, D.R. Penn, Phys. Rev. B 42 (1990) 8702.

60 [289] [290] [291] [292] [293] [294]

[295] [296] [297] [298] [299]

[300]

[301]

[302] [303] [304] [305] [306] [307] [308] [309] [310] [31 l]

B. Brandow IPhysics Reports 296 (1998)

1-63

CM. Place, P. Rhodes, Phys. Stat. Sol. B 47 (1971) 475. D.J. Kim, Solid State Commun. 34 (1980) 963; D.J. Kim, C. Tanaka, J. Magn. Magn. Mat. 58 (1986) 254. Z. Fisk, J.L. Smith, H.R. Ott, B. Batlogg, J. Magn. Magn. Mat. 52 (1985) 79. L.E. DeLong, Phys. Rev. B 33 (1986) 3556; and in: S.A. Wolf, V.Z. Kresin (Eds.), Novel Superconductivity, Plenum, New York, 1987, p. 253. L.E. DeLong, R.P. Guertin, S. Hassanain, T. Fariss, Phys. Rev. B 31 (1985) 7059. G.P. Meisner, A.L. Giorgi, A.C. Lawson, G.R. Stewart, J.O. Willis, MS. Wire, J.L. Smith, Phys. Rev. Lett. 53 (1984) 1829. For further discussion of uranium-compound systematics, as related to inter-atomic spacings, see S. Takagi, J. Phys. Sot. Japan 62 (1993) 1431. H. Rietschel, H. Winter, W. Reichardt, Phys. Rev. B 22 (1980) 4284. J.F. Janak, Phys. Rev. B 16 (1977) 255. C.S. Jacobsen, J. Phys. C 19 (1985) 5643. K. Levin, J.H. Kim, J.P. Lu, Q. Si, Physica C 175 (1991) 449. A.S. Edelstein, C.J. Tranchita, O.D. McMasters, K.A. Gschneidner, Jr., Sol. St. Commun. 15 (1974) 81; AS. Edelstein, R.E. Majewski, T.H. Blewett, in: R.D. Parks (Ed.), Valence Instabilities and Related NarrowBand Phenomena, Plenum, New York, 1977, p. 115; G. Remenyi, A. Briggs, J. Floquet, 0. Laborde, F. Lapietre, J. Magn. Magn. Mat. 31-34 (1983) 407; G. Remenyi, D. Jaccard, J. Floquet, A. Briggs, Z. Fisk J. Physique 47 (1986) 367; U. Rauchschwalbe, Physica B 147 (1987) 1; U. Rauchschwalbe, F. Steglich, A. de Visser, J.J.M. Franse, J. Magn. Magn. Mat. 63-64 (1987) 347; A. de Visser, A. Menovsky, J.J.M. Franse, ibid. 63-64 (1987) 365; D. Jaccard, R. Cibin, J. Sierro, Helv. Phys. Acta. 61 (1988) 530; T. Graf, J.M. Lawrence, M.F. Hundley, J.D. Thompson, A. Lacerda, E. Haanappel, M.S. Torikachvili, Z. Fisk, PC. Canfield, Phys. Rev. B 51 (1995) 15 053. See, for example, D.A. Bonn, J.D. Garrett, T. Timusk, Phys. Rev. Lett. 61 (1988) 1305; Z. Schlesinger, R.T. Collins, J.A. Calise, D.G. Hinks, A.W. Mitchell, Y. Zheng, B. Dabrowski, N.E. Bickers, D.J. Scalapino, Phys. Rev. B 40 (1989) 6862; K. Komelsen et al., in Ref. [176]; T. Timusk, D.B. Tanner, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors I, World Scientific, Singapore, 1989, p. 339; D.B. Tanner, T. Timusk, ibid. vol. III (1992), p. 363; I. Terasaki, T. Nakahashi, A. Maeda, K. Uchinokura, Phys. Rev. B 43 (1991) 551; Y. Watanabe, D.C. Tsui, J.T. Birmingham, N.P. Ong, J.M. Tarascon, ibid. 43 (1991) 3026; Y. Fujishima, Y. Tokura, T. Arima, S. Uchida, Physica C 185-189 (1991) 1001; D.A. Crandles, T. Timusk, J.D. Garrett, J.E. Greedan, Physica C 201 (1992) 407; 216 (1993) 94; P. Calvani, M. Capizzi, F. Donato, S. Lupi, P. Maselli, D. Peschiaroli, Phys. Rev. B 47 (1993) 8917; L. Degiorgi, E.J. Nicol, 0. Klein, G. Gruner, P. Wachter, S.-M. Huang, J. Wiley, R.B. Kaner, Phys. Rev. B. 49 (1994) 7012; I. Bozovic, J.H. Kim, J.S. Harris, Jr., C.B. Eom, J.M. Phillips, J.T. Cheung, Phys. Rev. Lett. 73 (1994) 1436; A. Yamanaka, N. Asayama, M. Sasada, K. moue, M. Udagawa, S. Nishizaki, Y. Maeno, T. Fujita, Physica C 263 (1996) 516. L.E. DeLong, in: L.C. Gupta, S.K. Malik (Eds.), Theoretical and Experimental Aspects of Valence Fluctuations and Heavy Fermions, Plenum, New York, 1987, p. 65, and in: S.A. Wolf, V.Z. Kresin (Eds.), Novel Superconductivity, Plenum, New York, 1987, p. 253. L.E. DeLong, G.W. Crabtree, L.N. Hall, D.G. Hinks, W.K. Kwok, SK. Malik, Phys. Rev. B 36 (1987) 7155. J.M. Vandenberg, B.T. Matthias, Science 198 (1977) 194. See also the reviews by Johnston, Braun, and by Shelton, in Ref. [l]. G. Vielsack, W. Weber, Phys. Rev. B. 54 (1996) 6614. R. Cywinski, Z.P. Han, R. Bewley, R. Cubitt, M.T. Wylie, E.M. Forgan, S.L. Lee, M. Warden, S.H. Kilcoyne, Physica C 233 (1994) 273. H. Takagi, R.J. Cava, H. Eisaki, J.O. Lee, K. Mizuhashi, B. Batlogg, S. Uchida, J.J. Krajewski, W.F. Peck Jr., Physica C 228 (1994) 389. Y.Y. Xue, Y. Cao, R.L. Meng, C. Kinalidis, C.W. Chu, Physica C 227 (1994) 63. C. Godart, L.C. Gupta, R. Nagarajan, S.K. Dhar, H. Noel, M. Potel, C. Mazumdar, Z. Hossain, C. Levy-Clement, G. SchilTmacher, B.D. Padalia, R. Vijayaraghavan, Phys. Rev. B 5 1 (1995) 489. R. Prozerov, E.R. Yacoby, I. Felner, Y. Yeshurun, Physica C 233 (1994) 367. P. Birrer et al., in Ref. [6]. M. Xu, B.K. Cho, P.C. Canfield, D.K. Finnemore, D.C. Johnston, Physica C 235-240 (1994) 2533; H. Michor, T. Holubar, C. Dusek, G. Hilscher, Phys. Rev. B 52 (1995) 16165; S. Oxx, D.P. Choudhury, B.A. Willemsen, H. Srikanth, S. Sridhar, B.K. Cho, P.C. Canfield, Physica C 264 (1996) 103.

B. Brandow I Physics Reports 296 (1998)

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[3 121 K.D.D. Rathnayaka, A.K. Bhatnagar, A. Parasiris, D.G. Naugle, P.C. Canfield, B.K. Cho, Phys. Rev. B 55 ( 1997) 8506. [3 131 E. Johnston-Halperin, J. Fiedler, D.E. Farrell, M. Xu, B.K. Cho, P.C. Canfield, D.K. Finnemore, D.C. Johnston, Phys. Rev. B 51 (1995) 12852. [314] R. Movshovich, M.F. Hundley, J.D. Thompson, P.C. Canfield, B.K. Cho, and A.V. Chubukov, Physica C 227 (1994) 381. See also H. Michor et al., in Ref. [311]. [315] A. Yatskar, N.K. Budraa, W.P. Beyermann, P.C. Canfield, S.L. Bud’ko, Phys. Rev. B 54 (1996) R3772. [316] A.A. Andreone, F. Fontana, M. Iavarone, R. Vaglio, F. Canepa, P. Manfrinetti, A. Palenzona, Physica C 251 (1995) 379. [3 171 M.S. Golden, M. Knupfer, M. Kielwein, M. Buchgeister, J. Fink, D. Teehan, W.E. Pickett, D.J. Singh, Europhys. Lett. 28 (1994) 369; A. Fujimori, K. Kobayashi, T. Mizokawa, K. Mamiya, A. Sekiyama, H. Takagi, H. Eisaki, S. Uchida, R.J. Cava, J.J. Krajewski, W.F. Peck Jr., Phys. Rev. B 50 (1994) 9660; J. Phys. Chem. Sol. 56 (1995); 1875; D.M. Poirier, C.G. Olson, D.W. Lynch, M. Schmidt, B.K. Cho, P.C. Canfield, ibid. 56 (1995) 1881. [318] D.D. Lawrie, J.P. Franck, Physica C 245 (1995) 159. [3 191 F. Borsa, Q. Hu, K.H. Kim, B.J. Suh, D.R. Torgerson, P. Canfield, M. Xu, B. Zhong, Physica C 235-240 ( 1994) 2547; M.E. Hanson, F. Lefloch, W.H. Hong, W.G. Clark, M.D. Lan, C.C. Hoellwarth, P. Klavins, R.N. Shelton, Phys. Rev. B 51 (1995) 674; T. Kohara, T. Oda, K. Ueda, Y. Yamada, A. Mahajan, K. Elankumaran, Z. Hossian, L.C. Gupta, R. Nagarajan, R. Vijayaraghavan, C. Mazumdar, ibid. 51 (1995) 3985; B.J. Suh et al., Phys. Rev. B 54 (1996) 15341. [320] T. Ekino, H. Fujii, M. Kosugi, Y. Zenitani, J. Akimitsu, Phys. Rev. B 53 (1996) 5640. [321] F. Bommeli, L. Degiorgi, P. Wachter, B.K. Cho, P.C. Canfield, R. Chau, M.B. Maple, Phys. Rev. Lett. 78 (1997) 547. [322] Particularly clean examples of this power-law bahavior (in single-crystal data) are in Ref. [314], but the T3 character can also be seen in several other studies: in Ref. [308], in H. Michor et al. in Ref. [311], and in S.A. Carter, B. Batlogg, R.J. Cava, J.J. Krajewski, W.F. Peck Jr., H. Takagi, Phys. Rev. B 50 (1994) 4216; N.M. Hong, H. Michor, M. Vybomov, T. Holubar, P. Hundegger, W. Perthold, G. Hilscher, P. Rogl, Physica C 227 (1994) 85. [323] S. Ikeda, K. Kumagai, J. Jiang, M.S. Lin, C.C. Lai, H.C. Ku, Physica C 263 (1996) 526. [324] R.J. Cava, H.W. Zandbergen, B. Batlogg, H. Eisaki, H. Takagi, J.J. Krajewski, W.F. Peck Jr., E.M. Gyorgy, S. Uchida, Nature 372 (1994) 245; H. Michor, R. Krendelsberger, G. Hilscher, E. Bauer, C. Dusek, R. Hauser, L. Naber, D. Werner, P. Rogl, H.W. Zandbergen, Phys. Rev. B 54 (1996) 9408. [325] W.H. Lee, H.K. Zeng, Y.D. Yao, Y.Y. Chen, Physica C 266 (1996) 138. [326] R. Brusetti, 0. Laborde, A. Sulpice, R. Calemczuk, M. Potel, P. Gougeon, Phys. Rev. B 52 (1995) 4481. See also P. Gougeon, M. Potel, J. Padiou, M. Sergent, Mat. Res. Bull. 22 (1987) 1087; 23 (1988) 453. [327] T. Hughbanks, R. Hoffmann, J. Am. Chem. Sot. 105 (1983) 1150. [328] H. Sugawara, 0. moue, Y. Kobayashi, H. Sato, T. Nishigaki, Y. Aoki, H. Sato, R. Settai, Y. Onuki, J. Phys. Sot. Japan 64 (1995) 3639. [329] H. Sugawara, H. Sato, T. Yamazaki, N. Kimura, R. Settai, Y. Onuki, J. Phys. Sot. Japan 64 (1995) 4849. [330] H. Sugawara, M. Higuchi, 0. moue, T. Nishigaki, Y. Aoki, H. Sato, R. Settai, Y. Onuki, A. Hasegawa, J. Phys. Sot. Japan 65 (1996) 1744. [331] G. Gladstone et al., in Ref. [42]. [332] K.H. Bennemann and J.W. Garland, in: D.H. Douglass (Ed.), Superconductivity in d- and f-Band Metals, American Institute of Physics, New York, 1972, p. 103. [333] D. Pines, Phys. Rev. 109 (1958) 280; P. Morel, P.W. Anderson, ibid. 125 (1962) 1263. [334] M.L. Cohen, in: R.D. Parks (Ed.), Superconductivity, vol I, Marcel Dekker, New York, 1969, p. 615. [335] N.M. Builova, V.B. Sandomirskii, Sov. Phys. Uspekhii 12 (1969) 64. [336] J.K. Hulm, M. Ashkin, D.W. Deis, C.K. Jones, in: C.J. Gorter (Ed.), Progress in Low Temperature Physics, vol VI, North-Holland, Amsterdam, 1970, p. 205. [337] H. Suzuki, H. Bando, Y. Ootuka, I.H. lnoue, Y. Yamamato, K. Takahashi, Y. Nishihara, J. Phys. Sot. Japan 65 (1996) 1529. [338] L.F. Mattheiss, Phys. Rev. B 6 (1972) 4718 and 4740. [339] Y. Takada, J. Phys. Sot. Japan 49 (1980) 1267. [340] L.J. Sham, Physica B 135 (1985) 451.

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[341] H. Murakami, W. Hattori, R. Aoki, Physica C 269 (1996) 83. [342] B.T. Matthias, E. Corenzwit, W.H. Zachariasen, Phys. Rev. 112 (1958) 89; B.T. Matthias, V.B. Compton, H. Suhl, E. Corenzwit, ibid. 115 (1959) 1597; B.T. Matthias, J. Appl. Phys. 31 Suppl. (1960) 23s; IBM J. Res. Dev. 6 (1962) 250; B.T. Matthias, T.H. Geballe, V.B. Compton, Rev. Mod. Phys. 35 (1963) 1; B.T. Matthias, C.W. Chu, E. Corenzwit, and D. Wohlleben, Proc. Nat. Acad. Sci. 64 (1969) 459; B.T. Matthias, Physica 55 (1971) 69; Int. J. Quantum Chem. Symp. 13 (1979) 467. Related developments may be found in C.P. Enz, B.T. Matthias, Science 201 (1978) 828, Z. Phys. B 33 (1979) 129; C.P. Enz, in: H. Suhl, M.B. Maple (Eds.), Superconductivity in d- and f-Band Metals, Academic Press, New York, 1980, p. 181; D. Fay, J. Appel, Phys. Rev. B 20 (1979) 3705. The Enz papers deal with a subtle competition between magnetism and superconductivity, but in contrast to Matthias et al. they have sought a phonon explanation for this. Fay and Appel found that the relevant phonon contribution is quite weak, and they suggested that the observed effect may instead come mainly from a change in the effective Coulomb interaction parameter. This latter suggestion is consistent with our VF picture. [343] H. Takagi et al., Physica C 228 (1994) 389. [344] S.A. Carter et al., Phys. Rev. B 50 (1994) 4216. [345] H. Takagi et al., Physica B 237-238 (1997) 292. [346] R. Tournier et al., J. Magn. Magn. Mat. 76-77 (1988) 552. [347] D.D. Lawrie et al., J. Low Temp. Phys. 107 (1997) 491. [348] A. Abrikosov, Phys. Rev. B 56 (1997) 446, 5112. [349] A. Carrington et al., Phys. Rev. B 54 (1996) R3788. [350] J.W. Radcliffe et al., J. Low Temp. Phys. 105 (1996) 903. [351] A.S. Alexandrov et al., Phys. Rev. Lett. 79 (1997) 1551. [352] J. Loram, private communication. [353] H. Ding, private communication. [354] S. Belin, K. Behnia, Phys. Rev. Lett. 79 (1997) 2125. [355] A.P. Ramirez, Phys. Lett. A 211 (1996) 59. [356] K. Ishida et al., Physica B 237-238 (1997) 304. [357] T. Ekino et al., Phys. Rev. B 56 (1997) 7851. [358] K. Ishida et al., Phys. Rev. B 56 (1997) R505. [359] R. Jin et al., J. Phys. Chem. Sol., in press.

Note added in proof We have several further comments and references: ( 1) Several investigators have plotted the Tc’s of many of these materials, and many other materials, against the specific heat coefficient y, with the result that no correlation corresponding to the Uemura relation was apparent [343-3451. (Nevertheless, a rough T, cc y-l relation was recognized for several of the exotic materials, and was attributed to a shared heavy-fermion charactor [346]. It is evident that &_ is more closely related to T, than is the Fermi-surface quantity y, for the exotic superconductors. This can be regarded as further evidence that in the exotic materials the pairing is broadly distributed over the Brillouin zone. (Compare with Section 3.1.) (2) A strongly upwards-curving f&(T), which fits the Bosecondensation theory [153], has been found for YBa2(Cul_,ZnX)306.94 with x = 0.09 and T, = 3.5 K, using a resistive criterion for Hc2 [347]. (3) Professor S. Foner has informed me of considerable f&(T) data from his laboratory showing conventional (linear) behavior near T,, for A-15 and Chevrel materials and also for NbSez and LiTi204, in contrast to the data quoted in Section 3.3. He noted however that this was relatively early data. The upwards curvature is seen in more recent data with presumably better samples. (4) Another theory of upwards curvature in Z&(T) has been presented, considering the saddle-point character of the band structure and also the Pauli paramagnetic limiting

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effect [348]. (5) The problem of properly identifying and understanding Hc2( T) is now recognized to be even more subtle, due to another surprising feature. In the cuprates, the magnetic field dependence of the specific heat anomaly is drastically different from that of the top of the resistive transition [349-3511. (6) We proposed in Section 3.7 that the pseudogap phenomenology of the cuprates might be due simply to the proximity of the van Hove singularity to the Fermi level. This idea is wrong, because there is an accompanying temperature dependence of the chemical potential which limits the effectiveness of the van Hove singularity [352]. (7) The evidence against a d-wave gap form in Ref. [250], from angle-resolved photoemission in overdoped Bi-2212, is suspect because this data is subject to the superlattice problem discussed in [231] (see Ref. [353]). (8) In the quasi-onedimensional organic superconductor (TMTSF)&104 there is now evidence against gap nodes, from a strong suppression of thermal conductivity at T < 0.5T, [354]. (9) The borocarbide LuNi2B2C shows an H’i2T component in the specific heat far below T, [345]. This behavior may be due to gap nodes, but it could also come from other sources [355]. ( 10) The C-15 material CeCo2 has been shown to exhibit a Hebel-Slichter peak [356]. Tunnelling data on CeRu2 shows an enlarged gap ratio of 4.1-4.4 and a BCS form of d(T), although there is much inner-gap structure [357]. (11) There is further evidence for exotic character in Sr2Ru04, beyond the resistivity and mass enhancement. The l/T, in NMR drops sharply below T,, and has a T + 0 behavior suggestive of gap nodes. The Pauli susceptibility deduced from the Ru Knight shift is enhanced by a factor of 5.4 compared to band theory [358]. There is also a sign change in the c-axis magnetoresistance at 70K, far below the temperature of the c-axis resistance maximum (150K) [359]. This sign change is similar to what is found in heavy-fermion materials [299].