1. Introduction

Ever since the earliest manifestations of high- superconductivity were found in 1986 [1], the whole theoretical power [2–22] has been applied to explain and describe various normal and superconducting properties of various oxide families with critical temperatures, , ranging up to 138 K to date [23–27]. Unfortunately, even conceptual understanding of the mechanisms and character of superconductivity in cuprates is still lacking. Strictly speaking, there is a number of competing paradigms, every of them pretending to be “the theory of superconductivity’’ (see, e.g., [2]) but not recognized as such by other respected experts in the field.

After the discovery of high- oxides, experimentalists found several other superconducting families with higher than 23.2 K reached by the precuprate record-holder, NbGe [28, 29]. For instance, one may refer to fullerides [30, 31], doped bismuthates [32–34], hafnium nitrides [35, 36], and magnesium diborides [37–40]. One should also mention more controversial cases of superconducting oxides HWO with [41] and SrLaPbO with [42]. Finally, an unexpected and counter-intuitive discovery of the iron-based oxypnictide [43, 44] or oxygen-free pnictide [45] layered superconductors with over has been made recently (see also reviews [46–49]).

Presumably, the latter materials with FeAs layers have been overlooked as possible candidates for high- superconductors, since Fe ions in solids usually possess magnetic moments, which promote magnetic ordering, the latter being detrimental to superconductivity, especially the spin-singlet one [50–55]. Strictly speaking, such an omission is of no surprise because superconductivity in oxides is rather gentle, sensible to impurities, including the excess or deficiency of oxygen [56] in these nonstoichiometric [57, 58] compounds. Recent discovery [59] of previously unnoticed high- superconductivity in parent compounds -CuO (R = Pr, Nd, Sm, Eu, Gd) is very symptomatic in this regard, since an accurate removal of apical oxygen from thin films raised from exact zero (those compositions were earlier considered by theoreticians as typical correlated Mott-Hubbard insulators) to for NdCuO. As for the ferroarsenide family, one of its members, EuFe(AsP), reveals a true superconducting transition at , followed by the ferromagnetic ordering of Eu magnetic moments below , coexisting with superconductivity [60], which is quite unusual in view of the antagonism indicated above between two kinds of cooperative phenomena.

What is more, none of the mentioned superconductors except BaKBiO [33] were discovered due to theoretical predictions. Hence, one may consider the theoretical discovery of BaKBiO as an accidental case, since, according to the well-known chemist Cava: “one of the joys of solid state chemistry is its unpredictability’’ [61]. The same opinion was expressed by the other successive chemist Hosono: “understanding the mechanism with respect to predicting the critical temperature of a material is far from complete at the present stage even for brilliant physicists. Such a situation provides a large opportunity including a good luck for material scientists who continue the exploration for a new material, not limited to superconductors, and a new functionality based on their own view points’’ [48]. That is why Pickett recently made a sad remark that “the next breakthrough in superconductivity will not be the result of surveying the history of past breakthroughs’’ [62]. It means that microscopic theories of superconductivity are incapable of describing specific materials precisely, although together they give an adequate overall picture. In this connection, the failure of the most sophisticated approaches to make any prediction of true or, at least “bare’’ (provided that the corresponding -value is not known a priori) despite hundreds of existing superconductors with varying fascinating properties, forced Phillips [63] to reject all apparently first-principle continuum theories in favor of his own percolative filamentary theory of superconductivity [64–67] (see also the random attractive Hubbard model studies of superconductivity [68, 69] and the analysis of competition between superconductivity and charge density waves studied in the framework of similar scenarios [70–72]). We totally agree with such considerations in the sense of the important role of disorder in superconductors with high on the verge of crystal lattice instability [73–83]. Nevertheless, it is questionable whether a simple one-parameter “master function’’ of [63, 67] would be able to make quantitative and practically precise predictions of . As for the qualitative correctness of the dependence versus weighted number of Pauling resonating valence bonds [63, 67, 84], it can be considered at least as a useful guideline in the superconductivity ocean. The phenomenological character of the master function (chemical trend diagram) is an advantage rather than a shortcoming of this approach, as often happens in the physics of superconductors (see, e.g., more or less successful criteria of superconductivity with different extent of phenomenology [85–98]).

On the other hand, attempts to build sophisticated microscopic theories of the boson-mediated Bardeen-Cooper-Schrieffer (BCS) attraction, treating the Coulomb repulsion as a single Coulomb pseudopotential constant , are incapable of predicting actual critical superconducting properties [63, 91, 99–101]. The same can be said [67] about Hubbard-Hamiltonian models with extremely strong repulsive Coulomb energy parameter , which is formally based on the opposite ideology (see, e.g., [102, 103]). As an example of the theories described above, one can indicate work [104], where the strong-coupling Eliashberg equations for the electron-phonon mechanism of superconductivity [105, 106] were solved numerically taking into account even vertex corrections and treating the dispersive Coulomb interaction not on equal footing, but as a simple constant . In this connection, it seems that the prediction of [104] that the maximal for new iron-based superconductors is close to is unjustified. Of course, the same is true for other studies of such a kind.

It is remarkable that, for hole- and electron-doped cuprates, there is still no clarity concerning the specific mechanisms of superconductivity [17, 107–115] and the order parameter symmetry [109, 116–130], contrary to the “official’’ viewpoint [131–133]) and even the very character of the phenomenon (in particular, there have been furious debates concerning the Cooper pairing versus boson condensation dilemma in cuprates [8, 134, 135]). The same seems to be true for other old and new “exotic’’ superconductors [46, 107, 108, 136–153], their exoticism being in essence a degree of our ignorance.

It would be of benefit to consider all indicated problems in detail for all classes of superconductors and show possible solutions. Unfortunately, it cannot be done even in the scope of huge treatises (see, e.g., [154–157]). The objective of this review is much more modest. Specifically, it deals mostly with high- cuprate materials, other superconductors being mentioned only for comparison. Moreover, in the present state of affairs, it would be too presumptuous to pretend to cover all aspects of the oxide superconductivity. Hence, we will restrict ourselves to the analysis of lattice instabilities and concomitant charge density waves (CDWs) in high- oxides. Their interplay with superconductivity is one of the fascinating and fundamentally important phenomena observed in cuprates and discussed by us earlier [158–160]. Nevertheless, in this rapidly developing branch of the solid state physics, many new theories and experimental data on various CDW superconductors appeared during last years. They are waiting for both unbiased and thorough analysis. This article discusses this new information, referring the reader to our previous reviews for more general and established issues, as well as some cumbersome technical details.

The outline of this review is as follows. In Section 2, for the sake of completeness, we briefly consider possible mechanisms of superconductivity in cuprates, the problem of the relationship between BCS pairing and Bose-Einstein condensation (BEC), and the multigapness of the superconducting order parameter. Section 3 is devoted to the experimental evidence for CDWs, the so-called pseudogaps, dip-hump structures, and manifestations of intrinsic inhomogeneity in cuprate materials. The original theory of CDW superconductors and the interpretation of CDW-related phenomena in high- oxides are presented in Section 4. At the end of Section 4, some recent data on coexistence between superconductivity and spin density waves (SDWs)—a close analogue of CDWs—are covered. This topic became hot once more after the discovery of ferropnictides [43–48, 161]. Short conclusions are made in Section 5.

2. Considerations on Peculiarities and Mechanisms of Superconductivity in Oxides

When BaPbBiO (BPB) was shown [162] to be a superconductor with a huge (at that time!) for , a rather low concomitant concentration of current carriers , and poor electric conductivity [56] (the phase diagram of BPB is extremely complex, with a number of partial metal-insulator structural transitions [56, 163–167]), it looked like an exception. Now, it is fully recognized that oxides with highest are bad metals from the viewpoint of normal state conductivity [168]. In particular, the mean free path of current carriers is of the order of the crystal lattice constant, so that the Ioffe-Regel criterion of the metal-insulator transition [169] is violated. Moreover, there exists an oxide superconductor SrTiO with a tiny maximal , attained by doping, but an extremely small [170]. Note that the undoped semiconducting SrTiO is so close to the metal-insulator border that it may be transformed into a metal by the electrostatic-field effect [171] (this technique has been successfully applied to other oxides [172]). Moreover, a two-dimensional metallic layer has been discovered [173] at the interface between two insulating oxides LaAlO and SrTiO, which was later found to be superconducting with [174, 175]. The appearance of superconductivity at nonmetallic charge carrier densities in oxides of different classes comprises a hint that it is not wise to treat various oxide families separately (see, e.g., [176]), all of them having similar perovskite-like ion structures [23, 25, 26, 177–181] and similar normal and superconducting properties [27], whatever the values of their critical parameters are. As for the apparent dispersion of the latter among superconducting oxides, it mostly reflects their conventional exponential dependences on atomic and itinerant-electron characteristics [9, 10].

The junior member of the superconducting oxide family, SrTiO, demonstrates (although not in a spectacular manner) several important peculiarities, which are often considered as properties intrinsic primarily to high- cuprates. Indeed, in addition to the low , this polar, almost ferroelectric [182, 183], material was shown to reveal polaron conductivity [184] and is suspected to possess bipolaron superconductivity [185–187], first suggested by Vinetskii almost 50 years ago [188]. It means that SrTiO might be not a Bardeen-Cooper-Schrieffer (BCS) superconductor [189] with a large coherence length , where is the crystal lattice constant, but most likely an example of a material with , so that a Bose-condensation of local electron pairs would occur at , according to the Schafroth-Butler-Blatt scenario [190] or its later extensions [8, 134, 135, 191–199].

The concept of bipolarons (local charge carrier pairs) has been later applied to BPB [200–203], BaKBiO (BKB, [204, 205]) [203, 206, 207] and cuprates [195, 199, 208–211]. It was explicitly shown for BPB and BKB by X-ray absorption spectroscopy [203] that bipolaronic states and CDWs coexist and compete, which might lead, in particular, to the observed nonmonotonic dependence [212]. At the same time, Hall measurements demonstrate that the more appropriate characteristics is monotonic [56, 213, 214], so that the expected suppression of at high as a consequence of screening of the electron-phonon matrix elements [99, 215, 216] is not achieved here as opposed to the curve [170] in reduced samples of SrTiO. As for cuprates, the bipolaron superconductivity mechanism, as well as any other BEC scheme, in its pure state would require an existence of the preformed electron (hole) pairs (bipolarons), which might be the case [177, 217], and a prior destruction of the Fermi surface (FS), the condition contradicting observations (see, e.g., [218]). Therefore, boson-fermion models for charge carriers in superconductors was introduced [134, 219–225] and, later on, severely criticized [226, 227]. In any case, the available objections concern the bipolaronic mechanism of superconductivity itself, the occurrence of polaronic effect in oxides with high dielectric permittivities raising no doubt [115, 177, 199, 228–232].

It is remarkable that the boson-fermion approach mentioned above is not a unique tool for describing superconductivity in complex systems. A necessary “degree of freedom’’ connected to another group of charge carriers has been introduced, for example, as the so-called -centers [233–235], earlier suggested by Anderson [236] as a phenomenological reincarnation of bipolarons in amorphous materials [188]. Independently, narrow-band nondegenerate charge carriers submerged into the sea of itinerant electrons were proposed for cuprates as another, not fully hybridized kind of the “second heavy component’’ [237, 238]. For completeness, we should also mention a quite different model involving a second heavy charge carrier subsystem (-electrons in transition metals [239] or heavy holes in degenerate semiconductors [240]), necessary to convert high-frequency Langmuir plasmons intrinsic to the itinerant electron component into the ion-acoustic collective excitation branch, in order that a high- superconductivity would appear. Those hopes, however, lack support from any evidence in natural or artificial systems (see the analysis of plasmon mechanisms [206, 241–247], the optimism of some authors seems to us and others [248] a little bit exaggerated). As can be readily seen from the References given above, all nonconventional approaches, rejecting or generalizing the BCS scheme and going back to the explanations of a relatively weak superconductivity in degenerate semiconductors [138, 191, 215, 249–252], have been applied to every family of superconducting oxides, including cuprates.

Strontium titanate became a testing ground [253] of one further attractive idea (based on the same concept of several interacting charge carrier components) of two-gap or multigap superconductivity, with the interband interplay being crucial to the substantial increase of and other critical parameters. The corresponding models came into being in connection with the transition s-d metals [254, 255]. They were subsequently applied to analyze superconductivity in multivalley semiconductors [256, 257], high- oxides [231, 258–266], MgB [40, 267–269], ZrB [270], VSi [271], MgIrB [272], YNiBC [273], NbSe [274, 275], FeSi (R = Lu,Sc) [276], ScIrSi [277], NaCoOO [278] as well as pnictides LaFeAsOF [279], LaFeAsOF [280], SmFeAsOF [281], and BaKFeAs [282], BaKFeAs with [283]. We did not explicitly include into the list such modifications of magnesium diboride as MgAlB or Mg(BC), and so forth.

Since, instead of one, two or more well-separated superconducting energy gaps, a continuous, sometimes wide, gap distribution is often observed (see results for NbSn in [284] and MgB in [285–289]), the original picture of the gap multiplicity in the momentum, , space loses its beauty, whereas the competing scenario [76, 290] of the spatial (-space) extrinsic or intrinsic gap spread becomes more adequate and predictive [77–79]. For the case of cuprates, it has been recently shown experimentally that the spread is really spatial, but corresponds to the pseudogap (CDW gap) rather than its superconducting counterpart, the latter most probably being a single one [291] (see also the discussion in [83] and below).

In accordance with what was already mentioned, the application of very different, sometimes conflicting, models to oxide families, including cuprates, means an absence of a deep insight into the nature of their superconducting and normal state properties. We are not going to analyze here the successes and failures of the microscopic approaches to high- superconductivity in detail; instead we want to emphasize that even the boson-mediators (we accept the applicability of the Cooper-pairing concept to oxides on the basis of crucial flux-quantization experiments [292, 293]) are not known for sure. Indeed, at the early stages of the high- studies, magnons were considered as glue, coupling electrons or holes. The very temperature-composition (doping) phase diagrams supported this idea, since undoped and slightly doped oxides were found antiferromagnetic [26, 103, 294–304]. However, a plethora of theories suggesting virtual spin fluctuations as the origin of superconductivity in high- oxides and leading to the symmetry of the superconducting order parameter have been developed [6, 11, 15, 103, 302, 305–310].

Fortunately for the scholars, it became clear that reality is richer for oxides than was expected, so that (i) the order parameter may include a substantial -wave admixture [109, 116–129]; and (ii) phonons still exist in perovskite crystal lattices, inevitably affecting or, may be, even determining the pairing process [4, 10, 112, 115, 311], not to talk about polaron and bipolaron effects discussed above. It should be noted that there are reasonable scenarios of -wave order parameter symmetry in the framework of the electron-phonon interaction alone [208, 312–316] (a similar conclusion was made for the case of plasmon mechanism [317]).

At the same time, if one adopts a substantial (crucial?) role of spin-fluctuation mechanism in superconductivity, the ubiquitous phonons can (i) be neutral to the dominant -wave pairing; (ii) act synergetically with spin fluctuations; (iii) or reduce , as it would have been for switched-off phonons. The existing theories support all three variants, although some authors cautiously avoid any direct conclusions [103]. For instance, Kulić demonstrated the destructive interference between both mechanisms of superconductivity [4]. Phononic reduction of the magnetically induced was claimed in [308, 318], whereas anisotropic phonons seem to enhance , thus obtained [319]. Finally, according to [228, 320, 321], spins and phonons act constructively in cuprates. Once again, the microscopic approach was incapable of unambiguously predicting a result for the extremely complex system.

One should bear in mind that the problem is much wider than the interplay between spin excitations and phonons. Namely, it is more correct to consider the interplay between Coulomb inter-electron and electron-lattice interactions [232, 322]. Of course, the latter is also Coulombic in nature, phonons being simply an ion sound, that is, ion Langmuir plasma oscillations [323] screened in this case by degenerate light electrons [324] (thus, acoustic phonons constitute a similar phenomenon as the acoustic plasmons in the electron system [239, 240] with an accuracy to frequencies). One of the main difficulties is how to separate the metal constituents in order that some contributions would not be counted twice [322, 325–333]. Since it is possible to do rigorously only in primitive plasma-like models [91, 92, 99, 251, 322], the problem has not been solved. Therefore, empirical considerations remain the main source of future success for experimentalists, as it happened, for example, in the case of MgB [37].

3. CDWs and CDW-Related Phenomena in Cuprates

The reasoning presented in Section 2 demonstrates that for the objects concerned, it is insufficient to rely only on microscopic theories, so that phenomenological approaches should deserve respect and attention. In actual truth, they might not be less helpful in understanding the normal and superconducting properties of cuprates, being generalizations of a great body of experimental evidence collected during last decades. In this section, we are going to show that two very important features are common to all high- families. Specifically, these are the intrinsic inhomogeneity of nonstoichiometric superconducting ceramic and single crystalline samples [334–343] and the persistence of CDWs [340, 344, 345] and other phenomena, which we also consider as CDW manifestations (dip-hump structures, DHS [339, 346–348], and pseudogaps below and above [349–358] in tunneling spectra and angle-resolved photoemission spectra, ARPES).

CDWs were seen directly as periodic incommensurate structures in superconducting BiSrCaCuO (BSCCO) using various experimental methods [12, 334, 359–370]. Photoemission studies reveal the charge-ordered “checkerboard’’ state in CaNaCuOCl [371], and tunnel measurements visualized the same kind of ordering in BSCCO [370]. Scanning tunnel microscopy (STM) measurements found CDWs in BiSrLaCuO () with an incommensurate period and CDW wave vectors depending on oxygen doping degree [340]. The same method revealed nondispersive (energy-independent) checkerboard CDWs in BiPbSrLaCuO ( for the optimally doped composition) [344]. In this case, substantially depends on doping, rising from in an optimally doped sample to for an underdoped sample with . It is easily explained by the authors taking into account the shrinkage of the hole FS with decreasing hole number, so that the vector that links the flat nested FS sections grows, whereas the CDW period decreases (see Figure 1). One should note that, in the presence of impurities (e.g., an inevitably non-homogeneous distribution of oxygen atoms), the attribution of the observed charge order (if any) to unidirectional or checkerboard type might be ambiguous [372].

Figure 1: (Color online) Fermi surface nesting; and tight-binding-calculated Fermi surface (solid black curve) of optimally doped BiSrCuO based on ARPES data [373]. The nesting wave vector (black arrow) in the antinodal flat band region has length . Underdoped BiSrCuO Fermi surfaces (shown schematically as red dashed lines) show a reduced volume and longer nesting wave vector, consistent with a CDW origin of the doping-dependent checkerboard pattern reported here (Taken from [344]).

A similar coexistence of CDWs and superconductivity was observed in a good many different kinds of materials with a reduced dimensionality of their electron system, so that the corresponding FS includes nested (congruent) sections [158–160]. For completeness, we will add some new cases discovered after our previous reviews were published. First of all, the analogy between CDWs in cuprates and layered dichalcogenides was proved by ARPES [352, 374–376]. It should be noted that CDW competition with superconductivity in cuprates was supposed as early as in 1987 on the basis of heat capacity and optical studies [377], whereas the similarity between high- oxides and dichalcogenides was first noticed by Klemm [378, 379]. Additionally, a new dichalcogenide system CuTiSe was found with coexisting superconductivity and CDWs (at ) [380, 381]. The coexistence between two phenomena was observed in the organic material -(BEDT-TTF)KHg(SCN), but superconductivity was attributed to boundaries between CDW domains, where the CDW order parameter is suppressed [382]. High-pressure studies of another organic conductor (Per)[Au(mnt)] revealed an appearance of superconductivity after the CDW suppression [383]. Still, it remained unclear, whether some remnants of CDWs survived in the superconducting region of the phase diagram. Application of high pressure also suppressed CDWs in the compound TbTe at about  GPa, inducing superconductivity with , enhanced to at  GPa [384], the behavior demonstrating the competition of Cooper and electron-hole pairings for the FS [385, 386]. The same experiments in this quasi-two-dimensional material revealed two kinds of CDW anomalies merging at  GPa, as well as antiferromagnetism, which makes this object especially promising. Finally, CDWs were found in another superconducting oxide NaCoO HO by specific heat investigations [387–389], showing two-energy-gap superconductivity for as-prepared samples and nonsuperconducting CDW dielectrized state after ageing of the order of days. The sample ageing is a situation widely met for superconductors [390, 391], whereas the dielectrization of as-synthesized superconducting ceramic samples accompanied by a transformation of bulk superconductivity into a percolating one with the CDW background was observed for BPB long ago [56, 392]. Nevertheless, such a scenario was not proved directly at that time, while the bulk heat capacity peak in NaCoO HO [387–389] unequivocally shows the emergence of CDWs instead of superconductivity.

We emphasize that CDWs compete with superconductivity, whenever they meet on the same FS. This is the experimental fact, which agrees qualitatively with a number of theories [385, 386, 393–397].

Returning to cuprates, we want to emphasize that the existence of pseudogaps above and below is one of their most important features. Pseudogap manifestations are diverse, but their common origin consists in the (actually, observed) depletion of the electron densities of states (DOS). It is natural that tunnel and ARPES experiments, which are very sensitive to DOS variations, made the largest contribution to the cuprate pseudogap data base (see references in our works [81–83, 158–160]). Recent results show that the concept of two gaps (the superconducting gap and the pseudogap, the latter considered here as a CDW gap) [82, 352, 353, 357, 377, 398–404] begins to dominate in the literature over the one-gap concept [211, 355, 405–416], according to which the pseudogap phenomenon is most frequently treated as a precursor of superconductivity (for instance, a gas of bipolarons that Bose-condenses below [413] or a -wave superconducting-like state without a long-range phase rigidity [416]). The main arguments, which make the one-gap viewpoint less probable, is the coexistence of both gaps below [349, 417], their different position in the momentum space of the two-dimensional Brillouin zone [351, 353, 356, 418, 419], and their different behaviors in the external magnetic fields [420], for various dopings [417], and under the effects of disordering [419].

Nevertheless, some puzzles still remain unresolved in the pseudogap physics. For instance, Kordyuk et al. [352] found that the pseudogap in Bi(Pb)SrCa(Tb)CuO revealed by ARPES is nonmonotonic in . Such a behavior, as they indicated, might be related to the existence of commensurate and incommensurate CDW gaps, in a close analogy with the case of dicahlcogenides [421]. Another photoemission study of LaBaCuO has shown [354] that there seems to be two different pseudogaps: a -wave-like pseudogap—a precursor to superconductivity—near the node of the truly superconducting gap and a pseudogap in the antinodal momentum region—it became more or less familiar to the community during last years [350, 351, 353, 356, 403, 418, 419] and is identified by us as the CDW gap.

Despite existing ambiguities, the most probable scenario of the competition between CDW gaps (pseudogaps) and superconducting gaps in high- oxides, in particular, in BSCCO, includes the former emerging at antinodal (nested) sections of the FS and the latter dominating over the nodal sections (see Figure 2, reproduced from [403], where BSCCO was investigated, and results for (Bi,Pb)(Sr,La)CuO presented in [356]). Since CDW gaps are much larger than their superconducting counterparts, the simultaneous existence of the superconducting gaps in the antinodal region might be overlooked in the experiments. This picture means that the theoretical model of the partial dielectric gapping (of CDW origin or caused by a related phenomenon—spin density waves, SDWs) belonging to Bilbro and McMillan [385] (see also [56, 158–160, 386, 397, 422–428]) is adequate for cuprates. On the other hand, the coexistence of CDW and superconducting gaps, each of them spanning the whole FS [429–432], can happen only for extremely narrow parameter ranges [433]. Moreover, as is clearly seen from data presented in Figure 2 [403] and a lot of other measurements for different classes of superconductors, complete dielectric gapping has not been realized. The reason is obvious: nested FS sections cannot spread over the whole FS, since the actual crystal lattice is always three-dimensional and three-dimensionality effects lead to the inevitable FS warping detrimental to nesting conditions formulated below.

Figure 2: (Color online) Schematic illustrations of the gap function evolution for three different doping levels of BiSrCaCuO. (a) Underdoped sample with . (b) Underdoped sample with . (c) Overdoped sample with . At above there exists a gapless Fermi arc region near the node; a pseudogap has already fully developed near the antinodal region (red curves). With increasing doping, this gapless Fermi arc elongates (thick red curve on the Fermi surface), as the pseudogap effect weakens. At a -wave like superconducting gap begins to open near the nodal region (green curves); however, the gap profile in the antinodal region deviates from the simple form. At a temperature well below (), the superconducting gap with the simple form eventually extends across entire Fermi surface (blue curves) in (b) and (c) but not in (a). (Taken from [403].)

It is interesting that pseudogaps were also observed in oxypnictides LaFeAsOF and LaFePOF by ARPES [434] and SmFeAsOF by femtosecond spectroscopy [435], where SDWs might play the same role as CDWs do in cuprates. At the same time, in iron arsenide BaKFeAs, photoemission studies detected a peculiar electronic ordering with a (, ) wave vector [436], a true nature of which is still not known, but which might be related either to the magnetic reconstruction of the electron subsystem (SDWs) and/or to structural transitions (when CDWs accompanied by periodic crystal lattice distortions emerge in the itinerant electron liquid near the structural transition temperature [437, 438]). The interplay between structural and magnetic instabilities is important for pnictides [161], since, for example, structural and SDW anomalies appear jointly at 140 K in BaFeAs [439]. It is not inconceivable that pnictides may be a playground for density waves as well as high- oxides, with a rich variety of attendant manifestations.

The DHS is another visiting card of cuprates, being a peculiarity in tunnel and photoemission spectra at low and energies much higher than those of coherent superconducting peaks [81–83, 160, 339, 347, 348, 440, 441]. It is remarkable that in the S-I-N tunnel junctions, where S, I, and N stand for a high- superconductor, an insulator, and a normal metal, respectively, a DHS might appear for either one bias voltage polarity only [347] or both [442, 443], depending on the specific sample. In S-I-N junctions, current-voltage-characteristics (CVCs) with two symmetrically located DHSs (one per branch) are also observed, but with amplitudes that can differ drastically [442, 443]. In S-I-S symmetric junctions, DHS structures are observable (or not) in CVC branches of both polarities simultaneously [347], which seems quite natural. It is very important that although the CVC for every in the series of S-I-N junctions with BSCCO as an superconducting electrodes was nonsymmetric, especially due to the presence of the DHS, the CVC obtained by averaging over an ensemble of such junctions turned out almost symmetric, or at least its nonsymmetricity turned out much lower than the nonsymmetricity of every CVC taken into consideration [443].

There is quite a number of interpretations concerning this phenomenon [347, 444–450]. We have discussed most of them in detail in our previous publications, whereas our theory and necessary reference to other models will be presented below.

STM mapping of high- oxide samples revealed substantial inhomogeneties of energy gap spatial distribution [334, 336, 338, 339, 341–343, 363, 370, 441, 451–459]. The same conclusion was made from the interlayer tunneling spectroscopy [460, 461], more conventional S-I-N tunnel (point-contact) studies [440, 442], optical femtosecond relaxation spectroscopy [337], and inelastic neutron scattering measurements [335]. It is quite natural that some inhomogeneity should exist, since the oxygen content is always nonstoichiometric in those compounds [304]. Indeed, correlations were found between oxygen dopant atom positions and the nanoscale electronic disorder probed by STM [336]. The problem has been recently investigated theoretically making allowance for electrostatic modulations of various system parameters by impurity atoms [462].

Nevertheless, the gap distributions occurred to be anomalously large, with sometimes conspicuous two-peak structures in BSCCO [451, 457, 463], BiSrGdCuO [338], (Cu,C)BaCaCuO [440], and TlBaCaCuO [442]. Nanoscale electronic nonhomogeneity on the crystal surface was shown to substantially affect the CDW-like DOS modulation observed by STM in BiSrLaCuO [340].

Large gap scatterings obviously do not correlate with sharp transitions into the superconducting state at any doping of well prepared samples (implying Cooper-pairing homogeneity), which was demonstrated, for example, by specific heat studies [464]. To solve the problem, one should bear in mind that the gaps measured by STM technique are of two kinds (in our opinion, superconducting gaps and pseudogaps—CDW gaps), which cannot be easily distinguished experimentally [81–83, 160, 337]. The guess was proved in [291], where contributions of both gaps in the STM spectra of (BiPb)SrCuO were separated by an ingenious trick. Namely, the authors normalized the measured local conductances by removing the larger-gap inhomogeneous background. Then, it became clear that the superconducting gap is more or less homogeneous over the sample's surface, whereas the larger gap (the pseudogap, i.e., the CDW gap) is essentially inhomogeneous.

The intimate origin of the pseudogap variations is currently not understood. At the same time, the inhomogeneity of electron characteristics is also inherent to the related solid solutions BPB, which was demonstrated by spatially resolved electron energy loss spectroscopy [465]. It is reasonable to suggest that this inhomogeneity both in BPB and high- oxides is strengthened near free surfaces in agreement with Josephson current measurements across BPB bicrystal tunnel boundaries [466].

Still, there is an interesting phenomenon, which might explain trends for electric properties in cuprates to be inhomogeneous. We mean a spontaneous phase separation, suggested long ago for antiferromagnets [467–470] and the electron gas in paramagnets [471–474]. This idea was later transformed into stripe activity in cuprate and manganite physics, where alternating conducting and magnetic regions constituted separated “phases’’ [12, 302, 475–480]. Recently, a lot of evidence for local lattice distortions, Jahn-Teller polaron occurrence, and other percolation and filamentary structure formation appeared [177, 217, 228, 481–485], supporting new sophisticated theoretical efforts in the science of phase separation [84, 230, 379, 486–493], mostly but not necessarily dealing with high- oxides. The electronic inhomogeneity in cuprates, as discussed above, belongs to the same category of phenomena. Whatever its origin, intrinsic inhomogeneity of cuprates and other oxides seems to be an important feature that needs explanation in order to understand superconductivity (much more homogeneous) itself. Note that electronic phase separation into magnetic and nonmagnetic domains was also found in the iron pnictide superconductor BaKFeAs [494], whereas disorder-induced inhomogeneities of superconducting properties was observed in TiN films [495].

Another high- oxide, YBaCuO, containing CuO chains in addition to CuO planes, was known for a long time as a material exhibiting one-dimensional CDWs [496]. However, the authors of more recent tunnel measurements [497] concluded that the would-be CDW manifestations might have a different nature, since the observed one-dimensional modulation wavelengths have rather a strong dispersion. Nevertheless, it seems that in view of the large CDW amplitude scatter in BSCCO discovered later, this conclusion is premature, with local variations of the FS shape being a possible origin of CDW wave vector modifications.

As one sees from the evidence discussed above, CDW modulations are observed in cuprates both directly (as patterns of localized energy-independent electron states in the conventional -space) and indirectly (as concomitant gapping phenomena). The pseudogap energy constitutes an appropriate scale for CDW gapping. Here, is the superconducting gap. On the other hand, at low energies , single-particle tunneling spectroscopy probes mixed electron-hole -wave Bogoliubov quasiparticles [498], which are delocalized excitations. In this case, it is natural to describe the tunnel conductance in the momentum, -space. The interference between Bogoliubov quasiparticles is especially strong for certain wave vectors () connecting extreme points on the constant energy contours [499–502]. The interference -space patterns involve those wave vectors [343, 416, 499, 503–505], this picture being distinct from and complementary to the partially disordered CDW unidirectional or checkerboard structures [344, 359, 365, 371, 458, 506–508].

It is remarkable that interference -space patterns on cuprate surfaces, the latter being in the superconducting state, are not detected, contrary to the clear-cut STM observations of electron de Broglie standing waves, induced by point defects or step edges, revealed in conductance maps on the normal metal surfaces [509, 510]. The latter waves are in effect Friedel oscillations [511] formed by two-dimensional normal electron density crests and troughs with the wave length , being the Fermi wave vector. On the other hand, spatial oscillating structures of local DOS in the -wave superconducting state are determined by other representative vectors , so that the characteristic oscillations can be denominated as Friedel-like ones at most [502, 512]. Nevertheless, the attenuation of both kinds of spatial oscillations due to superconducting modifications of the screening medium should be more or less similar. Namely, in the isotropic superconducting state, the electron gas polarization operator loses its original singularity at for gapped FS sections [513]. As a consequence, Friedel oscillations gain an extra factor [514, 515], where is the BCS coherence length [498]. For -wave superconductors, the attenuation will be weaker and will totally disappear in the order-parameter node directions. However, those distinctions are not crucial, since the nodes have a zero measure. The modification of screening by formation of Bogoliubov quasiparticles in -wave high- oxides explains the absence of conspicuous spatial structures in STM maps, which correspond to the wave vectors mentioned above.

We consider the observed CDWs in oxides as a consequence of electron-hole (dielectric) pairing on the nested sections of corresponding FSs [158–160, 516]. Such a viewpoint is also clearly supported by the experiments in layered dichalcogenides [374–376], the materials analogous to cuprates in the sense of superconductivity appearance against the dielectric (CDW) partial gapping background [378, 379]. At the same time, other sources of CDW instabilities are also possible [517, 518]. As for the microscopic mechanism causing CDW formation, it might be an electron-phonon (Peierls insulator) [519, 520] or a Coulomb one (excitonic insulator) [431, 521, 522], or their specific combination. Excitonic instability may also lead to the SDW state [522, 523], also competing with superconductivity for the FS [160, 524–529]. It should be noted that researchers asserted that they found plenty of Peierls insulators or partially gapped Peierls metals [158–160, 530–532]. At the same time, the excitonic phase, being mathematically identical in the mean-field limit [533] and physically similar [534] to the Peierls insulator, was not identified unequivocally. One can only mention that some materials claimed to be excitonic insulators, namely, a layered transition-metal dichalcogenide 1T-TiSe with a commensurate CDW [535, 536], alloys TmSeTe [537], SmLaS [538], and TaNiSe with a direct band gap at the Brillouin zone point in the parent high- state [539]. Therefore it is reasonable that precisely in the later case, the low- excitonic state is not accompanied by CDWs.

It is necessary to indicate that in many cases, the claimed “charge stripe order’’ and the more unpretentious “charge order’’ are an euphemism describing the old good CDWs: “Stripes is a term that is used to describe unidirectional density-wave states, which can involve unidirectional charge modulations (charge stripes) or coexisting charge and spin-density order spin stripes’’ [12]. We do not think it makes sense to use the term “stripes’’ in the cases of pure CDW or spin-density-wave (SDW) ordered states. At the same time, this term should be reserved for different possible more general kinds of microseparation [12, 477, 479, 540–542], having nothing or little to do with periodic lattice distortions, FS nesting, or Van Hove singularities. The need to avoid misnomers and duplications while naming concepts is quite general in science, as was explicitly stressed by John Archibald Wheeler, who himself coined many terms in physics (“black hole’’ included) [543].

In this connection, it seems that some experimentalists unnecessarily vaguely attribute the spatially periodical charge structure in the low-temperature tetragonal phase of LaBaCuO, revealed by X-ray scattering [544], to the hypothetical nematic structure or the checkerboard Wigner crystal. Indeed, quite similar spatial charge structures found in LaBaSrCuO by neutron scattering [545] were correctly and without reservation identified as CDW-related ones, whereas a checkerboard structure (if any) can be considered as a superposition of two mutually perpendicular CDWs. The same can be written about the “stripe’’ terminology used in [546], where X-ray scattering revealed a periodical charge structure in the low-temperature tetragonal phase of another cuprate LaEuSrCuO.

One should mention two other possible collective states competing with Cooper pairing. Namely, these are states with microscopic orbital and spin currents that circulate in the ground state of excitonic insulator (there can be four types of the latter [522]). The concept of the state with current circulation, preserving initial crystal lattice translational symmetry, was invoked to explain cuprate properties [547]. Another order parameter, hidden from clear-cut identification by its supposed extreme sensitivity to sample imperfection, is the so-called -density wave-order parameter [548, 549]. It is nothing but a CDW order parameter times the same form-factor the product being similar to that for -superconductors. Here, and are the wave-vector components in the CuO plane. To some extent, the dielectric order parameter of the Bilbro-McMillan model [159, 160, 385] and its generalizations—they are presented below—contains the same physical idea as in the -density-wave model: nonuniformity of the CDW gap function in the momentum space.

Although the destructive CDW action on superconductivity of many good materials is beyond question [56, 160, 380, 384, 545, 550, 551], it does not mean that maximal are limited by this factor only. For instance, falls rapidly with the hole concentration in overdoping regions of phase diagrams for different Pb-substituted BiSrCuO compounds, even in the case when the critical doping value corresponding to lies outside the superconducting dome [552]. A Cu-doped superconducting chalcogenide CuTiSe constitutes another example confirming the same trend [380]. Namely, CDW manifestations die out for , whereas starts to decrease for . As has been already mentioned, overdoping can reduce simply owing to screening of matrix elements for electron-phonon interaction [99, 215, 216].

4. Theory of CDW Superconductors and Its Application to Cuprates

The majority of our results presented below were obtained for -wave superconductors with CDWs. It is a case, directly applicable to many materials (e.g., dichalcogenides, trichalcogenides, tungsten brozes, etc.). On the other hand, as was indicated above, the exact symmetry of the superconducting order parameter in cuprates is not known, although the -wave variant is considered by most researchers in the field as the ultimate truth. Notwithstanding any future solution of the problem, our theory of CDW-related peculiarities in quasiparticle tunnel CVCs can be applied to cuprates, since we are not interested in small energies , where the behavior of a reconstructed DOS substantially depends on whether it is the - or -wave order parameter [553–555]. Here, is the elementary charge, and is the amplitude of the superconducting order parameter. As for the thermodynamics of CDW superconductors, we present both - and -cases, each of them having their own specific features.

4.1. Thermodynamics of s-Wave CDW Superconductors

The Dyson-Gorkov equations for the normal () and anomalous () temperature Green’s functions in the case of coupled superconducting and dielectric (CDW) matrix order parameters are the starting point of calculations and can be found elsewhere [160, 386, 397, 426, 427]. Greek superscripts correspond to electron spin projections, and italic subscripts describe the natural split of the FS into degenerate (nested, d) and non-degenerate (non-nested, n) sections. For the quasiparticles on the nested sections, the standard condition leading to the CDW gapping holds: where is the quasimomentum, is the CDW vector (see the discussion above), Planck's constant . This equation binds the electron and hole bands for the excitonic insulator [431, 522] and different parts of the one-dimensional self-congruent band in the Peierls insulator case [516]. At the same time, the rest of the FS remains undistorted below and is described by the electron spectrum branch . Such an approach was suggested long ago by Bilbro and McMillan [385]. We adopt the strong-mixing approximation for states from different FS sections. This means an appearance of a single superconducting order parameter for d and nd FS sections. The spin-singlet structure (-wave superconductivity and CDWs) of the matrix normal () and anomalous () self-energy parts (where ) in the weak-coupling limit is suggested. Here, is the Kronecker delta. The self-consistency equations for the order parameters obtained in accordance with the fundamentals can be expressed in the following form [386]: where Here, the Boltzmann constant , and are contact four-fermion interactions responsible for superconductivity and CDW gapping, respectively. The gap is a combined gap appearing on the nested FS sections, whereas the order parameter defines the resulting observed gap on the rest of the FS (compared with the situation in cuprates [344, 350, 356, 403]). The parameter characterizes the degree of the FS dielectrization (hereafter, we use this nonconventional term instead of “gapping’’ in some places to avoid confusion with the superconducting gapping), so that and are the electron DOSs per spin on the FS for the nested and nonnested sections, respectively. The upper limit in (3) is the relevant cut-off frequency, which is assumed to be equal for both interactions. If the cut-offs BCS and CDW are considered different, the arising correction, , is logarithmically small [385] and does not change qualitatively the subsequent results. Only in the case of almost complete electron spectrum dielectric gapping () does the difference between BCS and CDW become important for the phase coexistence problem [433]. This situation is, however, of no relevance for substances with detectable superconductivity, since tends to zero for . In this subsection, we confine ourselves to the case , , since the phase of the complex order parameter does not affect the thermodynamic properties, whereas tunnel currents do depend on [160, 556, 557], which will be demonstrated explicitly below.

Introducing the bare order parameters and , we can rewrite the system of (2) in an equivalent form, convenient for numerical calculations: where is the standard Mühlschlegel integral [558], the root of which is the well-known gap dependence for the -wave BCS superconductor [9], , and [385] However, (5) mean that both gaps and have the BCS form [386], namely: (i) , that is, the actual value of the superconducting gap of the CDW superconductor at is rather than , and the actual superconducting critical temperature is ; (ii) at the same time, , which determines . Here, is the Euler constant.

From (4), we obtain that, at , Replacing by its value (7), we arrive at the conclusion that in the model of -wave superconductor with partial CDW gapping, two order parameters coexist only if . Then, according to (7), ; that is, the formation of the CDW, if it happens, always inhibits superconductivity, in agreement with the totality of experiments [160, 375, 380, 382, 551]. Also, vice versa, according to (4), for , ; that is, superconductivity suppresses dielectrization.

In Figure 3, the dependences and are shown for various parameters of the partially dielectrized CDW -wave superconductor. It can be easily inferred from the data shown in both panels that, in agreement with the foregoing, curves coincide with the Mühlschlegel one for any values of the dimensionless parameters and . The novel feature, which has been overlooked in other investigations, is the possibility of such a strong suppression of for low enough that it becomes smaller than , although is larger than (see Figure 3(b)). This intriguing situation can be realized for the parameter close to unity. One should note that the actual gaps and (the former coincides with the superconducting order parameter) are monotonic functions of . However the dielectric order parameter is not.

Figure 3: Temperature dependences of the superconducting () and dielectric () order parameters for different values of the dimensionless parameters (the portion of the nested Fermi surface sections, where the charge-density-wave, CDW, gap develops) and (see explanations in the text). (Taken from [386].)

The magnitudes of and strongly depend on and , although the simple BCS-like scaling between them survives, that is, for CDW -wave superconductors . Although for, say, and reasonable [386], the demand of self-consistency between and becomes less important quantitatively. It justifies our previous approach with -independent [427] and the estimation of combined gap as with -independent made on the basis of interlayer tunneling measurements in BSCCO mesas [559]; self-consistency leads to new qualitative effects and cannot be avoided. As for the magnitude of the very , inferred from tunneling measurements, it was found in [559] to be substantially smaller than that of , whereas the opposite case turned out to be true both for BSCCO [349, 399, 560, 561], BiPbSrCaCuO [460], and (Bi,Pb)SrCaCuO [562]. Other tunnel measurement for BSCCO [417] revealed for underdoped samples and for overdoped ones. A marked sensitivity of to doping together with strong inhomogeneity, discovered in Bi-based ceramics [334–336, 338, 343, 359, 440, 441, 456–458, 563, 564] and CaNaCuOCl [565], may be responsible for the indicated discrepancies.

Since the BCS character of the gap dependences for the CDW -wave superconductor is preserved, the -dependence of the heat capacity for the doubly gapped electron liquid (i.e., below the actual ) equals to the superposition of two BCS-like functions: where