d-Wave Superconductivity and s-Wave Charge Density Waves: Coexistence between Order Parameters of Different Origin and Symmetry
A review of the theory describing the coexistence between d-wave superconductivity and s-wave charge-density-waves (CDWs) is presented. The CDW gapping is identified with pseudogapping observed in high-T c oxides. According to the cuprate specificity, the analysis is carried out for the two-dimensional geometry of the Fermi surface (FS). Phase diagrams on the σ 0 − α plane-here, σ 0 is the ratio between the energy gaps in the parent pure CDW and superconducting states, and the quantity 2α is
... e quantity 2α is connected with the degree of dielectric (CDW) FS gapping-were obtained for various possible configurations of the order parameters in the momentum space. Relevant tunnel and photoemission experimental data for high-T c oxides are compared with theoretical predictions. A brief review of the results obtained earlier for the coexistence between s-wave superconductivity and CDWs is also given. Symmetry 2011, 3 700 Classification: PACS 74.20.-z; 74.20.Rp; 71.45.Lr; 74.72.-h Introduction Superconductivity in high-T c oxides has been for a long time suspected to exhibit non-conventional order parameter symmetry  . Nevertheless, there is no consensus that it is really the case. Indeed, some phase-sensitive experiments show isotropic s-wave superconductivity (SC)     , whereas the majority of measurements reveal d x 2 −y 2 -wave Cooper pairing           or, may be, an extended d-wave gap with higher angle harmonics  . Moreover, a lot of phase-insensitive evidence can be regarded as a manifestation of the extended s-wave pairing       . (To reconcile the latter interpretation with the observed d-wave-like data    11, 13] , the author of Reference  supposed that the d-wave symmetry is inherent to "the degraded surfaces" rather than to the samples' bulk.) It should also be emphasized that various power-law bulk temperature, T , dependences cannot be regarded as a ponderable argument for the existence of nodes on the Fermi surface (FS), which are appropriate to non-conventional superconducting order parameters [23, 24] . Namely, a disordered multidomain structure of high-T c oxides might be the origin of the transformation of Bardeen-Copper-Schrieffer (BCS) exponential dependences for a number of gap-related properties into power-law ones due to the averaging over those domains with varying T c 's and corresponding energy gaps     . The treatment of high-T c superconductors as spatially inhomogeneous percolating conglomerates was earlier suggested in Reference  from other considerations (see also ). Note, that for thermodynamic properties, governed by energy gaps, the sign, as well as the phase, of superconducting order parameter is irrelevant, at least in the standard situation, when the actual order parameter is not a superposition of terms with different symmetries, the possibility, which can not been ruled out [35,36]. On the other hand, the existence and character of nodes matter (see a thorough account in References [23, 24] ), making the electron spectrum gapless and the T -dependences power-law ones, as was indicated above. The picture becomes richer, if superconductivity coexists with another long-range order, e.g., ferromagnetism or antiferromagnetism   , spin-density waves (SDWs)         or charge-density waves (CDWs) [46,          . In particular, following the seminal work  (see also Reference  based on the specific two-dimensional tight-binding model with first-and second-neighbor couplings taken into consideration), we have developed a theory of CDW superconductors for the s-wave superconducting order parameter            . In agreement with the statement made above, the thermodynamics does not depend on the phases of both order parameters, whereas quasiparticle and Josephson currents do              . There are a good many CDW superconductors, for which the model [61, 71] is suitable (see, e.g., References [46, 52, 60,       ). Therefore, we suggested a model of CDW superconductors with d x 2 −y 2 -symmetry on the basis of the electron spectrum peculiarities found in numerous experiments for high-T c oxides [93-101]. Their enigmatic properties are treated below in the framework of this model. Most likely, the actual truth for the materials concerned lies in between the ultimate cases of s-wave and d-wave CDW Unidirectional PLDs were observed in La 1.875 Ba 0.125 CuO 4 and La 1.875 Ba 0.075 Sr 0.05 CuO 4 by neutron scattering , La 1.8−x Eu 0.2 Sr x CuO 4 by X-ray diffraction [161,162], Ca 1.88 Na 0.12 CuO 2 Cl 2 and Bi 2 Sr 2 Dy 0.2 Ca 0.8 Cu 2 O 8+δ by STM , Bi 2+x Sr 2−x CuO 6+δ by electron diffraction and high-resolution electron microscopy , Bi 2−x Pb x Sr 2 CaCu 2 O 8+y by STM , and Bi 2 Sr 2 CaCu 2 O 8+δ by X-ray diffraction [165, 166] . It is crucial that CDWs were shown to exist both below and above T c . We also emphasize that various kinds of modulations were found for the same material, BSCCO [128,      165, 166] . A transition from unidirectional to checkerboard CDWs may be stimulated, e.g., by doping, as in the case of YBa 2 Cu 3 O 7−δ , where a Lifshits topological transition occurs at a hole concentration of 0.08 [167, 168] . One should note that in the presence of impurities (for instance, an inevitably non-homogeneous distribution of oxygen atoms) the attribution of the observed charge order (if any) to unidirectional versus checkerboard type might be ambiguous  . Another remark must be made concerning commensurability of PLDs-CDWs. Namely, their wave vectors Q are, in general, incommensurate and doping-dependent [97, 117] , so that the expressions like 4a 0 × 4a 0 are always approximate, although correctly reflecting the four-fold symmetry of the distortions concerned. Measurements of transport and photoemission properties in non-superconducting layered nickelates R 2−x Sr x NiO 4 (R = Nd, Eu), which are structurally similar to cuprates, revealed a correlation between the pseudogap emergence and charge ordering  . Pseudogaps appeared on the same Fermi surface (FS) sections as in cuprates, thus testifying the similarity between two classes of materials. Layered dichalcogenides constitute another group of materials with CDWs [122, 123, 170] similar to those in cuprates, as has been recently shown     (see also Reference ). In particular, a true pseudogap-a non-mean-field fluctuation precursor phenomenon    -is observed in 2H-TaSe 2 above the normal metal-incommensurate CDW transition temperature T N −IC ≈ 122 K . Such a behavior comprises a strong argument in favor of the CDW nature of pseudogapping in cuprates as well. As for pseudogaps, they were found in cuprates both above and below T c , which is one of their most important features. The pseudogap is a 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 bank (see also references in our works       ). Recent results show that the concept of two gaps (superconducting gap and pseudogap, the latter considered here as a CDW gap) [79, 80, 85, 96,             begins to dominate in the literature over the one-gap concept            , according to which the pseudogap phenomenon is most frequently treated as a precursor of superconductivity (for instance, as properties of bipolaron gas above T c that Bose-condenses below T c  ). The main arguments, which show that superconducting and pseudo-gaps are not identical, are the coexistence of both features below T c [106, 202] , their different position in the momentum space of the two-dimensional Brillouin zone [187,     , and their differing behavior in the external magnetic fields H  , for various dopings  , and under the effects of disorder  . Sometimes, evidence for CDW ordering may be rather indirect, although the very appearance of the phase transition is beyond any doubt. In cuprates, T -anomalies in the nuclear quadrupole resonance transverse relaxation rate in YBa 2 Cu 3 O 7−δ are the best example of such a behavior [208, 209] .