Coexistence of charge density waves and d-wave superconductivity in cuprates. Sharing of the Fermi surface Alexander M. Gabovich * and Alexander I. Voitenko Institute of Physics of the NASU, Nauka Ave. 46, Kyiv 03680 Ukraine Received June 16, 2010; accepted August 3, 2010 Charge density waves / d-wave superconductivity / Order parameter reentrance Abstract. A self-consistent theory that describes charge density waves in a partially dielectrically gapped supercon- ductor with d x 2 y 2 -pairing has been proposed. The depen- dences of dielectric, S, and superconducting, D, order parameters on the temperature and other problem para- meters have been considered, and the phase diagram has been built. The corresponding angular diagrams for gap distribution over the Fermi surface have been plotted. The developed theory is used for the explanation of properties of high-temperature oxides. The influence of mismatch an- gle between the lobes of order parameters S and D on the gap distribution in the momentum space and the reen- trance phenomenon for S with respect to temperature has been analyzed. Pseudogapping, which coexists with genuine supercon- ducting gapping in high-T c cuprates, is an enigmatic phe- nomenon that divides the scholars into two camps. Namely, this phenomenon is considered as either an above-T c precursor of Cooper pairing [1], which – below T c – gives way to a coherent superconducting state with the long-range order (one-gap scenario), or a manifestation of a hostile phase [2], which can coexist with supercon- ductivity only in certain situations (two-gap scenario). Several candidates may be responsible for pseudogapping, e.g., charge density waves (CDWs), spin density waves (SDWs) or d-density waves, which may be of CDW or SDW nature [3]. Some time ago we worked out a self-consistent theory describing the coexistence between CDWs and s-wave superconductivity [4] with important implications to quasi- particle tunneling [5–7]. However, the majority of experi- ments testify that the superconducting order parameter is of the d x 2 y 2 type for high-T c oxides [8]. Therefore, leav- ing aside all deep controversies over the actual intrinsic order parameter symmetry [9], we are going to present a new self-consistent phenomenological theory of the phase, appropriate to cuprates, where two mutually perpendicular CDWs (a checkerboard pattern) and d-wave superconduc- tivity coexist [10]. We suggest that CDW gaps should be identified with observed pseudogaps in various oxides. In the framework of our approach, CDWs are hostile to superconductivity and compete for the Fermi surface (FS). As stems from various experiments, especially from an- gle-resolved photoemission spectroscopic (ARPES) ones, relatively undistorted d-wave superconducting energy gaps are observed in nodal directions of two-dimensional mo- mentum plane, whereas pseudogaps appear in the anti-no- dal sectors of the FS [2]. In the framework of our ap- proach, we can consider a more generalized picture that could be inherent to cuprates. Specifically, our extended approach involves the mismatch angle b between the cen- ters of the CDW order parameter sectors and the d-wave lobe maxima. The corresponding diagram is shown in Fig. 1. Here Q 1 and Q 2 are the CDW wave vectors, 2a is the opening of each CDW sector, the functions DðT ; qÞ¼ DðT Þ cos 2q and SðT Þ denote superconducting and CDW order parameters, respectively, T is temperature (the Boltzmann constant k B ¼ 1), and q is reckoned from the direction of one of d-wave lobe maxima in the mo- mentum space (the abscissa axis). The order parameters emerge due to the Cooper pairing and dielectric (Peierls or excitonic) instabilities. The latter appears because the cup- rate parent quasiparticle spectrum has nested sections [11]. 492 Z. Kristallogr. 225 (2010) 492–494 / DOI 10.1524/zkri.2010.1289 # by Oldenbourg Wissenschaftsverlag, Mu ¨nchen * Correspondence author (e-mail: gabovich@iop.kiev.ua) Fig. 1. Superconducting (D, solid curve) and dielectric (S, dashed curve) order parameter profiles in two-dimensional momentum space for the bare phases of d x 2 y 2 -superconductor and partially gapped me- tal with charge-density waves (CDW), respectively, i.e. when the competitive pairing channel is switched off.