Mixed-state quasiparticle spectrum for d-wave superconductors.

Controversy concerning the pairing symmetry of high-Tc materials has motivated an interest in those measureable properties of superconductors for which qualitative differences exist between the s-wave and d-wave cases. We report on a comparison between the microscopic electronic properties of dwave and s-wave superconductors in the mixed state. Our study is based on self-consistent numerical solutions of the mean-field Bogoliubov-de Gennes equations for phenomenological BCS models which have s-wave and d-wave condensates in the absence of a magnetic field. We discuss differences between the s-wave and the d-wave local density-of-states, both near and away from vortex cores. Experimental implications for both scanning-tunnelingmicroscopy measurements and specific heat measurements are discussed. PACS: 74.60.Ec, 74.72.-h Typeset using REVTEX 1 Since shortly after the discovery of high-temperature superconductors (HTSC), there has been great interest in determining the pairing symmetry of the order parameter [1]. In the absence of disorder, low temperature electronic properties in the Meissner state of a d-wave superconductor differ qualitatively from those of a conventional s-wave superconconductor because of the existence of nodes in the gap function. These differences can in principle be used to identify the pairing symmetry, although strong anisotropy and the complicated nature of the materials have conspired to make conclusive experiments difficult. (Recent work is strongly suggestive of dx2−y2 pairing.) It is also of interest to study differences between the mixed-states of d-wave and s-wave type II superconductors. In the mixed-state magnetic flux will penetrate the superconductor and form an Abrikosov vortex lattice. Lowlying quasiparticle excitations will then exist for both pairings, although in the conventional case they must be bound to the vortex core where the order parameter vanishes. The existence of bound quasiparticle states was first predicted by Caroli, de Gennes and Matricon [2] when they studied an isolated vortex line in a conventional superconductor. Experimentally, these quasiparticles have been observed in scanning-tunneling-microscopy (STM) measurements [3]. For the d-wave case, progress has recently been made on both experimental and theoretical fronts. Volovik has used semiclassical approximations [4,5] to calculate the density-of-states (DOS) at the Fermi energy N(0) for the mixed state of a dx2−y2 superconductor in a weak magnetic field H ≪ Hc2. He found a finite N(0) in the absence of disorder proportional to H compared to the H behavior expected for conventional superconductors in the same approximation. This prediction appears to be in accord with recent measurements of the magnetic field dependence of the low-temperature specific heat [6,7] in high Tc materials. However, the short coherence length of high Tc materials raises some uncertainty about the detailed applicability of a semi-classical analysis and motivates a fully microscopic study of the same problem. In this Rapid Communication we report on such a study. Application of microscopic mean-field-theory to inhomogeneous states of superconductors gives rise to the Bogoliubov-de Gennes (BdG) equations [8]. Motivated by the STM 2 experiments of Hess [3], numerical solutions of the BdG equations have been obtained for both continuum [9,10] and lattice [11] models of a superconductor containing an isolated vortex. This work has recently been generalized to the case of isolated vortex in a d-wave superconductor [12]. According to Volovik, the DOS in the mixed state of a d-wave superconductor is dependent on the typical distance between vortices so that for the present study it is necessary to solve the BdG equations for the vortex-lattice state of a d-wave superconductor. To model decoupled CuO2 layers we consider single-band Hamiltonians on a 2D square lattice with nearest neighbour hopping and both on-site and nearest-neighbor interactions: H = H +H ′ , (1a)