Quasiparticle Transport and Localization in High-Tc Superconductors

We present a theory of the effects of impurity scattering in dx22y2 superconductors and their quantum disordered counterparts, based on a nonlinear-sigma-model formulation. We show the existence, in a quasi-two-dimensional system, of a novel spin-metal phase with a nonzero spin diffusion constant at zero temperature. With decreasing interlayer coupling, the system undergoes a quantum phase transition (in a new universality class) to a localized spin insulator. Experimental implications for spin and thermal transport in the high-temperature superconductors are discussed. [S0031-9007(98)07710-2] Over the last few years, experiments [1] have convincingly established that the superconducting state of the holedoped cuprate materials is characterized by spin singlet dx22y2 pairing. In such a superconductor, the gap vanishes at four points on the (two-dimensional) Fermi surface. The quasiparticle excitations at these “nodes” have a linear dispersion, and an associated density of states that vanishes linearly on approaching the Fermi surface. This leads to power law dependences of various physical quantities on temperature. Impurity scattering is expected to strongly modify these properties. Experimentally, the power law temperature dependences are rounded off, apparently approaching constant, temperature-independent behavior at the lowest temperatures. This fact is well reproduced by approximate, self-consistent treatments of impurity scattering which show that a constant finite density of states is generated at the Fermi energy for any arbitrarily weak impurity strength [2,3]. Quasiparticle transport properties have also been investigated [4,5] theoretically with such self-consistent approximations with some phenomenological success. Going further, Lee [2] has suggested, on the basis of calculations of the zero frequency microwave conductivity, that the quasiparticle eigenstates are strongly localized. In this paper, we reconsider the effects of disorder on the low energy quasiparticles in the dx22y2 superconductor. We point out that the problem of quasiparticle transport and localization in a superconductor is conceptually very different from the more familiar situation of noninteracting electrons in a random potential. This is because, unlike in a normal metal, the charge of the quasiparticles in the superconductor is not a conserved quantity. This immediately implies that the quasiparticle charge in the superconductor cannot be transported through diffusion. Indeed, the quasiparticle charge density is not a hydrodynamic mode in the superconductor. However, in a singlet superconductor (and in particular in the high-Tc superconductors), the condensate does not carry any spin, and consequently the spin of the quasiparticles is a good quantum number and is conserved. The quasiparticle energy is also conserved. Thus, there is the possibility of having spin and energy diffusion without charge diffusion. These differences in symmetry lead to interesting differences between the localization properties of quasiparticles in the superconductor, and in the normal metal. Such differences have been pointed out before [6] in the context of the random matrix theory of mesoscopic normal/superconducting systems. We address quasiparticle transport using a replica field theoretic formulation. As expected, the field theory is different from that describing Anderson localization in a normal metal. The properties of the theory are determined by a single coupling constant, which is the dimensionless spin conductance. This is the physically correct quantity whose behavior as a function of system size enables construction of a scaling theory of localization. By analyzing the properties of the field theory, we show the existence of a logarithmic “weak localization” correction in two dimensions suggesting localization at the largest length scales. This correction persists, in part, in the presence of an orbital magnetic field (unlike usual Anderson localization) or a Zeeman field, but is suppressed when both are present. In all cases, however, the quasiparticles are generically ultimately localized in two dimensions. Upon inclusion of interlayer coupling, there is the interesting possibility of a quantum phase transition between an extended spin metal and a localized spin insulator. The spin metal has diffusive spin correlations, a finite spin susceptibility, and an associated finite spin conductivity all at zero temperature. We are not aware of the existence of such a spin phase in any insulating Heisenberg spin model with or without randomness. The spin insulator is expected to exhibit local moments and spin-glass or random-singlet behavior at very low temperatures. The transition between these two phases is described by the critical point of the replica field theory (neglecting quasiparticle interactions), and is a new universality class for localization. Most of these results also go over unmodified to the quantum disordered version of the dx22y2 superconductor —the “nodal liquid” phase that has been analyzed recently [7] as a possible low-temperature theory of the pseudogap regime in the cuprate materials.