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arXiv:cond-mat/9401026 · Physical Review Letters

Disorder effects in two-dimensionald-wave superconductors

A. A. Nersesyan, Alexei M. Tsvelik, F. Wenger

Published Apr 18, 1994• 226 citations
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Abstract

We show that weak nonmagnetic impurities in a two-dimensional d-wave superconductor introduce a nontrivial exponent in the energy dependence of the single-particle density of states: \ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\omega}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\alpha}}}$. The exponent \ensuremath{\alpha}>0 is exactly calculated for two types of disorder. We argue that the appearance of a finite \ensuremath{\rho}(0) would be equivalent to breaking of a continuous symmetry, prohibited in two dimensions. The method we use is based on the Abelian and non-Abelian bosonization of the effective model of 1D interacting fermions.

DOI
10.1103/physrevlett.72.2628

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We show that weak nonmagnetic impurities in a two-dimensional d-wave superconductor introduce a nontrivial exponent in the energy dependence of the single-particle density of states: \ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\omega}\ensuremath∥\ensuremathα{\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\alpha}}}\ensuremath∥\ensuremathα. The exponent \ensuremath{\alpha}>0 is exactly calculated for two types of disorder. We argue that the appearance of a finite \ensuremath{\rho}(0) would be equivalent to breaking of a continuous symmetry, prohibited in two dimensions. The method we use is based on the Abelian and non-Abelian bosonization of the effective model of 1D interacting fermions.
We report on a numerical study of disorder effects in 2D d-wave BCS superconductors. We compare exact numerical solutions of the Bogoliubov-de Gennes (BdG) equations for the density of states rho(E) with the standard T-matrix approximation. Local suppression of the order parameter near impurity sites, which occurs in self-consistent solutions of the BdG equations, leads to apparent power-law behavior rho(E) approximately |E|(alpha) with nonuniversal alpha over an energy scale comparable to the single-impurity resonance energy Omega(0). We show that the novel effects arise from static spatial correlations between the order parameter and the impurity distribution.
We study how a randomness of either boson or fermion site energies affects the superconducting phase of the boson fermion model. We find that, contrary to what is expected for s-wave superconductors, the nonmagnetic disorder is detrimental to the s-wave superconductivity. However, depending on which subsystem the disorder is located, we can observe different channels being affected. Weak disorder of the fermion subsystem is mainly responsible for the renormalization of the single-particle density of states, while disorder in the boson subsystem leads directly to fluctuations of the strength of the effective pairing between fermions.
In this paper we present a method to compute the exact density of states induced by NNN non magnetic impurities in a system of two dimensional Dirac fermions in the unitarity limit. We review the case of the πππ-flux phase of the Heisenberg model and also treat the disordered d-wave superconductor. In both case we find additional states in the gap with δρ(ω)≃ni/∣ω(ln⁡2∣ω/Δ0∣+(π/2)2)∣δρ(ω) \simeq n_i/ |ω(\ln^2 |ω/Δ_0| + (π/2)^2)|δρ(ω)≃ni​/∣ω(ln2∣ω/Δ0​∣+(π/2)2)∣.
Effects of nonmagnetic impurities with unitarity scattering in two-dimensional d -wave superconductors are investigated by the numerical diagonalization of finite-size systems of up to 35×35 sites. It is shown that a localized state with almost zero excitation energy is formed around an impurity site if it is isolated. However, its localized state is extended in real space along the direction of the node of the d -wave gap, so that the localized states of adjacent impurities easily interfere with each other forming a kind of impurity band, i.e., resonance states around the zero energy. It is also pointed out that there exists a strong association between the resonance density of states due to impurity scattering in the unitarity limit and the so-called zero-bias anomaly of tunneling conductance.
Within numerically exact solutions of the Bogoliubov-de Gennes equations, we demonstrate that discrepancies between predicted low-energy quasiparticle properties in disordered 2D d-wave superconductors occur because of the unanticipated importance of disorder model details and normal state particle-hole symmetry. For the realistic case, which is best described by a binary alloy model without particle-hole symmetry, we predict density-of-state suppression below an energy scale which appears to be correlated with the corresponding single impurity resonance.
A lattice model for disordered d-wave superconductors with spin-rotational and time-reversal symmetries is reconsidered. Near the band center, the lattice model can be described by Dirac fermions with several species. Each fermion yields a Wess-Zumino-Witten (WZW) term for an effective action of the Goldstone mode, which is known to play a crucial role in critical phenomena in two dimensions. The WZW terms cancel out each other because of the fourfold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining a WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.