The exemplary metal-oxide perovskite structure of La2CuO4 with conducting CuO2 planes (shadowed) and a fragment of square lattice in such a plane. Arrows indicate AFM spin order at Cu sites.
DOS in a clean d-wave SC system (solid line). Dashed lines indicate the linear low-energy asymptotics, the logarithmic divergence at ∣ɛ∣ → Δ, and the tendency to constant value ρ N at ∣ɛ∣ ≫ Δ.
Low-energy resonance levels ±ɛres (arrows) in the DOS of d-wave superconductor containing a finite concentration c = 0.2ρ N Δ of impurity scatterers with VL ≈ 0.67/ρ N (corresponding to ≈ 2). Other distinctions from the pure crystal DOS in Fig. 2 (shown here by the dashed line) are the finite spikes at |ɛ| = Δ and the enhanced slope at |ɛ| < ɛres.
Suppression of the local SC order parameter at the very impurity site η = 1−Δ0/Δ in function of the perturbation parameter , calculated for d-wave case from Eq. (39) (solid line) vs its analytical s-wave form π2 2/(1 + π2 2) (Ref. 15) (dash-dotted line). Inset: comparison of the integrand functions in the numerator (solid line) and denominator (dashed line) of Eq. (39) at = 2 reveals a notable negative effect of the resonance level at ɛres that diminishes η in the d-wave case vs the s-wave case where no such resonance exist.
Extended perturbation over four nearest oxygen sites δ i to the impurity ion (its projection onto the CuO2 plane is shown by the dashed circle at the origin).
DOS in the d-wave superconductor with extended impurity centers (the solid line), for the choice of parameters W = 2 eV, μ = 0.3 eV, ɛ D = 0.15 eV, VL = 0.3 eV, c = 0.1. The arrow indicates the low-energy resonance by the A-channel impurity effect and the dashed line represents the pure d-wave DOS.
Local density of states on the nearest neighbor site to an extended impurity center, for the same choice of parameters as in Fig. 6 (but supposing c→0). Note the overall enhancement of electronic density compared to that on remote sites from impurity (dashed line).
The dimensionless function F(ɛ) (solid line) used in the numerator of Eq. (52), compared to the dimensionless integrand in its denominator (dashed line) to calculate the suppression parameter ηsup at the same conditions as in Fig. 4.
Effective magnetic perturbation for charge carriers on nearest neighbor sites to the nonmagnetic impurity substitute for Cu2+ in CuO2 plane.
Local density of states near magnetic impurity ρ δ (ɛ), given by Eqs. (57) and (58), at the choice of perurbation parameter J = 0.3 eV, slighly above the critical value eV, presents a sharp resonance just below the Fermi level, similar to that observed in the STM spectrum on Zn site in Bi2Sr2CaCu2O8 at ≈ −1.5 meV (Ref. 18) (inset).
The comparison between the behavior of dimensionless numerator (solid line) and denominator (dashed line) in Eq. (59).
Contour plots of the left hand side of Eq. (65), in function of complex self-energy Σ0 for two different energies ɛ at the choice of c = ρ N Δ and three different perturbation parameters: unitary limit = 10 (upper row), intermediate regime = 1 (middle row), and Born limit = 0.35 (bottom row). There are always two roots shown by white circles and denoted SCTMA1 and SCTMA2, and at ɛ→0 the first of them tends to zero, close to the real axis, while the other tends to a finite imaginary limit.
Residual self-energy for the SCTMA2 solution, calculated from Eq. (69) with the choice of in function of the perturbation parameter (open circles). The dashed lines show the limiting behaviors: exponential in the Born limit, γ0/Δ ≈ 4exp(−ρ N Δ/πc 2), and a constant value in the unitary limit.
Construction of the self-consistent DOS (solid line) adjusted to the two different SCTMA solutions beyond the region of impurity resonance ɛres of width Γres. The impurity parameters are chosen as = 1 and c = 0.2ρ N Δ. The SCTMA1 solution is shown by the dashed line, the SCTMA2 solution by the dash-dotted line, and the short-dash line shows the common T-matrix solution from Fig. 3.
Real and imaginary parts of the self-energy Σ(ɛ) (in units of Δ) obtained for two different SCTMA solutions at the choice of perturbation parameters c = ρ N Δ, = 1. Note the tendency of ReΣ( 2 )(ɛ) to ɛ (dashed line) at ɛ → 0.
Schematic of local coordinates near nodal points in the Brillouin zone of a d-wave superconductor.
the isolated poles of the integrand in Eq. (8) in function of pair separation vector r = (x,y), are located along the nodal axes in the direct space, where one of the pole conditions, g 1(r) = 0, holds identically. Another pole condition, , is reached at discrete points (here at the choice of parameters β = 0.05, ). The main panel shows the related behavior of g 3(r)−1 along the nodal direction of r.
Trajectories in the space of variables g 1(r), g 3(r), corresponding to the poles of GE denominator, Eq. (93), for M impurities at the choice of = 1 and δ = 0.1.
In the r-plane, the trajectories of zeroes of D r form continuous loops within the stripes of ∼ F /( Δ|δ|kF ) length and ∼1/(|δ|kF ) width along the nodal axes.
Low-energy d-wave DOS ρ(ɛ) in simultaneous presence of NM impurities (with cNM = 3% and NM = 1), producing two symmetric broad resonances, and M impurities (with c = 0.03%, = NM , and g as = 0.9), producing single sharp resonance at extremely low energy ɛres. Inset shows the mobility edges ɛ c and ɛ c′ around the shifted nodal point ɛ0, they separate localized states (shadowed area) with almost constant DOS, ρ(ɛ) ≈ ρ(ɛ0), from band-like states whose DOS is close to the T-matrix value (solid line).
Isoenergetic lines for d-wave dispersion law display elliptic structure near nodal points and hyperbolic structure near antinodal points. Inset shows how these lines near an antinodal point give rise to the hyperbolic coordinates E (solid lines) and t (dashed lines) in the “spacelike” (SL) and “timelike” (TL) sectors of the plane.
Narrow areas (seen as solid lines) in the Brillouin zone of a d-wave superconductor with low concentration of impurities, c ≪ cAN , where the quasiparticle states are no more described by the band dispersion law but localized on clusters of close impurities.
Article metrics loading...
Full text loading...