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arXiv:cond-mat/0507430 · Physical Review Letters

Broad Peak in the dx2−y2 Superconducting Correlation Length as a Function of Hole Concentration in the Two-Dimensional t−J Model

W. O. Putikka, Marco U. Luchini

Published Jun 19, 2006• 10 citations
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Abstract

We have calculated high temperature series to 12th order in inverse temperature for singlet superconducting correlation functions of the 2D $t\mathrm{\text{\ensuremath{-}}}J$ model with $s$, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$, and ${d}_{xy}$ symmetry pairs. Our calculations differ from previous work by removing disconnected pieces from the original four-point correlator and by treating the resulting pairing correlator as a matrix. We find the correlation length for ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing grows significantly with decreasing temperature and develops a broad peak as a function of doping around $\ensuremath{\delta}=0.25$ for $T/J=0.25$ at $J/t=0.4$. The correlation lengths for $s$ and ${d}_{xy}$ symmetry remain small and do not display peaks. Antiferromagnetic spin correlations at low doping act to suppress the ${d}_{{x}^{2}\ensuremath{-}{y}^{x}}$ and ${d}_{xy}$ superconducting correlation lengths.

DOI
10.1103/physrevlett.96.247001

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We have calculated high temperature series to 12th order in inverse temperature for singlet superconducting correlation functions of the 2D t\ensuremath-Jt\mathrm{\text{\ensuremath{-}}}Jt\ensuremath-J model with sss, dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​, and dxy{d}_{xy}dxy​ symmetry pairs. Our calculations differ from previous work by removing disconnected pieces from the original four-point correlator and by treating the resulting pairing correlator as a matrix. We find the correlation length for dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ pairing grows significantly with decreasing temperature and develops a broad peak as a function of doping around \ensuremathδ=0.25\ensuremath{\delta}=0.25\ensuremathδ=0.25 for T/J=0.25T/J=0.25T/J=0.25 at J/t=0.4J/t=0.4J/t=0.4. The correlation lengths for sss and dxy{d}_{xy}dxy​ symmetry remain small and do not display peaks. Antiferromagnetic spin correlations at low doping act to suppress the dx2\ensuremath−yx{d}_{{x}^{2}\ensuremath{-}{y}^{x}}dx2\ensuremath−yx​ and dxy{d}_{xy}dxy​ superconducting correlation lengths.
Based on the known phenomenology of high-Tc cuprates and the available numerical calculations of the t − J model, a two-dimensional effective fermionic model with the nearest neighbor attraction is proposed. Numerical calculations suggest that the model has the dx2−y2 superconductivity (SC) in the ground state at a low fermionic density. We argue that this model captures the important physics of the dx2−y2 superconducting correlations found earlier in the t − J model by the exact diagonalization approach. Within a self-consistent RPA diagrammatic study, the density and the coupling strength dependence of the critical temperature is calculated. We also investigate the influence of the impurities on our results and show that the suppression of the superconductivity is insignificant, when the retardation effects are accounted for as opposed to the Hartree–Fock approximation.
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The two-dimensional t-J model in the ground state is investigated by the power Lanczos method. The pairing-pairing correlation function for ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$-wave symmetry is enhanced in the realistic parameter regime for high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ superconductors. The charge susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}$ shows divergent behavior as ${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}$\ensuremath{\propto}${\mathrm{\ensuremath{\delta}}}^{\mathrm{\ensuremath{-}}1}$ near half filling for the doping concentration \ensuremath{\delta}, indicating that the value of the dynamical exponent z is four under the assumption of hyperscaling. The peak height of the spin structure factor Smax{\mathrm{S}}_{\mathrm{max}}Smax​(Q) also behaves as Smax{\mathrm{S}}_{\mathrm{max}}Smax​(Q)\ensuremath{\propto}${\mathrm{\ensuremath{\delta}}}^{\mathrm{\ensuremath{-}}1}$ near half filling, which leads to the divergence of the antiferromagnetic correlation length ${\ensuremath{\xi}}_{\mathrm{m}}$ as ${\ensuremath{\xi}}_{\mathrm{m}}$\ensuremath{\propto}${\mathrm{\ensuremath{\delta}}}^{\mathrm{\ensuremath{-}}1\mathrm{/}2}$. The boundary of phase separation is estimated on the basis of the Maxwell construction. Numerical results are compared with experimental features observed in high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ cuprates.
In the present work the Roth s two-pole approximation (Phys. Rev. 184, 451 (1969)) has been used to investigate the role of d− p hybridization in the normal and superconducting states of an extended d− p Hubbard model. Superconductivity with singlet dx2−y2 -wave pairing is treated by following Beenen and Edwards formalism (Phys. Rev. B 52, 13636 (1995)). In the first part of this work, the effects of the hybridization on the superconductivity, in the hole-doped regime, have been studied treating Roth s band shift within two different approximations. In the first one, the band shift has been calculated in the limit U →¥ (U is the Coulomb interaction), with zero temperature and without consider the superconducting effects. These regards, are restrict to the band shift. In the other parts of the problem, U, the temperature and the superconducting effects have been considered finites. In the second approximation, the Coulomb interaction, the temperature and the superconductivity have been considered in the calculation of some relevant correlation functions present in the Roth s band shift. The obtained results show that the hybridization acts in the sense of to suppresses the superconductivity. Also, it has been verified that the first approximation overstimates the effects of the hybridization on the superconductivity. In the second part of these work, hoppings to second-nearest-neighbors have been included in the model with the purpose of reproduces adequately the asymetries (mainly those related with the Fermi surface, band structure and phase diagram) between the hole- and electron-doped systems. Particularly, it is shown that the crossover from hole-like to electronlike Fermi surface is deeply affected by the d − p hybridization in the hole doping case. It has been verified that the effect of the hybridization is most pronounced around the saddle-points, where the superconducting gap is maximum in the particular case of dx2−y2 -wave symmetry. As a consequence, the critical temperature Tc is directly affected by the hibridization. Moreover, the obtained results suggest that in the hole doped regime, the hybridization may act on the transport properties of the system due to the sign changes of the Hall coefficient when the crossover of the Fermi surface occurs. In the electron doped case, the crossover in the Fermi surface is not verified. Nevertheless, as the hybridization suppresses the density of states near the Fermi level, the superconductivity is affected. The topology of the Fermi surface in the hole and electron doping regime agree with recent experimental ARPES results for La2−xSrxCuO4 (hole doping) and Nd2−xCexCuO4 (electron doping).
The two dimensional (2D) t - J model is studied in a Green's function decoupling scheme, which is an extension of the Kondo and Yamaji's theory (KY theory). The KY theory was originally proposed for spin systems, and is applicable to cases in which there is no symmetry breaking. We derive the lowest order corrections by the electron hopping energy t to self-consistent equations for the correlation function. Hole concentration dependences of the antiferromagnetic (AF) long-range-order (LRO) and spin correlation length at zero temperature are obtained. It is found that they decrease sensitively with doping holes even in a situation very favorable for the antiferromagnetism. The critical hole concentration at which AF LRO vanishes is estimated to be much smaller than in the three-dimensional case. A characteristic feature of 2D electron systems with strong correlations is discussed.
Recently we have shown [V. F. Elesin, A. V. Krasheninnikov, and L. A. Openov, Sov. Phys. JETP 79, 789 (1994)] that in the hole-doped Cu4{\mathrm{Cu}}_{4}Cu4​O8{\mathrm{O}}_{8}O8​ cluster pairing correlations with s∗{s}^{*}s∗ symmetry are nearly twice as strong as those with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry, while in the electron-doped cluster pairing correlations seem to be present in the s∗{s}^{*}s∗ channel only. In this paper the effect of on-site disorder on the pairing correlators with s∗{s}^{*}s∗ and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry in the Cu4{\mathrm{Cu}}_{4}Cu4​O8{\mathrm{O}}_{8}O8​ cluster is studied numerically through an exact diagonalization of the Emery-Anderson Hamiltonian. It is shown that at a nearly optimal doping level (0.25 excess holes per copper atom) the disorder weakens pairing correlations in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ channel and has much less influence on pairing correlations in the s∗{s}^{*}s∗ channel. Based on the results obtained, the "superconducting" components of pairing correlators are estimated to be 10-30% of their full values, with the remaining parts of correlators being contributions from a finite-size effect. These results indicate that cluster calculations may be relevant for the study of high-Tc{T}_{c}Tc​ superconductivity.