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Physical review. B, Condensed matter

Band structure, magnetic fluctuations, and quasiparticle nature of the two-dimensional three-band Hubbard model

Hong-Qiang Ding, G. H. Lang, William A. Goddard

Published Dec 1, 1992• 14 citations
Publisher

Abstract

A single hole doped into the ${\mathrm{CuO}}_{2}$ plane of a high-${\mathit{T}}_{\mathit{c}}$ superconductor is studied via exact calculations of the three-band Hubbard model (large-U limit) on a 4\ifmmode\times\else\texttimes\fi{}4 periodic cluster. The quasiparticle band has its bottom at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) and disperses strongly towards \ensuremath{\Gamma} and X with a dispersion 1.7${\mathit{J}}_{\mathit{d}\mathit{d}}$=0.22 eV. The quasiparticle has a nearly isotropic effective mass (\ensuremath{\sim}4${\mathit{m}}_{\mathit{e}}$) and the dressing effects are well localized (\ensuremath{\sim}8 \AA{}) with characteristic features of both the three-spin polaron and the Zhang-Rice singlet (although solutions of the t-J model are substantially different).

DOI
10.1103/physrevb.46.14317

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A single hole doped into the CuO2{\mathrm{CuO}}_{2}CuO2​ plane of a high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductor is studied via exact calculations of the three-band Hubbard model (large-U limit) on a 4\ifmmode\times\else\texttimes\fi{}4 periodic cluster. The quasiparticle band has its bottom at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) and disperses strongly towards \ensuremath{\Gamma} and X with a dispersion 1.7Jdd{\mathit{J}}_{\mathit{d}\mathit{d}}Jdd​=0.22 eV. The quasiparticle has a nearly isotropic effective mass (\ensuremath{\sim}4me{\mathit{m}}_{\mathit{e}}me​) and the dressing effects are well localized (\ensuremath{\sim}8 \AA{}) with characteristic features of both the three-spin polaron and the Zhang-Rice singlet (although solutions of the t-J model are substantially different).
A CuO chain with one or two holes doped is studied via the two-band Hubbard model in the large-U limit on a finite cluster. The quasiparticle band is Ek{\mathit{E}}_{\mathit{k}}Ek​\ensuremath{\sim}0.87J cos(2ka) with the bottom at ka=\ensuremath{\pi}/2. The quasiparticles are kinks which have the character of three-spin polarons. The copper spin excitations are incommensurate spin waves. The two doped holes repel each other very slightly, in contrast to the slight attraction in the t-J model. Spin-charge separation also occurs.
The momentum dispersion of the peak structure of the single-particle spectral weight A(p,\ensuremath{\omega}) of the two-dimensional Hubbard model is discussed. Using results obtained from Monte Carlo simulations on lattices up to 12\ifmmode\times\else\texttimes\fi{}12 in size, we determine the low-lying quasiparticle dispersion relation. This dispersion relation is anomalously flat near the (\ensuremath{\pi},0) and (0,\ensuremath{\pi}) points of the Brillouin zone, similar to the results of recent angular resolved photoemission measurements of the hole-doped cuprates. We argue that the generic nature of the quasiparticle dispersion relation observed in these materials arises from the strong Coulomb interaction and reflects the hole-spin correlations rather than the one-electron interactions which customarily determine the band structure.
A detailed numerical study of the three-band Hubbard (3BH) model in the underdoped region is carried out, both in the hole- as well as in the electron-doped case, by means of the variational cluster approach. The phase diagram and the low-energy single-particle spectrum are found to be very similar to recent results for the single-band Hubbard model with next-nearest-neighbor hoppings t '. In particular, we obtain a mixed antiferromagnetic+superconducting phase at low doping with a first-order transition to a pure superconducting phase accompanied by phase separation. Despite the fact that the physically relevant and generally accepted parameters ( t pp / t pd , t pd /Δ pd ) of our 3BH-model are not in the small regime, required for a 'downward' projection onto a single-band model, in the single-particle spectrum a clear Zhang–Rice (ZR) singlet band (with a projection on d- and p-orbitals consistent with the phase factor of the ZR singlet wave function) can be seen. It displays an incoherent and a coherent part, in which holes enter upon doping around (π/2, π/2). The latter is very similar to the coherent quasi-particle band crossing the Fermi surface in the t – t '– U single-band model, with generally accepted values of the parameters. Doped electrons go instead into the upper Hubbard band, first filling the regions of the Brillouin zone around (π, 0). This fact can be related to the enhanced robustness of the antiferromagnetic phase as a function of electron doping compared to hole doping.
On the basis of exact diagonalizations, a comparative study of two-particle optical and magnetic, as well as single-particle, excitations is presented for a two-dimensional (2D) multiorbital Hubbard model. For reasonable parameter sets appropriate for the cuprate superconductors, the single-particle excitations display strongly correlated states related to the Zhang-Rice Cu-O singlet construction. These states define the gap (to the upper Hubbard band) at half-filling and become partially occupied by doping holes in our 2\ifmmode\times\else\texttimes\fi{}2 unit-cell system. The optical results, which are the first quantitative calculations performed for realistic parameters of the three-band Hubbard model, clearly show three allowed optical transitions: (i) itinerant motion of the Cu-O singlets, having (for doping concentrations x\ensuremath{\ne}0) a spectral Drude distribution around \ensuremath{\omega}=0 with spectral weight proportional to x; (ii) unbinding of the O hole from the Cu spin in the singlet. This gives, in particular, a strong absorption peak due to singlet\ensuremath{\rightarrow}nonbonding oxygen transitions, again with relative weight \ensuremath{\sim}x.It is roughly centered at \ensuremath{\omega}\ensuremath{\sim}JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​, where JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​ is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+Upd{\mathit{U}}_{\mathit{p}\mathit{d}}Upd​. They show a pronounced excitonic effect due to the p-d interaction Upd{\mathit{U}}_{\mathit{p}\mathit{d}}Upd​ and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​, which should be detectable in experiment.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.