Correlation strength, gaps, and particle-hole asymmetry in high-${T}_{c}$ cuprates: A dynamical mean field study of the three-band copper-oxide model

Correlation strength, gaps, and particle-hole asymmetry in high- $T_{c}$ cuprates: A dynamical mean field study of the three-band copper-oxide model

Luca de’ Medici, Xin Wang, Massimo Capone, and Andrew J. Millis

Phys. Rev. B 80, 054501 – Published 4 August 2009

Abstract

The three-band model relevant to high-temperature copper-oxide superconductors is solved using single-site dynamical mean field theory and a tight-binding parametrization of the copper and oxygen bands. For a band filling of one hole per unit cell the metal/charge-transfer-insulator phase diagram is determined. The electron spectral function, optical conductivity, and quasiparticle mass enhancement are computed as functions of electron and hole doping for parameters such that at one hole per cell the paramagnetic phase is insulating and for parameters such that at one hole per unit cell the paramagnetic phase is metallic. The optical conductivity is computed using the Peierls phase approximation for the optical matrix elements. The calculation includes the physics of “Zhang-Rice singlets.” The effects of antiferromagnetism on the magnitude of the gap and the relation between correlation strength and doping-induced changes in state density are determined. Three-band and one-band models are compared. The two models are found to yield quantitatively consistent results for all energies less than about 4 eV including energies in the vicinity of the charge-transfer gap. Parameters on the insulating side of the metal/charge-transfer insulator phase boundary lead to gaps which are too large and near-gap conductivities which are too small relative to data. The results place the cuprates clearly in the intermediate correlation regime, on the paramagnetic metal side of the metal/charge-transfer insulator phase boundary.

Received 8 July 2009

DOI:https://doi.org/10.1103/PhysRevB.80.054501

Authors & Affiliations

Luca de’ Medici^1,2,3, Xin Wang⁴, Massimo Capone^5,3,6, and Andrew J. Millis⁴

¹Department of Physics and Center for Materials Theory, Rutgers–The State University of New Jersey, 136 Frelinghuysen Road, Piscataway, New Jersey 08854, USA
²Laboratoire de Physique des Solides, Université Paris-Sud–CNRS, UMR 8502, F-91405 Orsay Cedex, France
³Dipartimento di Fisica, Università di Roma “La Sapienza,” Piazzale Aldo Moro 2, I-00185 Rome, Italy
⁴Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA
⁵SMC Center, CNR–INFM, Piazzale Aldo Moro 2, I-00185 Rome, Italy
⁶ISC–CNR, Via dei Taurini 18, I-00185 Rome, Italy

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Vol. 80, Iss. 5 — 1 August 2009

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Figure 1
(Color online) Metal-insulator phase diagram in plane of interaction $U$ and $p − d$ level splitting $Δ$ for one hole per ${CuO}_{2}$ unit in paramagnetic phase. Dotted lines (blue online): phase boundaries from ED calculation at $T = 0$ ; solid lines (red online) indicate limit of stability of metallic phase from CT-QMC calculation at $T = 1 / 40 eV$ . In the region between the two lines metallic and insulating solutions coexist. Square (blue online), circle (black online), and triangle (red online): parameters studied by Refs. 10, 12, 13, respectively.Reuse & Permissions
Figure 2
(Color online) Electron spectral function per spin for paramagnetic three-band model (negative frequency: removal spectrum; positive frequency: addition spectrum) (dashed-dotted lines; magenta online) and projections onto $d$ (dashed lines; black online) and $p$ (dotted lines; blue online) states calculated with the ED solver. Parameters: $U = 9 eV$ and $Δ = 4 eV$ (upper panels) and $Δ = 2 eV$ (lower panels). Upper graphs: 0.15 hole doping ( $Δ = 4 eV$ : $ε_{d} = - 7.7 eV$ , $ε_{p} = - 3.7 eV$ ; $Δ = 2 eV$ , $ε_{d} = - 6.3 eV$ , $ε_{p} = - 4.3 eV$ ) middle graphs: undoped ( $Δ = 4 eV$ , $ε_{d} = - 8.8 eV$ $ε_{p} = - 4.8 eV$ and $Δ = 2 eV$ , $ε_{d} = - 7.9 eV$ $ε_{p} = - 5.9 eV$ ). Lower graphs: 0.15 electron doping ( $Δ = 4 eV$ , $ε_{d} = - 9.8 eV$ , $ε_{p} = - 5.8 eV$ ; $Δ = 2 eV$ ; $ε_{d} = - 9.6 eV$ , $ε_{p} = - 7.6 eV$ ).Reuse & Permissions
Figure 3
(Color online) Upper panel: comparison of CT-QMC and ED self-energies over a wide frequency range calculated for $U = 9 eV$ and $Δ = 4 eV \sim Δ_{c 1}$ at dopings indicated. Lower panel: doping dependence of $- \partial Σ / \partial ω ∣_{ω \to 0}$ estimated from lowest two Matsubara points obtained from $U = 9 eV$ ED calculations.Reuse & Permissions
Figure 4
(Color online) Doping dependence of velocity renormalization $v^{*} / v^{bare}$ and inverse of self-energy derivative $Z^{- 1} = 1 / (1 - \partial Σ / \partial ω)$ . Upper panel: $Δ = 2 eV$ ; lower panel $Δ = 4 eV$ .Reuse & Permissions
Figure 5
(Color online) Upper panel: optical conductivity in near-gap region calculated for antiferromagnetic phase of the three-band model from QMC calculation at $T = 0.1 eV$ , $U = 9 eV$ , carrier density of one hole per ${CuO}_{2}$ unit (undoped case) and $Δ$ values indicated. Parameters: $Δ = 4.5 eV$ , $ε_{d} = - 9.1 eV$ ; $Δ = 4 eV$ , $ε_{d} = - 8.8 eV$ ; $Δ = 2 eV$ , $ε_{d} = - 7.8 eV$ . Lower panel: doping dependence of paramagnetic conductivity for $Δ = 4 eV$ and $U = 9 eV$ from ED calculation (red dash-dotted lines: undoped; blue long dashed lines: $\pm 0.10$ doped; magenta short dashed lines: $\pm 0.18$ doped) along with antiferromagnetic conductivity in the undoped case from QMC calculation (black solid lines, parameters are the same as in the upper panel). ED parameters: 0.18 hole doping, $ε_{d} = - 7.7 eV$ ; 0.10 hole doping, $ε_{d} = - 7.9 eV$ ; undoped, $ε_{d} = - 8.8 eV$ ; 0.10 electron doping, $ε_{d} = - 9.6 eV$ ; 0.18 electron doping, $ε_{d} = - 9.9 eV$ .Reuse & Permissions
Figure 6
(Color online) Comparison between the three-band model and the one-band Hubbard model at dopings indicated. Only frequencies and energies relevant to one-band model are shown. The behavior of the three-band model at other frequencies and energies is similar to Fig. 2 and the lower panel of Fig. 5. Upper panel: spectral functions. Lower panel: optical conductivities. The one-band model is computed at nearest-neighbor hopping $t = 0.37 eV$ , $U_{eff} = 12 t = 4.44 eV$ , $T = 0.1 t = 0.037 eV$ using CT-QMC solver. Parameters: 0.10 hole doping: $ε_{d} = - 2.7 t = - 1.00 eV$ ; undoped: $ε_{d} = - 6 t = - 2.22 eV$ ; 0.10 electron doping: $ε_{d} = - 9.3 t = - 3.44 eV$ . The three-band model is computed at $U = 9 eV$ , $Δ = 2.5 eV$ , and $T = 0$ using ED solver. Parameters: 0.10 hole doping: $ε_{d} = - 6.8 eV$ ; undoped: $ε_{d} = - 8.4 eV$ ; 0.10 electron doping: $ε_{d} = - 9.5 eV$ .Reuse & Permissions

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