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arXiv:cond-mat/9509179 · Europhysics Letters (EPL)

Theory for superconducting properties of the cuprates: doping dependence of the electronic excitations and shadow states

Slawomir Janusz Grabowski, Marcel Langer, Jörg Schmalian, Karl H. Bennemann

Published Apr 20, 1996• 44 citations
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Abstract

The superconducting phase of the 2D one-band Hubbard model is studied within the FLEX approximation and by using an Eliashberg theory. We investigate the doping dependence of T c , of the gap function Δ( k ,ω) and of the effective pairing interaction. Thus we find that T c becomes maximal for 13% doping. In overdoped systems T c decreases due to the weakening of the antiferromagnetic correlations, while in the underdoped systems due to the decreasing quasi-particle lifetimes. Furthermore, we find shadow states below T c which affect the electronic excitation spectrum and lead to fine structure in photoemission experiments.

DOI
10.1209/epl/i1996-00442-2

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The superconducting phase of the 2D one-band Hubbard model is studied within the FLEX approximation and by using an Eliashberg theory. We investigate the doping dependence of T c , of the gap function Δ( k ,ω) and of the effective pairing interaction. Thus we find that T c becomes maximal for 13% doping. In overdoped systems T c decreases due to the weakening of the antiferromagnetic correlations, while in the underdoped systems due to the decreasing quasi-particle lifetimes. Furthermore, we find shadow states below T c which affect the electronic excitation spectrum and lead to fine structure in photoemission experiments.
A consequence of strong antiferromagnetic correlations in models of high-Tc cuprates is the appearance in photoemission (PES) calculations of considerable more weight above the Fermi momentum pF{\rm {\bf p}_F }pF​ than expected for non-interacting electrons. This effect, first discussed by Kampf and Schrieffer (Phys. Rev. {\bf B 41}, 6399 (1990)) under the name of ``shadow bands'', is here analyzed in the two dimensional Hubbard and t-J models using Monte Carlo and exact diagonalization techniques. It is concluded that for underdoped compounds weight above pF{\rm {\bf p}_F }pF​ could be observable in experimental PES data, while in the overdoped regime it will be likely hidden in the experimental background. In the intermediate region the signal is weak. Our results are thus compatible with recent experimental data by Aebi et al. (Phys. Rev. Lett. {\bf 72}, 2757 (1994)) for Bi2212 at optimal doping. However, to definitely prove the existence of this effect in the cuprates, experiments in the underdoped regime are necessary.
The two-dimensional Hubbard model is studied within the composite operator method (COM) with the self-energy computed in the self-consistent Born approximation (SCBA). The COM describes interacting electrons in terms of the new elementary excitations that appear in the system owing to strong correlations; residual interactions among these excitations are treated within the SCBA. On decreasing the doping (from the overdoped to underdoped region), anomalous features develop in the spectral function $A(\mathbf{k},\ensuremath{\omega})$, the Fermi surface, the momentum distribution function n(k)n(\mathbf{k})n(k), the dispersion, and the density of states $N(\ensuremath{\omega})$ in the intermediate-coupling regime (U=8)(U=8)(U=8) at low temperatures (T=0.01−−0.02)(T=0.01--0.02)(T=0.01−−0.02). At high doping (n=0.7)(n=0.7)(n=0.7), the system resembles an ordinary weakly interacting metal. At low doping (n=0.92)(n=0.92)(n=0.92), a pseudogap opens, hot and cold spots appear, and non-Fermi-liquid features develop. This behavior, together with the presence of kinks in the calculated electronic dispersion, is in agreement with angle-resolved photoemission spectroscopy data for high-Tc{T}_{c}Tc​ cuprates superconductors.
We study the three-band Hubbard model for the cuprate superconductors using Quantum Monte Carlo simulations and the fluctuation exchange approximation including a self-consistent many-body renormalization of the single-particle propagator. The energy dispersion of the low-lying one-particle excitations is in very good agreement with spectroscopic measurements for a hole doping concentration δ = 0.25 of the CuO2 planes. At the same doping, we examine the spin susceptibility yielding a non-ordered phase with short-range incommensurate fluctuations near the antiferromagnetic wave vector. In the superconducting regime, the interaction vertex for the pairing correlation function in the extended s-wave channel shows a maximum for δ = 0.20, resembling the dependence of transition temperature Tc on doping.
Using a new method for the solution of the fluctuation exchange approximation equations, which allows the determination of the self-energy ${\ensuremath{\Sigma}}_{\mathrm{k}}(\ensuremath{\omega})$ of the 2D one-band Hubbard model on the real frequency axis, we calculate the doping dependence of the quasiparticle excitations of high-Tc{T}_{c}Tc​ superconductors. We obtain new results for the shadows of the Fermi surface, their dependence on the deformation of the quasiparticle dispersion, an anomalous $\ensuremath{\omega}$ dependence of $\mathrm{Im}{\ensuremath{\Sigma}}_{\mathrm{k}}(\ensuremath{\omega})$, and a related violation of the Luttinger theorem. This sheds new light on the influence of short range magnetic order on the low energy excitations and its significance for photoemission experiments.
In recent years there has been significant work aimed at understanding what effect the variation of electronic doping has on material properties. In the high transition-temperature (high-Tc_cc​) cuprate superconductors hole doping has an impact on the superconducting transition temperature. In the underdoped regime, the cuprates exhibit anomalous properties due to a pseudogap which forms and is thought to be related to Mott insulating physics. While there is no general consensus as to the mechanism underlying high temperature superconductivity, the resonating valence bond (RVB) theory proposed by Anderson in 1987 with a Gutzwiller projected d-wave BCS wave function could give a first picture of the high-Tc_cc​ cuprates. We have calculated properties of the cuprates using the assumption that the pseudogap state acts as a normal state to an otherwise standard BCS mean field theory. We find that the phenomenological RVB spin liquid model proposed by Yang, Rice and Zhang (YRZ) is highly successful at describing the doping dependent features of the cuprates. Through application of the YRZ model and the tools of many-body theory we present results on anomalous properties observed in: electronic specific heat; Raman and angle-resolved photoemission spectroscopy (ARPES) data; effective mass renormalization; and thermal broadening seen in ARPES. We verify that the YRZ ansatz qualitatively describes these anomalies along with their doping dependent variations. We conclude from this work that the physics underlying the pseudogap, while distinct in origin from superconductivity, is likely to arise from an RVB wavefunction that is closely related to the BCS state. In graphene, variation in doping modifies the polarization function which describes a screened electron-electron interaction. This leads to additional features in the spectral function which are due to electron-plasmon coupling. In this work, we calculated the electronic density of states including this interaction along with its doping dependence with and without an electron-phonon interaction. We find clear features of electron-electron interactions in the density of states. These features are related to the energies of plasmaron bands in the spectral function and can be modified through doping so as to be distinct from the phonon energy scales.
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.