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arXiv:cond-mat/9508096 · Physical review. B, Condensed matter

Analysis of optical conductivities in cuprates

Hae‐Young Kee, Jongbae Hong

Published Mar 1, 1997• 14 citations
arXivHTMLPDF

Abstract

Analysis of optical conductivities measured for cuprates has been done via the doped Hubbard model in infinite dimensions using the analytical variant of the dynamic Lanczos method. We examine the frequency behavior in the low-frequency region at comparatively large doping, the midinfrared band, and the isobestic point in the optical conductivity of cuprates. We demonstrate that these features shown in infinite dimensions are the generic features of strong correlation.

DOI
10.1103/physrevb.55.5670

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Analysis of optical conductivities measured for cuprates has been done via the doped Hubbard model in infinite dimensions using the analytical variant of the dynamic Lanczos method. We examine the frequency behavior in the low-frequency region at comparatively large doping, the midinfrared band, and the isobestic point in the optical conductivity of cuprates. We demonstrate that these features shown in infinite dimensions are the generic features of strong correlation.
A Monte Carlo maximum-entropy calculation of the optical conductivity of the infinite-dimensional Hubbard model is presented. We show that the optical conductivity displays the anomalies found in the cuprate superconductors, including a Drude width which grows linearly with temperature, a Drude weight which grows linearly with doping, and a temperature and doping-dependent mid-IR peak. These anomalies arise as a consequence of the dynamical generation of a quasiparticle band at the Fermi energy as T\ensuremath{\rightarrow}0, and are a generic property of the strongly correlated Hubbard model in all dimensions greater than one.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the dc resistivity $\ensuremath{\rho}(T)$ within the extended $t\text{\ensuremath{-}}J$ model on a square lattice, as relevant to high-Tc{T}_{c}Tc​ cuprates, are reinvestigated using the exact-diagonalization method for small systems, improved by performing a twisted boundary condition averaging. The influence of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ is also considered. The behavior of results at intermediate doping is consistent with a marginal-Fermi-liquid scenario and in the case of ${t}^{\ensuremath{'}}=0$ for $\ensuremath{\omega}>T$ follows the power law $\ensuremath{\sigma}\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\sim}0.65$ consistent with experiments. At low doping ch<0.1{c}_{h}<0.1ch​<0.1 for T<JT<JT<J $\ensuremath{\sigma}(\ensuremath{\omega})$ develops a shoulder at $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}^{*}$, consistent with the observed midinfrared peak in experiments, accompanied by a shallow dip for $\ensuremath{\omega}<{\ensuremath{\omega}}^{*}$. This region is characterized by the resistivity saturation, whereas a more coherent transport appears at T<T∗T<{T}^{*}T<T∗ producing a more pronounced decrease in $\ensuremath{\rho}(T)$. The behavior of the normalized resistivity ${c}_{h}\ensuremath{\rho}(T)$ is within a factor of 2 quantitatively consistent with experiments in cuprates.
We calculate the optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of the Hubbard and t-J models on 10- and 16-site lattices, respectively, using a Lanczos method. Results are presented for various values of the couplings U/t, J/t, and band fillings, using free- and periodic-boundary conditions. We discuss the behavior of the kinetic energy in the ground state, the distribution of spectral weight of \ensuremath{\sigma}(\ensuremath{\omega}) between low and high energies and the sum rule. Indications of a Drude-like peak at low energy in the one-hole subspace are observed in both models. Assuming that holes behave like independent (mass-renormalized) particles in the normal state of the superconductors, our results also suggest that the optical midinfrared broadband observed experimentally can be explained by hole excitations of strongly correlated systems.
The optical conductivity for the Hubbard and t-J models is calculated numerically by use of a Lanczos algorithm. One- and two-dimensional systems of up to 16 sites with periodic and open boundary conditions are examined for the t-J model, as well as Hubbard-model systems of up to 9 sites. In the one-dimensional case, the low-frequency part of the conductivity away from half filling is shown to consist almost entirely of a single \ensuremath{\delta} function (Drude peak) even for J of the order of t. This is a direct consequence of the decoupling of charge and spin in one dimension. In two dimensions there is an additional broad absorption band above the Drude peak, consistent with the experimentally observed midinfrared band in the copper oxide superconductors. The importance of vertex corrections is demonstrated by comparing the exact conductivity with the convolution integral of the exact single-particle Green's function.
We study the Cu and O spectral density of states and the optical conductivity of CuO2{}_{2}2​ planes using an effective generalized one-band Hubbard model derived from the extended three-band Hubbard model. We solve exactly a square cluster of 10 unit cells and average the results over all possible boundary conditions, which leads to smooth functions of frequency. Upon doping, the Fermi energy jumps to Zhang-Rice states which are connected to the rest of the valence band (in contrast to an isolated band in the middle of the gap). The transfer of spectral weight depends on the parameters of the original three-band model not only through the one-band effective parameters but also through the relevant matrix elements. We discuss the evolution of the gap upon doping. The optical conductivity of the doped system shows a mid-infrared peak due to intraband transitions, a pseudogap, and a high-frequency part related to interband transitions. Its shape and integrated weight up to a given frequency (including the Drude weight) agree qualitatively with experiments in the cuprates for low to moderate doping levels, but significant deviations exist for doping x>0.3x>0.3x>0.3.
The optical conductivity (OC) of cuprates is studied theoretically in the low-density limit of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model taking into account the hole-lattice coupling. By developing a limited phonon basis exact diagonalization method capable of treating the lattice of largest size ever considered $(4\ifmmode\times\else\texttimes\fi{}4)$, we are able to discern the fine features of the mid-infrared (MIR) part of the OC revealing a three-peak structure. The two lowest peaks are observed in experiments and the highest one is tacitly resolved in moderately doped cuprates. Comparison of OC with the results of semianalytic approaches and detailed analysis of the calculated isotope effect indicate that the middle-energy MIR peak is of mostly magnetic origin while the lowest MIR band originates from the scattering of holes by phonons.