Se p 20 07 Two - electron bound state formation in the t − J − U model for exchange - coupled planes

An anisotropic t−J −U model Hamiltonian is used to model electron behaviour in quasi-2d materials in the dilute limit, and as a highly simplified representation of the weakly coupled CuO2 planes of the high-Tc cuprates we model the very poor out-ofplane conductivity via the complete suppression of interplanar hopping. However, we do include the very weak interplanar superexchange, and are thus considering a model of exchange-coupled planes. For an isotropic three-dimensional system in the dilute limit, we find that the formation of two-particle bound states requires Jc/t & 5.9. Also, it is known that Jc/t = 2 for a 2d square lattice. However, for our model of exchange-coupled planes any infinitesimal interplanar exchange (Jc = 0) is adequate to form bound states. Published as J. Phys.: Condens. Matter 19, 386216 (2007). PACS numbers: 71.10.-w,71.27.+a,74.20.-b Bound state formation in exchange-coupled planes 2 1. Motivation and Introduction The discovery of high-temperature superconductivity in copper-based transition metal oxides was made by Alex Müller and Georg Bednorz in 1986 (see [1] for more information). They found that the copper oxide compound (La,Ba)2CuO4 became a superconductor at Tc ≈ 30K, which was substantially higher than for other any other compounds known at that time. Presently, one can find transition temperatures at ambient pressure close to 138K for other cuprate-based systems. The mechanism for this novel behaviour is still a subject of spirited debate, but one idea that has been put forward repeatedly is the Heisenberg superexchange between neighbouring Cu sites, a biproduct of the strong repulsive electronic (Hubbard-type) correlations that are present on the transition metal sites. The structure of all of the cuprate materials is similar, in that CuO2 planes are stacked one on top of another to produce a quasi-2d crystallographic arrangement. This characterization of these materials is supported by their very poor interplanar conducting behaviour [2], at least in the weakly doped regime [3, 4]. The so-called c-axis puzzle has been studied extensively [5], and its relation to the superconductivity has been discussed [6]. Further, given the dependence of Tc on the number of CuO2 planes per unit cell [7], the possibility of an interplanar pairing mechanism cannot be ignored, and indeed previous work has shown [8, 9], within various approximations, that any interplanar interaction increases Tc. In this report we examine the two-electron problem (viz. the dilute limit) in the highly simplified situation of zero interplanar hopping. That is, our model is meant to be a very rough approximation to the extremely low out-of-plane conductivity, but leaving the residual interplanar superexchange found in the cuprates, the latter produced by strong electronic correlations. We refer to this situation as exchange-coupled planes. We solve for the conditions under which a two-particle bound state is formed, and in particular determine the minimum value of superexchange (Jc/t) for which bound states appear. Our results make evident the potential benefit of having electrons confined to individual planes that can only move via the spin-exchange ( 2 S i S − j ) process. To be specific, we find that while for a two-dimensional plane (Jc/t = 2) or an isotropic threedimensional system (Jc/t ≈ 5.9) one requires a superexchange much larger than that found experimentally (J/t ∼ 0.3), for exchange-coupled planes one requires only an infinitesimal (Jc = 0) interaction. Therefore, even the very small out-of-plane exchange coupling (estimated to be roughly 10 of the in-plane exchange) would suffice to form bound states. We note that similar results were found previously for exchange-coupled chains [10], emphasizing the potential importance of such electronic confinement. Bound state formation in exchange-coupled planes 3