Effective one–band electron–phonon Hamiltonian for nickel perovskites

Inspired by recent experiments on the Sr–doped nickelates, La2−xSrxNiO4, we propose a minimal microscopic model capable to describe the variety of the observed quasi–static charge/lattice modulations and the resulting magnetic and electronic–transport anomalies. Analyzing the motion of low–spin (s = 1/2) holes in a high–spin (S = 1) background as well as their their coupling to the in–plane oxygen phonon modes, we construct a sort of generalized Holstein t–J Hamiltonian for the NiO2 planes, which contains besides the rather complex “composite–hole” hopping part non–local spin–spin and hole– phonon interaction terms. PACS number(s): 71.27.+a, 71.38.+i, 75.10.Lp Typeset using REVTEX 1 Charge carrier doping of transition metal oxides with perovskite related structure induces remarkable phenomena, such as high–temperature superconductivity in cuprates (e.g., La2−xSrxCuO4), intrinsic incommensurate charge and spin ordering in non–metallic nickelates La2−xSrxNiO4 [LSNO(x)], metal–insulator transition and colossal magneto-resistance in Mn–oxides (e.g., La1−xCaxMnO3). All these phenomena are strongly concentration dependent and the experiments suggest the decisive role of interconnection between the spin– and charge correlations and the lattice and transport properties for their emergence [1,2]. In this paper we derive an effective Hamiltonian describing the interplay of charge–, spin– and lattice degrees of freedom in doped Ni–oxides. As revealed by recent neutron scattering of LSNO(x) [3], the stripe order of both charge and spin densities in general is found to be incommensurate in the low–density region (x < ∼ 0.3); commensurability is restricted to very special values of x, such as 1/3 and 1/2 [4]. The rich variety of charge and spin ordering accompanied by the transport anomalies in nickelates [4–6] deserves the attention not only by itself, but also with respect to understanding the superconducting state in isostructural cuprates. In fact, the incommensurate (stripe–like) spin–, charge– correlations and lattice structure modulations are observed also in the metallic cuprates but there they are of dynamical and very short–range character [1]. The parent compound of the LSNO(x) system is the antiferromagnetic (AF) insulator La2NiO4 with a Néel temperature TN ≈ 330 K and an in–plane exchange constant J ≈ 30 meV. The magnetic 3d Ni ions, having holes in 3dx2−y2 , 3d3z2−1 orbitals, are in the high–spin state (HSS) with S = 1 according to Hund’s rule. Doping this parent compound induces additional holes in the NiO2 plane, but, in contrast to the superconducting cuprates, the layered nickel oxides LSNO(x) become metallic only near x ≈ 1. An additional hole in the NiO2 plane quasi-localized at some Ni 2+ ion aligns its spin antiparallel to the S = 1 spin of the ion due to the strong effective on–site interaction originated by the crystal field effect (overcoming the Hund’s rule coupling) [7]. The resulting low–spin state (LSS) with total spin 1/2 is tightly bound to the moving charge carrier forming a so–called “Zhang–Rice doublet” [6], which is the counterpart to the usual Zhang– 2 Rice singlet [8] in the cuprates. Aiming at the construction of an one–band model, i.e., a sort of generalization of the t–J model for the NiO2 plane, we have to take into account the constraints put on the motion of the composite–hole LSS by the background of correlated HSS of Ni ions. The two configurations, corresponding to an extra hole trapped at one of the two nearest neighbor (NN) Ni ions of the bond 〈ij〉 (i, j label the sites of the square lattice built up by the Ni ions in the a–b plane), are connected by an effective transition constant determined by the second order effect of the intermediate configuration with the hole in the p–orbital of the central oxygen ion [9]. Assuming the orbital djx of the hole localized at the site j to be nearly the same as the orbital dx2−y2 of the Ni–ion coupled to this extra hole (see Table I; in the following the indices x, z stand for the orbitals dx2−y2 , d3z2−1, respectively), we shall take orbitals djx, djx to play equivalent roles in the hopping process (and likewise for analogical orbitals related to the site i). This assumption leads to two possible ways in which the two configurations (differing by the localization of the extra hole in the bond 〈ij〉) are connected (cf. Table I). Moreover, the hopping rate of the hole from j to i also depends on the spin states of both configurations leading to different prefactors in front of the effective transfer constant t (which is determined by the overlap of the d– and p–orbital functions ∝ tpd). According to Serber’s method [10] (generalizing the Dirac’s spin Hamiltonian), the transition matrix elements implying the effect of spin states associated with the initial and final configurations, are given by the matrix elements of the sum of the following operators H t = tQ j Q LSS i P I Q j Q i , (1) H t = −tQ j Q LSS i P 12 Q j Q i , (2) acting in the spin function space of two NN Ni ions and one extra hole. The identical permutation operator Ps I and the transposition operator Ps 12 , both acting on the spin variables with indices 1, 2, correspond to the direct–type and exchange–type hole transfers, respectively, distinguished in Table I. The operators QLSS k (QHSS k ) project the spin functions pertaining 3 to site k on the subspace of LSS (HSS). Consequently, the projection operators restrict the motion of the composite hole to the subspace of LSS (for the hole occupied sites) and HSS (for the hole unoccupied ones). The matrix elements of the transitions between configurations having the total spin projections MT = 1/2 and MT = 3/2 are given by the sum of matrices [Ht] = [H (d) t ] + [H t ] as follows [H t ] = |i0, j+〉 |i1, j−〉 |i1, j+〉 〈i+, j0| 〈i−, j1| 〈i+, j1| 