First-Principles Periodic Calculation of Four-Body Spin Terms in High-Tc Cuprate Superconductors

in High-Tc Cuprate Superconductors Ibério de P. R. Moreira, Carmen J. Calzado, Jean-Paul Malrieu, and Francesc Illas Departament de Quı́mica Fı́sica and Centre Especial de Recerca en Quı́mica Teòrica, Universitat de Barcelona and Parc Cientı́fic de Barcelona, C/ Martı́ i Franquès 1, E-08028 Barcelona, Spain Departamento de Quı́mica Fı́sica, Universidad de Sevilla, C/ Prof. Garcı́a González s/n, E-41012 Sevilla, Spain IRSAMC, Laboratoire de Physique Quantique, Université Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse-Cedex, France (Received 18 October 2005; published 25 August 2006)

The strong antiferromagnetic interactions observed in lamellar cuprates are fundamental ingredients of the high-T c (HTC) superconductivity microscopic mechanism [1,2]. Magnetic interactions arise from the particular crystal and electronic structure of these cuprates with Cu 2 ions arranged in edge sharing Cu 4 O 4 plaquettes. The electronic ground state involves a single d x2ÿy2 -type hole in each Cu3d shell leading to a network of effective spin S 1=2 particles. Nevertheless, these systems are strongly correlated in nature, making standard band theory techniques unable to accurately describe either their valence or low energy spectrum [3,4].
The low energy spectrum and collective properties of these compounds are assumed to be governed by a Heisenberg Hamiltonian as in the first term of Eq. (1) accounting for the magnetic coupling J ij between nearestneighbor (NN) centers i and j only. This is in agreement with the widely accepted general picture for HTC superconductivity involving a ''Heisenberg sea'' where holes are introduced by doping the perfect structures. However, to fully understand the magnetic excitations and the infrared and neutron scattering spectra of 2D [5][6][7][8][9][10][11] and spin ladder cuprates [12,13], it has been necessary to extend the spin Hamiltonian as in Eq. (1), with other two-body exchange constants J ij [hereafter referred as J for NN interactions in square lattices and J d for next-nearest-neighbor (NNN) interactions], four-body cyclic terms J ring , and higher order terms. The largest coupling is expected to be precisely J as found for almost all existing HTC superconductor cuprates parent com-pounds. Nevertheless, NNN interactions may be nonnegligible or even of the same order of magnitude as in the case of the S 1=2 CuGeO 3 system [14,15]. Likewise, J ring terms may be important in Cu 4 O 4 plaquettes since they are originated by the electrons cyclic circulation around the ring with a clear analogy to conjugated systems [16]. Similar four-body terms are crucial to describe the ground state properties of 3 He [17].
Although there is a general agreement about the role of J ring in determining the properties of cuprates [9][10][11][12], only very recently Toader et al. [18] provided definitive evidence of its existence in La 2 CuO 4 . They suggest J ring 0:5J, comparable to the pairing energies, and propose that the resulting circulating currents could have an important role in the mechanism of superconductivity. Notwithstanding, previous estimates of J ring for 2D and spin ladder cuprates, obtained from either indirect measurements or numerical simulations with an extended Heisenberg model, propose substantially smaller amplitudes with J ring 0:30J [7,9,10,12,19]. The J ring terms as evaluated by Toader et al. rely on NN and NNN coupling constants of J 111:8 meV and J d ÿ11:4 meV, respectively, extracted from one of the various fittings of the magnon spectrum [10]. However, it is important to point out that (i) this value for J is smaller than another experimental estimate of 135 6 meV obtained with a nearest-neighbor Heisenberg Hamiltonian [20] and (ii) the ferromagnetic nature of J d contradicts theoretical predictions [21] and measurements on materials with similar exchange paths [22]. For the SrCu 2 O 3 ladder compound the recent Raman response experiments by Schmidt et al. propose J r 140 meV, J ring =J r 0:2, and J l =J r 1:5 [23].
Clearly, a bottom-up accurate and independent determination of all important terms in the spin Hamiltonian in Eq. (1) for La 2 CuO 4 -or any other similar system-is highly desirable. For various cuprates, unrestricted Hartree-Fock (HF) cluster model predictions for J are in agreement with those resulting from equivalent periodic calculations, thus validating the finite models [24]. Subsequent configuration interaction (CI) calculations on embedded cluster models lead to results which are in very good agreement with available experimental data [25]. Cluster models were also used to extract J d and J ring in 2D and spin ladder cuprates [26 -28]. For the J ring =J ratio, CI calculations predict values in the 0.11-0.22 range. However, the embedded cluster derived J ring values are difficult to validate since (i) convergence check with respect to the cluster model size requires exceedingly large calculations and (ii) a general procedure to validate the cluster results for J ring from periodic calculations is still lacking. Martin and Illas [29] suggested that hybrid density functional (DF) theory could provide a convenient framework for the description of the local magnetic interaction in systems with localized spins. This can be equally applicable to periodic or cluster models, and a suitable choice of the hybrid exchange-correlation potential permits one to reproduce the experimental J values [4,30]. Up to date, this procedure has been restricted to the determination of twobody terms (J and J d ) only. In this Letter, we present a general procedure which enables the simultaneous and independent calculation of essentially all parameters defining the spin Hamiltonian of Eq. (1) from periodic DF calculations.
The generalization comes from a careful analysis of the procedure used to extract J in cluster and periodic models [24,31]. For a system with two localized S 1=2 spins, the Heisenberg Hamiltonian [Eq. (1)] has two spin eigenfunctions-a singlet (S) and a triplet (T)-with energies ÿJ and 0, respectively [32], and hence ES ÿ ET ÿJ. Using pure spin electronic ab initio CI wave functions to describe the corresponding electronic states of such a system permits one to establish a direct one-to-one mapping between the energies of the S and T electronic states and to compute J from this energy difference. However, in a band theory framework one is constrained to write the wave function or the electron density by means of a single Slater determinant, and this mapping cannot be used. A way to circumvent the problem is to rely on the expectation energy for the single Slater determinant ferromagnetic (jFMi) and antiferromagnetic (jAFMi) broken symmetry solutions. For the simple case above, one has jFMi j""i and jAFMi j"#i or j#"i. For these ''magnetic phases'' one can easily show that the expectation energies of the Heisenberg Hamiltonian are hFMjHjFMi 0 and hAFMjHjAFMi ÿJ=2, but hFMjHjFMi and hAFMjHjAFMi are precisely EAFM and EFM and hence EAFM ÿ EFM ÿJ=2. The same solution is obtained by making use of the Ising spin Hamiltonian [32], although one has to assume that the system is equally well described by a Heisenberg or a Ising model Hamiltonian.
The use of expectation values for the FM and various AFM broken symmetry solutions is the clue to extract the various parameters of the model Hamiltonian in Eq. (1) using a periodic approach. In fact, in a periodic system one can find self-consistent field (SCF) solutions-either for HF or DF potentials -representing jFMi j""" "i and various jAFMii j"# #"#i phases with different spin alignments along the relevant dimensions. The resulting SCF solutions can be related to a localized valence bond (VB) picture where nondynamical correlation effects are introduced partly through spin polarization [33,34]. A quasidegenerate perturbation theory expansion of the matrix representation of a Hubbard-type Hamiltonian in the neutral VB basis up to fourth order permits one to extract the leading interactions -J, J d , and J ring -involved in each neutral VB localized spin setting. Using cluster models for La 2 CuO 4 and SrCu 2 O 3 , Calzado and Malrieu [28] have shown that, apart from J, the dominant terms of Eq. (1) correspond precisely to J d and J ring . The latter appears at fourth order and implies a cyclic circulation of the electrons around a plaquette; notice that in a plaquette with all spins parallel (hS z 2i) such a current is impossible. However, for spin settings on the plaquette with hS z i 0; 1 these currents lead to additional off-diagonal and diagonal corrections (see Tables III-VIII in Ref. [28]). This is the kernel of the general mapping procedure proposed here to extract these terms from periodic calculations. This permits one to obtain the terms in Eq. (1) simultaneously, avoiding any possible bias due to the choice of a too limited material model.
The calculated energy of several magnetic solutions (either per formula unit or per Cu atom) can be mapped onto the diagonal part of the matrix representation of Eq. (1) in a localized VB basis set [28]. To obtain J in La 2 CuO 4 or Sr 2 CuO 2 Cl 2 , two magnetic solutions (FM and AFM) are enough (see Ref. [24]). However, several magnetic phases are needed to extract the relevant J d and J ring terms. Technically, the main difficulty is to obtain a sufficient number of magnetic SCF solutions so as to construct a large enough set of equations that permits one to extract the numeric values of the relevant terms from appropriate energy differences. The complete set of magnetic solutions together with the full justification and development of the set of equations necessary to carry out the mapping procedure exceeds the length of this Letter and will be reported elsewhere [35]. Relevant magnetic solutions and the corresponding equations for La 2 CuO 4 and Sr 2 CuO 2 Cl 2 are given in Table I and Fig. 1. For the SrCu 2 O 3 ladder compound ten magnetic solutions have been considered [35]; relative energies are in  [4]. Crystalline orbitals are built as linear combinations of Bloch functions, which in turn are built from atomic orbitals (AOs) optimized for the crystal environment. The AOs are contracted real solid spherical harmonic Gaussian-type functions. Extended allelectron basis sets have been used to describe the Cu 2 and O 2ÿ ions in the ionic environment, whereas core pseudopotentials have been used to represent inner electrons of the remaining ions [36]. Strict convergence criteria and a set of 105 points in the irreducible Brillouin zone have been used to ensure a numerical accuracy of 10 ÿ7 hartree per formula unit. Calculations have been carried out using the CRYSTAL03 code [37] and experimental cell parameters.
The amplitudes of J, J d , and J ring estimated from the energy differences corresponding to equations given in Tables I and II are reported in Table III (see also Fig. 1). For the La 2 CuO 4 and Sr 2 CuCl 2 O 2 2D systems, the set of equations is determined and the values in Table III are univocal. For the SrCu 2 O 3 ladder, a least square procedure has been used to solve the overdetermined set of equations. For the 2D systems, the J values predicted by the Fock-35 potential are close to experiment, as expected from previous work on similar systems [24,28,34], thus further confirming the accuracy of the present approach. Next, we come to the most important result of the present work: the first direct estimate of the amplitude of the J ring terms from ab initio periodic calculations. For La 2 CuO 4 , the outcome of the present periodic approach is in agreement with the only available theoretical results for this compound, although arising from cluster calculations [26 -28]. Second, the present estimate of  [19]. Notice, however, that the J ring =J 0:25 ratio predicted here is smaller than that reported by Toader et al. [18], which is of 0.5, but in excellent agreement with a more generally accepted ratio J ring =J 0:3 [9,10,19] for this compound. Also, the Fock-35 method predicts an antiferromagnetic J d 8:8 meV, again consistent with the values provided by embedded cluster calculations. Hence, it will be of great interest to repeat the fit in the experiments by Toader et al. [18] by using the present estimate of both J and J d . For Sr 2 CuO 2 Cl 2 , there are no previous values for the amplitude of the four-body exchange terms. Present calculations predict values that are slightly smaller than the corresponding ones in La 2 CuO 4 consistent with larger NN and NNN distances in the latter (3.809 vs 3.972 Å for NN and 5:38 vs 5.62 Å for NNN distances). There exists a good agreement between the estimate of the NN interaction (J 130 meV) and the experimental value (J 125 6 meV [38]); the J d coupling is also predicted to be antiferromagnetic and the J ring =J ratio 0:20 is somewhat smaller than that for La 2 CuO 4 but still consistent with previous estimates.
Finally, for the ladder compound the dominant interactions are the couplings across the rungs (J r ) and along the legs (J l ) which are nearly equal in magnitude, i.e., J r =J l 1 as expected from the similarity of the Cu-O-Cu exchange paths. On the other hand, J d is antiferromag-   netic as in the 2D cuprates and a non-negligible interladder ferromagnetic exchange is found, J i ÿ0:22J l . Hence, one should not consider these as noninteracting ladders.
Regarding J ring , both its amplitude and the J ring =J r ratio are larger than for 2D cuprates, but still close to 0.3 in agreement with various experiments [9][10][11][12]. In particular, calculated J r and J ring =J r are close to those reported by Schmidt et al. extracted from Raman response experiments [23], although they ignore J i and, to justify some differences between ladder and 2D magnetic systems, they propose J l =J r 1:5. The present results do not support such large anisotropy between J l and J r . Interestingly enough, we found that neglecting J ring results in a larger anisotropy in the fitting. These results strongly suggest revising previous fittings using the present estimates as starting points.
To summarize, the present work proposes a new, general, and unprejudiced scheme to predict the amplitude of the parameters defining a general spin Hamiltonian from DF periodic calculations. The procedure does not need to make assumptions on the relative amplitude of these terms. Instead, it relies on a mapping approach between the energy of pertinent magnetic solutions and the diagonal terms of the spin Hamiltonian in a local representation. Then, the unique possible bias arises from the quality of the exchange-correlation potential. In particular, the present work provides an independent confirmation of the importance of four-body terms in La 2 CuO 4 recently evidenced from the experimental work of Toader et al. [18] and supplies reliable values for spin Hamiltonian parameters of other key compounds. It is predicted that the importance of four-body terms is likely to be similar for most of the HTC related cuprates and even more important in ladder compounds in agreement with the recent experiments of Schmidt et al. [23]. It is also suggested that the fit to neutron scattering data should be revised by considering alternative values for both J and J d magnetic coupling terms. Finally, the present study adds further evidence that the ferromagnetic interladder exchange introducing spin frustration between legs and the four-body term in the