Ab initio systematic determination of the t–J effective Hamiltonian parameters for superconducting Cu-oxides

25 June 1999

Chemical Physics Letters 307 Ž1999. 102–108

Ab initio systematic determination of the t–J effective Hamiltonian parameters for superconducting Cu-oxides Carmen J. Calzado a , Javier Fdez. Sanz

a,)

, Jean-Paul Malrieu b, Francesc Illas

c

a

c

Departamento de Quımica Fısica, Facultad de Quımica, UniÕersidad de SeÕilla, E-41012 SeÕilla, Spain ´ ´ ´ b IRSAMC, UniÕersite´ P. Sabatier, 118 route Narbonne, F-31062 Toulouse Cedex, France Departamento de Quımica Fısica, UniÕersidad de Barcelona, C r Martı´ i Franques ´ ´ ´ 1, E-08028 Barcelona, Spain Received 2 February 1999; in final form 19 April 1999

Abstract A method for variationally computing the electron transfer integral t and the magnetic coupling constant J within a unified theoretical framework is reported. The method is used to estimate these quantities for the high-Tc La 2 CuO4 superconductor allowing for an ab initio determination of the Zhang and Rice t–J single-band effective Hamiltonian. It is shown that by using a suitable configuration interaction ŽCI. space, the magnetic coupling can be rationally and quantitatively determined. On the other hand, the analysis of the final CI eigenvectors shows that hole doping involves states in which the hole has a large oxygen contribution in agreement with experiment. q 1999 Elsevier Science B.V. All rights reserved.

Since the discovery of high-Tc Cu-oxide superconductors, the electronic structure and magnetic properties of these materials have been the subject of considerable theoretical and experimental work. As is well-known, La 2 CuO4 perovskites are anti-ferromagnetic ŽAF. insulators. However, by doping appropriately with holes or electrons, this solid becomes superconducting. The first attempt to unravel the key aspects of the electronic structure of these materials was that of Anderson w1x, who assumed that a single-band Hubbard Hamiltonian containing the strong on-site Coulomb interaction among a partially filled band of Cu 3d levels would essentially account for the physical problem. However, the spectroscopic data Žfor recent reviews, see e.g. Refs. w2,3x. revealed that in an electron-deficient material, like La 2y x Sr x CuO4 , the nature of the holes has a large oxygen contribution, without the formation of Cu3q ions. This led Emery w4x to include the oxygen 2p x, y orbitals in a three-band Hubbard Hamiltonian which has been found to successfully interpret experimental data such as XPS and EELS spectra w5,6x. Because of the relative complexity of such a three-band Žor more. model Hamiltonian, several attempts intended to recover the single-band model have been reported. In particular, starting from a two-band model, Zhang and Rice w7x developed an effective single-band Hamiltonian which implicitly accounted for the p–d hybridization. This so-called t–J model Hamiltonian strongly binds a hole on

)

Corresponding author. E-mail: [email protected]

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 5 0 6 - 0

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each square of O atoms to the central Cu2q ion to form a local singlet which then moves through the lattice of Cu centers as a hole. The applicability limits of such a type of single-band effective Hamiltonians Ž t–J or further refinements as the so-called w8x t–tX –J . has been the subject of some controversy. Thus, Emery and Reiter w9x reported that in some cases the single- and three-band models exhibit different behavior. On the other hand, it has been shown that t–J wavefunctions correspond to projections of the ‘exact’ three-band ones larger than 90% w10x. Despite this controversy, it seems well established that the t–J model constitutes an excellent basis for the wave functions of the low-energy states. The central underlying problem in the use of model Hamiltonians deals with the determination of parameters. In this sense, the main benefit of the t–J model is that it only involves two of them H s J Ý Ž Si S j y 14 n i n j . y t

Ý

cq i s Ž 1 y n iy s . Ž 1 y n jy s . c j s q h.c. ,

Ž 1.

² ij :s

² ij :

where the parameters t and J correspond to the electron transfer integral and the magnetic coupling constant between neighboring copper centers, respectively. The value of J can be obtained from experiment and, thus, for the prototypical cuprate La 2 CuO4 , it was found to be y128 meV from Raman measurements w11x and y134 meV from neutron scattering w12x. This is not the case for the hopping integral t, whose value is inferred by fitting a model to experimental data. For doped La 2 CuO4 , the accepted value is t f 0.5 eV w13x. The theoretical estimation of t and J faces two main difficulties. The first concerns the necessity of employing highly reliable wavefunctions in order to incorporate the strong interaction between Cu centers, as well as orbital relaxation and polarization effects. The second refers to the smallness of these quantities, usually of the order of a few hundred meV, which confers a large uncertainty on the estimates. In this Letter, we report on a unified computational strategy which allows for an accurate and systematic determination of t and J based on high level ab initio configuration interaction ŽCI. calculations. The procedure makes use of the quasi-degenerate perturbation theory ŽQDPT; for a monograph on effective Hamiltonians, see Ref. w14x. to select a list of configurations contributing to the second-order energy corrections which is subsequently diagonalized. The method is used to determine t and J for La 2 CuO4 material from energy differences within a two-state model. We will show how, using a Cu 2 cluster model, it is possible to obtain values for such a stringent and elusive quantity as J, which practically matches the experimental value. Let us start with the calculation of the magnetic coupling constant J between nearest-neighbor atoms appearing in the Heisenberg Hamiltonian H s yJ ݲ i j: Si P S j . For a two-center system, i.e. a cluster containing two Cu2q ions Ž S1 s S2 s 1r2., the different spin arrangements gives rise to a singlet and a triplet, and the magnetic coupling is given by the energy gap between these states: J s ES y E T s D EST . This is true whatever the degree of sophistication involved in the calculation of these states. At a zeroth-order level, these two states may be represented by single configurations:

c S0 s

1

'2


c T0 s

1

'2


Ž 2.

which in turn can be built up from appropriate combinations of two neutral determinants:

c 10 s
c 20 s
where a and b are the localized magnetic orbitals and essentially correspond to the Cu d x 2yy 2 orbitals.

Ž 3.

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Doping with a hole, and according to the basic grounds of the t–J model, gives rise to a localized singlet state which moves across the lattice, as does the hole between the Cu centers. Therefore, an equivalent description of the process consists of an electron transfer between Cu2q and Cu3q ions:

In the present case, the hopping integral t can be estimated using a two-state framework, since it represents the electronic coupling between the diabatic states corresponding to those having the hole localized on the right or on the left. Using the above defined magnetic orbitals, the zeroth-order diabatic wave functions are:

ca0 s
c b0 s
Ž 4.

0 and the transfer integral is then given by: t f Hab s ² ca0 < H < c b0 :. Alternatively, if the diabatic states are orthogonal, the transfer integral corresponds to half the difference between the adiabatic energies at the crossing seam. The calculation of both t and J can be carried out in the framework of the QDPT which allows one to bring partial exact information into a limited model space S0 . The wave functions given in Eqs. Ž3. and Ž4. span two model spaces, S0J and S0t , on which an effective 2 = 2 Hamiltonian is defined on each, and whose off-diagonal element can be related to t and J. Thus, to the second order, we have w15,16x:

1 2

eff Ž2. 0 J f H12 s H12 qÝ

K

t f Haeffb Ž2. s Ha0b q Ý L

² c 10 < H < K 0 :² K 0 < H < c 20 :

´ 0 y ´K

² ca0 < H < L0 :² L0 < H < c b0 :

´ 0 y ´L

,

,

Ž 5. Ž 6.

where  K 0 4 and  L0 4 are the sets of excited determinants simultaneously interacting in each case with the zeroth-order model functions. Including the second-order correction improves the estimates, however there still are inaccuracies, due to the smallness of the denominators. To circumvent this problem, instead of the second-order expansion, a diagonalization of the CI matrices restricted to the w SoJ q  K 0 4x and w S0t q  L0 4x spaces is performed that, on the other hand, introduces higher-order corrections. This procedure provides a rational guide for accurately computing energy differences through a difference dedicated CI ŽDDCI.. J is simply the energy difference between the two lowest roots Ži.e., D EST . w15x, and for the particular case in which the localized states ca and c b are mirror images, as in the present case, t is given by half the energy difference Eqy Ey w16x. Before continuing, two main points concerning the above sets of DDCI spaces will be addressed. The first deals with the nature of the excitations included in these spaces, which are, of course, much less numerous than those interacting with the model functions. For t, they correspond to excitations involving core, active Ž a and b . and virtual molecular orbitals ŽMOs. up to three degrees of freedom ŽDDCI-3., while for J the determinants entering the list, whose nature has been analyzed by De Loth et al. w17x, only involve excitations up to two degrees of freedom: DDCI-2. The second aspect to highlight is that these spaces are invariant under a unitary transformation of the MOs, and therefore, the CI result does not depend on using localized or delocalized unitarily-transformed MOs. This invariance leads to a significant reduction of the CI spaces since the model spaces can then be defined on the basis of delocalized states built up from delocalized active orbitals obtained by a rotation of the localized ones g s Ž a q b .r '2 and u s Ž a y b .r '2 , allowing for a convenient

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exploitation of the higher symmetry ŽD 2h instead of C 2v .. Hence, the zeroth-order states of Eqs. Ž2. and Ž4. may be written as:

c S0 s

1

< < < < '2 Ž core gg y core uu . ,

c T0 s

1

< < < < '2 Ž core gu y core ug .

Ž 7.

and

cg0 s
cu0 s
Ž 8.

Having stated the methodological approach, we will now report on the practical aspects. Taking into account the local character of both processes, two model clusters have been selected: Cu 2 O 7 and Cu 2 O 11 ŽFig. 1.. These clusters are embedded in an array of formal point charges to incorporate the Madelung electrostatic potential. However, in order to avoid an artificial polarization of cluster electrons toward the nearest positive charges, all Cu and La ions directly bound to cluster oxygens are replaced with effective total ion potentials ŽTIPs. represented through large core Hay–Wadt effective core potentials ŽECPs. w18x with formal residual charges. For Cu cluster atoms, the Stevens, Basch and Kraus ECPs are used w19x to describe the inner electrons while for the valence, the basis set was Ž8s8p6d. contracted to w4s4p3dx w20x. Because of the key role played by the bridging oxygen on both J w21x and t w22x, it is described through an all electron basis set including a polarization function, the contraction being Ž10s5p1d.rw3s2p1dx w23x. For the rest of cluster oxygen atoms, and in order to keep a reasonable size of the MO set, a Ž6s6p. contracted to w2s2px basis set was used for the valence electrons, while the core 1s ones were described through an ECP w24x. In order to start the computational procedure, we first need to choose the MO basis set to span the zeroth-order wave functions of Eqs. Ž7. and Ž8.. These orbitals are univocally determined from a restricted open-shell Hartree–Fock ŽROHF. calculation of the triplet state for the undoped system. This variational

Fig. 1. Cu 2 O 7 and Cu 2 O11 clusters used to represent La 2 CuO4 .

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calculation will introduce a certain degree of mixing between the d copper and p oxygen orbitals, and in fact, the d character of the magnetic orbitals is 93 and 90%, respectively. The results obtained for both the electronic and magnetic coupling constants are summarized in Table 1. Starting with the DDCI-2 estimates for J, one can see that cluster Cu 2 O 7 gives y96 meV, in agreement with previous findings w25x. Increasing the cluster to Cu 2 O 11 produces only a small improvement Žy99 meV., which in any case does not justify the computational effort involved. This result shows the two-dimensional nature of the interaction and that the main effects are played by the CuO 2 planes. Comparing these values with the experiment, it turns out that they are still higher than the experiment by ; 30%. In order to improve the result, and having ruled out this discrepancy being be attributed to the cluster size w26x, we have to look for a better incorporation of the electron correlation effects. Actually, the computations performed by Van Oosten et al. w27x using a non-orthogonal CI ŽNOCI. and similar clusters lead to a value of J quite close to the experimental value Žy120 meV.. The superexchange mechanism proposed by Anderson to explain the AF coupling comes from the mixing of the valence-bond, VB, neutral singlet configuration c S0 with its ionic counterpart c S1 s Ž1r '2 .
c S0X s l
ab q ba

'2

q Ž lym.

aa q bb

'2

.

A priori, a 2-electronr2-active-orbital CI incorporates this mechanism. However, other processes play a role, for instance the dynamical spin-polarization and the instantaneous repolarization of the ionic components. All these contributions are incorporated in the DDCI-2 list since it involves all determinants having at most 2 inactive holes or 2 inactive particles, or 1 inactive hole and 1 inactive particle. This is in general sufficient when the coupling constant is weak Ž; 100 cmy1 . and the weight of the ionic VB structures is low enough. However, in the La 2 CuO4 lattice, the coupling is much larger Ž) 1000 cmy1 . and, therefore, one may be tempted to include the ionic VB determinants c 30 s
Table 1 Values ŽmeV. for the antiferromagnetic coupling constant J and the transfer integral t computed from ab initio DDCI calculations Magnetic coupling, J

Transfer integral, t

DDCI-2

DDCI-3

DDCI-3

IDDCI-3

Cu 2 O 7ny

y96 Ž7 687.

y138 Ž411 091.

553 Ž255 520.

575 Ž255 520.

my Cu 2 O11

y99 Ž11 779.

y139 Ž804 835.

532 Ž526 767.

544 Ž526 767.

Exp.

y128"6 a y134"5 b

For J: ns10, ms18; for t: ns9, ms17. a Ref. w11x. b Ref. w12x. c Value generally accepted, Ref. w13x. The number of determinants diagonalized are included in parentheses.

500 c

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incorporates the indirect superexchange mechanism, going through charge transfers from the bridging ligand L to the metal M:

However, the contribution of this process will be underestimated if one does not introduce the instantaneous repolarization of the MyLq charge transfer states, brought by the 2-holer1-particle determinants belonging to the DDCI-3 list. The results of such a CI are reported in Table 1 under the DDCI-3 entry. It appears clearly that now our theoretical estimate practically matches the experimental value. Hence, we may conclude that using a DDCI-3 space we can compute accurately the magnetic coupling constant, and on the other hand it allows us to confirm the significant contribution to the exchange coupling arising from the orbital relaxation found by Van Oosten et al. w27x. Let us now analyze the results obtained for the transfer integral. As shown in Table 1, the DDCI-3 approach gives a value of 553 meV for cluster Cu 2 O 7 . Increasing the cluster size to Cu 2 O 11 only introduces minor modifications, the value of t dropping to 532 meV. This result indicates that, as for J, the main effects in the electron transfer involve the CuO 2 planes. Nevertheless, these values agree quite well both with those computed using other theoretical approaches ŽMartin w28x, 0.65 eV; Wang et al. w29x, 0.57–0.59 eV. and with the generally accepted value of 0.5 eV w13x. It should be noted that these calculations have been carried out using as active orbitals the magnetic orbitals determined for J, i.e. those resulting from the triplet state optimization, since in principle the t–J Hamiltonian uses a unique set of active orbitals. However, there has been a lot of discussion about the nature of the hole, which would be predominantly of an oxygen character Žactually, a natural orbital analysis of the DDCI-3 states shows that the oxygen nature of the hole is ; 60%.. In other words, one wonders whether the computed values would significantly change if, instead of using the magnetic orbitals to build up the model functions, some other hole adapted orbitals were used. In fact, the components of the DDCI-3 wave functions in the model space are dominant Žcoefficients ; 0.76, 0.77 on the zeroth-order determinants. but important contributions from the charge transfer L ™ M components are observed. That is why a second set of calculations, in which the active orbitals have been redefined, was performed. These were obtained by iterating the natural orbitals for the hole doped problem which provides optimized g X , uX Žor aX , bX . active orbitals used to define new model functions: caX s
108

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advantage of this ab initio approach concerns the prejudiceless determination of the active orbitals whose Cu and O character can be monitored by the computation of the natural orbitals after an extensive CI calculation. In this context, it is found that hole doping gives rise to states in which the hole appears to have a large oxygen character Ž) 60%. also in agreement with the non-mixed valence behavior experimentally observed w2,3x.

Acknowledgements This work was financially supported by the Spanish DGICYT, projects Nos. PB95-1125 and PB95-0847CO2-01. CJC thanks the Ministerio de Educacion ´ y Ciencia for supporting her stay at the Universite´ de Toulouse.

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