Electronic structure of Ca Cu2 O3: Spin ladder versus one-dimensional spin chain

Electronic structure of CaCu2O3: Spin ladder versus one-dimensional spin chain Esther Bordas, Coen de Graaf,*and Rosa Caballol Departament de Química Física i Inòrganica, Universitat Rovira i Virgili, Plaça Imperial Tàrraco 1, 43005 Tarragona, Spain Carmen J. Calzado Departamento de Química Física, Universidad de Sevilla, c/Prof. García González s/n, 41012 Sevilla, Spain 共Received 20 April 2004; revised manuscript received 14 September 2004; published 11 January 2005兲 Quantum chemical calculations on embedded cluster models have been performed to extract accurate estimates of the magnetic coupling Jand hopping parameters tof CaCu2O3. It is shown that this copper oxide compound is best described as a quasi-one-dimensional spin chain with weak interchain interactions within and between the Cu2O3planes. This magnetic structure is not reflected in the hopping parameters, since we find a large interplane hopping. Hence, the use of the simple second-order expression that relates J,t, and the on-site repulsion U共J=−4t2/U兲is not justified in all cases. DOI: 10.1103/PhysRevB.71.045108 PACS number共s兲: 75.30.Et, 74.25.Jb, 75.10.Dg, 75.50.Ee I. INTRODUCTION The spin ladders form a group of compounds with a wide variety of interesting phenomena. In relation to high-critical- temperature superconductivity, copper oxide spin ladders have received a lot of attention since the publication of the crystal structure of SrCu2O3and derivates by Hiroi and co-workers1in 1991. The even-legged ladders in this series of compounds show spin-gap behavior and finite spin-spin correlation length, while the odd-legged ladders behave as isolated spin chains.2,3 Beside these planar copper oxide spin ladders, other compounds with similar characteristics have been described in the literature 共La,Ca,Sr兲14Cu24O41 and LaCuO2.5 being the most important ones.4–6 The first one exhibits Cu2O3spin ladder planes similar to SrCu2O3com- bined with CuO2spin chain layers, whereas in the latter compound the ladders are oriented in such a way that large interladder coupling can be expected which gives rise to a three-dimensional magnetic network with long-range order below ⬃110 K.7 Recently, Kiryukhin and co-workers discussed the magnetic properties of the structurally related CaCu2O3.8The buckling of the spin ladder Cu2O3planes in this compounds reduces the magnetic interactions along the rungs of the ladder, and it was argued that CaCu2O3is actually not a spin ladder, but should be considered as a quasi-one-dimensional 共quasi-1D兲spin-1/2 chain. Consequently, a phase transition is observed at ⬃25 K, where magnetic ordering sets in. The dominant magnetic interaction is along the legs of the ladders 共Jleg兲in the bdirection of the crystal 共see Fig. 1兲. The high-temperature magnetic susceptibility was fitted with the theoretical expression for a 1D spin-1/2 chain based on the Bethe ansatz with a Jvalue of −170 meV. Weaker interactions of about 10 meV were assumed along the rung 共Jrung兲 and along the cdirection between different ladder planes 共Jc兲. The interladder interaction within the spin ladder planes 共a-bplanes兲was argued to be of less importance, being highly frustrated. The Néel temperature 共TN兲is significantly higher than in Sr2CuO3共5.4 K兲and Ca2CuO3共11 K兲, for which the interchain magnetic interactions are of the order of −1 meV.9,10 Based on x-ray absorption spectroscopy combined with density functional theory 共DFT兲calculations, Kim et al. make an attempt to quantify more precisely the magnetic interactions in CaCu2O3.11 The DFT calculations are performed within the local density approximation +U共LDA +U, where Uis the on-site repulsion energy for two electrons on the same Cu atom兲. By fitting the resulting band structure, the hopping parameters were determined along the rung 共trung⬃250 meV兲and in the cdirection between different spin ladder planes 共tc⬃125 meV兲. From the simple superexchange expression J=−4t2/U, the authors estimate the corresponding magnetic coupling parameters. Using U⬃3–5 eV, they arrive at Jrung⬃−50 meV and Jc ⬃±20 meV. With a ratio Jrung/Jleg=0.3, a pseudoladder magnetic structure is proposed and the authors ascribe the disappearance of the spin gap to the relatively large magnetic interaction between different ladder planes. Quantum chemical calculations on embedded clusters are used here to further quantify the magnetic interaction and hopping parameters in CaCu2O3. The methodology solves as accurate as possible the exact 共nonrelativistic兲Hamiltonian FIG. 1. 共Color online兲Crystal structure of CaCu2O3. Thick gray lines connect the cluster atoms used to compute the two different interplane interactions. The c,1 and c,2 interaction paths are schematically depicted on the left and right sides, respectively. PHYSICAL REVIEW B 71, 045108 共2005兲 1098-0121/2005/71共4兲/045108共6兲/$23.00 ©2005 The American Physical Society045108-1 within the material model defined by the embedded cluster. In this way, the important electron correlation effects present in many transition metal compounds can be treated very accurately, and electronic structure parameters can be derived without any further assumption. This ab initio computational strategy has been applied very succesfully over the last ten years and has reproduced, clarified, and even predicted the magnetic interaction and hopping parameters in many transition metal compounds 共see Refs. 12–21 and references therein兲. II. COMPUTATIONAL INFORMATION The clusters used in the calculation of the electronic structure parameters contain two or four copper atoms and all oxygens coordinating them. Figure 2 shows the two-center clusters that are used to determine the interactions along the rung and leg and between ladders in the a-bplane. The clusters used to extract information about the interactions between magnetic centers in different a-bplanes are depicted in Fig. 1. The four-center cluster contains four Cu atoms placed on the same ladder in a rectangular geometry and all the oxygens coordinated to them. All clusters are embedded in optimized point charges 共OPC’s兲that represent the Madelung potential due to the rest of the crystal. This Madelung potential is calculated by an Ewald summation assuming formal ionic point charges: i.e., 2+ for Cu and Ca and 2− for O. The optimized point charges reproduce the exact Madelung potential on a dense grid in the cluster region with a standard deviation less than 0.1 meV. The choice of formal ionic charges is consistent with the assignment of an integer number of electrons to the cluster to ensure overall charge neutrality. Furthermore, there exists by now substantial evidence in the literature that the electronic structure parameters considered here do not critically depend on the value of the charges to calculate the Madelung potential.22–24 This observation only holds for ionic transition metal compounds. In case of materials with covalent bonds such as CuGeO3, the embedding procedure with formal charges does not lead to meaningful results and alternative embedding schemes should be applied.25,26 In the results section, we will shortly come back to the influence of the Madelung potential on the properties studied here. To avoid artificial polarization of the electronic charge distribution due to the cluster atoms towards the point charges, total ion potentials27 共TIP’s兲replace the point charges on the boundary of the cluster. These TIP’s account for the Coulomb and exchange interactions between the cluster atoms and the atoms directly around it. The Ca2+ TIP corresponds to the large core potential of Durand and Barthelat28 and the Cu2+ TIP is a modification of the original large core potential to include the 3dshell in the potential.29 The geometry of the cluster has been taken from experiment.8 The validity of the embedded cluster model to extract electronic structure parameters in ionic transition metal compounds has been established in two ways. In the first place, several studies have been published that contrast embedded cluster results with periodic calculations. In all cases, the calculated values are very similar given that the approximation to the N-electron wave function is identical in both approaches.30–33 Second, cluster size effects have been studied before in other cuprates 共Li2CuO2,Sr 2CuO3, and La2CuO4兲and nickel compounds 共NiO, KNiF3, and K2NiF4兲 by comparing results obtained with two, three, four, and even five magnetic centers. In none of these cases significant effects were observed.10,18,19,34 Two different computational schemes have been applied to approximate the exact N-electron wave function of the cluster electrons. The first scheme is the complete activespace self-consistent field 共CASSCF兲, which simultaneously optimizes the spatial extent of the orbitals and the wave function expansion in the space spanned by the N-electron configurations that can be constructed by distributing the unpaired electrons in all possible ways over the active orbitals. For the undoped clusters, the active orbitals are mainly localized on the copper centers although they show some delocalization onto the neighboring oxygen ions. The twocenter cluster CAS contains two orbitals and two electrons. In the four-center cluster, there are four magnetic orbitals and four electrons in the CAS. For the doped clusters, we used active spaces with one electron less. The active orbitals turn out to be more delocalized with large contributions from the neighboring O 2porbitals.35 The CASSCF approach accounts for the direct exchange Kbetween the two magnetic centers and the kinetic exchange −4t2/U. However, this description of the electronic structure leads to a severe overestimation of Uand, consequently, too small magnetic coupling parameters. Second, we include the remaining electron correlation effects such as spin polarization, ligand to metal charge transfer configurations, etc. 共see Ref. 36 for a detailed discussion兲with the difference dedicated configuration interaction 共DDCI兲. This method is specially designed to obtain accurate relative energies of different electronic states37 and has been proven to give magnetic interaction parameters in close agreement with experiment. The molecular orbital set is obtained by means of the iterative DDCI 共IDDCI兲scheme38 to avoid any possible bias toward one of the electronic states. In this approach, an average density matrix is constructed from the FIG. 2. 共Color online兲Schematic representation of the twocenter clusters in the a-bplane. BORDAS et al. PHYSICAL REVIEW B 71, 045108 共2005兲 045108-2 DDCI density matrices of all electronic states. The vectors that are obtained from the diagonalization of this average density matrix serve as input for a new DDCI cycle. The procedure is repeated until convergence, what usually happens within four to six iterations. The basis set to expand the one-electron functions have the following characteristics: The Cu 共21s,14p,10d,4f兲 primitive set is contracted to 共6s,5p,4d,1f兲functions, and the O 共14s,9p,4d兲primitive set is contracted to 共4s,3p,1d兲 functions.39,40 Calculations have been performed with MOLCAS5.4 共Ref. 41兲and CASDI 共Ref. 42兲. III. RESULTS A. Magnetic coupling Table I lists the magnetic coupling and hopping parameters ordered by increasing Cu-Cu interatomic distance. Beside the usual in-plane interactions along the leg and rung and between copper atoms situated on different ladders 共Jinter and tinter兲, we also investigated two different interplane interactions. The interaction between copper atoms in different planes separated by 3.5 Å along the caxis 共see Fig. 1, on the left兲is labeled with the subscript c,1. A second possible interaction pathway is shown on the right in Fig. 1 and is referred to with the subscript c,2. Although the copper atoms are further separated in space, the relative orientation of the CuO4units with an oxygen atom connecting both metallic centers could be more favorable for magnetic interaction and hopping processes than along the c,1 path parallel to the c axis. To complete the discussion, we also mention the inplane interactions along the diagonal of the Cu4O12 plaquettes.43 The corner sharing CuO4squares along the legs of the ladders provide the optimum geometry for a strong magnetic coupling. Both for CASSCF and IDDCI, the coupling along the leg gives indeed the largest magnetic interaction. For CASSCF, we find −24 meV, but the inclusion of electron correlation effects strongly enhances the magnitude of the coupling. Our final IDDCI estimate of the coupling is −134 meV 共1610 K兲. The experimental estimate based on the fitting of the temperature dependence of the magnetic susceptibility 关 ␹ 共T兲兴 is −160±25 meV 共1950±300 K兲.8Our ab initio estimate is on the lower limit of this range, but in the fitting of the experimental data only the interaction along the leg was considered; the influence of other magnetic interactions on ␹ 共T兲was not accounted for. The interaction along the rung is much weaker because of the distorted Cu-Or-Cu bond 共Oris the oxygen located on the rung兲with an angle of 123°. The buckling of the ladder planes makes the ratio Jrung/Jleg calculated with IDDCI less than 0.1, whereas it is close to 1 in the structurally related spin ladder compound SrCu2O3. In a computational experiment, we gradually restore the linearity of the Cu-Cu linkage along the rung by varying the Cu-Or-Cu angle at fixed distances. The embedding is kept frozen. As expected, Jrung increases monotonicaly with increasing bond angle and Jrung/Jleg approaches 0.3, as proposed in Ref. 11, for 140°. For angles as large as 170°, the ratio is 0.7, still significantly smaller than 1. To study the stability of the results with respect to the Madelung potential, we varied the value of the embedding point charges around the formal ionic value of ±2 by increasing and decreasing them with 20%. In line with previous findings,22–24 Jrung is found to be rather stable with the variation of the Madelung potential. It varies between −13.2 meV and −9.5 meV for the reduced and increased point charges, respectively. The interladder interaction in the Cu2O3planes is ferromagnetic and about twice as large as the rung coupling. However, it has recently been mentioned that IDDCI possibly slightly overestimates ferromagnetic couplings,24,44 and the present value of 24 meV should be taken as an upper limit. The remaining magnetic coupling within the ladder planes Jdiag has been calculated from a four-center cluster. Calzado and Malrieu showed that the energy eigenvalues of the spin wave function no longer give sufficient information and Jdiag can only be determined with the help of effective Hamiltonians.18 The resulting value for Jdiag is much smaller than those found in La2CuO4共Ref. 18兲and SrCu2O3共Ref. 43兲, which again can be ascribed to the buckling of the ladder planes. The lack of a bridging ligand and the relative orientation of the CuO4units make the magnetic coupling along the c axis very weak. The CuO4plaquettes involved in Jc,1 are stacked in a parallel way, comparable to the interchain coupling in the above-mentioned 1D spin chains Sr2CuO3and Ca2CuO3. In the present case, however, the CuO4units are displaced with respect to each other by approximately 1.8 Å. This makes the overlap between the magnetic orbitals 共mainly of Cu 3dx2−y2character兲even less favorable and Jc,1 TABLE I. CASSCF and IDDCI magnetic coupling Jand hopping parameters t共in meV兲for CaCu2O3. d共Cu-Cu兲 Magnetic coupling Hopping CASSCF IDDCI CASSCF IDDCI Inter 2.8 Å 3.0 24.3 −124 −143 Rung 3.3 Å −1.9 −11.3 −250 −244 c,1 3.5 Å 0.02 0.09 −16 −2.8 Leg 4.1 Å −24.0 −134.1 −720 −622 c,2 4.8 Å 0.09 0.75 −122 −134 Diag 5.3 Å −0.04 −0.16 −14 −30 ELECTRONIC STRUCTURE OF CaCu2O3: SPIN…PHYSICAL REVIEW B 71, 045108 共2005兲 045108-3 共IDDCI兲is less than 0.1 meV 共⬃1K兲, which is significantly smaller than the ⬃−1 meV for Sr2CuO3and Ca2CuO3. The relative orientation of the CuO4units is rather different for Jc,2. Despite the larger Cu-Cu interatomic distance, Jc,2 is larger than Jc,1. This is in line with the observation that the magnetic orbitals are no longer parallel and the rung oxygen provides some type of bridge between the two copper atoms involved in the magnetic coupling parametrized by Jc,2. B. Hopping parameters The differences in CASSCF and IDDCI hopping parameters listed in Table I are less pronounced than for the magnetic coupling parameters. The insensitivity of the hopping parameter to the exact details of the electron correlation treatment has been observed before for other systems10,45 and implies that this parameter is essentially a one-electron property. This allows us to compare our cluster model IDDCI estimates to those obtained from the periodic calculations by Kim et al. within the LDA+Uscheme.11 The largest hopping is found along the legs of the ladders. The IDDCI value 共−622 meV兲is comparable to the hopping along similar Cu-O-Cubonds found in the two-dimensional antiferromagnets La2CuO4共Ref. 35兲and related cuprates 共Ref. 33兲. Whereas Jrung is more than 10 times smaller than Jleg, the corresponding hopping parameter is only smaller by a factor of 2.5. The IDDCI estimate of −244 meV is in remarkable good agreement with the LDA+Uvalue of ⬃250 meV proposed for the rung. The interplane hopping parameters can only be obtained from the mapping of the IDDCI wave functions onto an effective Hamiltonian. The lack of an inversion center in the Cu2O8clusters 共two CuO4plaquettes in different ladder planes兲makes that the energy eigenvalues of the two lowest doublet states are not sufficient to calculate t. The difference observed for the magnetic interactions along the pathways c,1 and c,2 is more pronounced for the hopping parameters: tc,1 is very small, only −3 meV, while tc,2 is an order of magnitude larger and similar to the in-plane interladder hopping tinter. These results suggest that the band dispersion in the cdirection observed by Kim et al. is due to the hopping of electrons 共or holes兲between CuO4plaquettes with a relative orientation as shown on the right in Fig. 1—i.e., the c,2 interaction path. Our interplane hopping 共tc,2=−134 meV兲is again very close to the LDA+Uvalue of ⬃125 meV. Finally, we find a hopping parameter of −30 meV along the diagonal of the buckled Cu4O12 plaquettes. This interaction should probably also be included in a simulation of the macroscopic properties of CaCu2O3. C. Validity of the superexchange relation With the ab initio values for tand Jat hand, the question arises to what extent the simple superexchange relation U =−4t2/Jcan be used to estimate one of the three parameters once the other two are known. Several examples can be found in the literature where assumptions about the relative size of different J’s have been made solely based on the magnitude of the hopping parameters. Beside for CaCu2O3, this strategy has also been applied for Li2CuO2and Li2VOSiO4.46,47 Table II recompiles the estimates of Ubased on the superexchange formula 共Upert兲using Jand tobtained with IDDCI. We add the values for La2CuO4,Li 2CuO2, and Sr2CuO3. The on-site repulsion parameter can also be determined variationally from quantum chemical calculations by means of the projection of the IDDCI wave functions onto an effective Hamiltonian as outlined in Ref. 48. We use Uvar for the values obtained by this more accurate procedure. It is readily seen that the applicability of the formula is not universal. For nearest-neighbor interactions along 共al- TABLE II. On-site repulsion energy 共in eV兲for different copper oxide compounds estimated from −4t2/J using IDDCI parameters 共Upert兲and calculated with IDDCI 共Uvar兲. Compound t共meV兲J共meV兲Upert Uvar CaCu2O3 Inter −143 24.3 3 5.8 Rung −244 −11.3 21 6.7 c,1 −3 0.09 3 Leg −622 −134.1 12 6.5 c,2 −134 0.75 96 La2CuO4a−598 150 9.5 7.3 Li2CuO2b NN −143 12.2 6.7 NNN 120 1.9 26 Sr2CuO3c In-chain 659 −246 7.1 Inter 30 −0.44 8.3 aValues taken from Refs. 33, 35, and 48. bValues refer to in-chain nearest-neighbor 共NN兲and next-nearest-neighbor 共NNN兲interactions 共Ref. 19兲. cValues taken from Ref. 10. BORDAS et al. PHYSICAL REVIEW B 71, 045108 共2005兲 045108-4 most兲lineal Cu-O-Cu bonds, Upert is at most of the right order of magnitude. For other type of interactions, the Upert estimates show a large dispersion and unphysical values as large as 96 eV are obtained for the interaction in CaCu2O3 along c,2. Uvar is related 共but not equal兲to the energy difference between the neutral state 共with mainly Cu 3d9-O 2p6-Cu 3d9contributions兲and the ionic state 共mainly Cu 3d10-O 2p6-Cu 3d8兲. The variational determined on-site repulsion parameters for CaCu2O3obtained with different clusters are more consistent than those calculated with the superexchange relation. We obtain Uvar=5.8, 6.7, and 6.5 eV for the in-plane interladder, the rung, and the leg clusters, respectively. The failure of the perturbative expression to relate twith J shows that the reduction of the magnetic coupling constant to its kinetic exchange compound is too crude an approximation. In the first place, it should be noted that within the Anderson model, the magnetic coupling Jis the sum of a ferromagnetic term JF, generally ascribed to direct exchange, and an antiferromagnetic term JAF due to the kinetic exchange. Hence, strictly speaking, the superexchange formula only relates twith the antiferromagnetic component and not to the magnetic coupling itself. Second, the hopping parameter tas determined from doped clusters does not have exactly the same meaning as the tin the superexchange relation. In the latter relation, tis related to the Hamilton matrix element between the neutral and the ionic states,48 while it parametrizes the mobility of the holes in the cluster calculations. The same holds for the values derived from the LDA band structure calculations. IV. CONCLUSIONS The IDDCI magnetic interactions parameters suggest that CaCu2O3is best described as a quasi-1D spin chain. The chains are coupled in the adirection through the distorted rungs and nonzero interplane interactions along the caxis, especially Jc,2. These interactions can be at the origin of the long-range ordering below the Néel temperature. The magnetic structure of the compound is, however, not as simple as the related Ca2CuO3. In that case the mean-field expression49 that relates the magnetic in-chain and interchain coupling parameters to TNgives very reasonable results for the DDCI parameters.10 The application of the same equation for CaCu2O3leads to an overestimation of TNby at least an order of magnitude. Obviously, other interactions than Jrung and Jc,2 also play a role in the magnetic structure. The role of the frustrated in-plane interladder interaction is expected to be small. The quantum Monte Carlo simulations of Johnston et al. for the planar spin ladder SrCu2O3suggest that this coupling does not influence the magnetic susceptibility up to Jinter/Jleg=0.2.50 The four-body cyclic exchange 共estimated with ab initio calculations43 to be 4 meV兲and Jdiag may play an important role in the magnetic structure. The close resemblance between the theoretical estimates of tfrom a periodic modelization of the crystal 共the LDA +Ucalculations of Kim et al.11兲and from a cluster model approach 共the present IDDCI estimates兲gives additional evidence of the appropriateness of the local model to extract this type of parameters. An interesting observation from the calculations is that the magnetic structure cannot be directly extracted from the size of the hopping parameters along the different interaction paths. The magnetic interaction along the caxis is much smaller than can be expected at first sight from the size of the hopping parameter. This behavior is not unique for CaCu2O3 but has also been observed for other cuprates—e.g., Li2CuO2. We conclude that only for similar interaction paths, the magnitude of tserves as a guide for the relative size of the corresponding magnetic couplings. 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