Four-spin cyclic exchange in spin ladder cuprates

Four-spin cyclic exchange in spin ladder cuprates Carmen J. Calzado,1Coen de Graaf,2Esther Bordas,2Rosa Caballol,2and Jean-Paul Malrieu3 1Departamento de Quı ´mica Fı ´sica, Universidad de Sevilla, 41012 Sevilla, Spain 2Departament de Quı ´mica Fı ´sica i Inorga `nica, Universitat Rovira i Virgili, 43005 Tarragona, Spain 3Laboratoire de Physique Quantique, IRSAMC, Universite ´Paul Sabatier, 31062 Toulouse-Cedex, France 共Received 8 January 2003; published 28 April 2003兲 The four-spin cyclic exchange term Jring of three spin-ladder cuprates (SrCu2O3,Sr 2Cu3O5, and CaCu2O3) has been calculated from ab initio quantum chemistry calculations. For the first two compounds, a nonnegligible cyclic exchange is found, aproximately 20% of the magnetic coupling across the rungs, J⬜, and always larger than the value obtained for two-dimensional La2CuO4system. In the case of CaCu2O3, the Jring value is quite small, due to the folding of the Cu-O-Cu rung angle, but the Jring /J⬜ratio is also 0.2 as in the two other systems. DOI: 10.1103/PhysRevB.67.132409 PACS number共s兲: 75.30.Et, 71.27.⫹a, 71.70.Gm, 75.50.Ee Spin-ladder cuprates constitute an active research field in the last decade.1,2 They can be viewed as intermediates between the one-dimensional 共1D兲antiferromagnets and the still controversial two-dimensional 共2D兲square lattices. Ladders composed of Cu and O are specially interesting due to their proximity to high-Tccuprates. Their magnetic properties depend on the number of legs. Even-legged ladders show a spin gap excitation, whereas odd-legged ladders are gapless and behave as a 1D spin chain.1,2 They also present different properties regarding hole doping. It has been suggested that even-legged spin ladders become superconductors upon hole doping, which has been confirmed experimentally3in the two-legged ladder Sr14⫺xCaxCu24O41 under high pressure. The magnetic properties of these compounds are controlled by the effective magnetic coupling constant J, related with the amplitude of the interactions between the spin moments of the Cu⫹2ions. Different Jconstants can be defined, as shown in Fig. 1. The two most important are the coupling along the legs, J 储 , and across the rungs, J⬜. The ratio J⬜/J 储 is controversial since the interpretation of different experimental data has led to estimates ranging from spatially isotropic, J⬜/J 储 ⫽1, to strongly anisotropic couplings, J⬜/J 储 ⫽0.5. The strong spatial anisotropy J⬜/J 储 ⫽0.5 is in contradiction with geometrical considerations. Since the Cu-O-Cu bonds are quite similar, the exchange pathways are expected to be equivalent, and so, J 储 ⬃J⬜. The theoretical calculations of Mizuno, Tohyama, and Maekawa;4and de Graaf et al.5are in agreement with these considerations. It should be noted that most of the available J⬜and J 储 values have been obtained by fitting the experimental data onto a model Heisenberg Hamiltonian, containing just twobody operators. As in the case of the 2D cuprates,6–10 some authors have recently suggested the necessity of introducing additional interactions in the model Heisenberg Hamiltonian to study the properties of the spin ladders. The most important are the diagonal coupling 共second-neighbor interactions兲, the interladder exchange and, especially, the four-spin cyclic exchange 共4SCE兲. In this context, de Graaf et al.5 have proposed that the omission of the interladder coupling in the analysis of experimental data for SrCu2O3may be the reason that a ratio J⬜/J 储 ⬃0.5 was obtained instead of J⬜/J 储 ⬃1. However, the quantum Monte Carlo 共QMC兲simulations of the temperature dependence of the magnetic susceptibility of Johnston et al.11 do not confirm this hypothesis. The inclusion of a ferromagnetic interladder coupling, (Jinter /J 储 ⫽⫺0.1), in their QMC simulations does not change the fitted J⬜/J 储 ⬃0.5 ratio obtained for SrCu2O3. Recently, Brehmer et al.12 have analyzed the role of the 4SCE on the determination of coupling constants from ladder spectra. The 4SCE is a fourth-order term in the Hubbard model, involving the circulation of the electrons around the plaquette and scales as 80t4/U3,tbeing the hopping integral and Uthe on-site Coulomb repulsion.13,14 The extended Heisenberg Hamiltonian containing the diagonal interactions and the 4SCE terms has the following form: H⫽兺 具 ij 典 legs J 储 冉 SiSj⫺1 4 冊 ⫹兺 具 ij 典 rungs J⬜ 冉 SiSj⫺1 4 冊 ⫹兺 具 ij 典 NNN Jdiag 冉 SiSj ⫺1 4 冊 ⫹兺 具 ijkl 典 Jring ijkl 冋 共SiSj兲共SkSl兲⫹共SiSl兲共SjSk兲 ⫺共SiSk兲共SjSl兲⫺1 16 册 ,共1兲 where the higher multiplet energy is set to zero, J 储 and J⬜ correspond to nearest-neighbor 共NN兲interactions, Jdiag to the next nearest-neighbor 共NNN兲coupling, and Jring ijkl to the 4SCE terms; and the superscript refers to the type of cyclic interaction. Actually the introduction of Jdiag 共especially, if it is antiferromagnetic兲implies that the NNN hopping tdiag is not negligible, and the circulation of the four electrons may involve the diagonal hopping. The physical content and origin of the three types of Jring is schematized in Fig. 2. It has been argued that a finite value of the ring exchange is necessary to reproduce the structure of the magnetic Raman spectrum for 2D insulating cuprates.7,9,10 In spin-ladder cuprates, Brehmer et al.12 have concluded that the cyclic exchange has a large influence on the spin gap and, consequently, on the exchange constant values J 储 and J⬜. A small amount of Jring (Jring⯝0.28J⬜) is consistent with J 储 ⯝J⬜as expected from the geometrical structure. A similar result has been obtained by Matsuda et al.15 for the two-legged ladder La6Ca8Cu24O41 . A reasonable fit to the experimental data is PHYSICAL REVIEW B 67, 132409 共2003兲 0163-1829/2003/67共13兲/132409共4兲/$20.00 ©2003 The American Physical Society67 132409-1 obtained when a finite cyclic exchange 共30% of J⬜) is included in the Hamiltonian, with J 储 ⫽J⬜⫽⫺110 meV. In the absence of this 4SCE, a fit of comparable quality is obtained with J⬜⫽⫺53 meV and J 储 ⫽⫺106 meV. So, it seems that the neglect of the 4SCE term could lead to the strong anisotropy found for spin-ladder cuprates. It is the aim of this report to simultaneously determine all the effective interactions, appearing in Eq. 共1兲, with special attention to the 4SCE term, by means of ab initio quantum chemical embedded cluster calculations. We report the amplitude of these operators for three spin-ladder compounds: SrCu2O3, CaCu2O3, and Sr2Cu3O5; and compare the results with the values obtained for the 2D La2CuO4system. A detailed analysis of the eigenvalues and wave functions of these systems enables us to determine the exchange interactions in a Cu4O12 plaquette: the NN interactions, J⬜and J 储 , the NNN interaction Jdiag , and the 4SCE term, Jring . This approach only depends on the quality of the approximation of the exact wave functions obtained from the calculations and the correctness of the modeling. The three systems here considered have different structural features. SrCu2O3is a two-legged ladder, with a spin gap of 680 K.16,17 Sr2Cu3O5is a three-legged ladder without spin gap. The structure of CaCu2O3is similar to that of SrCu2O3, but the Cu-O-Cu bond angle in the ladder rungs equals 123°, and, therefore, the magnetic interaction along the rung is expected to be much weaker than in SrCu2O3. In all systems, a Cu4O12 plaquette has been chosen, embedded in a set of optimized point charges placed at the lattice positions to model the crystalline environment 共see Fig. 1兲. The Cu ions directely bonded to the cluster have been described by total ion potentials 共TIP’s兲to avoid an artificial polarization of the oxygen orbitals. TIP’s have been also employed to represent the Sr and Ca ions in the neighborhood of the cluster. The comparison of the cluster model and periodic calculations on related compounds has shown that this representation of the crystal is sufficient to accurately describe the type of interactions, subject of the present study.18 Details concerning the type of configuration interaction 共CI兲calculations performed and the basis set used can be found in Ref. 19. The strategy to extract the effective 4SCE interaction in FIG. 1. Cu4O12 plaquettes and first-neighbor TIP’s environment models for 共a兲La2CuO4,共b兲SrCu2O3and CaCu2O3, and 共c兲 Sr2Cu3O5compounds. Gray, small black, and big dark circles correspond, respectively, to Cu, O, and counterions atoms (Sr⫹2, Ca⫹2,orLa ⫹3). The different types of exchange interactions in the spin ladders are shown in 共d兲. FIG. 2. Four-spin cyclic exchange couplings: 共a兲Jring1, circular movement of the electrons, 共b兲Jring2, simultaneous exchange along the legs, and 共c兲Jring3, simultaneous exchange across the rungs. TABLE I. The Heisenberg Hamiltonian on the basis of the model space for SrCu2O3and CaCu2O3.Jr1,Jr2, and Jr3correspond, respectively, to Jring1,Jring2, and Jring3共see text兲. 兩 a(↑)b(↓)c(↑)d(↓) 兩兩 a(↓)b(↑)c(↓)d(↑) 兩兩 a(↑)b(↑)c(↓)d(↓) 兩兩 a(↓)b(↓)c(↑)d(↑) 兩兩 a(↑)b(↓)c(↓)d(↑) 兩兩 a(↓)b(↑)c(↑)d(↓) 兩 ⫺J 储 ⫺J⬜1 2Jr11 2J 储 ⫹ ⫺Jr1⫺Jr2⫹Jr3 81 2J 储 ⫹ ⫺Jr1⫺Jr2⫹Jr3 81 2J⬜⫹ ⫺Jr1⫹Jr2⫺Jr3 81 2J⬜⫹ ⫺Jr1⫹Jr2⫺Jr3 8 ⫺J 储 ⫺J⬜ 1 2J 储 ⫹ ⫺Jr1⫺Jr2⫹Jr3 81 2J 储 ⫹ ⫺Jr1⫺Jr2⫹Jr3 81 2J⬜⫹ ⫺Jr1⫹Jr2⫺Jr3 81 2J⬜⫹ ⫺Jr1⫹Jr2⫺Jr3 8 ⫺J 储 ⫺Jdiag 1 2Jr21 2Jdiag⫹Jr1⫺Jr2⫺Jr3 81 2Jdiag⫹Jr1⫺Jr2⫺Jr3 8 ⫺J 储 ⫺Jdiag 1 2Jdiag⫹Jr1⫺Jr2⫺Jr3 81 2Jdiag⫹Jr1⫺Jr2⫺Jr3 8 ⫺J⬜⫺Jdiag 1 2Jr3 ⫺J⬜⫺Jdiag BRIEF REPORTS PHYSICAL REVIEW B 67, 132409 共2003兲 132409-2 the CuO2layers of La2CuO4has been reported in Ref. 19. For symmetry reasons, all the effective parameters in the plaquette (J,Jdiag , and Jring) can be evaluated from energy differences of the lowest states in the plaquette in La2CuO4. In the case of the ladders, the number of unknown parameters is larger and the spectrum is no longer sufficient. Let us consider the Cu4O12 fragment in some more detail. The four unpaired electrons are located in the in-plane dx2⫺y2-type orbitals centered on each Cu atom. Calling a,b, c, and dthe four magnetic orbitals 共the rungs being a-band d-c), the model space Sis constituted by six neutral determinants with Ms⫽0. Table I shows the extended Heisenberg Hamiltonian for four spins in a rectangular cluster on the basis of this model space. Jring1,Jring2, and Jring3concern the three types of 4SCE interactions present in the plaquette 共Fig. 2兲:Jring1⫽Jring abcd produces the circulation of all the spins in the plaquette, and Jring2⫽Jring adbc and Jring3⫽Jring abdc control, respectively, the simultaneous exchange of the spins in the two legs and across the two rungs. In the case of Sr2Cu3O5, we can distinguish between the internal J 储 int and the external leg J 储 ext , and then, in the diagonal elements of the matrix, J 储 must be replaced by (J 储 ext⫹J 储 int)/2. The diagonalization of this matrix gives six eigenstates of different spin-space symmetries. Figure 3 shows the spectrum written on the basis of the parameters of the model Hamiltonian. In all the cases, there are five energy differences. For SrCu2O3and CaCu2O3, there are six parameters; for Sr2Cu3O5there are seven. In order to avoid a bias in the determination of these sets of parameters, we use the effective Hamiltonian theory20 to evaluate all parameters, instead of neglecting beforehand the presumably small secondary four-spin interactions Jring2and Jring3. Our six eigenstates 兩 ␺ k 典 共with energies Ek) have the largest projections on the model space S, with PS ⫽兺I苸S 兩 ␾ I 典具 ␾ I 兩 the projector on the model space. The Bloch effective Hamiltonian20 can be written as HBloch 兩 PS ␺ k 典 ⫽Ek 兩 PS ␺ k 典 ,共2兲 that is, the eigenvectors of this effective Hamiltonian are projections of the exact eigenvectors on the model space and their eigenenergies are the ones of the CI space. The spectral representation of the Bloch effective Hamiltonian is HBloch ⫽兺k 兩 PS ␺ k 典 Ek 具 PS ␺ k † 兩 , where 兩 PS ␺ k † 典 ⫽S⫺1 兩 PS ␺ k 典 corresponds to the biorthogonal vectors, Sbeing the overlap matrix between the projections 兩 PS ␺ k 典 . Using this representation, it is possible to extract the values of the complete set of parameters. The values obtained for the three spin ladders are presented in Table II, together with those extracted for the 2D La2CuO4system.19 For SrCu2O3and Sr2Cu3O5, the J⬜/J 储 ratio is closer to 1 than to 0.5, consistent with the geometrical structure of the ladders, and in agreement with the values obtained for SrCu2O3from binuclear clusters.5The NN interactions are always larger than for the 2D La2CuO4compound. The diagonal interaction is antiferromagnetic, as in the 2D cuprates,8,21 with values around ⫺15 meV. Regarding the cyclic terms, the parameters Jring2and Jring3are small in all cases. They are never larger than 4 meV and are not explicitly reported 共hereafter, Jring refers to Jring1). Notice, however, that this is an a posteriori information. The 4SCE is around 35 meV, significantly larger than for La2CuO4. The Jring /J⬜ratio is 0.22 for both spin-ladder compounds, and it is consistent with that proposed by Matsuda et al.15 for La6Ca8Cu24O41 and the value suggested by Brehmer from numerical diagonalizations,12 but smaller than those obtained for SrCu2O3from the diagonalization of the d-pmodel Hamiltonian (Jring /J⬜⬃0.4).4 The results for the CaCu2O3system reflect the effect of the folding of the Cu-O-Cu rung angle. The coupling across the rungs is quite small, the bending of the Cu-O-Cu bond induces an unfavorable overlap of the active dx2⫺y2orbitals FIG. 3. Spectrum of the plaquette with one electron per Cu atom for SrCu2O3and CaCu2O3. For Sr2Cu3O5, the parameter J 储 must be replaced by (J 储 ext⫹J 储 int)/2. On the left, the corresponding symmetry of the different states in the D2hgroup. TABLE II. Exchange parameters for SrCu2O3,Sr 2Cu3O5, CaCu2O3, and La2CuO4. All parameters in meV, except Uin eV. J 储 J⬜Jdiag Jring J⬜/J 储 Jring /J⬜U共eV兲(Jring ladder/Jring 2D)pert (Jring ladder/Jring 2D)abinitio SrCu2O3⫺203 ⫺157 ⫺13 34 0.77 0.22 6.10 a2.49 2.43 Sr2Cu3O5⫺195 (ext)⫺177 ⫺14 39 0.91 共ext兲0.22 6.10 a2.78 2.79 ⫺208 (int) 0.85 共int兲 CaCu2O3⫺147 ⫺15 ⫺0.2 4 0.10 0.26 6.60 a0.16 0.29 La2CuO4⫺124 ⫺124 ⫺6.5 14 1.00 0.11 7.31 b1.0 1.0 aReference 24. bReference 23. BRIEF REPORTS PHYSICAL REVIEW B 67, 132409 共2003兲 132409-3 and the bridging oxygen ones. On the other hand, the J 储 value is ⫺147 meV, larger than the NN coupling in 2D cuprates, and in good agreement with the estimations coming from magnetic susceptibility and neutron diffraction22 (J 储 ⬃ ⫺167⫾25 meV). Both the NNN interaction and the 4SCE are also affected by the folding. However, the Jring /J⬜ratio is 0.26, similar to those obtained for the two other ladder compounds, and larger than the value reported for 2D cuprates. As mentioned above, the 4SCE is a fourth-order term, scaling as 80t4/U3. The perturbation theory second-order contribution to the magnetic coupling takes the form J ⫽⫺4t2/U. The perturbative expression for the 4SCE can be written as Jring pert⫽80t⬜ 2t 储 2/U3⫽5J⬜J 储 /U, and the perturbative Jring ladder/Jring 2Dratio is Jring ladder Jring 2D⫽J⬜J 储 J2D 2 U2D Uladder .共3兲 Table II reports the perturbative estimates of the Jring ladder/Jring 2Dratio, together with the on-site Coulomb repulsion, determined from ab initio quantum chemistry calculations on embedded binuclear clusters 共Ref. 23 for the 2D cuprates and Ref. 24 for the ladders兲. An excellent agreement between the perturbative and the ab initio ratios is observed. 共A similar behavior has been found for the perturbative estimates of Jring2and Jring3as will be shown elsewhere.24兲We can conclude that the larger values found for the 4SCE term in the spin-ladder cuprates reflect the enlargement of the NN coupling constants and the reduction of the on-site repulsion Uwith respect to the 2D cuprates. The NN coupling constant depends on tand U. Both parameters are affected by the changes in the Madelung potential and the different polarization effects in the ladder compounds in comparison to the 2D La2CuO4compound. A detailed analysis of these effects will be given in a forthcoming paper.24 Financial support has been provided by the Spanish Ministry of Science and Technology 共Project No. PB98-1216CO2-02兲, the DURSI of the Generalitat de Catalunya 共Grant No. SGR01-00315兲, and the Catalan-French scientific cooperation 共Grant No. PICS2001-13兲. 1E. Dagotto and T.M. Rice, Science 271, 618 共1996兲. 2T.M. Rice, Z. Phys. B 103, 165 共1997兲. 3H. Takahashi et al., Physica B 237-238,112共1997兲. 4Y. Mizuno, T. Tohyama, and S. Maekawa, Phys. Rev. B 58, R14 713 共1998兲; J. Low Temp. Phys. 117, 389 共1999兲. 5C. de Graaf et al., Phys. Rev. B 60, 3457 共1999兲. 6M. Takahashi, J. Phys. C 10, 1289 共1977兲. 7M. Roger and J.M. Delrieu, Phys. Rev. B 39, 2299 共1989兲. 8H. Schmidt and Y. 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Bordas et al. 共unpublished兲. BRIEF REPORTS PHYSICAL REVIEW B 67, 132409 共2003兲 132409-4