Magnetic structure of the edge-sharing copper oxide chain compound NaCu2O2
L. Capogna, M. Reehuis, A. Maljuk, R.K. Kremer, B. Ouladdiaf, M. Jansen, B. Keimer
MMagnetic structure of the edge-sharing copper oxide chain compound NaCu O L. Capogna,
1, 2
M. Reehuis, A. Maljuk,
3, 4
R.K. Kremer, B. Ouladdiaf, M. Jansen, and B. Keimer Consiglio Nazionale delle Ricerche, IOM-OGG, 6 rue J. Horowitz, F-38042 Grenoble, France Institut Laue Langevin, 6 rue J. Horowitz, F-38042 Grenoble, France Helmholtz-Zentrum Berlin f¨ur Materialen und Energie, Glienicker Str.100, D-14109 Berlin, Germany Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany (Dated: November 5, 2018)Single-crystal neutron diffraction has been used to determine the incommensurate magnetic struc-ture of NaCu O , a compound built up of chains of edge-sharing CuO plaquettes. Magnetic struc-tures compatible with the lattice symmetry were identified by a group-theoretical analysis, andtheir magnetic structure factors were compared to the experimentally observed Bragg intensities. Inconjunction with other experimental data, this analysis yields an elliptical helix structure in whichboth the helicity and the polarization plane alternate among copper-oxide chains. This magneticground state is discussed in the context of the recently reported multiferroic properties of othercopper-oxide chain compounds. PACS numbers: 75.25.-j, 75.85.+t, 75.50.Ee, 75.10.-b
I. INTRODUCTION
Research on compounds with coupled spontaneousmagnetic and electric polarization (multiferroics) has re-cently focused renewed attention on helical magnetism.A leading theory of multiferroicity predicts that helicalmagnetic order in insulators induces an electric polariza-tion of the form P ∝ (cid:88) (cid:104) ij (cid:105) ˆ n ij × ( S i × S j ) , (1)where ˆ n i,j is a vector connecting spins S i,j on exchangebond (cid:104) ij (cid:105) . Clearly, detailed information about themagnetic structure and crystal symmetry is required toevaluate the consequences of Eq. 1 for any given com-pound. The theory explains the observation of ferroelec-tricity in compounds such as TbMnO and CuO, whichexhibit helicoidal states with nonvanishing projectionof the propagation vector onto the polarization plane. While these compounds contain three-dimensional (3D)networks of exchange bonds, magnetic insulators withquasi-1D electronic structure have the potential to serveas particularly instructive model systems for multifer-roicity. The discovery of a macroscopic electric polariza-tion in the compounds LiCu O (Ref. 5) and LiCuVO (Ref. 6), which are built up of chains of edge-sharingCuO plaquettes, has therefore generated significant at-tention. Because of the competition between the anoma-lously small nearest-neighbor exchange coupling and thestronger next-nearest neighbor coupling of spin-1/2 cop-per ions along the chains, both compounds exhibit in-commensurate magnetic correlations, and interchain in-teractions induce helicoidal 3D long-range order at lowtemperatures. In addition to multiferroicity, these mate-rials are also of interest in the context of research on theinterplay between spin and charge correlations along thecopper-oxide chains. In spite of their simple electronic structure, the mul-tiferroic properties of LiCu O and LiCuVO are stillpoorly understood. Both compounds are orthorhom-bic, with copper-oxide chains running along the b -axis.Based on neutron diffraction experiments, the polariza-tion plane of the helix in LiCu O was originally reportedto lie within the ab plane, while P was found to be along c , in disagreement with Eq. 1. Later polarized-neutronexperiments indicated a bc -polarized helix, which isconsistent with the electrical polarization according toEq. 1. However, the behavior of P in an applied mag-netic field H was found to be difficult to reconcile withthis scenario, although interchain magneto-electriccoupling may offer a possible solution. Recent single-crystal neutron diffraction data have been interpretedas evidence of an elliptical helix state with polarizationplane tilted by an angle of 45 ◦ with respect to the bc -plane. The consequences of this structure for the electricpolarization remain to be assessed. In LiCuVO , the he-lix was reported to be polarized within the ab -plane for H = 0, which is consistent with the observation that P (cid:107) a . However, deviations from the theoretical predic-tions were again noted for H (cid:54) = 0. The apparent failureof Eq. 1 as a description of the magnetic field dependenceof the dielectric properties of LiCu O and LiCuVO has motivated an alternative scenario according to whichthe observed electric polarization is due to defects gen-erated by Li–Cu intersubstitution. While such de-fects are indeed quite common due to the similar ionicradii of Li and Cu, this scenario has been contestedbased on work on ostensibly stoichiometric LiCu O sin-gle crystals. The situation thus remains unresolved.Here we present an investigation of the magnetic struc-ture of NaCu O , a compound that is isostructuraland isolelectronic to LiCu O and, like LiCu O andLiCuVO , exhibits incommensurate magnetic order atlow temperatures. Unlike LiCu O and LiCuVO ,however, NaCu O is not intrinsically prone to disor-der because of the different sizes of Na and Cu ions, a r X i v : . [ c ond - m a t . s t r- e l ] A p r and Na–Cu intersubstitution can be ruled out with highsensitivity. A recent investigation came to the conclu-sion that ferroelectricity is absent in NaCu O . Detailedinformation about the magnetic structure is required inorder to assess the origin of the qualitatively differentelectric properties of LiCu O and LiCuVO on the onehand, and NaCu O on the other hand. However, whilea comprehensive set of single-crystal neutron diffractiondata is available for the former two compounds, information about the magnetic structure of NaCu O has thus far only been gleaned from more limited neu-tron powder diffraction, nuclear magnetic resonance(NMR), and resonant x-ray diffraction data. Moti-vated by the controversy about the origin of multiferroc-ity in copper-oxide chain compounds outlined above, wehave carried out a single-crystal neutron diffraction studyof the magnetic structure of NaCu O . We present theresults in the framework of a rigorous symmetry classifi-cation of possible magnetic structures. II. RESULTS AND ANALYSIS
Single crystals of NaCu O in the shape of small rect-angular plates a few millimeters long and wide and ∼ . X-ray diffraction, induction-coupled plasma atomic emission spectroscopy, andmagnetometry were used to fully characterize the struc-ture, purity, and quality of the crystals. Neutron diffrac-tion experiments were carried out at the Institut Laue-Langevin using the high-resolution powder diffractometerD2B, the Laue single-crystal diffractometer Vivaldi, andthe four-circle diffractometer D10. For the D2B measure-ments, we used a powder sample described elesewhere. -500050010001500 20 40 60 80 100 120 140 θ (deg) I n t e n s it y ( a . u . ) FIG. 1: Powder diffraction data on NaCu O taken on D2Bwith a neutron wavelength λ = 1 .
596 ˚A at T = 1 . Figure 1 shows powder diffraction data taken on D2B,which were used to accurately refine the nuclear struc-ture at temperatures 1.5 and 300 K. Rietveld refinementin the orthorhombic space group
P nma (No. 62) withparameters shown in Table I yields an excellent descrip-tion of the data (Fig. 1).
TABLE I: Lattice parameters and atomic positions in frac-tional coordinates, as derived from Rietveld refinement ofpowder diffraction data at temperatures T = 1 . P nma . All atoms are in Wyckoff position4 c . T (K) 1.5 300 a (˚A) 6.2001(1) 6.2148(1) b (˚A) 2.9310(1) 2.9361(1) c (˚A) 13.0337(2) 13.0731(3)Cu(I) x z x z x z x z x z R F In order to determine the magnetic structure ofNaCu O , a set of 40 magnetic reflections was acquiredon a single crystal at T = 1 . h, k, l ) M = ( h, k, l ) N ± k , where N stands fornuclear reflections. The propagation vector k = (0 . , ξ, ξ = 0 . ± .
002 resulting from refinement of thedata is consistent with prior neutron powder diffraction and resonant x-ray diffraction measurements. The tem-perature dependence of the magnetic Bragg intensity de-termined on Vivaldi (Fig. 2) showed a N´eel temperature T N = 12 K, again in agreement with prior work. Theintegrated intensities of the magnetic reflections were de-termined by rocking scans (inset in Fig. 2). For therefinement of the magnetic structure we used the mag-netic structure factors of 20 incommensurate magneticreflections, 10 of which are unique. The overall scale fac-tor as well as the extinction parameter were fixed to thevalues refined from a set of nuclear reflections collectedunder the same experimental conditions. The atomic po-sitions and the lattice parameters were fixed to the valuesrefined from the powder data (Table I).In order to examine the constraints set on the magneticstructure by the crystal symmetry, we used the represen-tation analysis method as implemented in the programBasIreps. The lattice symmetry (space group
P nma ) T(K)
12 13 14 ω (deg) I n t e n s it y ( c oun t s / m on it o r = ) I n t e g r a t e d I n t e n s it y ( a . u . ) FIG. 2: Temperature dependence of the intensity of the mag-netic Bragg reflection (0 . , − . , −
1) in NaCu O , taken onthe Laue diffractometer Vivaldi. The inset shows a rockingcurve taken at T = 2 K on the four-circle diffractometer D10. and atomic positions derived from the structural analy-sis described above, as well as the propagation vector k determined from the positions of the magnetic reflectionsserve as input for this analysis. Further, x-ray absorptionmeasurements indicate valence states of 1+ (with a full-shell configuration), and 2+ (with one unpaired hole inthe d -shell) for the Cu(I) and Cu(II) ions, respectively. Only the latter ions therefore carry a magnetic moment.They are in the following positions:Cu1: ( x, / , z )Cu2: ( − x, / , − z ) + (1 , , / x, / , / − z ) + ( − , , / − x, / , / z ) + (1 , , − x and z listed in Table I.The representation analysis yields a single, two-dimensional irreducible representation, Γ mag , for themagnetic structure. The possible spin configurations ofthe four Cu ions in the primitive unit cell are describedby the basis vectors Ψ n (with n = 1 ...
12) of Γ mag , whichare displayed in Table II. We denote ferro- and antifer-romagnetic alignment of spins along the crystallographicaxis i by the modes f i and a i , respectively. The modes f x , a y , and f z resulting from the representation analysis im-ply the sign sequences (++++), (+ − + − ), and (++ −− )for the x -, y -, and z -axis components, respectively, of themagnetization of atoms (Cu1 Cu2 Cu3 Cu4). Note thatthe two-dimensionality of Γ mag (which derives from thespin reversal along a implied by the propagation vector)permits a physically equivalent description in terms ofthe modes a x , f y , and a z of spins in an adjacent subcellof the magnetic lattice (Table II).The observed magnetic structure factors were thencompared to those generated by the complete set of fourmagnetic models compatible with the lattice symmetry,which can be specified by the possible permutations of TABLE II: Basis functions Ψ n of the irreducible representa-tion Γ mag for propagation vector k = (0 . , . ,
0) in theorthorhombic space group
P nma . Only the real parts of thebasis vectors are given, with α = cos 2 π ( k y /
2) = 0 . f i and a i characterize ferromagnetic and antiferromag-netic spin configurations, respectively, along crystal axis i .Cu1 Cu2 Cu3 Cu4Ψ (100) ( α f x (000) (000)Ψ (010) (0 − α a y (000) (000)Ψ (001) (00 α ) f z (000) (000)Ψ (000) (000) (100) ( α f x Ψ (000) (000) (010) (0 − α a y Ψ (000) (000) (00 −
1) (00 − α ) f z Ψ (000) (000) ( − α a x Ψ (000) (000) (0 −
10) (0 − α f y Ψ (000) (000) (001) (00 − α ) a z Ψ (100) ( − α a x (000) (000)Ψ (010) (0 α f y (000) (000)Ψ (001) (00 − α ) a z (000) (000) the real and imaginary parts, R i and I i , of the Fourier co-efficients of the magnetic moment along the crystal axes i .They include a sinusoidal modulation with only real com-ponents R x R y R z (model 1), and three elliptical heliceswith components I x R y R z , R x R y I z , and R x I y R z (mod-els 2–4). Note that replacing R ↔ I yields physicallyequivalent descriptions. In all cases, the representationanalysis fully constrains the spin sequences of the atomsinside the primitive unit cell. The model proposed byKobayashi and coworkers for LiCu O , in which thephase relation of these moments was chosen arbitrarily,is not compatible with the lattice symmetry and was notconsidered here. The description of more complex struc-tures such as conical helices would require more than onemagnetic representation. Such models are therefore alsoincompatible with our analysis, which yields a single rep-resentation.The program Fullprof was used to perform least-squares refinements of the three Fourier componentsof the magnetization in the four symmetry-compatiblestructures. The outcome is shown in Table III, along withthe average moment and the moment amplitude derivedfrom these quantities and the quality-of-fit parameters R F and χ . Low residuals R F were obtained for models1–3. For the sine-wave modulated structure (model 1 inTable III) the components along the b - and c -axes werefound to be very similar, while the component along a is slightly reduced. For the helical structures (models 2 TABLE III: Comparison of different models for the magnetic structure of NaCu O . The imaginary and real components of theFourier coefficients of the Cu magnetic moments, ( I x , I y , I z ) and ( R x , R y , R z ), as well as the resulting real part R and totalmoment M are given in units of µ B . For the sinusoidal structure, the magnetic moment amplitude is listed ( ∗ ). The agreementfactors of the least-square fits are defined as R F = (cid:80) ( | F obs | − | F calc | ) / (cid:80) | F obs | and χ = (cid:80) ( | I obs − I calc | /σ ) / ( n − p ), where n and p are the numbers of observations and refined parameters, respectively.Model 1 2 3 4 5 6 7 I x I y I z R x R y R z R ∗ M R F χ F obs , of magnetic satellite reflections ( h, k, l ) M andthe structure factor F calc calculated magnetic structure fac-tors of the spiral model 3 (see Table III).( h k l ) M F obs ± σ F calc ± . ± . ± . ± ± ± ± . ± ± . ± ± . ± . ± ± . .
773 2 0.078 ± .
773 2 0.094 ± . ± ± . ± . ± and 3 in Table III), we obtained nearly the same val-ues of the three magnetization components as calculatedfor the sinusoidal structure. In model 2, where the realcomponents are in the bc -plane and the imaginary com-ponent points along the a -axis, the copper moments arerotating in two different planes, which subtend angles of ± ◦ with the b -axis and encompass the moments ofCu1 and Cu4, and Cu2 and Cu3, respectively. In model3, where the real components are in the ab -plane and theimaginary component points along the c -axis, we also ob-tained two different rotation planes for the moments ofCu1 and Cu3, and Cu2 and Cu4, respectively, which werefound to subtend angles of ± ◦ with the a -axis. InTable IV, the magnetic structure factors calculated forthe latter model (which is the most likely ground stateof NaCu O , see Section III) are compared to the exper-imentally observed ones. Model 4 yields a substantiallylarger R F and can hence be excluded.Figure 3 provides pictorial representations of models 2and 3. Both structures are elliptical helices by symmetry,but since the real and imaginary Fourier coefficients areidentical within the standard deviation of the refinement,the ellipticity is small. For comparison, we also inves-tigated simple circular helices with polarization planescoincident with high-symmetry planes of the crystal lat-tice (models 5–7 in Table III), but these models yieldedunsatisfactory residuals. Model 2 Model 3
13 42 12 43 a b acb
FIG. 3: (Color online) Proposed magnetic structures forNaCu O . Model 3 yields competitive residuals in the re-finement of the neutron diffraction data (Table III) and isconsistent with other experimental data on NaCu O (see thetext). The arrows indicate the sense of rotation of helices ondifferent copper-oxide chains. III. DISCUSSION AND CONCLUSIONS
We first discuss possible limitations of the representa-tion analysis that underlies the choice of magnetic struc-tures we have selected for comparison with the data. Theanalysis is based on the assumption that the spin Hamil-tonian includes terms up to bilinear order in the spinoperators. While this is sufficient in the vast majorityof magnetic insulators, we note that strong charge and/ororbital fluctuations may lead to higher-order terms thatrequire a modified analysis.
In view of the large Mott-Hubbard gap of copper-oxide chain compounds andthe large crystal-field splitting of the Cu d -orbitals, suchfluctuations are expected to be negligible in NaCu O .Since recent research has focused attention on ring-exchange terms in the spin Hamiltonian of cuprate spin-ladder compounds, one might also consider three-spin interactions of moments on directly adjacent copperoxide chains in NaCu O . While no quantitative infor-mation on the magnitude of such interactions is available,they are expected to be substantially smaller than the bi-linear interactions and therefore have no major influenceon the magnetic structure. An analysis of the current setof neutron diffraction data based on bilinear exchange in-teractions therefore appears to be adequate. In any case,an analysis based on a choice of models without recourseto symmetry considerations seems inadequate.Three of the four magnetic structures revealed by therepresentation analysis yield good agreement with theneutron data, and none of them can be singled out basedon the diffraction study alone. We therefore discuss thesestructures in the light of data collected by other experi-mental probes. Amplitude-modulated states such as thesinusoidal structure of model 1 have been observed in compounds with doped edge-sharing copper oxide chains,which support low-energy charge fluctuations, but asmentioned above, such fluctuations are strongly sup-pressed in Mott insulators such as NaCu O . Collinearincommensurate structures with weaker amplitude mod-ulations, such as the soliton lattice observed in spin-Peierls systems in high fields, differ from model 1 by thecontent of higher harmonics, which could not be deter-mined in the present study because of the weakness of thecorresponding higher-order Bragg reflections. However,the strongly anisotropic behavior of the Na NMR line-shape for H applied along the different crystallographicdirections is difficult to reconcile with a collinear struc-ture and indicates helical order.The temperature dependence of the uniform magneticsusceptibility, χ , yields further insight into the magneticorder. Recent measurements on crystals from the samebatch as the ones investigated here indicate a suppres-sion of χ for H (cid:107) b and c , but not a , upon cooling below T N . Helical states such as model 2, in which the a -axis isin the plane of polarization, are inconsistent with thesedata. In model 3, on the other hand, the a -axis subtendsa larger angle with the polarization plane than both b -and c -axes, in qualitative agreement with the low-fieldsusceptibility data. The low symmetry of the polariza-tion plane in this model also explains the apparent ab-sence of spin-flop transitions for fields up to 7 T. Rather than a sharp spin-flop, an external magnetic fieldalong a is expected to induce a gradual rotation of thepolarization plane towards the bc -plane, which explainsthe good agreement of high-field NMR data with a modelbased on bc -polarized spirals. The elliptical helix structure with alternating polar-ization planes shown in Fig. 3b is more complex thanthe simple circular, bc -polarized helix previously identi-fied based on less complete powder neutron diffraction and NMR data. We stress, however, that these featuresare mandated by the lattice symmetry and the propaga-tion vector, which are accurately known for NaCu O .We also note the alternating sense of rotation of helicespropagating along different copper-oxide chains (arrowsin Fig. 3). According to Eq. 1, this implies an anti-ferroelectric state in which every chain generates a ferro-electric moment, but the macroscopic electric polariza-tion vanishes. This explains the absence of ferroelectric-ity in NaCu O . A small reduction of the dielectric con-stant below T N may be indicative of an anti-ferroelectricstate. Our results also cast light on the origin of the ferro-electric polarization in LiCu O , which exhibits the samelattice symmetry and an incommensurate helix propa-gation vector of the same form, k = (0 . , ξ, O . Our representation analysis therefore alsoapplies to LiCu O . Since according to Eq. 1 none of thefour magnetic states revealed by this analysis supports amacroscopic ferroelectric polarization, we conclude thatthe ferroelectricity observed in LiCu O cannot be ofintrinsic origin, and that defects generated by Li–Cu in-tersubstitution must play a central role. In this respect,our conclusion agrees with those of Ref. 18, but disagreeswith those of Ref. 12. We emphasize, however, that ourfindings do not invalidate models according to which heli-cal magnetism generates ferroelectricity. In the frame-work of this scenario, substitutional defects may locallylift the compensation of ferroelectric moments in differ-ent copper-oxide chains, and a description of the resultingmagneto-electric defect pattern may be the key to an ex-planation of the puzzling magnetic field dependence ofthe ferroelectric polarization in LiCu O . An assess-ment of the possible influence of defects on the ferroelec- tric properties of LiCuVO , as well as other multifer-roics with helicoidal order, is an interesting subject offurther investigation. ACKNOWLEDGMENTS
We acknowledge useful discussions with Ph. Leininger,G. McIntyre, and J. Rodriguez Carvajal. We also ac-knowledge financial support by the DFG under grantSFB/TRR 80. H. Katsura, N. Nagaosa, and A.V. Balatsky, Phys. Rev.Lett. , 057205 (2005). I. A. Sergienko and E. Dagotto, Phys. Rev. B , 094434(2006). M. Mostovoy, Phys. Rev. Lett. For a recent review, see K.F. Wang, J.M. Liu, and Z.F.Ren, Adv. Phys. , 321 (2009). S. Park, Y.J. Choi, C.L. Zhang, and S.-W. Cheong, Phys.Rev. Lett. , 057601 (2007). Y. Naito, K. Sato, Y. Yasui, Yu. Kobayashi, Yo.Kobayashi, and M. Sato, J. Phys. Soc. Jpn. , 023708(2007). R.V. Pisarev, A.S. Moskvin, A. M. Kalashnikova, A.A.Bush, and Th. Rasing, Phys. Rev. B , 132509 (2006). J. Malek, S.-L. Drechsler, U. Nitzsche, H. Rosner, and H.Eschrig, Phys. Rev. B , 060508(R) (2008). Y. Matiks, P. Horsch, R. K. Kremer, B. Keimer, and A.V.Boris, Phys. Rev. Lett. , 187401 (2009). T. Masuda, A. Zheludev, B. Roessli, A. Bush, M. Markina,and A. Vasiliev, Phys. Rev. B , 014405 (2005). S. Seki, Y. Yamasaki, M. Soda, M. Matsuura, K. Hirota,and Y. Tokura, Phys. Rev. Lett. , 127201 (2008). Y. Yasui, K. Sato, Y. Kobayashi, and M. Sato, J. Phys.Soc. Jpn. , 084720 (2009). C. Fang, T. Datta, and J. Hu, Phys. Rev. B , 014107(2009). Y. Kobayashi, K. Sato, Y. Yasui, T. Moyoshi, M. Sato,and K. Kakurai, J. Phys. Soc. Jpn. , 084721 (2009). B.J. Gibson, R. K. Kremer, A.V. Prokofiev, W. Assmus,and G.J. McIntyre, Physica B , E253 (2004). Y. Yasui, Y. Naito, K. Sato, T. Moyoshi, M. Sato, and K.Kakurai, J. Phys. Soc. Jpn. , 023712 (2008). A.S. Moskvin and S.-L. Drechsler, Europhys. Lett. ,57004 (2008). A.S. Moskvin and S.-L. Drechsler, Phys. Rev. B , 024102(2008); A.S. Moskvin, Y.D. Panov, and S.-L. Drechsler, ibid. , 104112 (2009). V. Prokofiev, I.G. Vasilyeva, V.N. Ikorskii, V.V. Malakhov, I.P. Asanov, and W. Assmus, J. Solid State Chem. ,3131 (2004). L. Capogna, M. Mayr, P. Horsch, M. Raichle, R. K. Kre-mer, M. Sofin, A. Maljuk, M. Jansen, and B. Keimer, Phys.Rev. B , 140402(R) (2005). M. Horvatic, C. Berthier, F. Tedoldi, A. Comment, M.Sofin, M. Jansen, and R. Stern, Prog. Theor. Phys. Suppl. , 106 (2005). A.A. Gippius, A.S. Moskvin, and S.-L. Drechsler, Phys.Rev. B , 180403(R) (2008). Ph. Leininger, M. Rahlenbeck, M. Raichle, B. Bohnenbuck,A. Maljuk, C. T. Lin, B. Keimer, E. Weschke, E. Schierle,S. Seki, Y. Tokura, and J. W. Freeland, Phys. Rev. B. ,085111 (2010). A. Maljuk, A. B. Kulakov, M. Sofin L. Capogna, J.Strempfer, C. T. Lina, M. Jansen, and B. Keimer, J. Cryst.Growth , 338 (2004). E. F. Bertaut, Acta Cryst. A , 217 (1968); J. Magn.Magn. Mater. , 267 (1981). J. Rodriguez Carvajal, Physica B , 55 (1993). B. Bohnenbuck, I. Zegkinoglou, J. Strempfer, C.S. Nelson,H.H. Wu, C. Sch¨ußler-Langeheine, M. Reehuis, E. Schierle,Ph. Leininger, T. Herrmannsd¨orfer, J.C. Lang, G. Srajer,C.T. Lin, and B. Keimer, Phys. Rev. Lett. , 037205(2009). M. Reehuis, C. Ulrich, P. Pattison, B. Ouladdiaf, M. C.Rheinst¨adter, M. Ohl, L. P. Regnault, M. Miyasaka, Y.Tokura, and B. Keimer, Phys. Rev. B , 094440 (2006). S. Brehmer, H.-J. Mikeska, M. M¨uller, N. Nagaosa, and S.Uchida, Phys. Rev. B , 329 (1999). A. G¨oßling, U. Kuhlmann, C. Thomsen, A. L¨offert, C.Gross, and W. Assmus, Phys. Rev. B , 052403 (2003). M. Raichle, M. Reehuis, G. Andr´e, L. Capogna, M. Sofin,M. Jansen, and B. Keimer, Phys. Rev. Lett. , 047202(2008). V. Kiryukhin, B. Keimer, J.P. Hill, and A. Vigliante, Phys.Rev. Lett.76