Helimagnetism and Weak Ferromagnetism in Edge-Shared Chain Cuprates

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 316 (2007) 306–312 www.elsevier.com/locate/jmmm

Helimagnetism and weak ferromagnetism in edge-shared chain cuprates

S.-L. Drechslera,Ã, J. Richterb, R. Kuzianc,J.Ma´lekd, N. Tristana,B.Bu¨chnera, A.S. Moskvine, A.A. Gippiusf, A. Vasilievf, O. Volkovaf, A. Prokoﬁevg, H. Rakotoh, J.-M. Brotoh, W. Schnellei, M. Schmitti, A. Ormecii, C. Loisoni, H. Rosneri

aLeibniz-Institut fu¨r Festko¨rper- und Werkstoffforschung IFW Dresden, Postfach 270116, D-01171 Dresden, Germany bInstitut fu¨r Theoretische Physik, Universita¨t Magdeburg, D-39016 Magdeburg, Germany cInstitute for Problems of Materials Science, Kiev, Ukraine dInstitute of Physics, ASCR, Prague, Czech Republic eUral State University, 620083 Ekaterinburg, Russia fMoscow State University, 119992 Moscow, Russia gInstitut fu¨r Festko¨rperphysik, Technische Universita¨t Wien, A-1040 Wien, Austria hLaboratoire National de Champs Magne´tiques Pulse´s, 31432 Toulouse, France iMax-Planck-Institut fu¨r Chemische Physik fester Stoffe, D-01187 Dresden, Germany Available online 1 April 2007

Abstract

The present understanding of a novel growing class of chain cuprates with intriguing magnetic properties is reviewed. Among them, several undoped edge-shared CuO2 chain compounds show at low temperature a clear tendency to helicoidal magnetical ordering with acute pitch angles and sometimes also to weak ferromagnetism. Our analysis is based on the isotropic 1D frustrated J12J2 Heisenberg model with ferromagnetic (FM) 1st neighbor and antiferromagnetic 2nd neighbor exchange. The achieved assignment is supported by microscopic calculations of the electronic and magnetic structure. We consider NaðLiÞCu2O2, LiVCuO4 as the best studied helimagnets, Li2ZrCuO4 and other systems close to a FM quantum critical point, as well as Li2CuO2 with FM inchain ordering. The interplay of frustrated inchain couplings, anisotropy and interchain exchange is discussed. r 2007 Elsevier B.V. All rights reserved.

PACS: 74.72.Jt; 76.60.k; 75.10.b; 75.10.Pq

Keywords: Edge-shared cuprate chain; Quantum helix; Weak ferromagnetism; High-ﬁeld magnetization

1. Introduction—brief historical remarks direction b below Th 3, 24, and 13 K, respectively, which represent the first long-sought [14] helices with acute pitch 1 Undoped edge-shared CuO2 chain compounds (see, e.g., angles for quantum localized spin-2 systems. Such a special Figs. 1–4) exhibit a surprisingly large variety of magnetic helical assignment was achieved and accepted only stepwise ground states. Thus, below T N 9 K the prototypical after the synthesis of these compounds 15–20 years ago and Li2CuO2 [1,2] shows a commensurate Neél state dominated first not very meaningful measurements. At that time, by antiferromagnetic (AFM) interchain coupling but guided by the observed maxima in the magnetic suscept- accompanied also by a ferromagnetic (FM) inchain ibility wðTÞ and the specific heat cp at relatively low T, ordering and below 2.5 K a not yet understood canted nearly ideal (quasi)-1D Heisenberg AFM behavior with a magnetic structure [2]. In contrast, the closely related standard Neél-ordering below due to remaining interchain LiVCuO4 [3,4], LiCu2O2 [5–9] NaCu2O2 [9,11–13] show coupling was suggested [15–20]. It can be described by the incommensurate (IC) magnetic structures along the chain Hamiltonian X ab a b ÃCorresponding author. Tel.: +49 351 4659 384; fax: +49 351 4659 490. H ¼ Jij Si Sj þ Dij ðSi SjÞ, (1) E-mail address: [email protected] (S.-L. Drechsler). ij;ab

0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.03.200 ARTICLE IN PRESS S.-L. Drechsler et al. / Journal of Magnetism and Magnetic Materials 316 (2007) 306–312 307

Fig. 1. Crystal structure of LiCuVO4 with ideal planar CuO2 chains k b. Atomic notation: large orange —Cu, small red —O, large green —V, and small blue —Li.

Fig. 2. The DC structure of NaðLiÞCu2O2. Crystal structure with a DC in the center (left panel). Atomic notation the same as in Fig. 1; large grey —the magnetically non-active Cuð1þÞ ions, nonequivalent O in small red and brown . Projected bilayer onto the ða; bÞ plane with two double- chain’s (right panel). E and denote Cu sites in different planes. The main exchange paths are denoted by off-chain lines (a shortened notation J3 J3y and J4 J4y is used in the text). where i and j run over pairs of nearest neighbor (NN) and next-nearest neighbor (NNN) CuO4 plaquettes and the last antisymmetric exchange term (Dzyaloshinski–Moriya Fig. 3. The crystal structure of Li2ZrCuO4 with buckled chains k c. Notation as in Figs. 1 and 2; Cu—bright olive-green Zr—bright inside (DM)) allowed by symmetry only for certain non-ideal the magenta corner-shared VO6 octahedra (upper panel). Middle panel: systems under consideration, will be at first ignored. The the same for the buckled chain compound Pb2½CuSO4ðOHÞ2. Pb(H)— general Hamiltonian H applied to a real chain compound large (small) blue , S—green inside the green SO4 tetrahedra. Lower exhibits a hierarchy of isotropic and anisotropic couplings panel: the isomorphic structure of the two strongly buckled and ð Þ with the leading terms given by isotropic NN and NNN in- asymmetric corner-shared CuO3 chain compounds Rb Cs 2Cu2Mo3O12. Mo—green inside the green MO4 tetrahedra, Rb(Cs)—small blue . chain exchange denoted as J1 and J2, respectively, below. At the next step the possible presence of IC magnetic ground states with helical structures and anisotropic not unique [21]. The corresponding alternative set to z 1 0 symmetric exchange [17,18,20] was realized. By neutron symmetric to 4: z ¼ 0:266, 0.273, and 0.3276, leads to 0 diffraction for LiCuVO4 and NaðLiÞCu2O2 [3–5,11] the obtuse pitch angles f ¼ 95:76 ð96:4Þ , etc., i.e. AFM J1 1 corresponding propagation vectors near 4: z ¼ 0:234, 0.227, values would also fit the neutron diffraction data. An and 0.1724, respectively, (in units of the reciprocal lattice analysis of additional data such as inelastic neutron vector 2p=b of a single chain) and nonequivalent Bragg scattering, wðTÞ, and high-field magnetization data or reflexes near integer values k z have been found. Thus, convincing theoretical arguments [4,6–8,12,22,23] are two of the related pitch angles f ¼ 84:2 ð83:6 0:6Þ ,81:7 , necessary to settle the FM sign (i.e. J1o0). In 62:1 are close to p=2. However, strictly speaking, the acute NaðLiÞCu2O2, in principle, a significant AFM double- pitch angle assignment from neutron diffraction alone is chain (dc) exchange J~ (see Fig. 2, right panel) might affect ARTICLE IN PRESS 308 S.-L. Drechsler et al. / Journal of Magnetism and Magnetic Materials 316 (2007) 306–312

0.60 widely accepted [22,23,10] as a reasonable starting point NaCu O 2 2 especially supported also by new results for the isomorphic LiCu2O2 0.40 compound NaCu2O2 [11–13] except recent ARPES data for LiCu2O2 described within an unfrustrated 1D spinon– holon picture [26] whose microscopic interpretation re- 0.20 (k) [eV]

mains rather unclear (see Section 2). For NaCu2O2 as a n ε slight modiﬁcation of model (ii): a frustrated FAAA J12J4 0.00 model with two added AFM long-range exchanges J3 and

Energy J4 has been proposed [11]. -0.20 Finally, based on a low-T ﬁt of wðTÞ Hase et al. [27] adopted the FA-sc-J1J2-model (ii) also for RbðCsÞ2Cu2Mo3O12. Above 2 K no magnetic ordering has been detected. But for -0.40 Γ X S Y Γ Z U R T Li2ZrCuO4 (see Fig. 3) [28] and for Pb½CuðSO4ÞðOHÞ2 (see Fig. 4) a yet unresolved magnetically ordered state might 0.30 occur below 6 K and 2.7–2.8 K, respectively suggested by a sharp speciﬁc heat anomaly and a maximum of 0.20 dðwðTÞTÞ=dT [29].

0.10 2. Crystal structure and basic electronic structure of (k) [eV]

ε helimagnetic chain cuprates with acute pitches 0.00 Before turning to electronic and magnetic properties we Energy -0.10 would like to remind the reader about common and specific structural features for each of the considered compounds. -0.20 The commonly used chemical notation is somewhat misleading because it does not reflect properly the cuprate Γ Y S X Γ Z T A R and the CuO2 chain character which, however, dominates the low-energy electronic and the magnetic properties. For instance, according to the orbital analysis of states near the Fermi energy using the FPLO–LDA band structure code [30], transition metal ions, except Cu, do enter magnetically inert complex cations to compensate the anionic cuprate units. Thus, LiCuVO4 (Fig. 1) is not a vanadate but a lithium vanadyl cuprate which should be written more 2þ 2 correctly as ½LiVO2 ½CuO2 . In this compound V is almost in a cationic V5þ state with unoccupied 3d and 4s shells and therefore it is magnetically silent. Similarly, as Zr and Mo in Li2ZrCuO4 and RbðCsÞ2Cu2Mo3O12, respectively, exhibit unfilled 4d and 5s shells and are therefore found in cationic 4+ and 6+ states. In LiðNaÞCu2O2 there are two Cu sites: one in the chains is 2+ and magnetically active whereas a second one in between (see Fig. 2)is1+ and magnetically silent. Finally, Pb in Pb2½CuSO4ðOHÞ2 is in the cationic 4+ state with a filled 5d shell. Taking into 2þ Fig. 4. Comparison of LDA–FPLO band structures for the isomorphic account the strong correlation effects for Cu compounds double chain compounds ACu2O2,A¼ Li, Na, (upper panel), it becomes clear that all these undoped cuprates are charge Li2ZrCuO4 (middle panel), and Pb½CuðSO4ÞðOHÞ2 (lower panel) transfer insulators with correspondingly large gap values in G; X; Y; Z; ... is the standard wave vector notation for symmetry points the order of D 324 eV or even larger. However, the ð0; 0; 0Þ; ð2p=a; 0; 0Þ; ð0; 2p=b; 0Þ; ð0; 0; 2p=cÞ;..., respectively. pd observed smaller optical gap Eg 1:95 eV in LiCu2O2 [26] results probably from transition into an empty Cu(1) the IC magnetic structure under consideration. In this derived band lying inside the larger charge transfer gap of context, for LiCu2O2 two alternative frustration models the Cuð2ÞO2 chains in contrast to the charge transfer have been employed interpreting the inelastic neutron assignment suggested in Ref. [26]. The large holon and scattering and other experimental data: (i) the ‘‘AF- spinon dispersions of 2teff 0:74 eV and 0:5pJeff M–AFM’’–DC (AA-dc) or dimer liquid model [5,24,25] 55 meV, respectively, found there cannot be understood and (ii) the isotropic FM–AFM single-chain J12J2 model neither from the relative small band width of the half-filled (FA-sc-J1J2) [6–8]. In the meantime model (ii) has been cuprate band at the Fermi energy (see Fig. 4) within the ARTICLE IN PRESS S.-L. Drechsler et al. / Journal of Magnetism and Magnetic Materials 316 (2007) 306–312 309

1D LDA which amounts for all edge-shared CuO2 compounds of wðTÞ together with the Curie–Weiss temperature YCW under consideration 0.5–0.8 eV, only, nor from a simple obtained from its high-temperature asymptotics yield internal self-consistency. In fact, adopting a standard value constraints for the strength of frustration and the for the effective one-band (!)Hubbard U324 eV instead of contribution of residual interchain exchange. Typical the U dd 8 eV (appropriate only for a multi(five)band-pd shapes of wðTÞ together with fits by the FA-sc-J1J2-model 2 Hubbard model), one would arrive at Jeff ¼ 4teff =Up for NaCu2O2 are shown in Fig. 5. A systematic evolution w 136 meV, a value exceeding by an order of magnitude the of the empirical maximum positions T max=J2 in units of J2 commonly accepted value for the most AFM edge-shared of the same model applied to related systems is shown in CuO2 compound CuGeO3. Fig. 6 [33]. Our analysis of wðTÞ and cp for polycrystalline Li2ZrCuO4 [28] yields a 0:2820:3, i.e. a value rather 1 close to the quantum critical point ac ¼ 4 (the spiral-FM 3. Low-field properties and weak ferromagnetism transition). To the best of our knowledge, at present Li2ZrCuO4 is the most critical system from the helical This disparity in transition temperatures T h, pitches, the side. It deserves more detailed investigation in future. w possible subsequent occurrence of weak ferromagnetism In Fig. 6 also the predictions for Tmax=J2 within the below T h, or the absence of any ordering at low T results from a complex interplay between the strength of frustration, the actual internal anisotropy of inchain and perpendicular transfer integrals governed by the chain geometry including the Cu–O–Cu bond angle g90, the exchange anisotropy affected by local distortions from the ideal flat chain-geometry, the strength of the crystal field affected by the position and the charge of the cations, as well as the nature and the strength of the interchain coupling. RbðCsÞ2Cu2Mo3O12 shows strongly distorted non-planar chains formed by an unusual asymmetric (non-diagonal!) corner-sharing of CuO4-plaquettes with a Cu–O–Cu NN bond angle of 103 (see Fig. 3). The arrangement of NNN plaquettes resembles that in edge- shared CuO2 chains pointing to an AFM J2, whereas the FM nature of J1 is similar to that for the strongly buckled rungs in the pseudo-ladder compound MgCu2O3 with an angle g 108 [31]. Then J110 to 30 meV can be estimated. Adopting the same order of J25210 meV as for edge-shared CuO2-chains one arrives at parameters comparable with Li2ZrCuO4 [28]. For this reason we Fig. 6. Maximum temperature of wðTÞ in units of the estimated NNN included RbðCsÞ2Cu2Mo3O12 in our phenomenological analysis too [32]. inchain exchange integral J2 of the FA-sc-J1J2-model vs. the inchain frustration parameter a ¼J =J for various edge-shared chain Susceptibility wðTÞ and magnetization measurements 2 1 cuprates. The numbers denote the compounds: 1—Li2ZrCuO4,2— combined with specific heat c ðT; HÞ data provide useful p Pb2½CuSO4ðOHÞ2,3—Rb2Cu2Mo3O12,4—Cs2Cu2Mo3O12, 5—LiCu2O2, insight in the basic spin–spin interactions. Thus, the shape 6—NaCu2O2, 7—LiCuVO4.

H II a ) B μ

emu/mol] 3 H II b -2

-3 H II c 2 theory

1 moment (10 susceptibility [10 1 0 0 100 200 300 0246 Temperature [K] Field (T)

Fig. 5. Theoretical spin-susceptibility wðTÞ for large periodic spin-1/2 FM-AFM Heisenberg chains with N ¼ 16 sites compared with experiments for NaCu2O2 taken from Ref. [12] (left panel). Magnetization vs. applied ﬁeld at T ¼ 2 K (right panel). ARTICLE IN PRESS 310 S.-L. Drechsler et al. / Journal of Magnetism and Magnetic Materials 316 (2007) 306–312

2þ FA-sc-J1J2-model are shown as derived from full diagona- shift of the Cu ion off from the plaquette center (Fig. 2) lizations of periodic rings with N ¼ 20 sites using own which might explain the origin of DM [37]. Alternatively, it results [28] as well as Ref. [34], both being in accord with might point to a secondary phase transition induced by the the transfer matrix renormalization group results of Ref. increasing spiral order parameter with decreasing T. In this [35] taken as estimates for N ¼1. Thus we may conclude way, strong enough local magnetic moments induced by that our FA-sc-J1J2-model yields an excellent starting point the spiral formation visualize finally the broken inversion for the considered chain cuprates. Despite small correc- symmetry of a single CuO2 chain at TpT oT h and allow tions due to possible exchange anisotropy and doping the smooth occurrence of strong enough DM exchange and effects caused by small deviations from stoichiometry in weak FM below T only. Finally, we note that, in real compounds the general good unified description principle, a symmetric exchange anisotropy [10] might clearly supports our point of view that the compounds explain the spiral orientation and according to Ref. [38] for under consideration in fact belong to a special subclass of ab1 (only roughly fulfilled here) also the occurrence of chain cuprates different from the ‘‘FM’’ Li2CuO2 and the chiral structures coexisting with ferromagnetism. This well-known AFM spin-Peierls system CuGeO3. might be of relevance in the context of a much weaker The staggered tilting of CuO4 plaquettes of all four non- induced FM moment at the limits of experimental error planar (buckled) chain compounds Li2ZrCuO4,Pb2½CuSO4 bars in the isomorphic LiCu2O2 [39] which exhibits a 1 ðOHÞ2, and RbðCsÞ2Cu2Mo3O12 as well as within the only [6–8]. bilayers of NaðLiÞCu2O2 may cause along a-axis oscillating terms in H and this way a staggered field induced 4. The high-field magnetization and the search for novel magnetization which yields a spin gap Ds for one of the commensurate quantum phases two acoustic branches near G, akin to the mechanism proposed for Cu-benzoate at the 1D Brillouine zone (BZ) Some details of the magnetic spiral, especially its boundary [36]. A field dependent Ds could explain the field evolution under applied external magnet fields, are still sensitivity of cp at moderate H seen in NaCu2O2. unclear. However, the small exchange integrals obtained Another interesting point is the weak hysteresis which for LiCuVO4 allow the determination of the field occurs in NaCu2O2 below T 829:5 K, well below the dependent magnetization up to the saturation field spiral ordering at Th ¼ 13 K (see Figs. 5, 7). The observed Hs40250 T [4] when all spins are aligned (at T ¼ 0). 3 local moment M0 4 10 mB at T ¼ 2 K is a typical Remarkably, for the FA-sc-J1J2-model even with some value for other weak FM cuprates. The weak ferromagnet- further extensions Hs can be found analytically [40], this ism can be ascribed to the presence of DM exchange way providing a useful constraint for the J-values. More- allowed in NaCu2O2. In fact, the inspection of the local over, these calculations point to a novel physical picture structure of the slightly tilted CuO4 plaquettes reveals a dominated by bound two-magnon states at least at high fields HpHs. In particular, it was found for large but finite 30 systems that there the magnetization determined by exact full diagonalizations (see Fig. 8) or applying the powerful DMRG-technique [34] changes in steps of DSz ¼ 2 unlike 25 H = 0.04 Tesla the usual one-magnon behavior where DSz ¼ 1. Based on H = 0.2 Tesla approximate mean-field (MF) calculations a related novel H = 0.4 Tesla commensurate period-4 collinear quantum phase above the H = 1 Tesla 20 H = 0.1 Tesla conic spiral phase for weak and medium fields has been emu/mole]

-3 predicted [41]. In Ref. [4], the classical saturation ﬁeld cl 2 H II b (chain-axis) gmBHs ¼ 2J2ð1 0:25=aÞ determined by the one-magnon instability of the fully aligned FM state has been ascribed 15 to the maximum of dM=dH which for Hkc amounts about 1 41 T (see Fig. 8). However, the true quantum spin-2 2 saturation ﬁeld gm Hs ¼ 2J2ð1 0:25=a 0:5=aÞ=ð1 10 B NaCu2O2 1=aÞ derived from the two-magnon instability of the fully aligned FM state [42,40] yields Hs 50 T using J1 and J2

magnetic susceptibility [10 from our wðTÞ-fit in reasonable accord with experiment. 5 Anyhow, within the numerical calculations no jump in the magnetization near 41 T could be detected. In accord with the statement given in Ref. [34] that for ab1 practically the 0 whole field region is determined by the two-magnon sector. 2468101214 Fig. 8 shows that this peculiar quantum effect holds also Temperature [K] already for a ¼ 2:39 and for a realistic symmetric Fig. 7. Low temperature spin-susceptibility wðTÞ for different magnetic anisotropy of the expected easy-axis type [43]. Thus, we fields H applied in chain direction. are forced to conclude that both exactly treated pure 1D ARTICLE IN PRESS S.-L. Drechsler et al. / Journal of Magnetism and Magnetic Materials 316 (2007) 306–312 311

1 commensurate phase estimated as DHq1D5 T from Fig. 8 isotropic exchange H s is realized. Interestingly, a failure of the 1D isotropic FA- J2=2.39, J1 =-1 anisotropic exchange sc-J1J2-model was mentioned by Lu et al. [35] for the xy z description of the high-ﬁeld magnetization of RbðCsÞ2Cu2 J =2.39; J = -1, J = -1.3 2 1 1 Mo3O12. They ascribed it to missing DM exchange Akima spline

max suggested by the low symmetry generic for buckled chains (see Fig. 3). Since in our case, with ﬂat CuO2 chains, there is no reason for a signiﬁcant DM exchange, we suggest that 0.5 T = 0 an interchain scenario as considered here might contribute also to that failure. In this context it should be noted that near the FM critical point the 1D contribution to the

Magnetization m/m saturation field Hs is small by definition. Hence, the MðHÞ N= 28 measurements at high fields might mainly probe the AFM Δ Sz= 2 interchain exchange. Thus, the situation is much different (two-magnon steps) from the cases of fitting wðTÞ and cpðTÞ which are dominated by strong sharp peaks at low T in accord with 0 024513 the predictions of the FA-sc-J1J2-model [28,34,35]. μ Magnetic field g BH/|J1| 5. Frustration from multiband Hubbard models and the role of interchain interaction H II c 2 T=1.6 K If one takes into account only NN and NNN exchange, T= 4.2 K for AFM coupling J240 along a single CuO2 chain, one is

) ? left with a frustration problem irrespective of the sign of J1. 1.5 T For Li2CuO2, where many experimental data are available, hel = 2.1 K te (2.1 - 2.8 K) we ﬁtted an extended ﬁve band Hubbard-model to describe phase(s the optical conductivity and O 1s X-ray-absorption data. 1D-like Then the low-energy states of small CunO2n clusters have LiVCuO4 1 <------> been mapped onto the corresponding clusters of the FA-sc-

commensura quantum one-magnon J1J2-model. As a result one arrives at a significant AFM sector MF-like Magnetization [a.u.] NNN exchange integral due to a non-negligible NNN <------> transfer integral t2y leading to a ¼ 0:7. However, such a 0.5 <------> large value of a clearly exceeds the well-known critical incommensurate <------> 1D 1 value of ac ¼ 4 for a spiral instability. This seeming (pre) conical helix contradiction with ENS data which show a FM inchain two-magnon sector ordering can be resolved taking into account the non- 0 0 10 20 30 40 50 negligible specific interchain coupling in Li2CuO2 [7,14].In Magnetic field H [Tesla] LiVCuO4 and NaðLiÞCu2O2, the interchain coupling only weakly affects ac since according to Refs. [4,6,12] it is not Fig. 8. Upper panel: magnetization vs. applied magnetic field at T ¼ 0 for frustrated. Here, the in-phase arrangement of chains a long periodic chain with N ¼ 28 sites within the easy-axis anisotropic slightly reduces the effective FM NN J1. and isotropic FA-sc-J1J2-models. Lower panel: experiment: LiVCuO4 and possible scenarios beyond 1D. 6. Summary

To conclude, we have shown that the FA-sc-J1J2 model models as well as simple MF-like approaches fail to supplemented with realistic interchain couplings from band describe the strong, nearly symmetric maximum in dM=dH structure calculations and small exchange anisotropies to near 41 T below Hs. On the other hand, there is no long- explain the observed weak ferromagnetism reveals a proper range spiral phase in 1D according to the Mermin–Wagner description at low magnetic fields of various edge-shared theorem. We expect that the present finite interchain CuO2 chain compounds. The detailed understanding of the exchange responsible for the observed finite Th near 3 K detected spiral states and the search of novel quantum [3,4] at H ¼ 0 is also responsible for a strong reduction of phases, especially under high magnetic fields, remain the two-magnon dominated collinear commensurate phase. challenging issues for future work. As a result, in the real quasi-1D situation a ‘‘crossover’’ Although all considered compounds exhibit rather between the predicted extremely broad 1D region different FM NN inchain exchange couplings: J1 ¼1:8 DH1D50 T and the narrow mean field region DHmf 1T to 40 meV, the NNN counter part differs only by a factor broad region for the novel quantum two-magnon collinear of two or three with J2 in between 4 and 11 meV. Li2CuO2 ARTICLE IN PRESS 312 S.-L. Drechsler et al. / Journal of Magnetism and Magnetic Materials 316 (2007) 306–312 is found to be very close to a FM–AFM helical in-chain Finally, the strong field dependence of the specific heat and ground state still prevented by a strong specific, frustrated the shape of the low temperature peak near 6 K reported in 1 interchain coupling, whereas the long sought ‘‘FM’’ spin-2 Ref. 28 can be explained by the vicinity of Li2ZrCuO4 close helix with an acute pitch is realized in LiCuVO4 and to the quantum critical point near the ferromagnetic-spiral- LiðNaÞCu2O2 and possibly also in Li2ZrCuO4 and transition. Pb2½CuSO4ðOHÞ2. Concerning RbðCsÞ2Cu2Mo3O12,a finite T h might be still observed around 2 K. Other members of this fascinating family will be considered References elsewhere. [1] S. Ebisu, et al., J. Phys. Chem. Solids 59 (1998) 1407. Acknowledgments [2] E.M.L. Chung, et al., Phys. Rev. B 68 (2003) 144410. [3] B.J. Gibson, et al., Physica B 350 (2004) E253. [4] M. Enderle, et al., Europhys. Lett. 70 (2005) 237. The DFG, 486 RUS 113, KL 1824 (Emmy-Noether- [5] T. Masuda, et al., Phys. Rev. Lett. 92 (2004) 177201. program), GIF (I-811-237.14103), INTAS, CRDF JSTC [6] A.A. Gippius, et al., Phys. Rev. B 70 (2004) R01426. 3501, and RFBR (06-02-16088, 07-02-91201, 07-02-00350) [7] S.-L. Drechsler, et al., Phys. Rev. Lett. 94 (2005) 039705. are acknowledged for financial support. [8] S.-L. Drechsler, et al., J. Magn. Magn. Mater. 290 (2005) 345. [9] A.A. Gippius, et al., J. Magn. Magn. Mater. 300 (2006) e335. [10] M. Mihaly, et al., Phys. Rev. Lett. 97 (2006) 067206. Note added in Proof [11] L. Capogna, et al., Phys. Rev. B 71 (2005) R140402. [12] S.-L. Drechsler, et al., Europhys. Lett. 73 (2006) 83. Since the original submission of the manuscript (June 24, [13] K.-Y. Choi, et al., Phys. Rev. B 73 (2006) 094409. [14] Y. Mizuno, et al., Phys. Rev. B 57 (1998) 5326. 2006) the research field of FM frustrated edge-shared CuO2 [15] M. Yamaguchi, et al., J. Phys. Soc. Japan 65 (1996) 2998. chain compounds has been rapidly developed. Hence, we [16] A.N. Vasil’ev, et al., Phys. Rev. B 64 (2001) 024419. would like to inform the reader about recently published [17] Ch. Kegler, et al., Eur. Phys. B 22 (2001) 321. results: (i) the observation of ferroelectricity in the spiral [18] H.-A. Krug von Nida, et al., Phys. Rev. B 65 (2003) 134445. phases of LiVCuO4 and LiCu2O2 by Y. Naito et al. J. Phys. [19] Y. Maeda, et al., Phys. Rev. Lett. 95 (2005) 037602. Soc. Jpn. 76 (2007) 023708 and S. Park et al., Phys. Rev. [20] Ch. Kegler, et al., Phys. Rev. B 73 (2006) 104418. [21] D. Dai, et al., Inorg. Chem. 43 (2004) 4026. Lett. 98 (2007) 057601, respectively, (ii) a new theoretical [22] T. Masuda, et al., Phys. Rev. Lett. 94 (2005) 039706. analysis of exchange integrals in LiCu2O2 by V. Mazur- [23] T. Masuda, et al., Phys. Rev. B 72 (2005) 014405. enko et al., cond-mat/0702276 (2007), (iii) an interpretation [24] S. Zvyagin, et al., Phys. Rev. B 66 (2000) 064424. of the same high-field magnetization data of LiVCuO4 as [25] K.-Y. Choi, et al., Phys. Rev. B 69 (2004) 104421. shown in Fig. by M.G. Banks et al., J. Phys. Cond. Mat. [26] M. Papagno, et al., Phys. Rev. B 73 (2006) 115120. [27] M. Hase, et al., Phys. Rev. B 70 (2005) 10426; 19(2007) 145227 as well as (iv) the observation of an M. Hase, et al., J. Appl. Phys. Lett. 97 (2005) 10B303. unusual strong field dependence of the specific heat in [28] S.-L. Drechsler, et al., Phys. Rev. Lett. 98 (2007) 077202. Li2ZrCuO4 [28]. [29] M. Baran, et al., Phys. Stat. Sol. (c) 3 (2006) 220. Due to the lack of space we may only briefly comment [30] K. Koepernik, H. Eschrig, Phys. Rev. B 59 (1999) 1743. some aspects directly related to the present paper. The [31] S.-L. Drechsler, et al., Physica C 408 (2004) 270. [32] Due to the incomplete structural data a band structure analysis of the microscopic reason for the observed effect (i) remains J’s cannot be performed for Cs2Cu2Mo3O12 at present. unclear especially with respect to uncertainties/discrepan- [33] We note that the J-values for RbðCsÞ2Cu2Mo3O12 have been slightly cies of the approximately known magnetic spin structure of changed compared with the values of low temperature fits reported in the helix [5]. Using LMTO and an LSDA+U calculations Refs. [27,28] using also the reported FM Curie–Weiss temperatures the authors of Ref. (ii) arrive at somewhat different values and adopting a weak interchain interaction of about 1 meV with two neighboring chains [28]. of the ferromagnetic J1 and the antiferromagnetic J2 [34] F. Heidrich-Meisneret, et al., Phys. Rev. B 74 (2006) 020403. exchange integrals as compared with our first analysis [6] [35] H.T. Lu, et al., Phys. Rev. B 74 (2006) 134425. (aimed at this early stage only to a proper prediction of [36] M. Oshikawa, et al., Phys. Rev. Lett. 79 (1997) 2886. their signs and the elucidation of the main origin of the [37] The nearly c orientation of the main DM-vectors might explain the frustration) and the later phenomenological analysis of the observed nearly ðabÞ orientation of the helix. In a rigorous sense, we expect a more complex spin structure with two alternatingly tilted wðTÞ data [8]. Anyhow, V. Mazurenko et al., do confirm helices from each bilayer boundary. Using the fitted g and J1 we our result that the interchain coupling within one double c estimate the NN DM vector as j Di;iþ1 j¼ ðDgc=gÞjJ1 j0:5 meV. chain of LiCu2O2 is very weak at variance with the result of For more details see Ref. [12]. the spin wave analysis of the neutron date given in Refs. [38] M. Zarea, et al., Eur. Phys. J. B 39 (2004) 1434. 22,23. Concerning the slight cusp seen near 8 K in the [39] R. Klingeler, private communication. [40] R. Kuzian, S.-L. Drechsler, Phys. Rev. B 75 (2007) 024401. magnetization data of LiVCuO4, the authors of Ref. (iii) [41] D. Dmitriev, V. Krivnov, Phys. Rev. B 73 (2006) 024402. suggest in accord with our ‘‘1D-like scenario’’ an upper [42] A. Chubukov, Phys. Rev. B 44 (1991) 4693. limit for the field induced instability of the ordinary helix. [43] V. Yushankai, R. Hayn, Europhys. Lett. 47 (1999) 116.