Spin-driven ferroelectricity and possible antiferroelectricity in triangular lattice A A antiferromagnets CrO ( = Cu, Ag, Li, or Na) 2 S. Seki1, Y. Onose1,2, and Y. Tokura1,2 1 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan 2 Multiferroics Project, ERATO, Japan Science and Technology Agency (JST), Tokyo 113-8656, Japan (Dated:) Correlation between dielectric and magnetic properties was investigated on thetriangular lattice antiferromagnets ACrO2 (A = Cu, Ag, Li, or Na) showing 120-degree spiral spin structure with easy-axis anisotropy. For the A = Cu and Ag compounds with delafossite structure, ferroelectric 8 polarizationemergesuponthespiralspinorder,implyingthestrongcouplingbetweentheferroelec- 0 tricityandspiralspinstructure. Ontheotherhand,fortheA=LiandNacompoundswithordered 0 rock salt structure, no polarization but only clear anomalies in dielectric constant can be observed 2 upon the spiral spin order. The absence of polarization can be interpreted as the antiferroelectric n state induced by the alternate stacking of Cr3+ layer with opposite spin vector chirality. These a resultsimplythatavastrangeoftrigonally stackedtriangular-lattice systemswith 120-degree spin J structurecan bemultiferroic, irrespective of theirmagnetic anisotropy. 4 PACSnumbers: 75.80.+q,77.22.Ej,75.40.Cx 2 ] l The correlationbetweenmagnetic anddielectric prop- The key issue is the microscopic mechanism of cou- e - erties has long been one of the important topics in the pling between ferroelectricity and magnetic order. Al- r condensed matter physics[1, 2]. The early attempts to though the symmetry analysis of spin structure can give t s realize material with both dielectric and magnetic or- go/no-gorule and predict the possible direction of spon- . t ders (multiferroics) met difficulties, since normally these taneous polarization[5], thorough understanding of the a m two features are mutually exclusive in their microscopic microscopicoriginisstilllacking. Sofar,oneofthemost origin. Even in the rare example of multiferroics, the successful schemes to explain the behavior of ferroelec- - d magnetic and dielectric phase transitions take place sep- tric spiral magnet is the spin-current model[6], in which n arately,resultinginweakcouplingbetweenbothfeatures. theelectricpolarizationP producedbetweenmutually- ij o In a recently discovered new class of multiferroics, how- cantedmagneticmomentsatneighboringsitesiandj(S c i [ ever, ferroelectricity arises simultaneously with the spin and Sj) is given as order[3],inwhichmagnetic(orelectric)controlofdielec- 1 v tric(magnetic)propertiesbecomepossible[4]. Thisgroup Pij =A0·eij ×(Si×Sj) (1) 7 of materials is now known to commonly show magnetic 5 frustration, which leads to complex spin order such as Here, eij is the unit vector connecting the site i and 7 helimagnetic structure. j, and A0 a coupling constant related to the spin-orbit 3 andspinexchangeinteractions. Thismodelpredictsthat . 1 a helimagnet with transverse spiral components can be 0 (a) delafossite (b) ordered rock salt ferroelectric, and well explains the ferroelectric behav- 8 iors observed for RMnO3 (R = Tb and Dy) [3, 7, 8], 0 : Ni3V2O8[9],CoCr2O4[10],MnWO4[11],LiCu2O2[12,13], v LiCuVO4[14, 15], and so on. i X c In contrast, the explanation of magnetoelectric cou- - a pling is notsostraightforwardfortriangularlattice anti- r x a is ferromagnet, the most typical example of geometrically frustrated spin system. With classical Heisenberg spins, ◦ this system generally favors120 spiralspin structure at the ground state. Depending on the sign of anisotropy O2- term H′ = DP(Sz)2, the spin spiral is confined in the A1+ i plane parallel (D > 0 : easy-plane type) or perpendic- ular (D < 0 : easy-axis type) to the triangular lattice Cr3+ plane[16]. Although in neither case can the spin-current modelpredictferroelectricity,theappearanceofpolariza- ◦ tion in 120 magnetic phase has recently been reported FIG. 1: (color online). Crystal structure of ACrO2 : (a) delafossite structure (A = Cu or Ag) and (b) ordered rock for RbFe(MoO4)2 with easy-plane anisotropy[17]. This salt structure (A= Li or Na). behaviorcanbejustifiedbythesymmetryconsideration, yet leaving its microscopic origin still unclear. 2 NnsteartTu).hciCsteuautrnCaeorrtg(OhFe2etirgao.enfxdta1hmAi(spag)Cple)ar.opOfeE2trr,acicraAhynCsgterualOellalm2irze(elanAittnt=tifcooerCtmahunes,tdiAtfeheglrea,rfotLomrsii,saaiontger-- [·c1 (emu/mol)0-3]667...050 (a) CuCrO2 0.5T [·c1 (emu/mol)0-3]667...050 (d) AgCrO2 0.5T gular lattice and stacks along the c-axis in the sequence 7.5 (b) 7.5 (e) Cr3+-O2−-A+-O2−-Cr3+. LiCrO2 and NaCrO2 crystal- lize into the ordered rock salt structure with the similar e 7.0 e 7.0 triangular lattice (Fig. 1(b)). Both belong to the space groupR¯3m,andonly adifference is the stackingpattern 6.5 0T 6.5 0T of O2−-A+-O2− layers. While the delafossite structure 40 (c) (f) 5 has the straight stacking, the ordered rock salt struc- 20 ture has the zigzag one. In both cases the rhombohedral 2C/m) 0 +300 kV/m 2C/m) 0 +300kV/m (ABCABC...) stackingisrealizedamongCr3+ layers,al- mP ( -20 -300 kV/m mP ( -300kV/m thoughthedistancebetweenthemismuchshorterinthe -5 -40 0T 0T latter case[18]. The magnetic properties are dominated 0 10 20 30 0 10 20 30 by Cr3+ ion with S = 3/2 spin, which is surrounded Temperature (K) Temperature (K) by octahedron of O2−. Because of the geometrical frus- tration of intra-plane antiferromagnetic exchange inter- FIG. 2: (color online). Temperature dependenceof magnetic ◦ susceptibilityχ,dielectricconstantε,andelectricpolarization action, the 120 spin structure is realized at the ground state. Based on several neutron studies as mentioned P for (a)-(c) CuCrO2 and (d)-(f) AgCrO2 with delafossite structure. In (c) and (f), the magnitude and sign of poling later,thesesystemsaregenerallyrecognizedtohaveeasy- electric field are also indicated. axis anisotropy along the c-axis. Inthispaper,wereportthediscoveryofthespin-driven ferroelectricity and also possible antiferroelectricity by 120◦ spinstructurewitheasy-axisanisotropy. Combined 120◦ spinstructure withspiralplaneincluding c-axisbe- with the case for RbFe(MoO4)2[17], we can predict that lowTN[20]. AtTN,dielectricconstantalsoshowsasharp a vast range of trigonally stacked triangular-lattice sys- anomaly,andspontaneouselectricpolarizationbeginsto tems with the 120◦ spin structure can be multiferroic, develop. Withoppositepolingelectricfield,thepolariza- irrespective of their magnetic anisotropy. tion direction can be reversed. These indicate the ferro- Powder specimen of CuCrO2, AgCrO2, LiCrO2, and electric nature of the magnetic ground state, and imply NaCrO2 were prepared by solid state reaction from sto- the coupling betweenthe ferroelectricityandspiralmag- ichiometric mixture of CuO, Ag, Li2CO3, Na2CO3 and netic order. ◦ Cr2O3. They wereheatedat1000 C for24 hoursinair, Figure 3(a) indicates the symmetry elements in the at 900 ◦C for 48 hours in O2, at 1200 ◦C for 24 hours in ACrO2 system with space group R¯3m; reflection mir- ◦ air,andat1100 Cfor30hoursinAr,respectively. Pow- ror (m), two-fold rotation axis(2), inversion center, and derx-raydiffractionmeasurementsshowednodetectable three-fold rotation axis along the c-axis. Because of the impurity, except slight Ag phase in AgCrO2 specimen ambiguity of spin structure, hereafter we examine two andslightCr2O3 phaseinNaCrO2 specimen. Theywere typesof120◦magneticorderwithspinspiraleitherinthe pressed into rod, sintered with additional heating, and (110) plane(Fig. 3(b)) or in the (1¯10) plane(Fig. 3(c)). cut into thin plate. The typical sample size is 4.5mm × The former case can be considered as the proper screw 4.5mm×0.7mm. As the electrodes,silverpaste wasput magnetic structure, whose spins rotate in the plane per- onthe widestfaces. Dielectric constantwasmeasuredat pendiculartothemodulationvector. Recently,somespe- 100kHz using anLCR meter. To deduce the electric po- cificspeculationwasgivenbyArima[21]forthissituation larization, we measured the pyroelectric current with a with the delafossite crystal structure and proper screw constantrateoftemperature sweep(2K/min∼20K/min) spinstructure. Withthe(110)spinspiralplane,onlya2′ andintegratedit withtime. To obtaina single ferroelec- symmetryelement, two-foldrotationaxis alongthe [110] tric domain, the poling electric field was applied in the directionwithtimereversaloperation,remainsunbroken. cooling process and removed just before the measure- SinceelectricpolarizationvectorPmustbeinvariantun- ments of pyroelectric current. Heat capacity was mea- derthesymmetryoperation,onlyPperpendiculartothe sured by the thermal relaxation method. Magnetization spin spiral plane (along the [110] direction) is allowed. was measured with a SQUID magnetometer. The problem to be solved next is its microscopic origin. Figures 2 (a)-(c) show the temperature dependence of Because any 120◦ spin structure gives the same S ·S i j magnetic susceptibility, dielectric constant, and electric for all bonds in the regular triangular lattice, conven- polarizationforCuCrO2. Thesusceptibilityshowsaclear tionalmagnetostrictioncannotcausethenetpolarization kink at TN ∼ 24K; TN is in accord with the previous with centrosymmetric crystal structure. Another candi- report[19]. A former powder neutron study suggests the date for the microscopic origin of P is the spin-current 3 model or inverse Dzyaloshinskii-Moriya mechanism rep- resented by Eq. (1). However, this also fails to explain the emergence of ferroelectricity for the regular triangu- lar lattice. Recently, Jia et al. pointed out that the spin-orbitinteractionbringsaboutsome modificationon d-p hybridization between ligand ion and 3d magnetic ion, which can cause the polarization along the bond direction[22, 23]. Although this term oscillates in the crystal and usually cannot cause macroscopic polariza- tion, some components along the modulation vector are proven not to be canceled out in the delafossite system with proper screw spin structure[21]. In CuFe1−xAlxO2 with the same crystal structure and the proper screw spin configuration (with incommensurate wave number q ∼ 0.22), the emergence of the polarization along the modulation vector is confirmed[24, 25, 26]. The similar situation is anticipated to occur in CuCrO2. In the case of (1¯10) spiralplane(Fig. 3 (c)), on the other hand, only areflectionmirrorcansurviveordisappeardependingon the spin direction. Therefore,fromthe symmetry, polar- izationcanbe allowedinanydirection. Thespincurrent FIG.3: (color online). (a) Symmetryelementsin theACrO2 modelpredictsthepolarization(1−α)P0alongthec-axis, systemwith spacegroupR¯3m: twofold rotation axis(2),re- whereαrepresentsthedifferenceofcouplingconstantA0 flectionmirror(m),andthreefoldrotationaxiswithinversion inEq. (1)betweenchainsalong[110]and[100](or[010]). center(trianglewithsmallcircle). O2−siteabove(below)the Giventheisotropiccouplingconstant(α=1),the polar- Cr3+ layer is indicated as closed (open) circle. (b)-(c) Sym- ◦ metry elements compatible to 120 spin structure with (b) izationshouldvanish,andhenceothermicroscopicorigin (110) spiral plane or (c) (1¯10) spiral plane. The thick bars would be required. Note that similar argument as above (left panel) indicate the spin spiral plane. Electric polariza- can be constructed for other centrosymmetric trigonal tion expected from the spin-current model along each chain systems. is also indicated, such as ±P0 and −(α/2)P0 (see text). Itisinterestingtoseeagenericfeatureofthedielectric response in other triangular-lattice Cr-oxides. Among them, the isostructural material AgCrO2 also shows the similar coupling between ferroelectricity and magnetic 20 LiCrO2 20 NaCrO2 order. Figures2(d)-(f) indicatethe temperatureprofiles mol)) (a) mol)) (d) mofatghneestaicmseuspcheypstiicbaillitqyuaisntoibtiseesrvfoedr AatgCslrigOh2t.lyTlhoweekrintkemin- (cid:215)C (J/(K 100 0T (cid:215)C (J/(K100 0T perature, TN ∼21K. Again, anomaly in dielectric con- 8.5 (b) 7 (e) stant and appearance of ferroelectric polarization P are observedatTN, althoughthe P value is reducedas com- e 8.0 e 6.5 paredwithCuCrO2. Aformerpowderneutronstudyhas ◦ proposedaslightlymodulated120 spinstructureforthe 0T 0T magnetic ground state below TN[27]. Mekata et al. ex- 7.5 (c) 6 (f) plained this modulation by the competition between the 5 5 intra-planeandinter-planeexchangeinteractions,andre- 2m) +250kV/m 2m) +250kV/m ported the shorter correlation length[27] and larger spin C/ 0 C/ 0 fluctuation[28] than in CuCrO2. Although the detail of mP ( mP ( -5 -5 magnetic structure, such as the direction of spin spiral 0T 0T plane, has not been determined yet, the smaller spon- 0 20 40 60 0 10 20 30 40 50 taneous polarization value in AgCrO2 (∼ 1/5 of that Temperature (K) Temperature (K) for CuCrO2) is consistent with these features. Since di- electric constantε reflectsthe fluctuation ofpolarization FIG. 4: (color online). Temperature profiles of specific heat ∆P in the form of ε−ε∞ ∝ h|∆P|2i/kBT, the weaker capacity C, dielectric constant ε, and electric polarization P anomaly in ε must come from the smaller polarization. for(a)-(c)LiCrO2 and(d)-(f)NaCrO2 withorderedrocksalt structure. Note that the ordinate scale in (c) and (f) is the In addition to the above delafossite crystals, we have same as in Fig. 2(f). also investigated LiCrO2 and NaCrO2 composed of the similar CrO2 sheet but with ordered rock salt structure 4 (Fig. 1(b)). The magnetic structure of LiCrO2 has been system. For the A = Li and Na compounds with or- investigated by the polarized neutron study on a single dered rock salt structure, by contrast, no polarization crystal[29], and below TN ∼ 60K[30] the proper screw butonlyanomaliesindielectricconstantwereobservedat ◦ type120 spinstructure(Fig. 3(b))wasreportedtogive TN. Consideringthe results ofthe formerneutron study, the best fit. For NaCrO2, only a powder neutron study this can be interpreted as the antiferroelectric state due was performed[31] and TN∼ 40K has been reported[32]. to the alternate stacking of magnetic layers with oppo- Figures 4 (a) - (f) indicate the temperature profiles of site spin vector chirality. Combined with the recent re- heat capacity, dielectric constant, and electric polariza- sults for RbFe(MoO4)2 with easy-planeanisotropy[17], a tion for LiCrO2 and NaCrO2. Although the anomaly in vastrangeoftrigonallystackedtriangular-latticesystems ◦ magnetic susceptibility is not clear[32], the heat capac- with120 spinstructurecanbe multiferroic,irrespective itymanifests magneticphasetransitions,asseeninFigs. of their magnetic anisotropy. 4(a) and (b), in accord with the former neutron stud- ies. At TN, dielectric constant shows a strong cusp like The authors thank T. Arima, Y. Yamasaki, H. Kat- anomalyasinthe twocompoundswithdelafossitestruc- sura, S. Tanaka, R. Kumai, S. Ishiwata, and N. Nagaosa ture. Thissuggeststhelargefluctuationofelectricdipole for enlightening discussions. This work was partly sup- around TN, and confirms the correlation between dielec- ported by Grants-In-Aid for Scientific Research (Grant tric and magnetic natures also in this system. However, No. 16076205,17340104)from the MEXT of Japan. unlike the case for delafossites of CuCrO2 and AgCrO2, the macroscopic polarization can hardly be observed for LiCrO2 or NaCrO2. One of the possible interpretations for the absence of P but the presence of sharp ε-peak is theantiferroelectricorderofelectricdipoles. ForLiCrO2, [1] M. Fiebig, J. Phys. D:Appl.Phys. 38, R123 (2005). [2] S. -W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 on the basis of the two magnetic modulation vectors (2007). q1 = (1/3,1/3,0) and q2 = (−2/3,1/3,1/2), alternate [3] T. Kimuraet al.,Nature(London) 426, 55 (2003). stacking of Cr3+ layer with opposite vector spin chiral- [4] Y. Tokura, Science 312, 1481 (2006). ity was suggested[29]. 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