Field Dependence of Magnetic Ordering in Kagomé-Staircase Compound Ni 3 V 2 O 8

Transcription

1 University of Pennsylvni SholrlyCommons Deprtment of Physis Ppers Deprtment of Physis Field Dependene of Mgneti Ordering in Kgomé-Stirse Compound Ni 3 V 2 O 8 Mihel Kenzelmnn A. Brooks Hrris University of Pennsylvni, hrris@ss.upenn.edu Amnon Ahrony Or Entin-Wohlmn Tner Yildirim University of Pennsylvni, tner@ses.upenn.edu See next pge for dditionl uthors Follow this nd dditionl works t: Prt of the Physis Commons Reommended Cittion Kenzelmnn, M., Hrris, A., Ahrony, A., Entin-Wohlmn, O., Yildirim, T., Hung, Q., Prk, S., Lwes, G. J., Broholm, C., Rogdo, N. S., Cv, R. J., Kim, K., Jorge, G. A., & Rmirez, A. P. (2006). Field Dependene of Mgneti Ordering in Kgomé-Stirse Compound Ni 3 V 2 O 8. Physil Review B, At the time of pulition, uthor Tner Yildirim ws ffilited with the Ntionl Institute of Stndrds nd Tehnology, Githersurg, Mrylnd. Currently, he is fulty memer in the Mterils Siene nd Engineering Deprtment t the University of Pennsylvni. This pper is posted t SholrlyCommons. For more informtion, plese ontt repository@poox.upenn.edu.

2 Field Dependene of Mgneti Ordering in Kgomé-Stirse Compound Ni 3 V 2 O 8 Astrt We present powder nd single-rystl neutron diffrtion nd ulk mesurements of the Kgomé-stirse ompound Ni 3 V 2 O 8 (NVO) in fields up to 8.5T pplied long the diretion. (The Kgomé plne is the plne.) This system ontins two types of Ni ions, whih we ll spine nd ross-tie. Our neutron mesurements n e desried with the prmgneti spe group Cm for T<15K nd eh oserved mgnetilly ordered phse is hrterized y the pproprite irreduile>representtion(s). Our zero-field mesurements show tht t T PH =9.1K NVO undergoes trnsition to predominntly longitudinl inommensurte struture in whih the spine spins re nerly long the -xis. At T HL =6.3K, there is trnsition to n elliptilly polrized inommensurte struture with oth spine nd ross-tie moments in the plne. At T LC =4K the system undergoes first-order phse trnsition to ommensurte ntiferromgneti struture with the stggered mgnetiztion primrily long the -xis nd wek ferromgneti moment long the -xis. A speifi het nomly t T CC =2.3K indites n dditionl trnsition, whih remrkly does not ffet Brgg peks of the ommensurte C struture. Neutron, speifi het, nd mgnetiztion mesurements produe omprehensive temperture-field phse digrm. The symmetries of the inommensurte mgneti phses re onsistent with the oservtion tht only one phse is eletrilly polrized. The mgneti strutures re explined theoretilly using simplified model Hmiltonin, tht involves ompeting nerest- nd next-nerest-neighor exhnge intertions, single-ion nisotropy, pseudodipolr intertions, nd Dzyloshinskii-Moriy intertions. Disiplines Physis Comments At the time of pulition, uthor Tner Yildirim ws ffilited with the Ntionl Institute of Stndrds nd Tehnology, Githersurg, Mrylnd. Currently, he is fulty memer in the Mterils Siene nd Engineering Deprtment t the University of Pennsylvni. Author(s) Mihel Kenzelmnn, A. Brooks Hrris, Amnon Ahrony, Or Entin-Wohlmn, Tner Yildirim, Qingzhen Hung, Sungil Prk, Gvin J. Lwes, C. Broholm, Nyriss S. Rogdo, Roert J. Cv, Keehoon Kim, Guillermo A. Jorge, nd Arthur P. Rmirez This journl rtile is ville t SholrlyCommons:

3 PHYSICAL REVIEW B 74, Field dependene of mgneti ordering in Kgomé-stirse ompound Ni 3 V 2 O 8 M. Kenzelmnn, 1,2 A. B. Hrris, 3 A. Ahrony, 4 O. Entin-Wohlmn, 4 T. Yildirim, 2 Q. Hung, 2 S. Prk, 2 G. Lwes, 5 C. Broholm, 1,2 N. Rogdo, 6 R. J. Cv, 6 K. H. Kim, 5 G. Jorge, 5 nd A. P. Rmirez 5,7 1 Deprtment of Physis nd Astronomy, Johns Hopkins University, Bltimore, Mrylnd 21218, USA 2 NIST Center for Neutron Reserh, Ntionl Institute of Stndrds nd Tehnology, Githersurg, Mrylnd 20899, USA 3 Deprtment of Physis nd Astronomy, University of Pennsylvni, Phildelphi, Pennsylvni 19104, USA 4 Shool of Physis nd Astronomy, Rymond nd Beverly Skler Fulty of Ext Sienes, Tel Aviv University, Tel Aviv 69978, Isrel nd Deprtment of Physis, Ben Gurion University, Beer Shev 84105, Isrel 5 Los Almos Ntionl Lortory, Los Almos, New Mexio 87545, USA 6 Deprtment of Chemistry nd Prineton Mterils Institute, Prineton University, Prineton, New Jersey 08544, USA 7 Bell Ls, Luent Tehnologies, 600 Mountin Avenue, Murry Hill, New Jersey 07974, USA Reeived 13 Otoer 2005; pulished 25 July 2006 We present powder nd single-rystl neutron diffrtion nd ulk mesurements of the Kgomé-stirse ompound Ni 3 V 2 O 8 NVO in fields up to 8.5 T pplied long the diretion. The Kgomé plne is the - plne. This system ontins two types of Ni ions, whih we ll spine nd ross-tie. Our neutron mesurements n e desried with the prmgneti spe group Cm for T15 K nd eh oserved mgnetilly ordered phse is hrterized y the pproprite irreduile representtions. Our zero-field mesurements show tht t T PH =9.1 K NVO undergoes trnsition to predominntly longitudinl inommensurte struture in whih the spine spins re nerly long the -xis. At T HL =6.3 K, there is trnsition to n elliptilly polrized inommensurte struture with oth spine nd ross-tie moments in the - plne. At T LC =4 K the system undergoes first-order phse trnsition to ommensurte ntiferromgneti struture with the stggered mgnetiztion primrily long the -xis nd wek ferromgneti moment long the -xis. A speifi het nomly t T CC =2.3 K indites n dditionl trnsition, whih remrkly does not ffet Brgg peks of the ommensurte C struture. Neutron, speifi het, nd mgnetiztion mesurements produe omprehensive temperture-field phse digrm. The symmetries of the inommensurte mgneti phses re onsistent with the oservtion tht only one phse is eletrilly polrized. The mgneti strutures re explined theoretilly using simplified model Hmiltonin, tht involves ompeting nerest- nd nextnerest-neighor exhnge intertions, single-ion nisotropy, pseudodipolr intertions, nd Dzyloshinskii- Moriy intertions. DOI: /PhysRevB PACS numers: z, Jm, G I. INTRODUCTION Quntupin systems with ompeting intertions n hve highly degenerte ground-stte mnifolds with unusul spin orreltions. Smll, otherwise insignifint perturtions n then eome deisively importnt y lifting the degenery of the low-energy spin flututions, nd led to unexpeted low temperture phses. The proximity to quntum phse trnsitions, whih seprte the vrious ground sttes, produe new types of instilities tht involve hrge nd lttie degrees of freedom. Exmples inlude exoti superondutivity, non-fermi liquid ondutors 1 nd ferroeletriity. 2 Frustrted low-spin mgnets re idel model systems in whih to study ompeting quntum phses euse they nturlly ontin ompeting intertions in lerly defined geometry. An importnt model system is the ntiferromgneti AF Kgomé lttie whih onsists of orner-shring tringles of spins with equl AF oupling etween nerest neighors. Theoretilly it is expeted tht the S= 1 2 Kgomé lttie does not hve long-rnge order t zero temperture, ut dopts quntupin liquid ground stte. 3,4 The most well-known Kgomé-relted mgnet is the S= 3 2 ompound SrCr 9 G 3 O 19. However, the struture is tully etter desried s Kgomé ilyers whih ontin intervening tringulr ltties. 5 This mteril hs spin-glss-like ground stte. 6 8 Work on jrosite systems showed vrious types of ommensurte long-rnge order stilized y interlyer nd Dzyloshinskii-Moriy DM intertions. 9,10 In oth SrCr 9 G 3 O 19 Ref. 8 nd the Jrosites, 11 there is nomlous slow dynmis in the low T stte. These results indite tht Kgomé relted mgnets re highly sensitive to reltively wek intertions, nd hene re likely venue for new nd exoti ordered sttes. Ni 3 V 2 O 8 NVO is system of wekly oupled Kgoméstirse plnes ontining two inequivlent mgneti spin-1 Ni 2+ sites see Fig. 1. It is thus vrint of the highly frustrted pure Kgomé lttie. However, the devitions from the idel Kgomé geometry introdue severl new intertions tht relieve the frustrtion of the underlying Kgomé AF in interesting nd unexpeted wys. For exmple, it hs een found 12 tht mgneti ordering in NVO genertes ferroeletriity. These smller intertions ply ruil role in this phenomenon nd n explin the mirosopi origins of multiferrois euse the ompeting intertions produe n inommensurte stte tht reks sptil inversion symmetry. 13 Mgneti suseptiility nd speifi het mesurements first reveled tht NVO undergoes series of mgneti phse /2006/741/ The Amerin Physil Soiety

4 KENZELMANN et l. PHYSICAL REVIEW B 74, FIG. 1. Color online Lttie nd mgneti struture d of vrious phses in NVO. The two different Ni 2+ sites re shown in red spine sites nd lue ross-tie sites. The size of nd signs orresponds to the omponents out of the pge nd into the pge, respetively. In pnel d the nting of the C phse is mgnified for legiility. For the HTI nd LTI phses, the omponent hs wvelength of pproximtely 1.4 so tht q0.27, s indited in Fig. 9. trnsitions versus temperture nd mgneti field. 14,15 Beuse of its importne throughout the pper, the phse digrm is reprodued in Fig. 2. The digrm first ppered in Ref. 15 whih we shll refer to s pper I in the following. Our zero-field neutron diffrtion study in pper I showed tht NVO dopts two different inommensurte phses ove 4 K. A minly longitudinl inommensurte phse ours t higher tempertures nd spirl inommensurte phse ours t lower temperture. These phses re denoted s the high-temperture inommensurte HTI nd lowtemperture inommensurte LTI phses nd re illustrted in Fig. 1 nd Fig. 1, respetively. We lso found evidene of ommensurte C nted AF phse elow 4 K, shown in Fig. 1d. One purpose of this pper is to present omprehensive review of neutron diffrtion dt tht enles us to hrterize nd understnd the ourrene of the HTI, LTI, nd C phses. We will not disuss in detil the C phse, whih exists for T2.3 K nd ppers to e more omplex thn previously ntiipted. The C phse will e the sujet of forthoming presenttion. A seond purpose of this pper is to show tht the phse digrm n e understood to e the result of ompeting nerest-neighor NN nd next-nerest-neighor NNN exhnge intertions, esy-xis nisotropy, nd nisotropi intertions, oth FIG. 2. Color online H-T phse digrm of NVO with H pplied long eh of the three rystllogrphi xes. This phse digrm is sed on speifi het dt indited y smll squres tken s funtion of T t onstnt H Ref. 15. The phses re leled P for the prmgneti mgnetilly disordered phse, HTI for the high-temperture inommensurte phse, LTI for the low-temperture inommensurte phse, C for the high-temperture nted AF phse, nd C for the low-temperture nted AF phse. The spin strutures of these phses re desried elow. The trnsition tempertures in order of deresing temperture re denoted T PH, T HL, T LC, nd T CC. For nonzero field long, the C nd P phses hve the sme symmetry, nd therefore there is no shrp phse trnsition etween them. Experiments were not performed t high enough field to lote the HTI-C phse oundry for H. pseudodipolr PD s well s DM intertions While the pulitions ited ove generlly del with eh of these effets in isoltion, it is their interply tht leds to the rih phse digrm of NVO nd whih shll e our fous of ttention in this pper. Understnding the intrite spin strutures of NVO is further motivted y the need to understnd the unusul multiferroi LTI phse. 12 The experimentl detils, dt nlysis, nd physil interprettion whih were only riefly presented in pper I, re fully explined in this pper. We performed zero-field powder nd single-rystl neutron diffrtion study to determine the mgneti strutures, nd we used group theory to identify the strutures tht re llowed y symmetry for the two oserved ordering wve vetors. Further we present the field dependene of the mgneti strutures y monitoring AF nd ferromgneti FM Brgg peks. Mgneti fields up to 8.5 T were pplied long the rystllogrphi diretion nd this ws found to fvor the AF C phse t the expense of the inommensurte phses. Thus the phse digrm otined y our neutron diffrtion experiments is onsistent with tht

5 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ inferred from mrosopi mesurements, some of whih re presented here nd provide dditionl informtion out the mirosopi intertions etween mgneti ions. Almost ll our results n e understood on the sis of theoretil models tht re nlyzed in detil in seprte pper, 30 whih we refer to s pper II. To void undue repetition we will here indite in qulittive terms how these models explin the dt nd refer the reder to pper II for quntittive detils. The pper is orgnized s follows. In Se. II we summrize the experimentl tehniques employed in this work. In Se. III we provide our determintion of the rystl struture whih onfirms erlier work y Suerrei et l.. 31 Setion IV ontins the mgneti struture determintions. Here we give rief summry of representtion theory, sine this is required to understnd the diffrtion dt. We lso present mgnetiztion nd suseptiility dt for mgneti fields pplied long the three rystllogrphi diretions. In Se. V we provide theoretil interprettion of the experimentl results. We otin rough estimtes of mny of the mirosopi intertion prmeters using the more detiled lultions presented in pper II. Finlly, our results re summrized in Se. VI. II. EXPERIMENT Powder smples of NVO were mde in ruile using NiO nd V 2 O 5 s strting mterils. 14 Single rystls were grown from BO-V 2 O 5 flux. 15 Neutron diffrtion experiments were performed using powder smple with totl mss of 10 g nd single rystl with mss of 0.13 g oriented with the h,k,0 or h,k,k rystllogrphi plnes in the horizontl sttering plne of the spetrometers. Powder neutron mesurements were performed using the BT-1 high-resolution powder diffrtometer t the NIST Center for Neutron Reserh, employing Cu 311 nd Ge 311 monohromtors to produe monohromti neutron em of wvelength Å nd Å, respetively. Collimtors with horizontl divergenes of 15, 20, nd 7 were used efore nd fter the monohromtor, nd fter the smple, respetively. The intensities were mesured in steps of 0.05 in the rnge of sttering ngles, 2, from 3 to 168. Dt were olleted t vrious tempertures from 1.5 K to 30 K to eluidte the temperture dependene of the rystl struture. The progrm GSAS Ref. 32 ws used to refine the struturl prmeters nd the ommensurte mgneti struture. Additionl diffrtion dt nd mgneti order prmeters were otined on the BT7 triple xis spetrometer to explore the mgneti sttering in more detil. For these mesurements, pyrolyti grphite PG002 doule monohromtor ws employed t wvelength of 2.47 Å, with 40 ollimtion fter the smple nd no nlyzer. The single-rystl neutron sttering mesurements were performed with the therml-neutron triple-xis spetrometers BT7 nd BT9, nd with the old-neutron triple-xis spetrometer SPINS. The BT7 experiment ws performed with 60 ollimtion fter the smple, PG002 nlyzer to reflet mev neutrons nd 180 em divergene fter PHYSICAL REVIEW B 74, the nlyzer. The BT9 diffrtion mesurements were performed with n inident energy of 30.5 mev nd ollimtion round the smple. The H-T mgneti phse digrm ws determined using SPINS with n inident energy of 5 mev, Be filter efore the smple nd ollimtion round the smple. The SPINS diffrtion ptterns were olleted with ollimtion round the smple, PG004 monohromtor omined with grphite filter in the inident em nd flt PG002 nlyzer set for 14.7 mev. The em divergene etween nlyzer nd detetor ws 240. III. NUCLEAR STRUCTURE A. Experimentl determintion of struture The NVO struture refinement from BT1 neutron powder diffrtion dt ws rried out suessfully using the previously reported struturl prmeters 31 s initil vlues. In greement with previous studies, we found the struture to hve Cm orthorhomi symmetry spe group No. 64 in the Interntionl Tles for Crystllogrphy 33. No struturl trnsition ws deteted for 1.5 KT300 K. The struturl prmeters nd seleted intertomi distnes t two tempertures re given in Tle I. The symmetry elements of the Cm spe group of NVO re given in Tle II. 34 Beuse V hs low oherent neutron sttering ross setion, the tomi positions nd temperture ftors of the vndium ions were fixed to the vlues otined y x-ry diffrtion. 31 To investigte the effet of mgneti ordering on the hemil struture, series of powder ptterns were tken elow 10 K. No signifint hnge for the -xis, unit ell volume, nd Ni-O ond distnes were oserved, s shown in Fig. 3 nd Tle I. However, reduing the temperture elow 10 K leds to n inresing lttie prmeter, while the lttie prmeter dereses without hnge in spe group symmetry. The strongest temperture dependene is ssoited with the isotropi men-squre displements for Ni nd O, whih hnge signifintly with the onset of ommensurte order t 4KFig. 4. This dt nd tht shown in Fig. 3 my indite wek oupling of the mgneti nd the hemil lttie nd ould therefore give evidene of the spin-phonon oupling tht must explin the pperne of ferroeletriity in NVO. 13 However, we did not oserve rystl distortion with the given neutron diffrtion resolution, ut this distortion is expeted to e quite smll. 13 Higher-resolution x-ry or neutron diffrtion would e needed to look for possile spe group symmetry reking ssoited with the onset of ommensurte order. B. Struturl properties Here we note some generl fetures of the struture. The struture of NVO onsists of Kgomé lyers of edge-shring NiO 6 othedr. The lyers re seprted y nonmgneti V 5+ O 4 tetrhedr. As shown in Figs. 1 nd 5, there re two inequivlent rystllogrphi sites for the mgneti Ni 2+ ions, denoted Ni s nd Ni. We will refer to these s spine nd ross-tie sites, respetively. At 15 K, the verge

6 KENZELMANN et l. TABLE I. Struturl prmeters nd seleted intertomi distnes for NVO, mesured using the BT1 spetrometer with the Ge 311 monohromtor nd = Å. Spe group: Cm No. 64 in Ref. 33. Atomi positions expressed s frtions of,, nd : Ni s :8e nottion s in Ref ,y, 1 4 ;Ni :4 0 00; V:8f 0yz; O 1 :8f 0yz; O 2 :8f 0yz; O 3 :16g xyz. B i 8 2 u 2 i, where u i is the displement of tom i from its equilirium position nd indites therml verge. Also R p = n i I i o I i i / i I o i where I o nd I i re the n oserved nd lulted intensities, respetively. R wp =w i I i o I i 2 /w i I i o 2 where the weight is given y w i =1/ 2 i. i is the error r of I i o. The sum of lest-squres is given y 2 = n i w i F i o F i 2 /n m, where m is the numer of fitted vriles. T=15 K T=1.5 K Å Å Å Ni s y BÅ Ni BÅ V y z BÅ O 1 y z BÅ O 2 y z BÅ O 3 x y z BÅ R p % R wp % distnes in Å Ni s -O Ni s -O Ni -O Ni -O Ni -O Ni s -Ni Ni -Ni PHYSICAL REVIEW B 74, TABLE II. Generl positions within the primitive unit ell for Cm whih desrie the symmetry opertions of this spe group. 2 is twofold rottion or srew xis nd m is mirror or glide - plne. E denotes the identity opertion. Er=x,y,z 2 r=x,ȳ+1/2,z+1/2 2 r=x,y+1/2,z +1/2 2 r=x,ȳ,z Ir=x,ȳ,z m r=x,y+1/2,z +1/2 m r=x,ȳ+1/2,z+1/2 m r=x,y,z Ni 2+ -Ni 2+ distne within the lyers is d 1 =2.94 Å, while the interlyer distne is d 2 =5.69 Å. Bsed on the reltively lrge interlyer to intrlyer rtio, d 2 /d 1 =1.9, strong twodimensionl mgneti hrter my e expeted in this ompound, with the mgnetism dominted y intrlyer Ni 2+ -Ni 2+ exhnge intertions. Unlike previously studied Kgomé lttie-sed mterils, 5,8,10 whih hve plnr mgneti lyers, the Ni-O lyers in NVO re ukled, resulting in the Kgomé-stirse struture. The symmetry of the superexhnge intertion medited y O ions shows tht there re two inequivlent superexhnge pths etween neighoring Ni 2+ ions within Kgomé plnes. 14 In prtiulr, there is superexhnge pth etween Ni s positions long the rystllogrphi -diretion whih is different from NN intertions etween Ni ions on neighoring Ni s nd Ni positions. All the Ni spine sites re relted y rystl symmetry s re ll the Ni ross-tie sites. The two types of Ni sites re shown in Fig. 5 nd their oordintes re given in Tle III. Our onvention for the oordinte xes is s follows: the spines lie long the 100, or-xis sometimes lled the x-xis, î, the sl plne inludes this xis nd the 001, or -xis sometimes lled the z xis, kˆ, nd the xis perpendiulr to this plne is denoted 010 or the -xis sometimes lled the y-xis, ĵ. FIG. 3. Lttie prmeters,, nd s funtion of temperture, determined from neutron powder diffrtion using BT1 with Cu 311 monohromtor nd wvelength Å. Also shown is the temperture dependene of the unit ell volume. The solid nd dshed lines re guides to the eye for the extrpoltions from higher temperture

7 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ PHYSICAL REVIEW B 74, FIG. 4. Isotropi men-squre displements, U iso =u i 2,ofNi nd O s funtion of temperture, otined from the sme powder spetr mentioned in Fig. 3. The spine nd ross-tie sites hve different lol symmetry nd will e seen to hve very different mgneti properties. Although NVO hs severl mgneti phses, ommon feture of ll of them is tht spins s1 nd s2 in the unit ell hve equl ut opposite moments, s do s3 nd s4, s shown in Fig. 6. Thus, within model of isotropi intertions, the ross-tie spins re sujet to zero men field, i.e., they re frustrted. In this regrd, this system is reminisent of Sr 2 Cu 3 O 4 Cl 2 Ref. 24 nd of vrious ldder systems whih hve reently een studied. 35 The ove struture hs severl implitions for the mgneti intertions. As explined in Ref. 14, the leding NN Ni-Ni mgneti oupling rises vi superexhnge intertions, medited y two Ni-O-Ni onds. For pir of NN spine spins, the ngles sutended y these two onds re 90.4 nd For spine-ross-tie pir, these ngles re 90.3 nd For the similr se of Cu-O-Cu onds, it hs een shown tht when these ngles re lose to 90 then the resulting exhnge energy is smll, 36 nd my even hnge its sign from FM to AF s the ngle dereses from 90. Sine oth Ni nd Cu involve d-holes in the high e g sttes within the oxygen othedron surrounding Ni or Cu, we expet similr results to pply for the Ni se. Aordingly, we do not neessrily expet tht NN intertions J 1 in Fig. 6 dominte seond neighor NNN intertions J 2 in Fig. 6. Similr lultions for the relted NNN oupling, vi Cu-O-O-Cu, gve AF intertions. 36 Similr Ni-O-O-Ni FIG. 5. Color online Ni sites in the onventionl unit ell. The numering of the ross-tie nd spine s sultties is pproprite for the primitive unit ell. The spine sites irles red re displed from the unukled plne y y= ±0.13 nd the rosstie sites squres lue hve zero displement. The sis vetors for the primitive unit ell re v 1 =/2î+/2ĵ, v 2 =/2î /2ĵ, nd v 3 =kˆ. intertions ould ompete with the NN intertions, nd give rise to inommensurte strutures, s explined elow. The ft tht ordering ours t temperture of order 10 K indites tht this should e the order of mgnitude of J 1 /k. TABLE III. Positions of Ni 2+ rrying S=1 within the primitive unit ell illustrted in Fig. 5. Eh omponent is expressed s frtion of the respetive lttie onstnt, so tht r 1 s =0.25î 0.13ĵ+0.25kˆ. Lttie positions r n s re spine sites nd r n re the ross-tie sites. NVO orders in spe group Cm, so there re six more toms in the onventionl unit ell whih re otined y i i trnsltion of sites r s nd r y 0.5, 0.5, 0. r 1 s = r 2 s = r 3 s = r 4 s = r 1 = r 2 = 0.25, 0.13, , 0.13, , 0.13, , 0.13,0.25 0, 0, 0 0.5, 0,

8 KENZELMANN et l. PHYSICAL REVIEW B 74, F = 1 v r,rs rs r. 2 r,r, 2 In view of trnsltionl invrine this free energy n e written in terms of Fourier mplitudes s F = 1 2 Q,,,, v ; QS Q,S Q,, 3 FIG. 6. Color online Simple model of isotropi exhnge intertions inditing the frustrtion of the ross-tie sites. The sites re represented s in Fig. 5. Here J 1 nd J 2 represent NN nd seond neighor intertions etween spine sites. Sine sites s1 nd s2 hve opposite spins, the men field t the ross-tie site is zero. Sites s3 nd s4 hve opposite spins whih re not shown euse their vlues reltive to s1 nd s2 re different for different mgneti phses. As result of the rystl symmetries, there is limited numer of independent NN intertion mtries. If we write the intertion etween spine spins Si nd Sj s H ij = M i, js is j, where =,, is omponent lel, then one we hve speified the mtrix M s1,s4 in the nottion of Fig. 5, we n express the intertion mtries for ll other NN pirs of spine sites in terms of M s1,s4. Symmetry lso ples some restritions on the form of M s1,s4. These results re otined in pper II. Similrly, we only need to speify single intertion mtrix, e.g., M s1,2, for NN pirs of spine nd ross-tie sites or for intertions etween NNN s in the sme spine. IV. MAGNETIC STRUCTURE A. Representtion theory Sine group theoretil onepts re entrl to our nlysis, we summrize here the results whih we will invoke. Additionl detils re ville in pper II nd in Appendix A. For reders who might wnt to skip this setion we summrize its min result: S Q,, the Fourier trnsform of the -omponent of the spin on sulttie should rise from single irreduile representtion in the HTI phse nd from this nd one dditionl representtion in the LTI phse. The llowed vlues of S Q, for eh representtion re listed in Tle VIII of the Appendix nd involve reltively smll numer of fitting prmeters. In generl, Lndu theory indites tht the free energy F in the disordered phse is dominntly qudrti form in the spin mplitudes S r t site r. Thus 1 where Q is the wve vetor 37 nd lels sites within the unit ell. Here the Fourier mplitudes re given y S Q, = N u 1 R S R + r e iq R+r, where r is the position of the th site within the unit ell nd the sum over R is over trnsltion vetors for system of N u unit ells. As our dt indite, the ordering trnsition is ontinuous one whih is signled y one of the eigenvlues of the qudrti free energy pssing through zero s the temperture is redued. This ondition will e stisfied y some wve vetor, or more preisely, y the str of some wve vetor q, whih is in the first Brillouin zone of the primitive lttie. This is usully lled wve vetor seletion. The ritil eigenvetor, i.e., the eigenvetor ssoited with this instility, will indite the pttern of spin omponents within the unit ell whih forms the ordered phse. In view of the symmetry of the prmgneti rystl, whih dittes the form of Eq. 3, we see tht this eigenvetor must trnsform ording to one of the irreduile representtions of the symmetry group whih leves the wve vetor q invrint. 38 This group is usully lled the little group. The ssumption tht the instility towrd the ondenstion of long rnge order involves only single irreduile representtion irrep is sed on the ssertion tht there n e no identl degenery whih would orrespond to higher order multiritil point. Here we re interested in the representtions of two types of wve vetors, nmely zero wve vetor in whih ntiferromgnetism rises euse of AF intertions within the unit ell nd n inommensurte wve vetor q,0,0 t some nonspeil point on the x-xis. The representtions for the wve vetor 0, 0, 0 nd q,0,0 re desried in Appendix B. Note tht ll spine spins re relted to one nother y rystl symmetry, s re ll ross-tie sites. As result, within given representtion one hs the prmeters m s, m s, nd m s whih ompletely fix the,, nd omponents, respetively, of the Fourier mplitudes S Q,sn of ll the spine spins within the unit ell nd the prmeters m, m, nd m whih similrly ompletely fix the Fourier mplitudes S Q,n of ll the ross-tie spins within the unit ell. For some representtions some of these mplitudes my not e llowed to e nonzero. For n inommensurte wve vetor these prmeters re omplex vlued, lthough, of ourse, the resulting spin omponents must e rel, euse we should invoke oth ssoited with q nd * ssoited with q. Hppily, s the ove disussion impliitly ssumes, the sitution is quite simple in tht for mny systems, suh s NVO, ll the representtions re one dimensionl. Wht tht

9 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ mens is tht under ny group opertion, the llowed eigenvetors trnsform either into themselves or into phse ftor of unit mgnitude times themselves. To summrize the results of Appendix B, if O p is n opertion tht leves the inommensurte nonzero wve vetor invrint, then we my write O p m g = p m g, 5 where g ssumes the vlues s for spine nd for ross-tie, =,, or, nd p is the hrter for the symmetry opertion O p in the representtion. For zero wve vetor relevnt for the AF phses these hrters re given in Tle V nd for the inommensurte phses they re given in Tle VII. The disussion up to now took ount only of those opertions whih leve the wvevetor invrint. However, the free energy must e invrint under ll the symmetry opertions of the prmgneti phse. 39,40 Thus fr the symmetry properties we hve disussed pply to ny rystl whose prmgneti spe group is Cm. Now we disuss the onsequenes of restriting the mgneti moments to the spine nd ross-tie sites whih hve higher site symmetry thn n ritrry lttie site nd in prtiulr we will study the inommensurte phses. We onsider the effet of sptil inversion on the spin wve funtions. To illustrte the onepts we ssume wve vetor q in units of 2/ long the -xis nd onsider spin onfigurtion whih trnsforms ording to 4 nd whih therefore hs the omponents 4 given in Tle VII. Sine the mgneti moment is n xil vetor, sptil inversion I tkes the moment into itself ut moves it to the sptilly inverted lttie site. Let us onsider the spin stte t spine site No. 3 in the unit ell t R X,Y,Z, ording to irrep 4. Before pplying sptil inversion the spin vetor t tht site ording to Tle VII is S 3 R; 4 = â ˆ + ĉe 2iqX+x s3 / + â ˆ + ĉ * e 2iqX+x s3 /, 6 where x s3 is the x-oordinte of spine site s3 within the unit ell nd the representtion lel is impliit. After inversion indited y prime the spin t this site will e tht whih efore inversion ws t R x s3 â, whih is site of sulttie No. 1. Thus S 3 R; 4 = â + ˆ + ĉe 2iqX+x s3 / + â + ˆ + ĉ * e 2iqX+x s3 /. 7 By ompring Eqs. 6 nd 7 we see tht for = or = we hve tht I = *, nd for the -omponent we hve I = m * s. 9 One n hek tht for irrep 4 the spin omponents of the other spine sites trnsform this sme wy. Furthermore, this 8 PHYSICAL REVIEW B 74, type of nlysis indites tht ll the oordintes of Tle VII for the ross-tie sites oey Eq. 8. Aordingly, for this irreduile irrep we now introdue symmetry-dpted oordintes m g whih oey Eq. 8. We write m g = m g, 10 exept for g=s nd =. For this se, to trnsform oordintes so tht Eq. 9 is trnsformed into the desired form of Eq. 8, we write m s = i. 11 For the other irreps the nlysis is similr, ut the omponents whih must trnsform s in Eq. 11 my e different. The omplex-vlued symmetry dpted oordintes whih trnsform ording to eh of the irreps re olleted in Tle VIII nd ll of these re onstruted to oey Eq. 8. When we ssume the ondenstion of single irrep,, the Lndu expnsion in terms of the ove symmetry dpted oordintes is of the form F = 1 v 2 gg m * g m g, 12 gg where m g is shorthnd for m g nd the relity of F requires tht v gg = v * g g. 13 Beuse these oordintes trnsform ording to single one-dimensionl irrep we know tht this expression for the free energy is indeed invrint under the opertions of the little group. But the free energy is lso invrint under sptil inversion. This dditionl invrine provides dditionl informtion. This sitution hs een reviewed reently y Shweizer, 41 ut the pproh we use elow seems simpler in the present se. Here, euse of the speil trnsformtion property of Eq. 8 we hve tht F = IF = 1 v 2 gg Im g * Im g gg = 1 v 2 gg m g m g *. gg Tking ount of Eq. 13 this implies tht 14 v g g = v gg = v * g g. 15 In other words, for the present symmetry, the oeffiients in the qudrti free energy expressed in terms of symmetrydpted oordintes re ll rel vlued. Wht this mens is tht the omplex eigenvetor of the qudrti form n e expressed s single overll omplex phse ftor times vetor with rel-vlued omponents. This ondition ensures tht ll the mplitudes whih mke up the spin eigenvetor hve the pproprite phse. The ove disussion is sed on the qudrti free energy of Eq. 2. In ddition, one n onsider the effets of qurti nd higher-order terms in the Lndu free energy s

10 KENZELMANN et l. PHYSICAL REVIEW B 74, FIG. 8. Top: Intensity t the sttering ngle 2, lose to the AF peks 1, 1, 2 nd 1, 3, 0. The intensity ove 4 K is relted to the 0.73, 3, 1 mgneti refletion ssoited with the inommensurte mgneti strutures tht ours t similr sttering ngle 2. Bottom: Intensity t the sttering ngle 2 for the inommensurte mgneti pek 0.73, 1, 0 s funtion of temperture. FIG. 7. Low-ngle portions of the BT1 neutron powder diffrtion pttern olleted t 15 K, 5 K, nd 1.6 K. Nuler struture fitting. Nuler struture fitting only, inommensurte mgneti peks oserved. Both nuler nd mgneti strutures were inluded in the fit. The differenes etween oserved nd lulted intensities re shown t the ottom of eh figure. The vertil lines indite the positions of the possile Brgg peks. The nlysis of diffrtion spetr t zero field requires onsidertion of verging over domins. This sujet is disussed in Appendix A. Figure 7 shows the low-ngle neutron powder diffrtion pttern mesured t 1.6 K, 5 K nd 15 K. The pperne of new Brgg peks upon ooling indites tht the ompound undergoes trnsitions to mgneti order elow 9.1 K. The Brgg peks elow 4 K n e indexed with ordering wve vetors tht re ommensurte, wheres no suh identifition is possile for higher tempertures, suggesting inommensurte mgneti strutures t higher tempertures. Figure 8 shows the temperture dependene of the intensity of n inommensurte mgneti pek ner 2=21.49, nd sttering ssoited with the ommensurte order oserved t 2= This shows tht the inommensurte phse exists in finite temperture window etween 4 nd 9.1 K, nd tht the low T stte hs ommensurte mgneti order. The mgneti order ws further investigted with experiments on single rystl in whih sttering ws monitored only in the h,k,0 nd h,k,k plnes. Figure 9 shows the elsti neutron sttering t three different tempertures for wve vetors of the form Q x,1,0. Upon entering the mgneti phse, Brgg pek is formed with mximum inten- well s flututions not inluded y men field theory. These orretions re nlyzed in pper II, ut do not invlidte the result of Eq. 15. B. Mgneti order t zero field FIG. 9. Neutron diffrtion intensity mesured s funtion of Q x for wve vetor Q=Q x,1,0. Here the pek position determines the vlue of 1+q

11 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ PHYSICAL REVIEW B 74, sity for Q x =Q x r.l.u.. This result indites weight in the Fourier trnsform of the spin SQ t wve vetor Q x 0,1,0 whih is outside the first Brillouin zone of the primitive unit ell. We dedue tht Brgg sttering is llowed t wve vetors G + q G ± q,0,0, 16 where q0.27 nd G is reiprol lttie vetor of the form G=l+m,l m,n, where l, m, nd n re integers. The ordering wve vetor in the HTI nd LTI phses is thus v =q,0,0. The pek in Fig. 9 is in the l=1,m=n=0 zone. In this formultion the wve vetor q indites tht the spin funtion vries s expiq R s the position is displed through trnsltion vetor R of the lttie, s defined in the ption to Fig. 5. Thus, for given irrep, the tul spin mplitudes re determined y the vlue of q nd the vlue of the spin oordintes within the unit ell s given in Tle VII or VIII. Note tht this wve vetor q does not give the phse ftor introdued upon moving from one spine site to its nerest neighor. In view of the intrell struture given in Tle VII, different omponents of spin will involve different phse ftors. If we onsider the omponent of spin in irrep 4 this is the tive representtion for the HTI phse nd let the trnsltion vetor R e X,Y,Z, then we see from Tle VII tht S s1, X = e iq2/x+/4, S s4, X = e iq2/x+3/4, 17 where s1 nd s4 re site lels s in Fig. 5. Thus S s4, X/S s1, X= e iq. Similrly one finds S s1, X +/S s4, X= e iq. Our onlusion is tht trnsltion long spine y /2 introdues phse ftor e iq+1. So, the wve vetors for the -omponent long single spine re 37 q ± =1±q. As we shll see lter in Se. V A 1, elow, other wve vetors re needed to reprodue the position dependene of other spin omponents within this representtion. The inommensurtion q is wekly temperture dependent, s shown in Fig. 10, inditing ompeting intertions in the spin lttie. Dt were tken y deresing lk irles nd inresing grey irles the temperture. Although we oserve sttering from the inommensurte wve vetors 1 q nd 1+q, whih the dt in pnel of Fig. 10 show re equivlent s expeted, their integrted intensities, shown in pnels nd d, re not equl euse the mgneti struture ftor nd its omponent perpendiulr to the wve vetor re different for the two stellite peks. The temperture dependene of these integrted intensities, shown in pnels nd d, indites the onset of mgneti order t T PH =9.1 K nd suggests n dditionl seond order trnsition t out T HL =6.3 K. These trnsition tempertures re onsistent with speifi het mesurements, whih show shrp peks t these tempertures, with mgnetiztion dt, nd re onfirmed y the mgneti struture determintion of Se. IV F, elow. The existene of two different inommensurte mgneti phses is further indition of ompeting intertions in NVO. FIG. 10. Temperture dependene of mgneti Brgg peks in zero field, mesured with deresing lk irles nd inresing grey irles temperture. The low-temperture field ws rehed y zero-field ooling. Integrted intensities were otined y integrting diffrtion intensities mesured s funtion of h wvevetor trnsfer. Temperture dependene of h-integrted intensity of the AF 1, 1, 0 refletion. Temperture dependene of h-integrted intensity of the inommensurte 1 q, 1,0 refletion. Temperture dependene of the wve vetor 1±q, 1,0 of the inommensurte mgneti refletions. The dt points elow T =4 K represent nonequilirium domins of the LTI phse emedded in the C phse. Suh peks re only present fter ooling through the LTI phse. d Temperture dependene of h-integrted intensity of the inommensurte 1+q, 1,0 refletion. Figure 10 shows tht the inommensurte Brgg peks ruptly lose most of their intensity t T LC =4 K, elow whih temperture ommensurte mgneti order eomes dominnt. Commensurte Brgg peks were oserved t h,k,0 for h+k=even, so the ommensurte struture is ssoited with n ordering wve vetor v=0,0,0. The mgneti unit ell in the C phse is thus identil to the hemil unit ell. In the zero-field ooled smple, we oserved wek inommensurte pek whih is not present in the 8 T field ooled smple. This is evidene tht the inommensurte phse elow T LC is metstle possily refletion of how lose the ommensurte nd inommensurte mgneti order lie in energy. However, Fig. 11 shows tht the ground stte of NVO n e nneled through the pplition of field long the -xis whih mkes the inommensurte Brgg pek vnish. The inommensurte Brgg peks do not repper when lowering the field, ut the intensity is insted trnsferred to the ommensurte pek tht grows in strength. C. Field dependene of Brgg refletions The field dependene of the mgneti Brgg refletions ws investigted only with fields long the -xis. Figure 12 shows the mgneti refletions t 2 T s funtion of temperture. Upon heting, the 1, 1, 0 refletion disppers in

12 KENZELMANN et l. PHYSICAL REVIEW B 74, FIG. 11. Integrted intensity of the 1, 1, 0 nd the 0.72, 1, 0 refletions otined fter zero-field ooling of the smple nd mesuring s funtion of inresing full irles nd deresing field grey irles. first-order trnsition s the inommensurte 1±q,1,0 refletions pper. The ommensurte phse survives to higher tempertures thn t zero field, nd the ordered moment inreses with field, oth inditions tht the mgneti field stilizes the ommensurte order. The LTI mgneti struture oupies reltively nrrow temperture rnge while the temperture oundries for the HTI mgneti struture re nerly independent of field. FIG. 13. As for Fig. 12, ut for H=5 T. The LTI mgneti struture is further suppressed with inresing mgneti field long the -xis. At 5 T, s shown in Fig. 13, the LTI struture does not our, nd s the temperture is inresed, the ommensurte struture gives wy diretly to the HTI mgneti struture. The phse trnsition etween the prmgneti nd HTI phse ours t temperture somewht elow its zero-field ritil temperture T PH =9 K. It is for this field diretion tht the phse oundries depend most strongly on the field. A field long the xis in the HTI phse leds to suppression of the inommensurte Brgg pek t 1 q,1,0 nd n inrese in the intensity of the 1, 1, 0 refletion, s shown in Fig. 14 for T=8.4 K. The temperture dependene of the intensity of the inommensurte Brgg peks suggests tht FIG. 12. Temperture dependene of mgneti Brgg peks for H=2 T, pplied long the rystllogrphi -xis. The lowtemperture stte ws rehed y ooling the smple in field of 8 T. Integrted intensities were otined y integrting the intensity of the mgneti Brgg refletion when oserved y sn over the h-omponent of the wve-vetor trnsfer. Temperture dependene of integrted intensity of the AF 1, 1, 0 refletion. Temperture dependene of integrted intensity of the inommensurte 1 q,1,0 refletion. Temperture dependene of the inommensurtion q. d Temperture dependene of integrted intensity of the inommensurte 1+q,1,0 refletion. FIG. 14. Field dependene of mgneti Brgg peks t T =8.4 K. Integrted intensities were otined s for Fig. 12. Field dependene of the integrted intensity of the AF 1, 1, 0 refletion, the inommensurte 1 q,1,0 refletion, nd d the inommensurte 1+q,1,0 refletion. shows the field dependene of the inommensurte wve vetor q

13 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ PHYSICAL REVIEW B 74, FIG. 15. Color online Field dependene of effetive stggered moment t three different tempertures, otined from the 1, 1, 0 refletion y tking the squre root of the pek intensity. the HTI phse disppers in ontinuous phse trnsition t ritil field H, giving wy to ommensurte field driven AF phse t higher fields. This ontrsts with the first order nture of the phse oundry etween the LTI nd AF phses. As shown in Fig. 14, oth the inommensurte wve vetor, q, nd the integrted intensity of the 1, 1, 0 AF Brgg pek inrese progressively more rpidly s the field inreses. All the mgneti phse oundries otined from these neutron mesurements re onsistent with the phse digrm otined fropeifi het mesurements with the field long the -xis. D. Phse digrm The zero-field phse oundries t T PH, T HL, nd T LC dedued from the diffrtion experiments re onsistent with those oserved with speifi het mesurements. In ontrst, the intensity of the 1, 1, 0 refletion does not show ny nomly t T CC, to level of 0.5%. This suggests tht the CC phse trnsition leves the mgneti struture of the C-phse desried y ordering wve-vetor v=0,0,0 unltered. Figures 12 nd 13 lredy show tht the C-LTI nd the C-HTI trnsitions re first order t smll H. Similr results were otined when we vried H t fixed T. Speifilly, Fig. 15 shows the jumps in the stggered moment of the C phse s one moves from the LTI or from the HTI phses into the C phse, for T8 K. In ontrst, t higher tempertures nd fields the trnsition from the HTI to the P phse is ontinuous, s n e seen from Fig. 14. In ft, t finite field long one nnot relly distinguish etween the P nd the C phses, nd we lredy sw in Figs. 12 nd 13 tht the HTI-P trnsition is ontinuous for H5 T. We thus onlude tht the HTI-C trnsition hnges from eing ontinuous to eing first order somewhere round the top of the oundry of the HTI phse in Fig. 2. Suh triritil points, whih seprte etween line of first order trnsitions nd line of FIG. 16. Temperture dependene of h-integrted intensity of the AF 1, 1, 0 refletion in field of H=8.5 T, pplied long the -xis, showing the sene of seond order phse trnsition when ooling from the prmgneti phse to the low-temperture ommensurte phse. The dt insted reflet rossover phenomenon for T8 K. seond order trnsitions, re undnt in nisotropi ntiferromgnets sujet to mgneti fields. 42 The hnge in slope of the 110 intensity versus T urve t the HTI-P trnsition see Fig. 14 indites the oupling etween the ommensurte nd inommensurte order prmeters, whih will e disussed in Se. V E. The oservtions whih re relevnt for the trnsition from the C to the P phse in the presene of mgneti field re the speifi het dt t 9 T whih show no nomly nd the lk of ny detetle nomly in the T-dependene of the 110 pek intensity see Fig. 16. These oservtions imply the sene of rel shrp trnsition etween these two phses. Indeed, s will e shown in Se. V D 1, the P phse in field llows the development of order hrteristi of only the irrep 7 in whih there is uniform moment long nd stggered moment long. This suggests tht the mgneti struture in the C nd P phse in field exhiit the sme symmetry, nd tht the symmetry t high field, t low nd high temperture, is the sme s tht of the C phse t zero field euse it n e rehed without rossing seond order phse trnsition. A more intuitive wy to understnd this effet is presented in Se. V D: the DM intertion genertes iliner oupling etween the uniform mgnetiztion long the -xis nd the stggered moment long the -xis. Therefore, mgneti field long genertes n effetive stggered field long, whih smers the trnsition etween P nd C. E. Mgneti Strutures Even more detiled informtion out the spin intertions n e otined y determining the symmetry of the ordered mgneti strutures. In this pper, we will fous our ttention on the HTI, LTI, C, nd P phses, nd we will leve disussion of the C phse for lter presenttion. In our nlysis it is essentil to use representtion theory to restrit fits to one for the HTI, C, nd P phses or two for the LTI phse representtions, in order to ensure tht the mgneti strutures do not violte fundmentl symmetry properties. 1. High-temperture inommensurte (HTI) struture For tempertures etween T HL nd T PH, Brgg refletions our t the 2n+1±q,2m+1,0 nd 2n+1±q,2m+1,2m

14 KENZELMANN et l. +1 positions. The intensities of 170 mgneti Brgg refletions were mesured nd n e explined with mgneti struture elonging to representtion 4 given in Tle VIII. The mgneti struture t T=7 K is given y m 4 s = 1.91,0.21,0.22 B, m 4 = 0,0.01,0.22 B, 18 where the numer in prentheses is the unertinty in the lst digit quoted. The nottion 0 without n unertinty indites tht omponent is not llowed in tht representtion. These struture prmeters re llowed to e omplex, ut the rguments of Se. IV A indite tht prt from n overll phse, they my e hosen to e rel. The qulity of the fit is given y R p =0.15 nd 2 =12 see I for the definition of these quntities. The spin rrngement t T=7 K is illustrted in Fig. 1. It is modulted struture in whih the only lerly nonzero omponents re the nd omponents of the spine spins. In irrep 4 one sees from Tle VII tht these omponents of spine spins in djent Kgomé plnes e.g, sites No. 1 nd No. 4, or sites No. 2 nd No. 3 re ntiprllel. The moments on the ross-tie sites either vnish or re very smll. If the ross-tie moments in the diretion re nonzero, they should e out of phse with the spine moments euse of the phse ftor introdued y Eq. 11. To determine the HTI mgneti struture in mgneti field long the -xis, we mesured the intensity of 28 mgneti Brgg peks in the h,k,0 plne for T=8 K nd H =5 T. The dt re est desried y the sis vetors of the irrep 4, nd the mgneti struture is given y m 4 s = 1.604,0.083,02 B, m 4 = 0,0.025,02 B. 19 The qulity of the fit is given y R p =0.14 nd 2 =6.7. The effet of the field long the -xis is thus to redue the ordered moment long the -xis ompred to the zero-field struture t somewht lower temperture. This my e euse the field indues uniform moment long the -xis whih redues the mount of moment ville long the -xis. 2. Low-temperture inommensurte (LTI) struture For tempertures etween T LC nd T HL, Brgg refletions were oserved t the 2n+1±q,2m+1,0 nd 2n +1±q,2m+1,2m+1 positions. These re the sme Brgg peks s oserved in the HTI phse, ut their reltive intensity hs hnged, inditing tht the spins undergo spin reorienttion t T HL. The present diffrtion dt re onsistent with either or struture. We hoose the former spin struture, not euse it hs smller vlue of 2, ut euse it is onsistent with the pperne of ferroeletriity, s is disussed in Se. IV F, elow. With tht ssumption the mgneti struture t 5 K is given y m 4 s = 1.61,0.0310,0.017 B, PHYSICAL REVIEW B 74, m 1 s = 0.05i,1.31i,0.11i B, m 4 = 0,1.41, B, m 1 = 2.21i,0,0 B. 20 The qulity of the fit is given y R p =0.19 nd 2 =7. The experiment ws not sensitive to the phse etween eigenvetors of 1 nd 4. The fixed spin length onstrint fvored y the qurti terms in the Lndu expnsion requires reltive phse of /2, leding to purely omplex numers for the eigenvetors of 1. The orresponding spin struture is shown in Fig. 1 nd onsists of elliptil - plne spirls on spine nd ross-tie sites. The struture t T=5 K thus onsists of spirls on the spine nd ross-tie sites, propgting long the -xis with moments in the - plne, s shown in Fig. 1d. The spine spins form elliptilly polrized spin density wves suh tht spins in djent Kgomé plnes whih re nerest neighors e.g., sites No. 1 nd No. 4 or sites No. 2 nd No. 3 re ntiprllel, s n e dedued from Tle VII. Similrly, the ross-tie spins form elliptilly polrized spin density wves suh tht spins in djent Kgomé plnes whih re nerest neighors in the sme plne perpendiulr to the -xis re ntiprllel, s n e dedued from Tle VII. For oth types of sites the spins point in the - plne nd spins whih re loted in the sme - plne re olliner. An inspetion of the spin struture suggests tht NN intertions etween Ni spins on djent spines re AF, oth within nd etween Kgomé plnes. The moment in the -diretion is zero within the error r, inditing the presene of spin nisotropy whih fores the spin into the plne. To determine the LTI mgneti struture in mgneti field long the -xis, we mesured the intensity of 28 mgneti Brgg peks in the h,k,0 plne for T=6 K nd H = 2 T. We otined est greement with the experimentl dt for sis vetors elonging to the irreps 1 nd 4. The struture is given y m 4 s = 2.51, 0.12,02 B, m 1 s = 0.53i,1.11i,02i B, m 4 = 0,0.01,02 B, m 1 = 0.068i,0,0 B, 21 The qulity of the fit is given y R p =0.29 nd 2 =9.3. As t zero field, we ssumed tht the fixed spin length onstrint fvored y the qurti terms in the Lndu expnsion requires reltive phse of /2, leding to purely omplex numers for the eigenvetors of 1. Qulittively, these prmeters re similr to those of the zero-field struture. The high-field LTI phse thus onsists of spirl on the spine sites nd no moment on the ross-ties, possily euse trnsverse field hs strong effet on the ross-ties whih re more wekly oupled ntiferromgnetilly. In ontrst to the HTI phse, however, the ordered moment t nonzero field is lrger thn t zero field

15 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ PHYSICAL REVIEW B 74, P nd C phse The C phse hs the sme symmetry s the P phse, nd for fields long the -xis, there is no phse oundry etween the high-field phse nd the zero-field phse for T CC TT LC. The symmetry of the C mgneti struture n thus e determined in high mgneti field long the -xis. We mesured set of mgneti Brgg peks in the h,k,0 plne t T=0.1 K nd H=8 T, nd we found tht the dt re est desried y irrep 7 nd the prmeters 7 = 2.41,0,0.05 B, m 7 = 0,0.81,01 B. 22 The qulity of the fit is given y R p =0.23 nd 2 =17. When we llowed the fit to inlude lso 1 prmeters, the omponents of the moments were found to e 0.11 B nd therefore sttistilly insignifint. A mgneti field long the -xis indues mgneti order even in the prmgneti phse. At T=10 K nd H=8 T, the spin struture is est desried y the irrep 7 with the following prmeters: 7 = 0.612,0,0.02 B, m 7 = 0,0.373,0.04 B. 23 The qulity of the fit is given y R p =0.31 nd 2 =2.49. Thus the field-indued mgneti struture t T=10 K is desried y the sme irrep s t T=0.1 K nd H=8 T. This is orroorted y the sene of phse trnsition upon ooling in high field, s shown in the temperture dependene of the 1,1,0 Brgg pek shown in Fig. 16. F. Ferroeletriity Reently NVO hs een shown to hve remrkle ferroeletri ehvior. 12 A spontneous polriztion P hs een found to pper only in the LTI phse. In other words, upon ooling into the LTI phse, the ferroeletri order prmeter develops simultneously with the LTI mgneti order prmeter. In Ref. 12 it ws proposed tht the multiomponent order prmeter ssoited with this phse trnsition requires the triliner oupling, V, where V = * LTI HTI + * LTI HTI P. * 24 Here P is the omponent of the spontneous polriztion nd the s re omplex-vlued order prmeters whih desrie the inommensurte long rnge order ssoited with irrep 4 for the HTI phse nd with irrep 1 for the dditionl order prmeter ppering in the LTI phse. Of ourse, V must e invrint under the symmetry opertions of the prmgneti phse. 39,40 As we hve seen, the mgneti order prmeters stisfy Eq. 8, whih here is I A = A *, 25 where A denotes either LTI or HTI. Using this reltion, we see tht the invrine of V under sptil inversion implies FIG. 17. M versus H long the three rystllogrphi xes for sequene of tempertures. tht is pure imginry: =ir, where r is rel vlued. Thus we my write V =2r P LTI HTI sin HTI LTI, 26 where the phses re defined y A = A e i A. To e invrint under the opertions of the little group P must trnsform like 4 1. Referring to Tle VI, wesee tht this mens tht P must e odd under 2 nd m. The first of these onditions mens tht r n only e nonzero for = or =. The seond ondition mens tht r n only e nonzero for =, in greement with the oservtion 12 tht the spontneous polriztion only ppers long the diretion. Hd we hosen the irrep 2 for the new LTI representtion in Eq. 20, we would hve inorretly predited the spontneous polriztion to e long the diretion. G. Mgnetiztion nd suseptiility mesurements Although neutron diffrtion enles one to fix mny detils of the mgneti struture, it is hrd to otin the ulk mgnetiztion from these mesurements euse the signl ssoited with ulk mgnetiztion is uried in the nuler Brgg peks. Aordingly, we summrize here the results for the zero wve-vetor mgnetiztion, M, mesured with SQUID mgnetometer, s funtion of field for fields long eh of the rystllogrphi diretions shown in Figs. 17 nd 18. The following fetures re noteworthy. In Fig. 17 one sees tht for H long the -xis, there is rnge of temperture orresponding to the C phses in whih M vs H does not extrpolte to zero for H 0. This is the est mesurement of the wek FM moment in this phse. The C phse my lso hve finite remnnt mgnetiztion though further me

16 KENZELMANN et l. FIG. 18. for mgneti field pplied long eh of the rystllogrphi diretions in Bohr mgnetons per Ni ion per Tesl versus T, where is defined y =M /H t H=0.1 T. surements re needed there. From these dt one sees onfirmtion of the phse oundries otined fropeifi het mesurements shown in Fig. 2 whih here re signled y disontinuities in the mgnetiztion s funtion of H when phse oundry is rossed. In Fig. 18 we show M /H, mesured for smll field H=0.1 T. This quntity will e nerly equl to the zero field suseptiility exept when it proes the spontneous mgnetiztion, s when H is long 0,0,1 nd T4 K nd the system is in the C or C phse. It is remrkle tht there re no visile nomlies ssoited with the phse trnsitions involving the HTI phse. V. THEORETICAL INTERPRETATIONS Here we disuss the simplest or miniml model tht n explin the experimentl results for NVO. In generl, the higher the temperture the simpler the model needed to explin experimentl results. Aordingly, we disuss the theoretil models for the phses in the order of deresing temperture. Tht wy, we will see tht the miniml model is onstruted y sequentilly inluding more terms s lower temperture phses re onsidered. A. The high temperture inommensurte (HTI) phse 1. Competing nerest nd next nerest neighor intertions Inommensurte phses usully result from ompeting intertions. 17,18 In the HTI phse we found spin ordering predominntly on the spine sites see Eq. 18, so the miniml model will del only with the spine spins. Sine the inommensurte wve vetor is long the spine xis, it is resonle to infer tht this inommensurility rises from ompetition etween NN nd NNN intertions long spine. We lredy indited tht suh ompetition is plusile, in view of the nerly 90 ngle of the NN Ni O Ni onds. Thus the miniml model to desrie the inommensurte phse is H spine = 1 2 n=1 2 J n r, n S rs r + n + H A + H ss, 27 PHYSICAL REVIEW B 74, where J 1 nd J 2 represent the priori nisotropi NN nd NNN exhnge intertions, is summed over the xes,, nd, r is summed over only spine sites, nd n =±n/2â re the first nd seond neighor displement vetors for the spine sites rememer tht there re two spine spins long eh spine xis in the unit ell, so the NN distne is /2. Also H ss represents proly wek AF intertions etween NN in djent spines: J for spins t distne /2 in the -diretion nd J for spins t distne /2 in the -diretion. In this miniml model, for the purpose of Fourier trnsformtion, we ple the lttie sites on n orthorhomi lttie for whih the - plnes re not ukled, ut we retin the sme intertions etween spins s for the ukled lttie. In this wy our model gives the orret thermodynmis, even though it should not e used to otin sttering intensities. To hve ompetition, we must hve J 2 0 t lest for the relevnt, whih turns out to e =. Given tht t low temperture we end up with ommensurte AF ordering of the -spin omponents, it is lso resonle to ssume tht J 1 0 t lest for =. If the spines did not intert with one nother, they would form n rry of independent onedimensionl systems for whih therml nd/or quntum flututions would destroy long-rnge order. To understnd the oupling etween djent spines, note tht displement /2from spine in one Kgomé lyer to spine in n djent Kgomé lyer tkes site No. 1 in one unit ell into site No. 4 in nother unit ell nd vie vers. Similrly this displement tkes site No. 2 in one unit ell into site No. 3 in nother unit ell nd vie vers. From Tle VII or Tle VIII one sees tht for the tive irreps No. 4 nd No. 1 suh nerest neighor displement long orresponds to hnge in the signs of nd. The ft tht these re the dominnt order prmeters for the spine sites therefore strongly suggests tht the NN interspine intertions long re ntiferromgneti. The evidene for ntiferromgneti intertions long is lmost s ompelling. For these NN pirs sites No. 1 nd No. 2, or sites No. 3 nd No. 4 the lrgest spine-spin order prmeters of the LTI phse see Eq. 20, i. e., the x omponents whih order ording to irrep No. 4 nd the y omponents whih order ording to irrep No. 1, re oth ntiferromgneti long. While it is true tht the ordering of the x omponent ording to irrep No. 1 is ferromgneti long, the mgnitude of this omponent of the order prmeter is indistinguishle from zero. Thus we onlude tht the NN intertions J 1, J, nd J re ll ntiferromgneti. Also, in Eq. 27 H A is single ion nisotropy, H A = A S r 2. r 28 The ontinuum men-field phse digrm of this Hmiltonin is known, for oth exhnge nd single ion nisotropies. 17,18 To determine whih phse orders s the temperture is lowered from the prmgneti phse, it is suffiient to look t the qudrti terms in the expnsion of the free energy per spin in the Fourier omponents of the spins,

17 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ where F 2 spine = 1 2 p s, p 1 S ps p, 29 s, p 1 = T/C + Ĵ p 2A 30 is the omponent of the inverse suseptiility of the spine sites ssoited with the Fourier omponent S p = S re ip r /N 31 r with the sum over ll the N=4N u spine spins in the lttie, while Ĵ p is the Fourier trnsform of the exhnge intertions. In Eq. 30, C is the Curie onstnt, C = SS +1/3 = 2/3. 32 We mesure energy in temperture units, whih mounts to setting the Boltzmnn onstnt k=1. For our simple model 27, Ĵ p =2J 1 osp /2 + J 2 osp + J osp /2 + J osp /2. 33 As T is lowered, the first phse to order will involve the order prmeter S p for whih s, p 1 first vnishes. For J 1 4J 2, this hppens for p = Q 0 21/,1/,1/, 34 implying simple two sulttie ntiferromgnet long eh spine hin with the orthorhomi unit ell ontining four spins in eh sulttie, nd with the spins inside the unit ell vrying with e iq 0 r. However, for J 2 J 1 /4, the minimum in Ĵ p ours t the inommensurte wve vetor p 0 =2q 0 /,1/,1/. 35 The modultion wve vetor q 0 for the spin omponent in r.l.u. is given y 17,18 osq 0 = J 1 /4J 2, nd thus the suseptiility for this wve vetor eomes 36 s, p 0 1 = T/C 2J 2 + J 2 1 /8J 2 + J + J + A. 37 Experimentlly, we know tht the leding order prmeter in the HTI phse onerns S p 0. In this phse, the neutron diffrtion dt lso give wve vetor whih vries slightly with temperture, lose to q 0 =1±q0.72 or Rell Fig. 9 nd the disussion of Eq. 17. To void onfusion, from now on we use the nottion q 0 =0.72. Using this pproximte vlue, Eq. 36 gives J J Thus we onlude tht t this men field level one hs PHYSICAL REVIEW B 74, T PH =2CJ 2 + J 2 1 /8J 2 + J + J + A, 39 nd we end up with longitudinlly modulted spin struture, with S r=s r=0 nd, long single spine, S x = S os2q 0 x/ + n, 40 where the phse n my depend on the spine index n. The n-dependene of n would e determined y the interspine oupling. For our model the neutron diffrtion dt indites tht djent spines re ntiferromgnetilly rrnged, so tht for generl spine site we my write S r = S osp 0 r + 0, 41 where the trnsverse omponents of p 0 re fixed s in Eq. 35 nd we dopt the nottion p 0 p 0. When one goes eyond ontinuum men-field theory, it is found tht insted of q 0 eing ontinuous funtion of J 1 nd J 2 one otins devil s stirse dependene of the wve vetor on the ontrol prmeters. 43,44 This tretment lso shows tht 0 nnot e fixed ritrrily s it would e in ontinuum men-field theory. Furthermore, ritil flututions will redue the tul T PH y ftor whih depends on the sptil nisotropy. To summrize: the leding spine moments, s oserved in the experiments, n e explined using isotropi ompeting NN nd NNN intertions. The rtio of NNN to NN intertions etween spine spins is losely fixed in Eq. 38 y the experimentl vlue of the inommensurtion. The mgnitude of J 1, given in Eq. 78 elow, is less well determined. The mgneti struture indites tht the intertion etween djent spins in the two trnsverse diretions is ntiferromgneti. For the spines the diretion is oviously the esy diretion. Although theory indites tht the inommensurtion should proeed vi sequene of ommensurte longperiod phses, we oserve, within experimentl resolution, tht the inommensurte wve vetor vries ontinuously with temperture. 2. Dzyloshinskii-Moriy intertions In ddition to the leding omponent of the spine spins, Eq. 18 lso indites smll, ut non-negligile, spine moments long the -xis nd long the -xis. Suh moments follow diretly from the Dzyloshinskii-Moriy DM intertions 26,27 etween spine spins. For simpliity, we onsider here only the NN DM intertions long the spine, H DM = Dr Sr Sr + î/2, 42 r nd the symmetry of the lttie dittes tht the DM vetors ehve s Dr = 0,D e iq0 r,d e ip r, 43 where Q 0 is the wve vetor for the two-sulttie ommensurte wve vetor Eq. 34, while P =0,0,2/ represents AF ordering long the -xis. Next nerest neighor DM intertions, disussed in pper II, do not hnge the qulittive results presented here. The Fourier representtion of H DM yields free energy of the form

18 KENZELMANN et l. F DM = D osp /2S ps p Q 0 id sinp /2 p S ps p P + h.. 44 Introduing qudrti terms in the trnsverse spine spin omponents, s in Eq. 29, we n now minimize the free energy nd find these trnsverse omponents: nonzero S p 0 genertes S p 0 + Q 0 2D s, p 0 + Q 0 osq 0 S p 0, S p 0 + P 2iD s, p 0 + P sinq 0 S p 0, 45 onsistent with the reltive signs nd phses of m s in Tle VIII. In ft, for ll the group representtions, ll the internl struture within the orthorhomi unit ell n e reprodued using ftors like e ip r, with the three wve vetors p 0, p 0 +Q 0 nd p 0 +P with possily different p 0 s for different representtions. Using the vlues from Eq. 18, nd p =2q 0 / with q , Eq. 45 yields D 0.07 s, p 0 + P 1, with PHYSICAL REVIEW B 74, H s = H S 2, H = s4 J is i, s where the mtrix Ji whih reltes to the oupling etween the spins t si nd t 2 ontins oth symmetri pseudodipolr, PD nd ntisymmetri DM off-digonl terms, whose signs depend on i see pper II. Ignoring the intertions mong ross-tie spins, eh suh spin will follow its lol field, S 2 =, H, 49 where, is the -omponent of the ross-tie suseptiility. The ove nlysis yields the spin omponents on eh ross-tie site in terms of the four surrounding spine spins. Tking lso into ount the vrition of the mtries Ji for different plquettes, s we did for Dr in Eq. 43, nd ssuming only liner response for the ross-tie spins, we end up with D 0.04 s, p 0 + Q S p 0 = 4, j d S p 0 sinq 0 /2, Ner T PH, s, p 0 1 is smll, ut s, p 0 +P 1 nd s, p 0 +Q 0 1 remin finite. Sine T PH =9.1 K, it is resonle to ssume tht ll these inverse suseptiilities re of order 10 K. Thus we hve the estimtes, D 0.4 K nd D 0.7 K. To summrize: the next to leding omponents of the oserved spine moments re explined s due to the DM spinespine intertions. This identifition llows us to estimte the DM vetor ssoited with these intertions. Below we onjeture set of exhnge prmeters, whih give p 0 +P 1 13 K nd p 0 +Q K, whih led to the estimtes D 0.5 K nd D 3 K. 3. The spine-ross-tie intertions The representtion 4 lso llows some smll inommensurte moments long the nd xes on the ross-tie sites. Indeed, the experimentl vlues in Eq. 18 lso llow for suh moments, ls with lrge error rs. Here we disuss the possile theoretil origin for these moments, nmely the nisotropi spin intertions etween the spine nd the rosstie spins. In ll the phses, the two spine hins whih re nerest neighors to given row of ross-tie sites in n - plne in this setion we ignore the ukling, whih does not ffet the present onsidertions hve ntiprllel -omponents of spins. Thus even in the inommensurte phses, the isotropi NN spine-ross-tie intertion is frustrted, nd one needs to dd symmetri nd ntisymmetri nisotropi spine-ross-tie intertions. In the simplest pproh, these intertions n e written in terms of n effetive internl field produed y the four spine spins s1 to s4 surrounding ross-tie spin 2 see Fig. 5, so tht S p 0 + P =4, j + d S p 0 sinq 0 /2, 50 where j nd d re the symmetri PD nd ntisymmetri DM elements of the mtrix J1. Using the experimentl vlues of the ross-tie moments from Eq. 18, we end up with j + d, = 0.06 ± 0.06, j d, = 0 ± To summrize: the moments whih re oserved on the ross-tie spins re explined s due to their PD intertion with the neighoring spine spins. Ignoring the very wek intertion mong the ross-tie spins, we n use the free spin Curie suseptiility,, 1/T0.2/K, hene j +d 0.33 K, j d However, the unertinties of these estimtes re very lrge. B. Temperture evolution of phses In this setion we will disuss the sequene of phses whih pper s the temperture is lowered t zero mgneti field. 17,18 For this purpose we will hve reourse to onedimensionl model whih inorportes ompetition etween NN nd NNN exhnge intertions J 1 nd J 2, respetively, nd lso single ion esy-xis nisotropy K. Suh model is oviously not to e used quntittively. However, it does eluidte some of the physis of the rel threedimensionl system for whih men field theory does not differ very muh from the men field tretment of the onedimensionl model, whose Hmiltonin is

19 FIELD DEPENDENCE OF MAGNETIC ORDERING IN¼ PHYSICAL REVIEW B 74, FIG. 20. Qulittive ehvior of r K versus K t T=0, showing the limit of stility of the AF phse. FIG. 19. Color online Numerilly otined men-field phse digrm for the Hmiltonin of Eq. 52. The dshed line illustrtes vlue of K t whih one hs the oserved sequene of phses whih ws oserved t zero field for NVO. H = K n S n 2 + J 1 Sn Sn +1 n + J 2 Sn Sn n For this Hmiltonin the men field Hmiltonin is H MF = n Hn, where Hn = KS n 2 + J 1 Sn 1 + Sn +1 + J 2 Sn 2 + Sn +2 Sn, 53 where Sn is the self-onsistently determined therml verged vlue, Tr Sne Hn/kT Sn = Tr e Hn/kT. 54 We solved this set of self-onsistent equtions itertively for the rtio J 1 /J 2 =2.56 to get the oserved inommensurte wve vetor nd found the phse digrhown in Fig. 19. The phses hve the properties expeted within this simple model: in the HTI phse the spins re ligned long the -xis with sinusoidlly modulted mplitude. In the LTI phse sinusoidlly modulted trnsverse omponent of spins ppers to yield n elliptilly polrized inommensurte struture nd in the AF phse we hve two sulttie ntiferromgnet with spins ligned long the -xis. This phse digrm n e understood in simple terms. Consider the qudrti terms in the Lndu expnsion, s in Eq. 29. As the temperture is lowered, the first ordered phse will hve the spins ligned only long the esy xis. As the temperture is further lowered, the qurti terms in the Lndu expnsion ome into ply nd they progressively enfore the onstrint of fixed spin length. This onstrint is perfetly stisfied only t zero temperture. To stisfy this onstrint, nd yet still stisfy the inommensurility enfored y the ompetition etween J 1 nd J 2, the system develops long-rnge trnsverse sinusoidl order, whih requires entering new phse, the LTI phse. The rnge in temperture over whih the HTI phse is stle will inrese s the nisotropy K is inresed, euse to overome lrger K requires stronger enforement of the fixed length onstrint, whih in turn requires going to lower temperture. To omplete the piture we now onsider the sitution for T ner zero. Here it is onvenient to onsider two limits, the first with K=0 the isotropi limit nd the seond with K =, the Ising limit, in whih the system is the ANNNI model. 44 In the first se, one finds tht the system is ntiferromgneti for rj 2 /J 1 r K=0, where r K=0=1/4 nd is modulted phse for rr 0. For the ANNNI limit the system is ntiferromgneti for rr K==1/2 nd is n up, up, down, down stte for rr K=. Aordingly, we my guess tht r K hs qulittively the ehvior shown in Fig. 20. Sine 1/4r1/2, one sees tht this digrm explins tht t zero temperture, for our model with r = 1/ , the AF phse will pper t some ritil vlue of K whih we find numerilly to e K/J One n go further to onsider the effets of temperture. We need to ompre the renormliztion with temperture of K to tht of the J s. For this estimtion we onsider the se of J 2 =0. Here the Hrtree deoupling shows tht the exhnge integrls renormlize proportionl to the internl energy, 45 UTU0+T 4, wheres the nisotropy renormlizes with power of the stggered mgnetiztion, NTN0 T 2 whih hs fster vrition with temperture thn does the internl energy. Thus, s the temperture inreses we expet tht the effetive vlue of K/J 1 dereses. This mens tht s the temperture is inresed, the system vlue of KT =0/J 1 T=0 hs to inrese in order to reh the AF phse, s the numeril result shows. Note the dshed line of Fig. 19 whih leds to the sequene of phses oserved for NVO s the temperture is lowered, P HTI LTI AF. We reiterte tht this model hs t est qulittive vlidity. Although we use it to eluidte the physis, it nnot e used to estimte the numeril vlue of K/J 1. C. The low temperture inommensurte (LTI) phse We now onsider wht men-field theory 17,18 hs to sy out the LTI phse. Eqution 20 indites growing