Mean-field theory for ferroelectricity in Ca 3 CoMnO 6

Transcription

1 Mn-fild thory for frrolctricity in C 3 CoMnO 6 Y. J. Guo, 1 Shui Dong, 1,2,3 K. F. Wng, 1 nd J.-M. Liu 1,4, * 1 Ntionl lbortory of Solid Stt Microstructurs, Nnjing Univrsity, Nnjing , Chin 2 Dprtmnt of Physics nd Astronomy, Univrsity of Tnnss, Knoxvill, Tnnss 37996, USA 3 Mtrils Scinc nd Tchnology Division, Ok Ridg Ntionl Lbortory, Tnnss 32831, USA 4 Intrntionl Cntr for Mtrils Physics, Chins Acdmy of Scincs, Shnyng, Chin Rcivd 30 Dcmbr 2008; publishd 4 Jun 2009 An lstic Ising modl for CoMnO 6 chin is proposd to xplin th frrolctricity inducd by collinr mgntic ordr in C 3 CoMnO 6, nd thn mn-fild thory with intrchin spin intrctions bsd on this modl is dvlopd. With inclusion of th dynmics of polriztion domins t finit tmprtur, w ddrss th rtionlity of our thory by th good grmnt of th clcultd lctric polriztion nd dilctric suscptibility with th rportd dt on C 3 Co 2 x Mn x O 6 x 0.96 Y. J. Choi, H. T. Yi, S. L, Q. Hung, V. Kiryukhin, nd S.-W. Chong, Phys. Rv. Ltt. 100, , typicl ditomic Ising spin chin systm, whil th prdictd mgntic suscptibility shows som diffrnc from xprimnt, th rson of which is discussd. DOI: /PhysRvB PACS numbr s : q, Bh, Kz, Pq I. INTRODUCTION Mgntolctric ME coupling btwn frrolctric nd mgntic ordrs, th so-clld multifrrocity, hs drwn incrsing ttntion rcntly. 1 4 Th microscopic mchnism tht cuss th multifrrocity hs not yt bn wll pprochd, lthough significnt progrsss hv bn md in th pst fw yrs. 5 9 To dscrib this coupling btwn th frrolctric nd mgntic ordrs for noncollinr nd spirl spin-ordrd systms, two mjor microscopic thoris hv so fr bn proposd. On is bsd on th spin currnt scnrio 5,6 nd th othr dls with th Dzyloshinskii- Moriy intrction DMI Rfs. 10 nd 11 which is blivd to ply th crucil rol. 7 9 Th two thoris illustrt consistntly th frrolctricity idntifid xprimntlly for quit numbr of spirl spin-ordrd systms. Nvrthlss, rcnt xprimnts rvld tht th spirl spin ordr is not th prrquisit for th ME coupling nd frrolctricity gnrtion Th idntifiction of frrolctricity in C 3 Co 2 x Mn x O 6 x 0.96 Rfs. 14 nd 15 confirmd convincingly tht th collinr spin ordr cn gnrt frrolctricity too. C 3 CoMnO 6 cn b viwd s dopd C 3 Co 2 O 6 mtril whr th Co ions r prtilly rplcd by Mn ions. It is wll stblishd tht th bs systm, C 3 Co 2 O 6 CCO, is modl spin-chin compound consisting of prlll on-dimnsionl Co 2 O 6 infinit chins of fc-shring CoO 3 trigonl prisms nd octhdrl prisms long c xis. Ech chin is surroundd by six qully spcd chins, which mks tringulr rrngmnt on th b pln. Th intrchin intrctions long th c xis r much strongr thn th intrchin ons btwn th Co 2 O 6 chins. Similr to CCO, C 3 Co 2 x Mn x O 6 CCMO hs lso much strongr intrchin intrctions thn th intrchin ons nd cn b structurlly viwd s consisting of prlll Ising spin chins with wk intrchin intrctions. It is good modl for undrstnding th physics of qusi-on-dimnsionl spinchin systms. At x=1, in ch chin, th Mn nd Co ions will occupy th oxygn-octhdron nd trigonl-prism sits, rspctivly, nmly, th Mn nd Co r ltrntly rrngd instd of rndomly distributd, which is th ky ingrdint for th multifrroicity. 16,17 Both th Mn nd Co r t th high-spin sits, 16,17 nd ch Ising chin hs n up-up-downdown mgntic ordr in th ground stt Diffrnt from spirl mgnts, for CCMO th xchng striction from th ntifrromgntic AFM intrction btwn nrst-nighbor NN spins long th chin, which shortns th bonds btwn th ntiprlll spins nd strtchs th bonds btwn prlll spins, is rsponsibl for th lctric polriztion. This xchng striction mchnism is rltivly simpl nd cn b dscribd by Ising-lik modl, to b don in this work, whil rl CCMO hs vry complictd nd dlict multifold intrctions nd such rltivly simpl scnrio bcoms prticulrly usful for dsigning novl multifrroics. Although quit fw works hv bn don to illustrt th frrolctricity in CCMO, 14 17,24 th microscopic mchnism is still confusing, nd th rsults from ths works r not fully consistnt with ch othr. Hr tnttiv pproch is ncssry to illumint th confusion from both xprimntl nd thorticl works. In this ppr, to xplin quntittivly th frrolctricity inducd by xchng striction in CCMO, w strt from CoMnO 6 chin, for which n lstic Ising modl will b proposd. W first dl with singl ditomic spin chin for th frrolctricity in th ground stt. Thn, with inclusion of both th intrchin nd intrchin intrctions of ths CoMnO 6 chins, mn-fild thory MFT is constructd for prdicting th lctricl nd mgntic proprtis t finit tmprtur T. Subsquntly, th xciting dynmics of polriztion domins is lso introducd to ccount for th dficincy of th MFT du to th brokn long-rng spin ordr for this qusi-on-dimnsionl systm t nonzro T. Th rtionlity of our modl is tstd by pplying it to prdict th frrolctricity in CCMO nd rltd proprtis s function of T, rspctivly, including mgntic nd dilctric suscptibilitis in comprison with xprimntl dt for CCMO x=0.96. Th mjor objctiv of this work is to propos microscopic thory for th ME coupling in Ising spin chin systms /2009/79 24 / Th Amricn Physicl Socity

2 GUO t l. II. ELASTIC ISING MODEL AND FERROELECTRICITY W strt from th ditomic modl for singl Ising spin chin, s th frrolctricity is inducd by th up-up-downdown spin ordr t T=0. 14 At T=0, such spin ordr cn b sn s indpndnt of th intrchin intrctions which is wk comprd to th intrchin intrctions. For such chin with ltrnting mgntic ions Mn nd Co, w considr th NN AFM intrction J Mn-Co nd th nxtnrst-nighbor NNN AFM intrctions J Mn-Mn nd J Co-Co S1 S2 S3 S4 S1 S2 both long th chin dirction c xis. In Fig. 1 w drw schmticlly two typs of ditomic spin chins lignd long th c xis. As xpctd, th ground stt hs mgntic ordr if th strngth of ths intrctions stisfis th conditions J Mn-Co J Mn-Mn +J Co-Co nd J Mn-Mn J Mn-Co J Co-Co, ssuming tht th spin momnts for ions Mn nd Co r qul thir diffrnc is vry smll. 17 Such Ising chins form tringulr lttic on th b pln, nd th configurtion is shown in Fig. 1 b with four conscutiv lyrs. With th considrtion of lctric-dipol intrctions, th Hmiltonin for th Ising modl is H = ij mn c () Co Mn J 1,i,m S i,m S j,m ij J AFM S i,m S i,n mn (b) J 2,i,m S i,m S j,m h B g l S i,m i,m J FM S i,m S i,n E P i,m i,m + 1 c P 2 i,m, 1 i,m I II III IV FIG. 1. Color onlin Two configurtions of th up-updown-down Ising chin with ltrnting Mn smll circl nd Co big circl ions. Th dshd circl rprsnts th originl positions of th ions nd th rrows indict th spin dirctions. b Four conscutiv lyrs of CCMO on th b pln. Th rl circl rprsnts spin up nd opn circl rprsnts spin down long th c xis. b whr subscript i rprsnts th ith sit in CoMnO 6 chin, subscript m rprsnts th mth CoMnO 6 chin on th b pln, ij rfrs to th NN spin pir nd ij to th NNN pir,orj Co-Co long th c xis, J 1,i =J Mn-Co, J 2,i =J Mn-Mn for th intrction btwn two NNN Mn ions or two NNN Co ions, S i,m rprsnts th spin on sit i in th mth CoMnO 6 chin, h dnots th mgntic fild, B is th Bohr mgnon, g l is th Lnd fctor, mn rfrs to th NN spin pir with n AFM xchng intrction J AFM nd mn to th NNN pir with frromgntic FM xchng intrction J FM on th b pln, 25,26 E is th lctricl fild, c is th dilctric suscptibility, nd P i is th lctric polriztion on sit i in th mth CoMnO 6 chin. It is blivd tht th intrchin intrctions J AFM nd J FM r much smllr thn th intrchin ons J Mn-Co, J Mn-Mn, nd J Co-Co. 14,17 Th lst two frrolctricity trms r lso xtrmly smll undr norml-msuring lctric fild du to th wk frrolctric polriztion; thus thy cn b nglctd sfly. Ths fcts indict tht th systm cn b trtd s n Ising modl. As shown in Fig. 1, CCMO hs spin ordr long th c xis nd on th b pln th spins form tringulr lttic. Th spin ordr is minly dtrmind by th prdominnt intrchin intrction, s th modl cn b sn s ditomic Ising chin for ground-stt polriztion. Howvr, if only th intrchin intrction is considrd, th ground-stt spin ordr hs two qul forms with opposit polriztion th lft chin nd right chin in Fig. 1. Th intrchin intrctions on th b pln, in spit of bing much wkr thn th intrchin ons, kp ll th spin ordrs to b th sm form. Thrfor, on is llowd to rgu tht th intrchin intrctions r minly th origin for th short-rng spin ordr, whil th intrchin ons tk rsponsibility for th long-rng ordr of th systm. A. Elstic intrction Diffrnt with th suggstion of Choi t l. in Rf. 14, w ssum tht th NN intrction is AFM rthr thn FM, which ws dmonstrtd in Rf. 17 ccording to th firstprincipls clcultion. Consulting to th xchng striction ffct, for th AFM intrction btwn NN spin pirs, rpulsion forc btwn th two ions with prlll spins, nd n ttrction forc btwn th ons with ntiprlll spins r xpctd. As shown in Fig. 1 whr th dshd circls rprsnt th originl ion positions without xchng striction, th spin ordr with ltrnting Mn nd Co ions which brks th invrsion symmtry hs two configurtions th lft nd right chins inducing opposit polriztions. 14 Tk th first configurtion, for xmpl th lft chin in Fig. 1, ll th Mn ions hv n upwrd displcmnt whil ll th Co ions displc downwrd. W hv th totl lctric polriztion t th ground stt T=0 for th ditomic Ising spin chin with 2N ions, 2N P gs T =0 = P i = N P Mn + P Co, 2 i=1 whr N is th numbr of Mn-Co coupls in unit volum

3 MEAN-FIELD THEORY FOR FERROELECTRICITY IN C Hr P Mn nd P Co r th lctric polriztion dfind with rspct to th shifts of Mn nd Co ions, rspctivly. Th xchng forc compting with th rstoring forc from th crystl lttic cuss th displcmnt of th ions. Along this lin, n lstic Ising modl is introducd to dscrib th xchng striction nd th xchng nrgy trms J 1,i nd J 2,i in Eq. 1 should contin long-rng lstic trm which is too complictd to b xprssd s n xplicit form. W suggst Lnnrd-Jons potntil hr for simpl tnttiv, 27 J = J 0 r 0 /r ij 12 r 0 /r ij 6, 3 whr r ij is th distnc btwn th ions on sit i nd sit j, nd r 0 is th quilibrium distnc btwn two intrcting spins. W mrk th displcmnt of th ion on sit i s r i, thn th distnc r ij cn b rprsntd s r ij =r 0 +r i +r j. Sinc r i is rltivly smll quntity with rspct to r 0, th lstic nrgy is cutoff only to th first ordr. Th lstic intrction bcoms wkr s th distnc btwn spins grows, so only NN nd NNN trms r tkn into ccount. Considring th lstic xchng ssocitd with th NNN spin pir J 2,i, no symmtry brking is found to cus ffctiv xchng striction, which indicts tht th lstic intrction should only b pplid to th NN intrction J Mn-Co. Including th lstic nrgy from crystl lttic, w xprss th lstic Hmiltonin s prturbtion to Eq. 1, H 1 = J Mn-Co i,j 6 r i + r j r 0 S i S j +1/2 k i r 2 i, i whr k i is th stiffnss fctor for ion on sit i. B. Frrolctricity t ground stt From th lstic Hmiltonin, Eq. 4, on obtins th ion quilibrium position by minimizing H 1 with rspct to r i. Thrfor th displcmnt r i bcoms r i = 6J Mn-CoS i S i+1 6J Mn-Co S i 1 S i = 12J Mn-CoS i S i+1. r 0 k i r 0 k i 5 Th lctricl polriztion in th ground stt T=0 in Eq. 2 cn b xprssd s 2N P gs = N P Mn + P Co = Q i r i i=1 = N Q Mn 12J Mn-CoS Mn S Co r 0 k Mn Q Co 12J Mn-CoS Mn S Co, 6 r 0 k Co whr Q i is th lctric chrg of ion on sit i Q Mn,Q Co, S Mn nd S Co r th spins of ions Mn nd Co, nd k Mn nd k Co r th stiffnss fctors of ions Mn nd Co, rspctivly. For CCMO, th prlll Mn-Co chins long th c xis form tringulr lttic on th b pln. Th intrchin distnc btwn th NN chins is =5.24 Å nd th intrchin 4 distnc btwn Mn nd Co ions is r 0 =2.65 Å So w obtin N= m 3. W lso hv th chrgs for Mn nd Co ions: Q Mn =4 nd Q Co =2. Th spin for Mn ion is S Mn =1.305 nd for Co ion is S Co = Furthrmor, six Mn-O bonds form n octhdrl prism, whil six Co-O bonds form trigonl prism. Thr of th six Mn-O or Co-O bonds hv n ngl Mn or Co from th Mn-Co chin long th c xis whil th othrs hv th ngl Mn or Co. W trt th covlnt bond quivlnt to Coulomb intrction with quivlnt chrgs Q Mn = Mn Q Mn, Q Co = Co Q Co, nd Q O = Mn,Co Q O for th O ions in Mn-O bonds nd Co-O bonds, rspctivly whr Q O =2 is th chrg of oxygn ion. Only considring th componnt of intrction long th c xis, th Coulomb intrction from ths six bonds cn b rgrdd s hrmonic oscilltor whn th displcmnt of th ion Mn or Co is smll, so tht th stiffnss fctors for ths bonds cn b vlutd. W hv th Coulomb intrction F for th Mn-O bonds nd Co-O bonds F = 3 0 Q Q O 2 cos 4 R r /cos 2 + Q Q O 2 cos 2 R r /cos 3Q 2 Q O 3 r, = Mn,Co, 7 0 R whr 0 is th dilctric constnt in vcuum, r is th displcmnt of Mn or Co ions long th c xis, th lngth of Mn-O bond is R Mn =1.905 Å, nd th lngth of Co-O bond is R Co =2.140 Å From th dfinition of hrmonic oscilltor, th stiffnss fctor cn thn b xprssd s k = 3Q 2 Q O 3, = Mn,Co. 8 0 R As th bond strngth for Mn-O is nd kj/mol for Co-O bond, 28,29 w obtin k Mn =221 N/m, th stiffnss fctor for Mn-O bonds, nd k Co =168 N/m, th fctor for Co-O bonds. III. MEAN-FIELD THEORY AT FINITE TEMPERATURE Givn th lctric polriztion in th ground stt, w invstigt th frrolctric nd mgntic proprtis t nonzro T. In this sction, MFT for Eq. 1 is dvlopd to vlut th T dpndncs of lctricity polriztion, mgntic suscptibility, nd dilctric suscptibility. As discussd bov, th lstic Hmiltonin is smll quntity, so th modl usd in th following cn b simplifid into simpl Ising modl with th intrchin intrctions. A. Ordr prmtrs nd ffctiv filds Givn th spin lignmnt with ltrnting Co ion nd Mn ion long th c xis t th ground stt, th Co ions nd Mn ions on th b pln form tringulr lttic. Viwing th tomic configurtion on th b pln, th smllst structur unit undr considrtion consists of four lyrs. Figur 1 b dpicts on lyr lyr I on th b pln nd th thr lyrs II, III, nd IV blow it, whr solid circls

4 GUO t l. rprsnt spin up, opn circls rprsnt spin down, smll circls rprsnt Mn ions, nd lrg circls rprsnt Co ions Fig. 1 b. For this structur unit, four ordr prmtrs should b includd to construct n ffctiv fild = S, whr =1, 2, 3, nd 4, nd S mns th vrg of S. Assuming tht spin chin hs th configurtion s shown in th lft chin of Fig. 1, with which S r mrkd out. In th ground stt, =1, 2, 3, nd 4 rprsnts Mn ions with spin up, Co ions with spin up, Mn ions with spin down, nd Co ions with spin down, rspctivly. In th ordrd phs, 1 =S Mn, 2 =S Co, 3 = S Mn, nd 4 = S Co. Clrly, for spindisordrd phs, ll th four ordr prmtrs r zro. Th mn-fild thory constructd blow is bsd on Hmiltonin Eq. 1, nglcting th lst two frrolctric trms. Th ffctiv filds from th intrchin intrctions r F 1 = J Mn-Co J Co-Co 3 F 2 = J Mn-Co J Mn-Mn 4, F 3 = J Mn-Co J Co-Co F 4 = J Mn-Co J Mn-Mn 2 9 whr J Mn-Co is th intrchin NN spin-pir intrction, J Mn-Mn nd J Co-Co r th NNN pir intrctions, nd F rprsnts th intrchin ffctiv filds on th spin momnt S =1,2,3,nd4. Givn th spin configurtion on th b pln Fig. 1 b, w considr th intrchin intrction btwn NN spins on th b pln to b AFM nd th intrction btwn NNN spins on th b pln to b FM. 25,26 Th ffctiv filds from th intrchin intrctions cn b xprssd s F 1 =2J AFM J FM 1 1 F 2 =2J AFM J FM 2, F 3 =2J AFM J FM F 4 =2J AFM J FM 4 10 whr F rprsnts th intrchin ffctiv filds on th spin momnt S =1,2,3,nd4. All th AFM intrctions J AFM btwn th NN spin pirs or th FM intrctions J FM btwn th NNN spin pirs on th b pln r supposd to b qul for diffrnt ions pirs. Following th stndrd procdur of mn-fild pproximtion, th ordr prmtrs cn b clcultd from st of slf-consistnt qutions = S tnh F + F + g l B H S, = 1,2,3,4, 11 whr S =S Mn t =1 nd 3, S =S Co t =2 nd 4, nd k B is th Boltzmnn constnt. Thr r four dgnrt solutions s T=0:,,, nd. Hr w 3 find tht th spin ordr in th ground stt is th sm s th on w prdictd in Sc. II, nd th singl chin lstic Ising modl nglcting intrction btwn chins is vlid for th frrolctricity in th ground stt. B. Frrolctric polriztion nd spin ordr With th MFT givn in Eq. 11, w cn driv out th dpndnc of polriztion P on nd thus th T dpndnc of P. In th ground stt, th spin ordr long th c xis will b grdully brokn with incrsing T du to th spin xcittion, thus th frrolctricity will fll down. First, considr spin-flip vnt whos probbility cn b dfind s = 0 / 2 0 0, =1, 2, 3, nd 4, whr is th ordr prmtr in th ground stt ordrd phs. In th mn-fild frmwork, dirct lgorithm on th configurtion of on spin chin llows th following proprtis: i in th ground stt, 1 = 2 = 3 = 4 =0 sinc = 0 nd P is givn by Eq. 2 ; ii in th disordrd phs, 1 = 2 = 3 = 4 =0.5 sinc =0, nd P=0; iii whn prmtrs 1 nd 3,or 2 nd 4, r disordrd, which mns 1 = 3 =0.5 or 2 = 4 =0.5, P is lso zro; iv for 1 = 3 =1 nd 2 = 4 =0 or 2 = 4 =1 nd 1 = 3 =0, th spin configurtion chngs from th lft chin of Fig. 1 to th right on nd P bcoms P; v if ll spins flip, i.. = 0, which mns 1 = 2 = 3 = 4 =1, th spin configurtion of th chin rmins th sm nd thus P rmins unchngd. To illustrt in clrr mnnr ths proprtis, w considr th polriztion of locl ion, for xmpl, Co ion. In this cs, w tk cr of thr-spin subsystm. Look t on Co 2+ spin nd its two NN Mn 4+ spins. For such subsystm, ight spin configurtions r countd, which cn b dividd into thr groups: i th spin lignmnt is or, gnrting polriztion P I,II = P Mn + P Co, s shown in Fig. 2 ; ii th spin lignmnt tks th form or, rsulting in polriztion P III,VI = P Mn + P Co, opposit to P I,II, s shown in Fig. 2 b ; iii th spin lignmnt tks ithr FM ordr or AFM ordr:,,,or, which dos not giv ris to ny nonzro polriztion, s shown in Fig. 2 c. Thrfor, four of th ight spin configurtions xhibit nonzro polriztion. As n xmpl, on configurtion, with th thr spins dnotd s S 1, S 2, nd S 3, rspctivly, is mrkd out by cors llips in Fig. 2. Th ground stt for this S 1 S 2 S 3 subsystm hs spin ordr with polriztion P I. In this subsystm, th probbility W for th four spin configurtions of nonzro P P I,II,III,IV cn b xprssd s function of th ordr prmtrs in th following wy: i for th ordr, W=W I = , ii for th ordr, on hs W=W II = 1 2 3, iii for th ordr, W=W III = , nd iv for th ordr, W =W VI = Th sm procdur cn lso b pplid to th othr thr subsystms, countd s S 2 S 3 S 4, S 3 S 4 S 1, nd S 4 S 1 S 2. Obviously, th vrgd polriztion P MFT for 2N-spin ditomic chin is P MFT = N P = N 4 =1,2,3,4 P I W I + P II W II + P III W III + P VI W VI = N P Mn + P Co ,

5 MEAN-FIELD THEORY FOR FERROELECTRICITY IN C S1 S2 S3 S4 ( ) ( b) () b 10 2 = 2/( 2+ 1 ) = 1/ 6 (c) S1 S2 c (b) P (d) Co Mn ( c) FIG. 2. Color onlin Eight spin configurtions for th thr spins S 1,S 2, nd S 3. Thy r clssifid into thr groups with diffrnt polriztions: P= P Mn + P Co, b P= P Mn + P Co, c P=0. which is th min rsult of th MFT, dling with th T dpndnc of polriztion P. It is found tht Eq. 12 fully mts th condition dfind bov for th rltionship btwn P nd =1,2,3,nd4. For instnc, in th ground stt, 1 = 2 = 3 = 4 =0, thn P MFT =N P Mn + P Co, sm s Eq. 2. C. Polriztion domins () c FIG. 3. Color onlin d Four xmpld polriztion domins viwd on th b pln. In b th bonds on th domin wll nd bonds insid th domin r shown, by which prmtr is clcultd. =0, b =1/6, c =2/9, nd d =1/3. shows th c-pln projction of th domin shown in b. f shows th thr-dimnsionl pttrn of th domin shown in d. As in th discussion bfor, th intrchin intrctions kp th polriztion dirction of th CoMnO 6 chins cohrnt, but such long-rng ordr is not stbl t finit tmprtur sinc th intrchin intrctions r much wkr thn th intrchin ons. It is known tht MFT cnnot work wll on th systm whil long-rng ordr is brokn, nd thn corrction to this dficincy should b ddd to our modl. Thrfor, th formtion of polriztion domins is prfrrd. In fct similr considrtion of th polriztion domins ws suggstd rlir. 17 This ffct my not influnc significntly th trnsition tmprtur for frrolctricity but ply n importnt rol in modulting th mgnitud of polriztion. Th rtionlity of this corrction will b provn blow by th quntittiv consistncy of th modl clcultion with xprimntl dt. Furthrmor, from Eq. 11 it is sily infrrd tht th ground stt t T=0 hs four spin lignmnts which r dgnrt in nrgy:,,, nd. Th formr two nd, shown s th lft chin in Fig. 1, hv n upwrd polriztion whil th lttr two nd hv downwrd polriztion, shown s th right chin in Fig. 1. It is sy to conclud tht th rvrsl of polriztion domin should mt th condition: ithr 1,3 1,3 whil 2,4 rmins invribl or 2,4 2,4 whil 1,3 rmins invribl. Bcus of ths dgnrt spin-ordr stts, th xistnc of polriztion domins t nonzro T is rsonbl rgumnt. Thrfor, w nd to tk into ccount th polriztion domin-wll nrgy. W focus on th domins with downwrd polriztion nd vlut th nrgy for thm to rvrs from th upwrd polriztion mtrix. Two trms hv contribution to th xcitd nrgy: on is th chng in th intrchin intrctions nd th othr is th Zmn nrgy from mgntic fild. Clrly, th smllst polriztion unit is CoMnO 6 chin of four spins long th c xis, ithr tking th form / or / t th ground stt. To chrctriz ny polriztion domin consisting of on or mor such units, w nd thr prmtrs. First, w dnot by prmtr L th numbr of th units long th c xis i.., domin hight. Scond, prmtr 0,1 is th clustring dgr of th domins on th b pln. Third, prmtr dscribs th totl chng in spin momnt for th rvrsl of this domin. It is sy to s tht th xcitd nrgy is proportionl to L which is supposd to b constnt givn fild h. Th clustring dgr is supposd to msur th ffct of th intrchin intrctions nd it scls roughly th domin siz on th b pln. In Fig. 3 svrl simpl polriztion domins on th b pln r shown ch circl on th b pln rprsnts polriztion unit, which r, rspctivly, mbddd in th mtrix of th opposit domin. In Fig. 3 b n xmpl to vlut prmtr is givn, nd w only considr th ffct from th NN intrchin intrction. W dfin =2/12=1/6. Similrly, on hs =0 for th domin in Fig. 3, =2/9 for th domin in Fig. 3 c, nd =1/3 for th domin in Fig. 3 d. With such dfinition, L P (f) b c

6 GUO t l. w s tht th xcitd nrgy ssocitd with rvrsl of nw domin from th prnt domin is proportionl to 1 normlizd by th numbr of th in-pln polriztion units insid this nw domin. For prmtr, th chng in spin momnt is from 0 to 2g l B for domins of up polriztion nd from 0 to 2g l B for domins of down polriztion. So chngs btwn 0 nd 2 upon ny domin rvrsl. Consquntly, th xcitd nrgy for rvrsl of th two typs of domins up nd down polriztion cn b xprssd s E 1 = L 1 2 F F Lh g l B 1 + 3, E 2 = L 1 2 F F Lh g l B 2 + 4, 13 whr is constnt for spcific domin. Following th Boltzmnn xcittion, th probbility for rvrsl of th two typs of domins r W d1 = xp E 1 / 1 + xp E 1 W d2 = xp E 2 / 1 + xp E D. Polriztion, mgntic suscptibility, nd dilctric suscptibility W considr th probbility for domin rvrsl ovr th whol lttic s th vrg of W d1 nd W d2 in Eq. 14, i.., W domin = W d1 +W d2 /2. Th finl polriztion givn by th MFT plus polriztion domin corrction dnotd by MFT +DC is P MFT+DC, P MFT+DC = P MFT 1 2 W domin. 15 Whn H is vry smll, th mgntic suscptibility cn b dfind s MFT = M/h = g l B /h. 16 With th polriztion domin corrction, th mgntic suscptibility is MFT+DC = MFT 1 W domin. 16b Similr to th clcultion of P in Eq. 12, th vrg of P 2 is P 2 = 1 4 =1,2,3,4 P I 2 W I + P II 2 W II + P III 2 W III + P VI 2 W VI = P Mn + P Co Noting tht th rvrsl of polriztion domins, which mks P into P, dos not chng th vlu of P 2, th dilctric suscptibility cn b clcultd from sttistic fluctution 24 J Mn-Co J Co-Co J Mn-Mn J AFM J FM TABLE I. Prmtrs chosn in th modl clcultions. mv 0.5 S Mn B mv 0.55 S Co B 1.24 mv 0.45 L h=0,4,7t 12, 8, 5 V 2 h=0,4,7t 0.7, 0.6, 0.55 V g L 2 rltiv unit 40 c = + P2 P 2, 18 T whr is th dilctric suscptibility contributd from othr ions in CCMO systm, nd it is suggstd to b constnt. IV. RESULTS AND DISCUSSION A. Comprison with xprimnt In this sction, w prsnt our modl clcultion on th T-dpndnt polriztion, mgntic suscptibility, nd dilctric suscptibility, nd compr thm with th msurd dt tkn from Rf. 14. Th prmtrs chosn for our clcultion r shown in Tbl I. By tking J Mn-Co = 0.5 mv into Eq. 6, on obtins th lctric polriztion in th ground stt P gs 92 C/m 2. With th ssumption of no structur chngs whil x is vry clos to 1.0 x 1, w trt th C 3 Co 2 x Mn x O 6 s x C 3 CoMnO 6 1 x C 3 Co 2 O 6 for simplicity. It is rough but still rsonbl sinc th Co prfrs to occupy th trigonl prism sit whil th Mn prfrs th octhdron sit du to th chmicl stbility, s formntiond. Thrfor, whn thr s tiny non-stoichiomtry-lik x = 0.04, th min upup-down-down spin structurs nd Co-Mn ltrntion should b kpt lthough thr r som C 3 Co 2 O 6 insrtions. Sinc it is known tht no lctricl polriztion in C 3 Co 2 O 6 systm is vilbl, th ground-stt polriztion cn b xprssd s vibl to x, P=N P Mn + P Co x. For x=0.96, w obtin P 88 C/m 2, nd this vlu is vry clos to th msurd dt P 90 C/m 2 t T=2 K. 14 Of cours, to fully ccount this nonstoichiomtry issu, th corrltion btwn th xcssiv Co should b considrd, which is vry complx spcilly whn x is lrg nd crtinly byond th currnt work. Figurs 4 4 c prsnt th clcultd polriztion P opn circl dots: MFT; opn squr dots: MFT+DC, s function of T undr svrl filds h. Th msurd dt tkn from Rf. 14 r lso plottd in th figurs solid circl dots: xprimnt for comprison. It is clrly shown tht th tmprtur dpndnc of P from th MFT hs grbl T c bout 16 K of frrolctricity with xprimnt dt, nd th polriztion from th MFT+DC shows quit good grmnt with th msurd on. This prfct grmnt suggsts tht th prsnt mn-fild thory cpturs th ssnc of frrolctricity gnrtion in CCMO. Figurs 5 nd 5 b prsnt th clcultd mgntic nd dilctric suscptibilitis nd s function of T for

7 MEAN-FIELD THEORY FOR FERROELECTRICITY IN C FIG. 4. Color onlin Elctric polriztion P s function of T, clcultd from th MFT opn circl, MFT+DC opn squr, nd tkn from xprimnt full circl, Rf. 14. h=0 T, b h =4 T, nd c h=7 T. comprison with xprimntl dt. It is sn tht th clcultd undr h=0.2 T nd undr h=0 r on th sm ordr of mgnitud with msurd dt. For dtils, th msurd T hs nonzro vlu t T=0 K, whil th clcultd on is zro for th AFM intrction btwn th NN spin pirs both long th c xis nd on th b pln. At low T bout 2 5 K th msurd T hs U-typ-lik bhvior, whil th clcultd on incrss monotonously. Howvr both th two T curvs hv th sm Nl point. W shll discuss th possibl rson for th diffrnc. It should b ddrssd tht th clcultd T shows similr bhvior to th msurd on, nd thy both xhibit th singl pk pttrn ginst tmprtur. B. Discussion Our modl focuss on n Ising modl with ditomic chins long th c xis nd tringulr lttic on th b pln. Rl CCMO smpls r much mor complictd. W ddrss th rtionlity of our MFT thory nd lso th rson for th diffrnc btwn xprimnt dt nd our MFT clcultion. Prmtrs in Tbl I r chosn to fit th xprimnt dt nd r lso vlidtd by th rtionlity of th choic if on consults to rlir works Howvr, ths rlir works gv quit scttrd rsults nd crful choic of th prmtrs is bsd on our discussion prsntd blow. First, ccording to th first-principls clcultion, 17 th NN intrchin intrction J Mn-Co is AFM rthr thn FM, FIG. 5. Color onlin Mgntic suscptibility nd b dilctric suscptibility s function of T, clcultd by th MFT opn circl, MFT+DC opn squr, nd tkn from xprimnt full circl, Rf. 14. Th inst in b shows from T=0 90 K. which is inconsistnt with th suggstion by Choi t l. 14 Howvr, th bonds btwn prlll Mn nd Co spins r still shortnd vn with th AFM intrction btwn th NN spin pirs in Rf. 17, nd th displcmnts r ttributd to th dirct mtl-mtl bonding, which is strongr btwn ions with idnticl spin thn th on with opposit spins. 17,30 Sinc th clcultd polriztion in Rf. 17 is unrsonbly lrg bout 200 tims lrgr thn th xprimntl on, w qustion th rlibility of this clcultion. W still bliv tht it is th xchng striction to tk rsponsibility for th frrolctricity in CCMO systm in this ppr. In Tbl I, J Mn-Co is chosn fvoring th AFM ordr, s th clcultion with th FM intrction dos not gr with xprimnt dt. With FM intrction btwn th NN spin pirs, high h =4 nd 7 T my induc FM ordr rthr thn ordr. This is lso disdvntg for th ssumption of FM intrction btwn th NN spin pirs. Scond, it is lso found tht th chosn xchng intrctions J Mn-Co, J Mn-Mn, nd J Co-Co in Tbl I 0.5, 0.55, nd 0.45 mv r much smllr thn thos tkn for firstprincipls clcultion in Rf , 2.09, nd 1.63 mv. W prsnt our clcultd ordr prmtrs s functions of T in Fig. 6. It is sn tht th clcultd bhviors r quit wll consistnt with xprimntl obsrvtions givn in Rf. 14, in svrl wys blow: i xprimnt finds no significnt chng in mgntic structur t T=8 K from th ground stt, whil th clcultd ordr prmtrs in Fig. 6 rmin invribl from T=0 to T=8 K. ii Exprimntlly, frrolctricity pprs blow T 16 K whil ordr prmtrs 2 nd 4 bcom fully disordrd t T=16 K. iii Exprimntlly, no mgntic or

8 GUO t l. FIG. 6. Color onlin Four ordr prmtrs s function of T t h=0, nd b probbility W domin for polriztion domin rvrsl s function of T undr vrious mgntic fild h =0,4,nd 7T. dr bov T=20 K ws obsrvd, whil ll th ordr prmtrs r zro t T 21 K. Ths good consistncis indict tht th xchng intrctions w choos r mor suitbl thn th prdictd vlus in Rf. 17. Third, for prmtrs L nd, w find tht mgntic fild h rducs th dimnsion of th domins long th c xis nd th clustring dgr on th b pln. Th mn thicknss of th domin long th c xis is rducd from bout Å h=0 T to 53 Å h=7 T. As w s, th ssumption w md for polriztion domins is simpl but rsonbl for CCMO. Fourth, w find tht xtrnl mgntic fild h supprsss th frrolctricity t low T 6 K but nhncs it t high T 6 K, which ws rrly obsrvd in othr multifrroics but it is tru for CCMO. Th wk intrchin intrctions, comprd with th intrchin intrctions, r suggstd to b th microscopic mchnism for th polriztion domin rvrsl t high T, which invrsly brks th long-rng spin ordr long th c xis. It is tnttiv for us to ttribut th rspons of polriztion ginst h to th brking of th long-rng spin ordr. Our clcultion, s shown in Fig. 6 b, indicts tht th probbility for polriztion domin rvrsl with incrsing h is rducd t T 6 K, whil it is nhncd t T 6 K. This mns tht mgntic fild h stbilizs th long-rng spin ordr t T 6 K nd dstbilizs th ordr blow 6K, noting tht h lwys unfvors th short-rng AFM spin ordr for ll tmprturs. It is concludd tht th comptition btwn th short-rng ordr nd long-rng ordr undr mgntic fild cuss th diffrnt h dpndncs of polriztion t low T nd high T. Fifth, wht should b mntiond nd lso shown in Fig. 5 is tht th clcultd T dos not coincid with xprimnt dt in th low-t rng. In fct, th cln nd compound of CCMO, i.., CCO, ws invstigtd crfully s tringulr Ising modl systm, 25,26,31 nd th modl prdictions hv similr diffrnc with obsrvd mgntic suscptibility in th low-t rng. It is suggstd tht th mgntic proprty of CCMO cn not b simply xplind by th Ising spin of Mn nd Co ions lon nd th low-t quntum fluctutions sms to b importnt. Finlly, w look t th dilctric suscptibility shown in Fig. 5 b. Th clcultd nd msurd dt hv similr T tndncy nd good consistncy btwn thm in th high-t rng is idntifid. Th diffrnc in th low-t rng rmins unclr to us nd my b cusd by th dficincy of th MFT. V. CONCLUSION In summry, w hv proposd n lstic Ising modl which hs bn dmonstrtd to prdict proprly th frrolctricity gnrtion in C 3 CoMnO 6 s multifrroic systm with collinr spin ordr. A mn-fild thory to this modl hs lso bn dvlopd. With th considrtion of th dynmics of polriztion domins, w hv improvd th consistncy btwn th MFT clcultion nd msurd rsult in dilctric nd frrolctric proprtis. Th mgntic ordr with ltrnting Co nd Mn ions nd th lstic xchng nrgy r considrd to b rsponsibl for th ion displcmnts nd, thus, th frrolctricity. Th comptition btwn th stbility of th short-rng nd short-ordr spin ordrs is suggstd to hv impct on polriztion, but only th stbility of th short-rng spin ordr dtrmins th frrolctric trnsition. ACKNOWLEDGMENTS This work ws supportd by th Nturl Scinc Foundtion of Chin Grnt Nos , , nd nd th Ntionl Ky Projcts for Bsic Rsrch of Chin Grnt Nos. 2009CB nd 2009CB *Corrsponding uthor: liujm@nju.du.cn 1 D. I. Khomskii, J. Mgn. Mgn. Mtr. 306, Y. Tokur, Scinc 312, S. W. Chong nd M. Mostovoy, Ntur Mtr. 6, M. Fibig, J. Phys. D 38, R H. Ktsur, N. Ngos, nd A. V. Bltsky, Phys. Rv. Ltt. 95, C. Ji, S. Onod, N. Ngos, nd J. H. Hn, Phys. Rv. B 76, I. A. Srginko nd E. Dgotto, Phys. Rv. B 73,

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