Magnetically driven ferroelectric order in Ni$_3$V$_2$O$_8$

a r X i v :c o n d -m a t /0503385v 1 [c o n d -m a t .m t r l -s c i ] 16 M a r 2005Magnetically driven ferroelectric order in Ni 3V 2O 8

https://www.wendangku.net/doc/781667127.html,wes,1,+A.B.Harris,2T.Kimura,1N.Rogado,3,?

R.J.Cava,3A.Aharony,4O.Entin-Wohlman,4T.Yildirim,5M.Kenzelmann,5,6C.Broholm,5,6,and A.P.Ramirez 1,71Los Alamos National Laboratory,Los Alamos,New Mexico 875452Department of Physics and Astronomy,University of Pennsylvania,Philadelphia,PA,191043Department of Chemistry and Princeton Materials Institute,Princeton University,Princeton,New Jersey 085444School of Physics and Astronomy,Raymond and Beverly Sackler Faculty of Exact Sciences,Tel Aviv University,Tel Aviv 69978,Israel 5NIST Center for Neutron Research,Gaithersburg,MD 208996Department of Physics and Astronomy,Johns Hopkins University,Baltimore,MD 21218and 7Bell Labs,Lucent Technologies,600Mountain Avenue,Murray Hill,NJ 07974(Dated:February 2,2008)Abstract We show that for Ni 3V 2O 8long-range ferroelectric and incommensurate magnetic order appear simultaneously in a single phase transition.The temperature and magnetic ?eld dependence of the spontaneous polarization show a strong coupling between magnetic and ferroelectric orders.We determine the magnetic symmetry of this system by constraining the data to be consistent with Landau theory for continuous phase transitions.This phenomenological theory explains our observation the spontaneous polarization is restricted to lie along the crystal b

The coupling between long-range magnetic and ferroelectric order has been studied since the1960s.[1,2,3]Although a number of systems which are ferroelectric at high temperatures become magnetically ordered at a lower temperature,[2,4]the simultaneous appearance of both kinds of order at a single phase transition is much less common.For the most part, studies of these multiferroic materials have focussed on commensurate magnets.The magne-toelectric properties of these systems are typically discussed in terms of magnetic symmetry groups.The onset of magnetic ordering requires breaking time-reversal symmetry,while ferroelectric order breaks spatial inversion symmetry.Therefore,only magnetic groups hav-ing the proper symmetries allow the possibility of simultaneous magnetic and ferroelectric order.

Recent studies of systems underoing simultaneous ferroelectric order at a magnetic tran-sition have identi?ed new multiferroic compounds,including the perovskite Rare Earth manganites TbMnO3and DyMnO3[5],and TbMn2O5[6].However,the magnetic structures of these materials are complex[7,8,9],which makes an investigation of their magnetoelectric properties based on symmetry analysis problematic.In contrast,the magnetic structure of Ni3V2O8(NVO),which we have identi?ed as a multiferroic material,is better determined. There are extensive neutron data on this compound[10],and a symmetry analysis of the data(discussed below)constrains the symmetries which are consistent with a continuous transition.Here we take advantage of our analysis of the magnetic symmetry of NVO to de-velop a phenomenological Landau theory which provides an explanation of our observations of ferroelectric order induced by magnetic ordering.This provides an alternate route to understanding the magnetoelectric coupling in multiferroic materials beyond the traditional magnetic symmetry group analysis.

We brie?y review earlier results[10,11,12]for the structure(magnetic and crystal)of (NVO)since its symmetry is crucial to the development of our model.NVO is a magnetic insulator consisting of planes of spin-1Ni2+ions arranged in a Kagom′e staircase lattice.Fig. 1a shows the positions of the two kinds of Ni+2spins which we call“spine”and“cross tie”https://www.wendangku.net/doc/781667127.html,petition between several weak magnetic interactions and anisotropies yields the complex phase diagram of Fig.2and the variety of magnetic structures illustrated in Fig 1b-d.Cooling at H=0,one?rst enters a high temperature incommensurate[13](HTI)phase at T H=9.1K(Fig.1b),then a low-temperature incommensurate(LTI)phase at T L=6.3K (Fig.1c).Below3.9K the system displays two slightly di?erent canted antiferromagnet

(CAF)phases(Fig.1d).The transitions involving the HTI phase are continuous,whereas that from the LTI to one of the CAF phases is discontinuous.In the HTI phase the long-range order is mostly on the“spine”sites with their spins parallel to a,while in the LTI phase the spine and cross tie spins rotate within an a-b plane as shown in Fig.1b-c. We used representation analysis[14]to determine the symmetry of these phases rather than attempting to acquire the vast amount of data which would be necessary to experimentally?x the several complex order parameters[e.g.see Eq.(2),below]for these magnetic structures. Since the irreducible representation associated with magnetic ordering are one dimensional, each magnetic order parameter is characterized by a phase factor of unit magnitude which speci?es how the order parameter transforms under the symmetry operations which leave the incommensurate wavevector invariant.Thus the HTI order parameter,σH is odd under a two-fold rotation about the a

-b

is discontinuous.

The isothermal data shown in Fig.3b and3d,corroborate the above picture.Fig.3d shows P b versus the magnetic?eld along c,H c,at T=5K.Atlow H c,P b is insensitive to the external magnetic?eld.As H c is increased,the sample undergoes CAF ordering,which completely suppresses the spontaneous polarization.On decreasing H c,P b returns to the initial value.The?eld-hysteresis is attributed to the?rst order LTI to CAF transition.Fig. 3b shows P b at T=2K,versus H||a.At small H a,there is no ferroelectric order in the CAF phase.Increasing H a however produces a spontaneous polarization in the LTI phase which is independent of the sign of the magnetic?eld.The crucial observation is that we can magnetically gate NVO to either suppress or promote ferroelectric order,depending only on the direction of the applied?eld and the temperature.

We now give a phenomenological explanation of our results.In the HTI phase the exis-tence of incommensurate magnetic order does not induce a nonzero value of P in the HTI phase because the HTI phase is inversion symmetric.This symmetry is not easy to establish directly by analyzing the neutron di?raction data.If one assumes the spin amplitudes are restricted according to a representation analysis,[14]one still has to determine the complex-valued order parametersσH;i(k)=σH;i(?k)?which characterize the HTI phase,where k denotes the wavevector of the incommensurate ordering and i=1,...,6,corresponds to the six spin components(the a,b,and c components of the spine and cross tie spins).In the LTI phase the additional order parametersσL;i(k)=σL;i(?k)?appear.Representation analysis is based on the so-called“little group,”G k,which leaves the wavevector invariant.But this analysis does not take into account any additional symmetries(such as spatial inversion I) which the crystal possesses.We use the inversion symmetry of the crystal lattice to de?ne the order parameters so that they obey

IσX;i(k)=σX;i(k)?,(1)

where X denotes either H or L.The quadratic terms in the Landau free energy which describe the appearance of the HTI phase are

F= i,j F ijσH;i(?k)σH;j(k),(2) where F i,j=F?j,i.Then Eq.(1)implies that the coe?cients F ij are all real valued.Since the spin components are those of the critical(i. e.the one whose eigenvalue becomes negative

as T is lowered through T H)eigenvectorof the real symmetric matrix F,the amplitudesσH,i are all real valued,apart from a common overall phase:

σH;i(k)=c i e iφHσH,(3)

where the c i,obtained from the critical eigenvector,are real.Thus the HTI phase is described by a single complex valued order parameter e iφHσH,whose overall phase,φH,is not ?xed at this level of analysis.The overall phaseφH can be eliminated by rede?ning the origin of coordinates.The fact that the resulting Fourier componentsσH;i(k)are all real means that the magnetic structure is invariant under spatial inversion.Because the relative phases of the spin components are di?cult to determine from an analysis of the di?raction data, this formal argument is essential to establish that the HTI phase is inversion symmetric and therefore that the magnetic ordering can not induce a spontaneous electric polarization. Since the CAF phaseis also inversion symmetric(Fig.1d),ferroelectricity is not induced by magnetic order in that phase either.These results con?rm that this model based on our extension of representation analysis successfully predicts the experimental observation that there is no ferroelectric order in the HTI or CAF phases.We can also understand why when a su?ciently large magnetic?eld is applied so as to enter the CAF phase(Fig.2b),the spontaneous polarization abruptly disappears.

We now introduce a phenomenological model to explain the symmetry of the magneto-electric coupling which induces the spontaneous polarization.In principle,this symmetry might be deduced from a study of magnetic symmetry,[1]but we have found it much simpler to directly obtain these results using Landau theory combined with representation analysis. While the speci?c details of this discussion apply to NVO,the general approach is applicable to other systems.To describe the incommensurate phases and ferroelectric behavior in a single model,we write the Landau free energy as

F=a(T?T H)σ2H+b(T?T L)σ2L

+O(σ4)+(2χE)?1P2+V,(4)

where a,and b are constants andχE is the electric susceptibility.(We assume that the wavevectors of the two incommensurate phase are locked to be equal.)In a conventional ferroelectric P becomes nonzero whenχE becomes in?nite,often associated with a structural

phase transition.HereχE is?nite and the appearance of a nonzero P is due to the term V coupling magnetic and ferroelectric orders whose form we discuss below.Note that this expansion is expressed in the disordered phase,so that all terms in this equation must be invariant under the complete set of symmetry operations of the disordered phase.In particular,this expression must be invariant under spatial inversion.

We now discuss the form of the magnetoelectric coupling V.To conserve wavevector this coupling must be at least trilinear,[15]being proportional to one order parameter at wavevector k,another at wavevector(?k),and P which is associated with zero wavevector. As we have already argued,this coupling is zero with only a single magnetic order parameter σH.A similar argument made for the LTI variables shows that ferroelectricity can not be induced if we only invoke the single order parameterσL.Accordingly,and in analogy with what has been done for the theory of second harmonic generation,[16]we posit the following interaction which involves two di?erent symmetry order parameters

V=? ijγ[a ijγσH;i(k)σL;j(?k)

+a?ijγσH;i(?k)σL;j(k)]Pγ.(5)

Using IσX;i(k)=σX;i(?k)and IPγ=?Pγ,we see that the inversion invariance of V implies that the coe?cients a ijγmust be purely https://www.wendangku.net/doc/781667127.html,ing Eq.(3)and its analog for the L order parameters we then have

V LT I= γaγσHσL sin(φH?φL)Pγ,(6) where aγis a real valued coe?cient.At fourth order in the Landau expansion it can be shown thatφH?φL=π/2,but to have Pγ=0it is only essential thatφH=φL.(A simple way to understand the e?ect of the fourth order terms is to recall that Wilson in his original formulation of the renormalization group,used these terms to mimic the constraint of?xed length spins.[17]Here a?xed length of the spins is most nearly achieved by having the two incommensurate types of order be out of phase byπ/2).We now insert Eq.(6)into Eq.(4) and minimize with respect to P.We then?nd a spontaneous polarization given by

Pγ∝aγχelσLσH.(7)

Note that this prediction can be checked by measuring all the quantities which appear in Eq.(7)as a function of magnetic?eld and temperature.This result on the dependence of

the spontaneous polarization on the magnetic order parameter could not be obtained from magnetic symmetry group analysis alone.

To analyze the consequences of the trilinear coupling in Eq.(6),it is necessary to know how the order parameters transform under the symmetry operations of the crystal.As stated above,σL is even andσH is odd under a two-fold rotation about the a

-b

[+]Current A?liation:Department of Physics,Wayne State University,Detroit,MI48201.

[1]R.R.Birss,Symmetry and Magnetism,2nd Ed.,(North-Holland,Amsterdam,1966).

[2]G.A.Smolenskii and I.E.Chupis,https://www.wendangku.net/doc/781667127.html,p.25,475(1982).

[3]H.Schmid,Ferroelectrics162,317-338(1994).

[4]T.Lottermoser,et al.,Nature430,541(2004).

[5]T.Goto,et al.,Phys.Rev.Lett.92,257201(2004).

[6]N.Hur,et al.,Nature429,392(2004).

[7]R.Kajimoto,et al.,Phys.Rev.B70,012401(2004).

[8]L.C.Chapon et al.,Phys.Rev.Lett.93,17402(2004).

[9]S.Kobayashi et al,J.Phys.Soc.Jpn73,3439(2004).

[10]https://www.wendangku.net/doc/781667127.html,wes,et al.,Phys.Rev.Lett.93,247201(2004)and to be published.

[11]N.Rogado,et al.,https://www.wendangku.net/doc/781667127.html,mun.124,229-233(2002).

[12] E.E.Sauerbrei,F.Faggiani,and C.Calvo,Acta Cryst.B29,2304-2306(1973).

[13]Mathematically“incommensurate”means that the modulation vector,k,can not be repre-

sented as the quotient n/m of two integers.By“incommensurate”we mean that m is large enough to betreated as in?nite.Experiment shows that the temperature dependenceof k is smooth at the level of about1/2%,which implies that m≥50.

[14] A.S.Wills,Phys.Rev.B63,064430(2001).

[15]The coupling described in Eq.(5)is also sometimes referred toas“bilinear”in the magneto-

electric literature.

[16] D.Frohlich,et al.,Phys.Rev.Lett.81,3239-3242(1998).

[17]K.G.Wilson,Phys.Rev.B4,3174(1971).

[18]T.Kimura,https://www.wendangku.net/doc/781667127.html,wes,and A.P.Ramirez,Phys.Rev.Lett.(in press).

[19]M.Kenzelmann,et al.,in progress.

FIG.1:Crystal and magnetic structures of NVO.a Crystal structure showing spin-1Ni2+spine sites in red and cross tie sites in blue.b-d Simpli?ed schematic representation of spin arrangement in the antiferromagnetic HTI,LTI,and CAF phases.[10]“ ”indicates the direction of uniform

magnetization distributed over spine and cross tie sites in the CAF phase.Only the HTI and CAF phases have inversion symmetry relative to the indicated central lattice point.Panel e illustrates the center of spatial inversion for the lattice does not coincide with the inversion centre of the magnetic structure.

FIG.2:Phase diagram of NVO versus T and H for H||a and H||c in panels a and b respectively. The data points indicate anomalies in speci?c heat(C),magnetization(M),dielectric permittivity (ε),and electric polarization(P)traces versus H and T.Solid lines are guides to the eye.The phases are described in the text and illustrated in Fig.1b-d

FIG.3:Promotion and suppression of electric polarization by applying magnetic?elds in NVO. Temperature and magnetic-?eld dependence of electric polarization along the b axis for H along the a(frames a and b)and c(frames c and d)axes.

a

b

HTI

P c

LTI

d

CAF

0 T

1 T

2 T

3 T

4 T μ

5 T