m. a. van der vegte, c. p. van der vegte and m. mostovoy- electromagnons and instabilities in...

8/3/2019 M. A. van der Vegte, C. P. van der Vegte and M. Mostovoy- Electromagnons and instabilities in magnetoelectric ma

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arXiv:0907.3055

v1

[cond-mat.mtrl-sci]17Jul2009

Electromagnons and instabilities in magnetoelectric materials with non-collinear spin

orders

M. A. van der Vegte, C. P. van der Vegte, and M. MostovoyZernike Institute for Advanced Materials, University of Groningen,

Nijenborgh 4, 9747 AG Groningen, The Netherlands(Dated: July 17, 2009)

We show that strong electromagnon peaks can be found in absorption spectra of non-collinearmagnets exhibiting a linear magnetoelectric effect. The frequencies of these peaks coincide with thefrequencies of antiferromagnetic resonances and the ratio of the sp ectral weights of the electromagnonand antiferromagnetic resonance is related to the ratio of the static magnetoelectric constant andmagnetic susceptibility. Using a Kagome lattice antiferromagnet as an example, we show thatfrustration of spin ordering gives rise to magnetoelastic instabilities at strong spin-lattice coupling,which transform a non-collinear magnetoelectric spin state into a collinear multiferroic state witha spontaneous electric polarization and magnetization. The Kagome lattice antiferromagnet alsoshows a ferroelectric incommensurate-spiral phase, where polarization is induced by the exchangestriction mechanism.

PACS numbers: 75.80.+q,71.45.Gm,76.50.+g,75.10.Hk

I. INTRODUCTION

The recent renewal of interest in multiferroic materialsled to discovery of many novel compounds where electricpolarization is induced by ordered magnetic states withbroken inversion symmetry.1,2,3 The electric polarizationin multiferroics is very susceptible to changes in spin or-dering produced by an applied magnetic field, which givesrise to dramatic effects such as the magnetically-inducedpolarization flops and colossal magnetocapacitance.4,5,6

Magnetoelectric interactions also couple spin waves topolar phonon modes and make possible to excite magnonsby an oscillating electric field of light, which gives rise tothe so-called electromagnon peaks in photoabsorption.7

Electromagnons were recently observed in twogroups of multiferroic orthorombic manganites, RMnO3(R = Gd,Tb,Dy,Eu1xYx) and RMn2O5 (R =Y,Tb).8,9,10,11 Ferroelectricity in RMnO3 appears in anon-collinear antiferromagnetic state with the cycloidalspiral ordering and the magnetoelectric coupling origi-nates from the so-called inverse Dzyaloshinskii-Moriyamechanism.12,13,14,15,16 In Ref. [17] it was noted thatthe same mechanism can couple magnons to photonsand that an oscillating electric field of light can exciterotations of the spiral plane. However, the selectionrule for the electromagnon polarization resulting fromthis coupling does not agree with recent experimentaldata10,18,19,20 and, moreover, the inverse Dzyaloshinskii-Moriya mechanism of relativistic nature is too weak to ex-plain the strength of the electromagnon peaks in RMnO3.

These peaks seem to originate from the exchange stric-tion, i.e. ionic shifts induced by changes in the Heisen-berg exchange energy when spins order or oscillate.21

This mechanism explains the experimentally observedpolarization of electromagnons. Since the Heisenberg ex-change interaction is stronger than the Dzyaloshinskii-Moriya interaction, it can induce larger electric dipoles.In Ref. [21] it was shown that the magnitude of the spec-

tral weight of the giant electromagnon peak in the spiralstate of rare earth manganites is in good agreement withthe large spontaneous polarization in the E-type antifer-romagnetic state,22 which has not been reliably measuredyet but is expected to exceed the polarization in the spi-ral state by 1-2 orders of magnitude.23,24

From the fact that the mechanism that couplesmagnons to light in rare earth manganites is differentfrom the coupling that induces the static polarizationin these materials we can conclude that electromagnonscan also be observed in non-multiferroic magnets. In thispaper we focus on electromagnons in materials exhibit-ing a linear magnetoelectric effect, i.e. when an appliedmagnetic field, H, induces an electric polarization, P,proportional to the field, while an applied electric field,E, induces a magnetization, M. This unusual couplingtakes place in antiferromagnets where both time reversaland inversion symmetries are spontaneously broken.25,26

It is natural to expect that when an electric field ap-plied to a magnetoelectric material oscillates, the inducedmagnetization will oscillate too. Such a dynamical mag-netoelectric response, however, requires presence of ex-citations that are coupled both to electric and magneticfields. They appear when magnons, which can be excitedby an oscillating magnetic field (antiferromagnetic reso-nances), mix with polar phonons, which are coupled toan electric field. Thus in materials showing a linear mag-netoelectric effect, for each electromagnon peak there is

an antiferromagnetic resonance with the same frequency.

This reasoning does not apply to all magnetoelectricsand the dc magnetoelectric effect is not necessarily re-lated to hybrid spin-lattice excitations. As will be dis-cussed below, in materials with collinear spin orders elec-tromagnons either do not exist or have a relatively lowspectral weight. In this paper we argue that electro-magnons should be present in non-collinear antiferromag-nets showing strong static magnetoelectric response. Asa simple example, we consider a Kagome lattice antifer-
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FIG. 1: (Color online) The Kagome magnet with the KIT-Pite crystal structure, in which the ligand ions (open circles)mediating the superexchange between spins are positioned

in a way that gives rise to a strong linear magnetoelectricresponse in the 120 spin state. Here, J1 and J2 denote,respectively, the nearest-neighbor and next-nearest-neighborexchange constants, the solid arrows denote spins, while theempty arrows denote the shifts of the ligand ions.

romagnet with the 120 spin ordering, shown in Fig. 1.Such an ordering has a nonzero magnetic monopole mo-ment, which allows for a linear magnetoelectric effectwith the magnetoelectric tensor ij = ij for electricand magnetic fields applied in the plane of the Kagomelattice.27 A relatively strong magnetoelectric responsewas recently predicted for Kagome magnets with theKITPite crystal structure, in which magnetic ions arelocated inside oxygen bipyramids.28 In this structure theoxygen ions mediating the superexchange in basal planesare located outside the up-triangles forming the Kagomelattice and inside the down-triangles or vice versa (seeFig. 1), in which case magnetoelectric responses of alltriangles add giving rise to a large magnetoelectric con-stant.

This paper is organized as follows. In Sec. II we analyzethe symmetry of magnon modes and the magnetoelectriccoupling in the Kagome lattice magnet with the KITPitestructure and show that the dc magnetoelectric effectin this system is related to presence of electromagnonmodes. The common origin of the dc and ac magneto-electric responses implies existence of relations betweenstatic and dynamic properties of magnetoelectric mate-rials, derived in Sec. III. In Sec. IV we discuss softeningof (electro)magnons and the resulting divergence of thecoupled magnetoelectric response. In Sec. V we discussthe transition from a magnetoelectric to a multiferroicstate at a strong spin-lattice coupling and plot the phasediagram of our model system. In section VI we discussthe importance of non-collinearity of spins for dynamic

magnetoelectric response and possible electromagnons inknown magnetoelectric materials. In Sec. VII we con-clude.

II. SYMMETRY CONSIDERATIONS

The coupling of magnetic excitations in the Kagome

magnet to electric field, resulting from the Heisenberg ex-change striction or any other non-relativistic interaction,can be found using the method outlined in Ref. [21]. Tosimplify notation, we consider a single up-triangle, whichhas the same point symmetry as the whole Kagome lat-tice with the 120 spin ordering shown in Fig. 1. Theform of the magnetoelectric coupling is constrained bythe 3z and mx symmetry operations:

29

Hme =

Ex2

[(S2 S3) (S1 S3)]

+Ey

6[(S1 S3) + (S2 S3) 2 (S1 S2)]

. (1)

We then replace Si by Sni + Si, where the unit vec-tors (n1,n2,n3) =32 x 12 y,

32 x 12 y, y

describe

the 120 spin ordering in the xy plane and Si ni isthe oscillating part, which is the superposition of the or-thogonal magnon modes in the triangle (the zero wavevector magnons for the Kagome lattice):

Si =

(qi + Spi) , (2)

where = 0, x , y labels the magnon,

0i = z

13

(1, 1, 1) ,

xi = z

1

6 (1, 1,2) ,yi = z 12 (1, 1, 0) ,(3)

are the out-of-plane components of the magnons andi = ini are the in-plane components (see Fig. 2),0 =

13

12 x

32 y,

12 x +

32 y,x

,

x =16

12

x 32

y, 12

x +32

y, 2x

,

y =12

12 x +

32 y,

12 x +

32 y, 0

.

(4)

The single-magnon excitation by an electric field isdescribed by the terms linear in S, while the termsquadratic in S give rise to the photoexcitation of a

two-magnon continuum. Since spins order in plane, thepolarization oscillations are induced by the in-plane os-cillations of spins and the coupling of electric field tomagnons, obtained from Eq.(1), has the form:

Hme = gE (qxEx + qyEy) , (5)where gE =

32S. This magnetoelectric coupling is

only nonzero in the magnetically ordered state with bro-ken time reversal symmetry, which is why the couplingconstant is proportional to S.


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FIG. 2: (Color online) The magnon modes in a triangle withthe 120 spin ordering. The thin arrows indicate the direc-tions of average spins, while the thick arrows show the in-plane components of the magnon modes.

We note that Eq.(5) can also be obtained by a conven-tional symmetry analysis of magnetic modes. The vector0 and, hence, the corresponding amplitude, q0, formsa one-dimensional representation 1, while

x,y

andqxqy

form a two-dimensional representation 3 (see Ta-

ble I). The direct product, 2 3, where 2 is thesymmetry of the spin ordering, transforms as the dou-

blet of the in-plane components of the electric fields, 4,which leads to Eq.(5). This general symmetry analysis is,however, insensitive to the microscopic mechanism of themagnetoelectric coupling, whereas the derivation staringfrom Eq.(1) only applies to non-relativistic mechanisms.

3z mx T

1 q0 +1 +1 1

2 S +1 1 1

3

qx

qy

!,

Hx

Hy

! 1

2

32

+32

12

! +1 0

0 1

! 1 0

0 1

!

4

Ex

Ey

! 1

23

2

+32

12

! 1 0

0 +1

! +1 0

0 +1

!

TABLE I: The transformation properties of several irreduciblerepresentations of the space group of the Kagome lattice andtime reversal operation T.

Table I shows that the coupling to the in-plane com-ponents of magnetic field has the form,

Hm = gH(qxHx + qyHy) , (6)

while the Hamiltonian describing magnon modes withzero wave vector in the Kagome layer is,

H(p,q) =p20

2m0+

1

2m

p2x +p

2y

+

0q20

2+

2

q2x + q

2y

.

(7)If we consider, for example, the microscopic spin Hamil-tonian describing the antiferromagnetic nearest-neighborand next-nearest-neighbor Heisenberg exchange interac-tions with the exchange constants, respectively, J1 and

J2 and the easy plane magnetic anisotropy ,

H = J1ijSi Sj + J2ij

Si Sj + 2

i

(Szi )2

, (8)

for which the 120 spin ordering shown in Fig. 1 is aclassical ground state,30,31 we get

m1

0 = [6(J + J) + ] S2

, m1

= S2

,0 = 0, = 3 (J1 + J2) .

(9)

The linearized equations of motion for spins in appliedelectric and magnetic fields are obtained from Eqs.(5),(6),and (7), if we impose the commutation relations for theamplitudes of the in-plane and out-of-plane parts of S:

[q, p] = i,. (10)

From these equations we find the frequencies of the threemagnon modes with zero wave vector: 20 = 0m

10 = 0

and 2x = 2y = m

1 = 3 (J1 + J2) S2.Minimizing the spin energy with respect to qx and qy

in the static limit, we obtain an effective magnetoelectriccoupling,

Hme = (HxEx + HyEy) . (11)where the magnetoelectric coefficient = gEgH2 . Fur-thermore, the qx-mode can be excited by both electricand magnetic field oscillating with the frequency x inthe direction parallel to the x axis, while the qy-modecan be excited by both Ey and Hy, which shows that thestatic linear magnetoelectric effect in this non-collinearmagnet is related to the presence of electromagnon andantiferromagnetic resonance peaks with equal frequenciesin the optical absorption spectrum.

III. RELATIONS BETWEEN STATIC AND

DYNAMIC MAGNETOELECTRIC RESPONSE

The common origin of the static and dynamic mag-netoelectric response of non-collinear magnets leads toquantitative relations between dc susceptibilities andspectral weights of peaks in the optical absorption spec-trum. These relations simplify when the coupling ofmagnons to electric field is mediated by a single opticalphonon. The description of magnons in terms of con-

jugated coordinates and momenta is very convenient for

derivation of these relations, since the coupled magnonand optical phonon are in this approach just a pair ofcoupled oscillators:

H =1

2m

p2x +p

2y

+

2

q2x + q

2y

+

1

2M

P2x + P

2y

+

K

2

Q2x + Q

2y

(qxQx + qyQy) f (QxEx + QyEy)gH(qxHx + qyHy) , (12)


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where (Qx, Px) and (Qy, Py) are the coordinates and mo-menta of the optical phonons coupled to magnons, whichalso form a two-dimensional representation.

The magnetoelectric response of such a system is easyto calculate. The result can be expressed in termsof observable quantities, such as the dressed magnonand phonon frequencies, mag and ph, and the spec-tral weights of the magnon and phonon peaks excited by

an electric and magnetic field. We denote the spectralweight of the electromagnon peak by

SEmag = 8

de (), (13)

where e () is the imaginary part of the dielectric acsusceptibility, while the spectral weight of the antiferro-magnetic resonance is,

SHmag = 8

dm(), (14)

where m() is the imaginary part of the magnetic acsusceptibility. The integration in Eqs.(13) and Eq.(14)goes over an interval of frequencies around mag. The twospectral weights for the phonon SEph and SHph are definedin a similar way. We assume that the magnon and phononpeaks are sufficiently narrow and can be separated fromeach other. The four spectral weights satisfy a relation,

SEmagSHmag = S

EphS

Hph, (15)

following from the fact that an electric field only interactswith the bare phonon, while a magnetic field is onlycoupled to the bare magnon.

The relations between the dc and ac magnetoelectricresponses of the coupled spin-lattice system have theform,

=SEmag2mag +

SEph2ph ,

=SHmag2mag

+SHph2ph

,

4|| =

SEmagSHmag

12mag 12ph ,

(16)

where () is the increase of the real part of thedielectric constant (magnetic permeability) at zero fre-quency resulting from the magnon and phonon peaks(we use the Gaussian units). The first two equationsare the Kramers-Kronig relations for the real and imag-inary parts of dielectric and magnetic susceptibilities,

while the last relation follows from equations of motion.Combining Eqs.(15) and (16) we can express the ratio ofthe spectral weights of the electromagnon and the anti-ferromagnetic resonance through the ratio of the staticmagnetoelectric constant and magnetic susceptibilitym =

m( = 0):

SEmag

SHmag=

m

21 +2mag2ph

SEmagSEph

1 2mag2ph

2

(17)

For 2mag 2ph, the ratio of the spectral weights is justthe square of the ratio of the dc magnetoelectric constantand magnetic susceptibility.

Due to the spin-lattice coupling, the dressed magnonfrequency, mag, is lower than its bare value,

m

(as-suming that the bare magnon frequency is smaller thanthe bare phonon frequency). As the spin-lattice cou-pling increases, mag vanishes at a critical value of the

coupling. According to Eq.(16), this results in the simul-taneous divergency of , and , indicating an insta-bility towards a multiferroic phase, which is both ferro-electric and ferromagnetic. Another manifestation of thisinstability is the fact that as the spin-lattice coupling ap-proaches the critical value, the magnetoelectric constant tends to its upper bound equal the geometric meanof the dielectric and magnetic susceptibilities,

em,

imposed by the requirement of stability with respect toapplied electric and magnetic fields.32

IV. MAGNON SOFTENING

To study the transition from the magnetoelectric stateof the Kagome magnet to the multiferroic state in moredetail, we consider a simple microscopic model, in whichpositions of magnetic ions are fixed, while ligand ionsare allowed to move. The spin-lattice coupling originatesfrom the dependence of the exchange constants on dis-placements of ligand ions mediating the superexchange inthe direction perpendicular to the straight line connect-ing two neighboring spins. We denote the positions of thethree ligand ions outside up-triangles by u1, u2, and u3,while the position of a ligand ion inside a down-triangleis denoted by v. Then, for example, the exchange con-

stant for the spins S1 and S2 is J1 + J1 (u3)y , while forthe spins S4 and S5 it is J1 + J1vy (see Fig. 1). Further-more, we assume that phonons are dispersionless and thelattice energy for a pair of the up- and down-triangles is,

Ulat =K

2

3i=1

u2i + v2

, (18)

where K is the spring constant.For the 120 structure shown in Fig. 3(a), the magne-

toelectric constant in Eq.(11) is given by

=

1

(1 g)3Q

J1

2(J1 + J2)Kv , (19)

where Q = 2e is the charge of the oxygen ion, =2BS is the average magnetic moment on each site, vis the unit cell volume, and

g =15

8

(J1S)2(J1 + J2)K

(20)

is the dimensionless spin-lattice coupling constant.


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An estimate for the magnetoelectric constant, 103, for the model parameters appropriate for the KIT-Pite structure (S = 2, J1 3meV, K 6eV A2,J1J1 3.5A1, and v = 177A3) agrees well with the result

of ab initio calculations.28 Furthermore, m 2 104,so that

m

2 25. Thus, the electromagnon peak in

KITPite should be much stronger than the antiferromag-

netic resonance peak, which is also the case for rare earthmanganites with a spiral ordering.21

At g = 1, the magnetoelectric constant diverges andso do the dielectric and magnetic susceptibilities:

e, m 11 g . (21)

Since in our model there are two polar phonons coupledto a magnon with a given polarization (one in the up-triangle and another in the down-triangle), the magneto-electric constant comes close to its upper bound but doesnot reach it at g = 1:

em

g=1

0.985. (22)

Surprisingly, the softening of the q0 magnon mode,which is not coupled to polar lattice distortions, occursat a lower value g0 < 1. Since 0 = 0, the softeningin this case means vanishing velocity of the q0-mode.The velocity vanishes, because away from the -pointin the magnetic Brillouin zone magnons with differentsymmetry become mixed and the q0-mode is coupled tothe electromagnon modes. As the spin-lattice couplinggrows and the electromagnon frequency decreases, thelowest-frequency magnon branch is pushed down, which

ultimately reduces the velocity of the q0-mode to zero.

V. MAGNETOELASTIC INSTABILITIES

Although KITPite, for which g 0.05, is far away fromthe instabilities discussed in the previous section, it is in-teresting to study behavior of magnetoelectric materialswhen the spin-lattice coupling becomes strong, in partic-ular, in view of the dramatic magnetoelectric effects re-cently observed in multiferroics. As the magnetoelectricconstant becomes large close to the transition betweenmagnetoelectric and multiferroic states, it is importantto understand possible scenarios of such a transition.

In this section we present the analytical and numeri-cal study of the phase diagram of the KITPite layer forstrong spin-lattice couplings. In particular, we show thatnone of the continuous transitions involving magnon soft-ening, discussed in Sec. IV, actually takes place, as thestrong spin-lattice coupling makes the frustrated Kagomemagnet unstable towards a first-order magnetoelastictransition that relieves the frustration. This frustration-driven instability is similar to the one found in spinels,where a collinear ordering of spins appears together with

FIG. 3: (Color online) The minimal-energy spin configura-tions of the Kagome magnet for three different values of thespin-lattice coupling: (a) the 120-state with zero wave vec-tor, (b) the incommensurate ferroelectric state, and (c) the

collinear multiferroic state. The short solid arrows show thespin directions, while the short empty arrows denote the shiftsof the ligand ions in these states. The long solid and emptyarrows show the direction of, respectively, the spontaneousmagnetization and polarization.


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a lattice deformation.33,34,35,36 We show that the trans-formation of a non-collinear magnetoelectric state intoa collinear multiferroic state can involve two transitionsand an intermediate phase, which is ferroelectric but notferromagnetic.

To understand the origin of magnetoelastic instabili-ties at strong spin-lattice coupling, we first consider asingle up-triangle and integrate out the lattice degrees of

freedom, ui (i = 1, 2, 3). Then the total energy of thetriangle takes the form of an effective spin Hamiltonianwith quadratic and bi-quadratic interactions:

E =i,j

J1Si Sj (J

1)2

2K(Si Sj)2

. (23)

The bi-quadratic interactions favor collinear spins and

for g = 158(J1S)

2

J1K= (J1+J2)

J1g > 5

6 , the lowest-energyspin configuration is a collinear state of the type(the spins lie in the lattice plane). Similarly, an effec-tive spin Hamiltonian for a down-triangle, where the ex-change along all bonds is mediated by a single ligand ion

located inside the triangle, has the form,

E =i,j

J1Si Sj (J

1)2

2K(Si Sj)2

+(J1)

2

2K

i=j=k

(Si Sj) (Sj Sk) . (24)

In this case the critical coupling is lower: g = 1532 .Due to the inequivalence of the up- and down-trianglesin the KITPite structure the transition from the 120-state shown in Fig. 3a to a fully collinear state shown inFig. 3c goes in two steps via an intermediate state, where

only the spins in the down-triangles are collinear whilethe spins in the up-triangles are still non-collinear (seeFig. 3b).

The transition to the intermediate state does not fullylift the frustration: the number of combinations of thedown-triangles with collinear spins and up-triangles withthe 120-angle between spins grows exponentially withthe system size. In our model this degeneracy is re-moved by the next-nearest-neighbor interactions betweenspins, which select the state shown in Fig. 3b. The next-nearest-neighbor interactions in the vertical direction in-duce a small spin canting, as a result of which spins indown-triangles are not strictly collinear and the angles

between spins in up-triangles deviate from 120 by theangle J2J1

. Furthermore, the nearest-neighbor in-teractions in the remaining two directions are frustratedin the commensurate spin state with the wave vector

Q = 2k (1, 0) + k12 ,

32

, where k = 3a (a being the

lattice constant). This frustration is lifted, when thespin ordering becomes incommensurate with the lattice:k = 3a + , where J2J1 .

The incommensurate state has zero net magnetization,but its electric polarization is nonzero. For the state

FIG. 4: (Color online) Plotted is the zero-temperature phasediagram of the Kagome layer for large values of the spin-

lattice coupling g = 158(J1S)

2

J1K= (J1+J2)

J1g. For relatively

small J2J1

the magnetoelectric (ME) phase with the 120 spinordering [see Fig. 3(a)] undergoes a first-order transition into

the incommensurate (IC) spin state, which is ferroelectric(FE) [see Fig. 3(b)], as the coupling constant g increases. Fur-ther increase of g results in the transition into the collinearmultiferroic (MF) phase [see Fig. 3(c)], with the parallel spon-taneous polarization and magnetization, P M. For largerJ2J1

, the ME state undergoes a direct transition into the fullycollinear MF state. Also plotted are the dot-dash line, wherethe q0-mode in the ME state softens and the dotted line, atwhich the magnetoelectric response of the ME state diverges(both phenomena do not occur because of the interveningfirst-order transitions to the FE and MF states).

shown in Fig. 3(b) the polarization vector is parallel to

the y axis. This polarization originates not from the in-verse Dzyaloshinskii-Moriya mechanism, which makes in-commensurate spiral states in e.g. RMnO3 ferroelectric,but from the fact that all bonds connecting parallel spinsare parallel to each other, which forces the ligand ions inall down-triangles to shift in the same direction, since

vx = 12

(S3 S5 S3 S4) ,

vy = 16

(2 (S4 S5) S3 S4 S3 S5) ,(25)

where =

32J1K

and the labeling of spins is the same as

in Fig. 1. For the state shown in Fig. 3(b), where bondsconnecting (nearly) parallel spins are oriented along thex axis, vx = 0, while vy and hence the polarization Py isnonzero.

We note that the coupling of exchange interactionsto strains, which gives rise to magnetoelastic transitionsin frustrated spinels,33,34,35 also favors the ferroelectricstate. The absence of inversion symmetry in the KIT-Pite layer allows for the piezoelectric coupling,

2uxyEx + (uxx uyy)Ey , (26)


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where uij is the strain tensor, so that the ligand displace-ment, v, the electric polarization, P, and the strains arecoupled to each other.

The zero-temperature phase diagram of the Kagomelayer with g and J2

J1along the horizontal and vertical

axes is shown in Fig. 4. For small J2J1

, the incommensu-

rate ferroelectric (IC FE) state, discussed above, inter-venes between the magnetoelectric (ME) state with the

120 spin ordering [see Fig. 3(a)] and the fully collinearmultiferroic (MF) state shown in Fig. 3(c), in which thespontaneous polarization and magnetization are parallelto each other, P M. As the ratio J2

J1grows, the interval

of the coupling constant g where the intermediate stateis stabilized shrinks and above the tricritical point theME state undergoes a direct transition into the collinearMF state along the critical line J2

J1= 53 g1. Also plotted

are the dot-dash line, at which the q0-mode would softenand the dotted line, at which , e, and m would di-verge, if the ME state would survive at strong spin-latticecouplings.

VI. DISCUSSION

We showed that the static magnetoelectric response ofnon-collinear antiferromagnets can be related to hybridmagnon-phonon modes coupled to both electric and mag-netic fields. Such magnetoelectric materials are analogsof displacive ferroelectrics the dielectric response of whichis governed by optical phonon modes. If spins in anordered state are collinear, the exchange striction can-not couple an electric field to a single magnon, as theexpansion of scalar products of parallel or antiparallelspins begins with terms of second order in S, which

give rise to photoexcitation of a two-magnon continuum(the so-called charged magnons37). Electromagnons incollinear magnets can still originate from mechanisms in-volving relativistic effects, such as the exchange strictioninduced by the antisymmetric Dzyaloshinskii-Moriya in-teraction, which is proportional to the vector productof two spins. In 3d transition metal compounds suchcouplings are weak compared to the exchange strictiondriven by the Heisenberg exchange, so that the spectralweight of electromagnons in collinear magnets should berelatively low.

Magnetoelectric materials with collinear spin ordersmay rather be analogs of order-disorder ferroelectricswith the static magnetoelectric response originating fromthermal spin fluctuations. Cr2O3 seems to be an exampleof such a material: its magnetoelectric coefficient passesthrough a maximum below Neel temperature and thenstrongly decreases when temperature goes to zero andspin fluctuations become suppressed.38,39

We note that the rotationally invariant coupling Eq.(1)may also originate from purely electronic mechanisms,such as the polarization of electronic orbitals inducedby a magnetic ordering.23,24,27,29,40 Ab initio calcula-tions suggest that in rare earth manganites the electronic

mechanisms of magnetoelectric coupling are as importantas the exchange striction.24 On the other hand, the in-crease of the spectral weight of the electromagnon peaksin RMnO3 below the spiral ordering temperature oc-curs largely at the expense of the strength of the opticalphonon peak at 100cm1, suggesting the dominantrole of the spin-lattice coupling.19,21 If electronic mech-anisms dominate and an electromagnon gets its spectral

weight from frequencies much higher than those of opti-cal phonons, Eq.(16) should to be modified in an obviousway, while Eq.(17), where

magph

should be replaced by 0,

is still valid.We note that the non-collinearity of spins by itself does

not guarantee strong magnetoelectric effect and electro-magnon peaks the crystal structure is equally impor-tant. Thus, in the layered Kagome antiferromagnet, theiron jarosite KFe3(OH)6(SO4)2,

41 which has the spin or-dering shown in Fig. 1, the ligand ions are located out-side of both up- and down-triangles, which cancels themagnetoelectric effect due to the Heisenberg exchangestriction. The cancellation also occurs in triangular mag-

nets with the 120 spin ordering, as they contain threedifferent spin triangles, such that spins in one triangleare rotated by 120 with respect to spins in two othertriangles28 (more generally, the linear magnetoelectric ef-fect can only be induced by a spin ordering with zero wavevector). We note, however, that the lattice trimerizationin hexagonal manganites42 makes the three types of spintriangles inequivalent and destroys the cancellation. Thiscan be also seen from the symmetry properties of the A1,2and B1,2 phases of hexagonal manganites

43 allowing forthe magnetoelectric term ExHyEyHx in the A1-phase,which has a toroidal moment, and the term ExHx+EyHyin the A2-phase, which has a magnetic monopole mo-ment. Whether electromagnons in these phases can beobserved, depends on the magnitude of the trimerizationand remains to be explored.44

VII. CONCLUSIONS

In conclusion, we showed that magnets with non-collinear spin orders resulting in a linear magnetoelec-tric effect may also show electromagnon peaks in opticalabsorption spectrum. While electromagnons should bepresent in many non-collinear magnets, the specific fea-ture of magnetoelectric materials is that some magnonmodes can be excited by both electric and magneticfields, i.e. electromagnons are also antiferromagnetic res-onances. We derived a simple relation Eq.(17) betweenthe ratio of the spectral weights of the electromagnonand antiferromagnetic resonance peaks and the ratio ofthe static magnetoelectric constant and magnetic suscep-tibility, which can be used to estimate the strength ofelectromagnon peaks on the basis of dc measurements.

To make our consideration more specific, we consid-ered a Kagome lattice magnet with the KITPite struc-ture, where the ligand ions are positioned in a way that


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gives rise to a relatively strong linear magnetoelectriceffect.28 Using the symmetry analysis we identified themagnon modes that are coupled to both electric and mag-netic fields and give rise to the linear magnetoelectric ef-fect. We showed that the softening of these modes at astrong spin-lattice coupling results in the divergence ofthe magnetoelectric constant as well as of magnetic anddielectric susceptibilities, signaling the instability of the

magnetoelectric state towards a multiferroic state withspontaneously generated P and M. However, the de-tailed study of the phase diagram of this model revealedthat the electromagnon softening does not actually takeplace, since the first-order transition to the collinear mul-tiferroic state occurs at a lower value of the spin-latticecoupling. In some region of model parameters a ferroelec-tric incommensurate-spiral phase intervenes between themagnetoelectric and multiferroic phases. While in knownspiral magnets, ferroelectricity is likely induced by the

inverse Dzyaloshinskii-Moriya mechanism, in our modelit results from the stronger exchange striction mecha-nism due to the collinearity of spins in half of the tri-angles. These magnetoelastic instabilities are typical forfrustrated magnets, where non-collinear spin orders usu-ally occur. We also discussed the possibility to observeelectromagnons in known magnetoelectric materials. Wehope that our study will stimulate experimental work in

this direction.

Acknowledgments

This work was supported by the Zernike Institute forAdvanced Materials and by the Stichting voor Funda-menteel Onderzoek der Materie (FOM).

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m. a. van der vegte, c. p. van der vegte and m. mostovoy- electromagnons and instabilities in...

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