markku p. v. stenberg and rogerio de sousa- model for twin electromagnons and magnetically induced...

8/3/2019 Markku P. V. Stenberg and Rogerio de Sousa- Model for twin electromagnons and magnetically induced oscillatory

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arXiv:0906.3059v2

[cond-mat.str-

el]24Sep2009

Model for twin electromagnons and magnetically induced oscillatory polarization in

multiferroic RMnO3

Markku P. V. Stenberg and Rogerio de Sousa

Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia, Canada V8W 3P6(Dated: September 24, 2009)

We propose a model for the pair of electromagnon excitations observed in the class of multiferroicmaterials RMnO3 (R is a rare-earth ion). The model is based on a harmonic cycloid ground stateinteracting with a zone-edge magnon and its twin excitation separated in momentum space bytwo times the cycloid wave vector. The pair of electromagnons is activated by cross couplingbetween magnetostriction and spin-orbit interactions. Remarkably, the spectral weight of the twinelectromagnon is directly related to the presence of a magnetically induced oscillatory polarization inthe ground state. This leads to the surprising prediction that TbMnO3 has an oscillatory polarizationwith amplitude 50 times larger than its uniform polarization.

PACS numbers: 75.80.+q, 78.20.Ls, 71.70.Ej, 75.30.Et

I. INTRODUCTION

The coexistence of magnetic and ferroelectric phases inmultiferroic materials gives rise to important effects re-

lated to cross correlation between order parameters andexternal fields.1,2 A notable consequence is that the el-ementary excitations are not purely magnetic or ferro-electric in character. The spin waves are admixed withvibrational modes of the electric polarization, giving riseto electric dipole active magnons, the so-called electro-magnons. Electromagnons were postulated to exist in1969 by Baryakhtar and Chupis3 but their existencewas confirmed only recently by sensitive optical experi-ments in the far infrared4,5,6,7,8,9 combined with neutronscattering.10

Apart from its fundamental importance, the electro-magnon spectra gives invaluable information on how

magnetism couples to ferroelectricity. The class of per-ovskite manganites RMnO3 (RMO) is playing a majorrole in this respect. Here R is a rare-earth ion such asGd, Tb, Dy, or mixtures between them (e.g., GdxTb1x,etc.). At low temperatures (T 20 K) the Mn spinsare ordered in a spiral state with period incommensu-rate with the lattice.11 The spin spiral is of the cycloidtype. This breaks space inversion and gives rise to aferroelectric moment along one of the directions in thecycloid plane.12 At the microscopic level, this ferroelec-tric moment originates from Dzyaloshinskii-Moriya (DM)coupling.12,13,14,15

Nevertheless, contrary to predictions based on DMcoupling,16 electromagnons in RMO are observed onlywhen the electric field of light is along the crystallo-graphic direction a.7,8,9,17 Moreover, for all ions R, twoelectromagnons are always observed.18 Recently, the ori-gin of one of these electromagnons was explained byAguilar et al.,7,8 who pointed out that the high fre-quency electromagnon was a zone-edge magnon activatedby pure magnetostriction. However, the origin of the low-frequency electromagnon remains unknown.

A recent theory19 shows that the anharmonicity of adistorted cycloid ground state gives rise to multiple elec-

tromagnons that were observed using Raman scatteringin BiFeO3.

6 However, the anharmonicity mechanism can-not explain the pair of electromagnons in RMO becausethe degree of anharmonicity detected by neutron diffrac-

tion is too low in these materials.11Here we propose a model that is able to explain the

double electromagnon feature observed in RMO. Weshow that a simple harmonic cycloid ground state withwave vector Q has a special pair of twin electromagnonexcitations, located at the zone edge and 2Q away from it.Remarkably, the activation of the latter electromagnon isdirectly related to the presence of an oscillatory polariza-tion with wave vector 2Q in the ground state.

I I. M ODEL

The usual phenomenological Landau theory is based ona free-energy expansion into powers of the spatial deriva-tives of the order parameters.20 Therefore, the Landauapproach3,12,19,21 cannot describe large wave vector ex-citations such as zone-edge magnons. Here we adopt amodel that is more microscopic than the Landau theory.Our model Hamiltonian consists of three distinct contri-butions, H = HS + Hph + Hme. The first contributiondescribes spin frustration in RMO,7,8,10,11,22

HS =n,m

Jn,mSn Sm + Dn

(Sn a)2. (1)

Here Jn,m is the exchange coupling between Mn spins atpositions Rn and Rm, and D is a single-ion anisotropyalong the crystallographic direction a (we assume D > 0,i.e., easy-plane anisotropy). RMO has an orthorhombic

lattice, with an orthogonal system of crystal axis a, b, cand unequal bond lengths a = b = c . There are fourMn spins per unit cell; spins 1 and 2 lie in the ab planes,with spins 3 and 4 a distance c/2 above them. The cou-pling between nearest neighbor spins in the ab planes isferromagnetic, and will be denoted by J0; the couplingbetween ab layers along the c direction is antiferromag-netic and is denoted by Jc. Spin frustration arises due
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FIG. 1: (Color online) Coupled spin waves and opticalphonons in the cycloid ground state. In-plane spin fluctu-ations are denoted by , while out-of-plane spin fluctuationsare denoted by .

to coupling between next-nearest neighbors along the bdirection; this is antiferromagnetic and denoted by J2b.When the stability condition J2b > J0/2 is satisfied,the ground state is a bc cycloid13,14,15 as observed inexperiments,1,11

S0(R) = S[cos(Q R)b + sin(Q R)c]. (2)

Here Q = Qb is the cycloid wave vector, withcos(Qb/2) = J0/(2J2b). The upper (+) sign appliesto ab layer spins 1 and 2, whereas the lower () signapplies to spins 3 and 4 in the neighboring ab layers im-mediately above and below them. Figure 1 depicts thecycloid ground state. The second ingredient of our modelis the lattice fluctuation Hamiltonian,

Hph =1

2m

n

x2n +

20x

2n

e

n

xn E, (3)

where xn is a relative displacement between cations andanions within each unit cell. Here e is the Born ef-fective charge, m is an effective mass, and 0 is the(bare) phonon frequency. The modes xn are directlyrelated to the polarization order parameter in Fourierspace, Pq() =

e

Nv0

dt

n ei(qRnt)xn, where v0

is the volume of the unit cell. The electric field oflight, Eeit, couples linearly to the q = 0 polarization,P0() = 0()E, with electric susceptibility 0() =(e2)/[(mv0)(20

2)].The third contribution Hme models the coupling be-

tween the spin degrees of freedom and the lattice. Fol-lowing the experimental observation that electromagnonsare only excited when the electric field is along a,7,8,9,17

we consider the interaction,

Hme = en

xan[gc(Sc1,n S

c1,n+b)(S

c2,n + S

c2,n+a)

+gb(Sb1,n S

b1,n+b)(S

b2,n + S

b2,n+a)

+(1 3, 2 4)], (4)

which is invariant under the Pbnm space-group opera-tions of RMO. Here gc = gb are coupling constants withdimension of electric fields; we neglect the components

proportional to ga because these play no role in the dis-cussion below. Remarkably, Eq. (4) is not rotationallyinvariant in spin operators. Therefore it differs in animportant way from the pure magnetostrictive couplingconsidered previously.7 The absence of rotational invari-ance in magnetoelectric coupling is a direct consequence

of cross coupling between magnetostriction and spin-orbit

interactions, and should be common property of all multi-

ferroics with orthorhombic lattice (this follows from sym-metry since a = b = c precludes rotations that take oneaxis into the other). Later we will explain why the ex-perimental observations favor this coupling over othersymmetry-allowed explanations.

It turns out that Eq. (4) gives rise to a remarkablestatic effect. Minimizing Hph with respect to xn, andplugging in the cycloidal spin order we get

exnv0

= PIOP sin[(2n + 1)Qb]a. (5)

Hence the local polarization per unit cell is oscillatory

with wave vector 2Q; the amplitude for the incommen-surate oscillatory polarization (IOP) is given by

PIOP = 4 (gb gc) 0S2 sin

Qb

2

. (6)

Remarkably, the presence of the IOP is directly relatedto the lack of rotational invariance in Eq. (4), i.e., thefact that gb = gc.

It is well known that TbMnO3 has a magnetically-induced lattice modulation with wave vector 2Q in thecycloidal phase.1 However, we are not aware of any claimsthat such a modulation leads to oscillatory electric polar-

ization. The main result of this paper is to show that thisoscillatory polarization is directly related to the spectralweight of the twin electromagnon; hence it can be de-tected by optical experiments.

III. TWIN ELECTROMAGNON EXCITATIONS

We now consider the elementary excitations of the cou-pled spin-phonon system. The spin excitations S =S S0 are parametrized as follows,

Si,n = i,nai,n[cos(Q Ri,n)csin(Q Ri,n)b], (7)

where the sign convention is the same as in Eq. (2). Here

the operators describe spin fluctuations out of the cy-cloidal plane, and denotes in-plane (tangential) fluctu-ations (Fig. 1). We carry out a mean-field expansion ofthe Hamiltonian H about its equilibrium value by keep-ing only terms quadratic in the fluctuation operators,e.g., P2, 2, P, etc. Using the canonical commu-tation relations [j,n, k,m] = ijknmS, we derive the


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FIG. 2: (Color online) Typical dispersion curves along the

cycloid direction b for the cyclon (blue solid) and extra-cyclon(red dashed) modes in a bc cycloid. Our theory gives rise totwin electromagnons depicted by a circle and a square. Whenthe magnetoelectric interaction is rotationally invariant [gb =gc in Eq. (4)], only the one denoted by a square is excited; inthe absence of rotational invariance, both electromagnons areexcited. Here Q is the magnitude of the cyclon wave vectorand b the lattice constant along b axis.

coupled equations of motion for spin and polarization,

2 2C,q

(1q + 2q + 3q + 4q) = qk0 , (8a)

2 2C,q+k0

(1q 2q + 3q 4q) = q, (8b)

2 2EC,q

(1q + 2q 3q 4q) = 0, (8c)

2 2EC,q+k0

(1q 2q 3q + 4q) = 0. (8d)

Here iq are Fourier transforms of the fields in; iden-tical equations hold for the fields in.

23 There are fourspin-wave modes, each transforming according to fourdifferent 1d representations of the Pbnm space-group.The second mode [Eq. (8b)] is related to the first mode

by q q + k0, with k0 = (2/b)b the Brillouin zone

edge (the lattice has periodicity b/2 along b). This corre-sponds to the fact that anti-phase fluctuations of theneighboring spins with the wave vector q are equivalentto in-phase fluctuations at q + 2/b. Their dispersionsatisfies C,q0 0, reflecting the phase sliding symme-try of the cycloid. Adding a constant phase to Eq. (2)yields a different ground state with the same energy. Atq = 0 the first mode is a pure phase fluctuation (in isthe same for all i and in = 0). For this reason thismode will be referred as a cyclon, with dispersion C,q.The second mode is naturally called a zone-edge cyclon.Similarly, the fourth mode can be obtained from the thirdby the same translation in qspace, q q+k0. However,the third and fourth modes are gapped (gap proportionalto Jc). These modes will be denoted extra cyclons. The(bare) magnon dispersions are shown in Fig. 2. For q

FIG. 3: Diagrams for twin electromagnon excitation by light,involving momentum exchange with the cycloidal groundstate. (a) Zone-edge electromagnon at q = k0 and (b) itstwin at q = k0 2Q. The fact that momentum is not con-served by k0 = 2/b reflects the underlying lattice symmetry,which produces spin-wave modes connected by q q + k0.

along b they are given by

2C,q =4S2

2

J0 + 2J2b cos

qb

2

2+ 2J2b(D + 2Jc)

4cos2 qb4

J20

J2

2b

cosqb

2 sin2 qb

4

, (9a)

2EC,q =4S2

2

J0 + 2J2b cos

qb

2

2+ 2J2bD

4cos2

qb

4

J20J22b

cosqb

2

sin2

qb

4+

JcJ2b

.(9b)

Typical values for the exchange couplings are J0 0.4meV, J2b 0.3 0.4 meV, and Jc 0.5 1 meV.8 In-terestingly, only the zone-edge cyclon is coupled to thepolarization. This occurs through the dynamical magne-toelectric field,

q =

8v0

2 S

2

sinQb

2

2J0

cosqb

2

+ cosQb

2

J2b [cos(qb) cos(Qb)] (D + 2Jc)}

2(gb + gc)Paq ()

(gb gc)

Paq2Q() + Paq+2Q()

. (10)

Similarly, the equation of motion for the polarizationonly couples Pa0 to the zone-edge cyclon at q = 0 andq = 2Q. Hence we have a set of twin electromagnonsat wave vectors q = k0 and q = k02Q. For gb = gc, onlyone electromagnon (the zone-edge cyclon) is activated.7

This excitation corresponds to a scattering process within and out-going momenta equal to the cycloid wave vec-tor Q [Fig. 3(a)]. However, when gb = gc, the twin elec-tromagnon (a cyclon at q = k02Q) appears, correspond-ing to two outgoing momenta adding up to 2Q [Fig. 3(b)].

IV. DIELECTRIC FUNCTION AND

COMPARISON TO EXPERIMENTS

In order to relate to optical experiments, we calculatethe dielectric function () = + 4Pa0 ()/E

a and


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Gd0.5Tb0.5 Gd0.1Tb0.9 Tb Tb0.41Dy0.59 DyAB

Exp

0.30 0.39 0.40 0.36 0.37c,k02Qc,k0

Th

0.55 0.53 0.52 0.52 0.54SASB

Exp

0.08 0.11 0.13 0.41 0.51h(gb+gc)

2

(gbgc)2Sc,k02Q

Sc,k0

iTh

0.14 0.16 0.16 0.16 0.18

TABLE I: This table compares our theoretical calculations to experiments in RMnO3. A and B are, respectively, the lower-and higher-energy electromagnon modes detected in [8].

study its resonances. When the resonance frequenciesare not too close to each other is a sum of Lorentzians,

() =Sk0

2C,k0 2k0

2+

Sk02Q2C,k02Q

2k02Q

2

+Sph

20 + 2ph

2+ . (11)

The electromagnon frequencies are seen to be down-shifted from the bare magnon frequency C,q to

2C,q 2q, while the phonon frequency 0 gets up-

shifted to

20 + 2ph. The spectral weights Sq share a

simple relationship with the frequency shifts: For elec-

tromagnons, Sq = [(0) ]2q, while for the phonon

Sph = [(0) ](20

2ph). The frequency shifts are re-

lated by 2ph = 2k0

+ 2k02Q, ensuring the satisfactionof the oscillator strength sum rule, Sk0 + Sk02Q + Sph =200. The electromagnon spectral weights are given by

Sk0 =4S20v0

40(gb+gc)

2

(20

2c,k0

)2tan2 Qb2

2C,k0J2b

, (12a)

Sk02Q =2S20v0

40(gbgc)

2 tan2 Qb2

(20

2c,k02Q

)2[cos4 Qb2 +sin4Qb2 ]2C,k02Q

J2b

.

(12b)

An important experimental result is that the spec-tral weights are nearly the same for the ab and bccycloids.7,9,17 It is easy to see that this result followsfrom our model when |gb| |ga|, |gc|. Note that in this

limit the magnetoelectric coupling Eq. (4) is invariantunder the flip of the cycloid plane, in spite of the factthat it lacks rotational symmetry. Table I compares ourtheoretical calculations to the measured values A/Band SA/SB, where A and B label the lower- and higher-energy electromagnons observed in RMO.8 We used thesame model parameters as the ones in Fig. 4(b) of Ref. [ 8](filled symbols). The variations in the ratio SA/SB in-dicate differences in the couplings gc, gb for different ionsR. For Tb our theory matches the observed values whengc/gb = 0.05; for Dy we get gc/gb = 0.25.

V. DISCUSSION AND CONCLUSIONS

We now consider other possibilities for the activation ofthe low-frequency electromagnon. Interactions contain-ing crossed terms such as xanS

bi S

cj are also allowed by the

Pbnm symmetry. For a bc cycloid, this interaction givesrise to an electromagnon at q = k0 2Q. However, whenthe cycloid is flipped to the ab plane, this interactionleads instead to an electromagnon at q = k0 Q, with noelectromagnon at q = k02Q. This result implies a large

shift in electromagnon frequency (more than 50% as seenin Fig. 2), that is in contradiction to experiments.7,8,9,17

Therefore the interaction xanSbi S

cj can not explain the ori-

gin of the low-frequency electromagnon. Similarly, in-teractions of the form xanSai Scj only give rise to electro-magnons at k0 Q; this leads to ratios of electromagnonfrequencies A/B that are twice as large as obtained

experimentally (Table I).8 Hence xanSai S

cj is also ruled

out.

Aguilar et al.7 suggested that an elliptical spiral struc-ture such as the one in BiFeO3 (Refs. 6 and 19) may ex-plain the low-frequency electromagnon. This possibilityis also ruled out because the degree of ellipticity observedin neutron-scattering experiments11 is too low to explainthe large spectral weight of the low frequency electro-magnon. Furthermore, Ref. 7 suggested that purely mag-netic interactions such as Sai S

cj would mix extra-cyclon

magnons at q = Q to the zone-edge electromagnon, pro-viding an alternative explanation for the low frequencyresonance. However, the extra-cyclon magnon energy atq = Q is approximately constant for different Rs, incontradiction to the trend observed in experiments [Atends to decrease with increasing ionic radius; A forGd0.5Tb0.5 is 40% higher than for Dy (Ref. 8)]. Inter-estingly, our theory gives a natural explanation for thistrend since the value of the cyclon energy at q = k0 2Qalso decreases appreciably as the ionic radius increases.


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We now discuss the implications of our model for theunderstanding of multiferroic order. One remarkable con-sequence of the presence of the q = k0 2Q electro-magnon is that its spectral weight can be directly re-lated to the presence of the IOP in the ground state. Us-ing Eqs. (6) and (12b) we derive an important relationbetween the amplitude of the IOP and the twin electro-magnon spectral weight,

P2IOP =8S3J2b

v0cos2

Qb

2

cos4

Qb

2+ sin4

Qb

2

Sk02Q2k02Q

.

(13)Hence optical experiments are a direct probe of the IOPamplitude. For TbMnO3, the measured twin electro-magnon frequency is C,k02Q = 25 cm

1, with spectralweight Sk02Q = 1.7 10

3 cm2.8 Using Eq. (13) we ob-tain PIOP = 4 C/cm2, a value that is 50 times largerthan the uniform polarization P0 = 8 102 C/cm2

present in TbMnO3.In conclusion, we introduced the concept of the twin

electromagnon in order to explain optical experimentsin the RMnO3 family of multiferroics. Our symmetryanalysis shows that an incommensurate oscillatory po-larization coexists with the well-known cycloid phase inthese materials. Remarkably, there exists a direct re-lation between the twin electromagnon spectral weightand the amplitude of this oscillatory polarization. Hencewe showed that TbMnO3 has an oscillatory polarization

with amplitude 50 times larger than its uniform polar-ization. This surprising conclusion underlines the impor-tance of electromagnons in the characterization of multi-ferroic order.

VI. ACKNOWLEDGMENTS

We thank N. Kida and A. Pimenov for useful discus-sions. This research was supported by NSERC discoveryand the UVic Faculty of Sciences.

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23 There exists a linear relation connecting the fields in tothe fields in; therefore the in are completely specifiedonce the in are set.
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markku p. v. stenberg and rogerio de sousa- model for twin electromagnons and magnetically induced...

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