Solid State Communications 149 (2009) 1557–1560

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Solid State Communications

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New collective modes in multiferroic TbMnO3: The electromagnonsShruti Shukla ∗, Debanand SaDepartment of Physics, Banaras Hindu University, Varanasi -221005, India

a r t i c l e i n f o

Article history:Received 27 February 2009Received in revised form22 May 2009Accepted 29 May 2009 by Y.E. LozovikAvailable online 2 June 2009

PACS:78.20.Ls75.80.+q75.30.Fv77.80.-e

Keywords:A. MultiferroicsD. Magneto-optical studiesD. Spin density wave

a b s t r a c t

Using a phenomenological Landau theory as well as equations of motion, we look for new excitationsin the multiferroic TbMnO3. Coupling these excitations to an electromagnetic field, we derived themagneto–polariton dispersion relation. Further, the frequency as well as temperature dependentdielectric susceptibility, and hence the refractive index, are obtained in this formulation. The imaginarypart of the latter is computed and compared with the existing experimental data.

© 2009 Elsevier Ltd. All rights reserved.

Multiferroic materials are those where more than one ferroicorder such as magnetism and ferroelectricity coexist together andare coupled to each other. From the application point of view,coupling between ferroelectricity and magnetism would be usefulin multistate memory devices with mutual electric and magneticcontrol or as magnetically switchable optical devices among oth-ers. The coexistence of ferroelectric and magnetic order presents afundamental challenge since the origins of both the orders are verydifferent:magnetism is related to the ordering of spins of electronsin incomplete ionic shells, whereas ferroelectricity results from therelative shifts of negative and positive ions.Much of the earlier work onmultiferroics was directed towards

bringing them together in one material [1], which was proved tobe difficult because of their mutually exclusive nature [2]. Thelong sought control of electric properties by magnetic fields wasachieved in a class ofmultiferroics [3–6] called frustratedmagnets,namely RMnO3, GdMnO3, RMn2O5, CoCr2O4, MnWO4, etc. Thesematerials become unique not due to the strength of their coupling(electric polarization is two to three orders of magnitude smallerthan in the typical ferroelectrics) but the high sensitivity of theirdielectric properties to an applied magnetic field [7–13]. This is inthe heart of complex spin structure in thesematerialswhichmakesthe ferroelectricity of magnetic origin. For example, the multi-ferroic TbMnO3, which has orthorhombically distorted perovskite

∗ Corresponding author.E-mail address: shruti12912@gmail.com (S. Shukla).

0038-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2009.05.044

(space group Pbnm) structure at high temperature, is identical tothe colossal magnetoresistive manganites in terms of the elec-tronic configuration of the Mn3+ ion. In LaMnO3, staggered or-bital order becomes responsible for the layered antiferromagnetic(AF) order, whereas the spin structure in TbMnO3 is of sinusoidalAF type (of Mn3+ moments) below a temperature TN ≈ 41 Kwith a wave vector Eq (0, qs, 1). The wave vector qs is incommen-surate (∼0.295) at TN , decreases with decreasing T and takes aconstant value (0.28) below Tl ≈ 28 K. Neutron scattering mea-surements [14] revealed that the second transition correspondsto a noncollinear spiral magnetic structure where a simultaneousstatic polarization appears along the z-axis. This is confirmed byX-ray diffraction measurements [15], where the modulated mag-netic phase is accompanied by amagnetoelastically induced latticemodulation.The existence of new elementary excitations in such mag-

neto–electric (ME) coupled systems called electromagnons was al-ready predicted long ago [16], and the first possible observationswere made [17] in GdMnO3 and TbMnO3. The dielectric data ob-tained for TbMnO3 without a magnetic field and with the ac com-ponent E ‖ x exhibit a broad peak with characteristic frequencyω = (20 ± 3) cm−1. The dielectric contribution of this modeincreases with decreasing temperature, and below Tl its widthdecreases. Similar effects are observed for GdMnO3 also. The ap-plication of a static magnetic field along the z-axis suppresses theimaginary part of the dielectric function by more than a factor of2, and also the real part. This provides a clear indication that theobserved mode is related to the modulated magnetic structure.

1558 S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560

In this study, we considered a Landau free energy (fromsymmetry considerations) to address both the magnetic phasesin multiferroic TbMnO3. Using equations of motion, in additionto Maxwell’s equation, we coupled the electromagnetic radia-tion to the excitations in these system and obtained the mag-neto–polariton dispersion. In order to compare our finding withthe available experiments, we derived an expression for the fre-quency and temperature dependent dielectric constant. The imag-inary part of the dielectric function is computed and is comparedwith the observed spectrum.In almost all the multiferroics listed above, complex magnetic

structures and phase diagrams are observed which show stronginterplay between magnetic and dielectric phenomena [18]. Thus,the key question would be how the magnetic ordering inducesferroelectricity. This question has been addressed by Katsuraet al. [10], which is based on the idea of the spin current inducedbetween noncollinear spins yielding an electric dipole moment.This can be regarded as the effect of either Dzyaloshinskii–Moriya(DM) interaction [19,20] or spin–lattice interaction [13]. In thepresent work, we consider the phenomenological treatment ofthis mechanism. The coupling between ferroelectric polarizationand magnetization is governed by the symmetries of theseorder parameters. The minimal coupling which could yieldferroelectricity is nonlinear [12]. Thus, we start with a free energywhich has magnetic and ferroelectric components, and with thecoupling between them given as

F = V−1∫dEr(Fs + Fl + Fs−l) (1)

where

Fs =12a EM2 +

12wM2z +

14u EM4

+12γ (∂yMy)2 + (∂yMz)2 +

12α(∂2yMy)

2+ (∂2yMz)

2, (2)

Fl =12bP2z − EzPz (3)

and

Fs−l = ηPz(Mz∂yMy −My∂yMz). (4)

In constructing the free energy, the details of the unit cell forTbMnO3 have not been taken into account. Here, the parametersa = a0(T − T0), a0, u, b, α are positive and γ is negative. The pa-rameters γ and α are responsible for the modulation of the mag-netic order parameter. The anisotropy parameter w is consideredto be positive. The parameter η represents the coupling betweenthe ferroelectricity and the magnetization.Considering the equilibrium AF states in TbMnO3 to be My0 =

M1 cos ky andMz0 = M2 sin ky, the above free energy is minimizedwith respect to the order parametersM1,M2, Pz and themodulationvector k. Thus, one finds the sinusoidal collinear (I) and thenoncollinear spiral (II) magnetic phases, which are given asfollows:

I :M2 = 0, Pz0 = 0, M21 = −4L1/3u,L1 = a− ac, ac = γ 2/4α, k2 = −γ /2α.

(5)

II :

M21 = (L2 − 3L1 + ε1)/2u,M22 = (L1 − 3L2 + ε2)/2u,L2 = a− ac + w,ε1,2 = 2k2η2(3L1,2 − 5L2,1)/ub,Pz0 = ηkM1M2/b.

(6)

In the minimization, we have assumed that the magnetoelectriccoupling is weak; that is, η

2k2

ub 1. The transition temperatureTN and Tl are obtained as TN = T0 + ac

a0and Tl ≈ TN − 3w

2a0+

6 η2k2wuba0

such that Tl < TN . Thus, the above minimization schemeyields a phase diagram where, on lowering T , one obtains thesinusoidal AF phase below TN , and on further lowering (below Tl)it is the noncollinear spiral magnetic phase which prevails. Theferroelectricity of magnetic origin appears below Tl (see the lastexpression in Eq. (6)).Since the system under consideration is ME coupled, the

excitations in such a system are investigated using equations ofmotion method. In order to do this, we consider the fluctuationsin the AF vector ( Em = EM − EM0) and the electric polarization(Ep = EP − EP0) with respect to their equilibrium values. Further,these excitations can be coupled to the z-component of theelectromagnetic field (Ez), propagating along the y-direction. Thisis taken care of by Maxwell’s equations since its electric fieldcomponent Ez excites polarization along the z-direction. Thus, theMaxwell’s equations, along with the equations of motion in thelinear approximation, give rise to a coupled set of equations for pzandmz which are written as

λpz + δ1pz +δFδpz= 0

µmz + δ2mz +δFδmz= 0

(7)

where µ and λ are the density parameters which characterize thekinetic energy of the system. In the limit of no damping (δ1 = 0= δ2), these equations can explicitly be written as

λpz + bpz − Ezpz + η(mz∂yMy0 −My0∂ymz) = 0

µmz + (a+ w + uM2y0 − γ ∂2y + α∂

4y )mz

+ η(2pz∂yMy0 +My0∂ypz) = 0.

(8)

In order to get themodes of excitations, these equations are solvedin Fourier space, and are taken to bemz(y, t) =

∑lml exp[i(lk+q)

y − ωt] and Ez(pz)(y, t) = E0(p0) exp[i(qy − ωt)]. Here, q is thewave vector associatedwith the electromagnetic field propagationand is assumed to be much smaller than that of the wave vector ofmagnetic order parameter which has sinusoidal variation (q k).In such an approximation, the higher harmonics in l except the firstone can be neglected. Moreover, using Maxwell’s equation, E0 canbe eliminated from the above equation by expressing in terms ofp0; that is, 4πp0 = E0(n2 − 1), n(= qc/ω) being the refractiveindex of the material. Thus, the above equations can be reduced toa coupled equation in terms of p0,m1 andm−1, written as( B −iDk −iD−kiD−k −R L+iDk L− −R

)( p0m−1m1

)=

(000

)(9)

where B = λω2 − b+ 4π/(n2 − 1), L± = µω2 − a− w − γ (q±k)2 − α(q ± k)4 − uM21/2, Dk = ηM1(k − q/2) and R = uM21/4.As already mentioned above, q k, so we take q = 0 (opticmodes only), which simplifies the parameters as D−k = −Dk,L+ = L− and m−1 = −m1. Thus, the above equation simplifiesto a 2 × 2 matrix equation in terms of p0 and m (m = m1 =−m−1) whose solution is obtained by solving the determinant.These yields the energies/frequencies of new q = 0 excitationscalled electromagnons, which are given as

ω21,2 =12

[ω2p + ω

20 ∓

√(ω2p − ω

20)2 + 8η2k2M21 (µλ)−1

], (10)

where ω0 and ωp are respectively the magnetic and dipolefrequencies, defined as ω20 =

1µ[w+ 2(a− ac)/3] =

2a03µ (T − Tl)+

4 η2k2

ubwµand ω2p = b/λ. It should be noted here that the magnetic

frequencyω0 depends on temperature and it takes the lowest valueat T = Tl.

S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560 1559

Fig. 1. Variation of the imaginary part of the dielectric function with frequency.The calculated value of ε2(ω, T ) is fitted with the dielectric data of Pimenov et al. at20 K (filled circles).

The frequency and temperature dependent dielectric function(ε(ω, T )) (which is the square of the refractive index n) in such aformulation is calculated to be

ε(ω, T ) =(ω2 −Ω21 )(ω

2−Ω22 )

(ω2 − ω21)(ω2 − ω22)

, (11)

where the expression for Ω21,2 is the same as that of ω21,2 with

ωp replaced by Ωp = (b+4π)λ. This result is contrasted with

the polariton problem in conventional ferroelectrics where thelongitudinal and the transverse optic mode frequencies appearas zeros and poles respectively in the frequency dependentdielectric function. One of the transverse optic mode softenstoward the phase transition and is condensed according tothe Lyddane–Sachs–Teller (LST) relation [21]. This conventionalviewpoint is not relevant in the case of multiferroics since thelattice displacement is not essential to the electronic polarization.Therefore, it is important to identify the relevant collectivemodes [22] that are responsible for ferroelectricity of magneticorigin and to study the dynamical magneto–electric effect.The dielectric function ε(ω, T ) can in general be a complex

quantity. This is realized from Eq. (10) by making ω → ω + iδ(δ being the imaginary part). Thus, writing ε(ω, T ) = ε1(ω, T ) +iε2(ω, T ) (ε1 and ε2 are respectively the real and imaginary partof the dielectric function), one can compute ε2(ω). The computedvalue of ε2(ω, T ) is fitted with the dielectric data of Pimenovet al. [17] at 20 K (see Fig. 1), which yields the frequencies asω1 = 23.2 cm−1, Ω1 = 26.3 cm−1, ω2 = 78.5 cm−1 andΩ2 = 231.2 cm−1. From this, the magnetic and lattice frequenciesare estimated to be ω0 (T = 20 K) = 67.5 cm−1, ωp = 75.7 cm−1

andΩp = 230.6 cm−1. In the process of fitting, the dimensionless

ME coupling came out to be small, that is, η2k2

ub = 0.031, whereasthe width parameter δ turned out to be considerably large (δ =11.0 cm−1), which is a case of overdamping. However, on loweringT , the Tb spins order at T = 7 K, and it is unclear at thismoment whether the narrowing of excitations results from suchan ordering. Using the above parameters, the variation of theimaginary part of the dielectric function with temperature at ω =6 cm−1 andω = 4 cm−1 is shown in Fig. 2. It is clear from the figurethat ε2(ω, T ) shows a peak at Tl, and its width decreases once thefrequency is lowered.From the above discussions, it is obvious that the magnetic

frequency (ω0) is smaller than the lattice frequencies (ωp,Ωp).Of both the electromagnon modes ω2 and ω1, the latter takes aminimum value (∼(η2k2/ub)2) at T = Tl. This is the signatureof a soft mode (see Fig. 3) which is associated with the second

Fig. 2. Variation of the imaginary part of the dielectric function with temperatureat ω = 6 cm−1 and ω = 4 cm−1 .

Fig. 3. Variation of electromagnon frequencies ω1 and ω2 with temperature,showing the soft mode.

transition (T = Tl), which is from the collinear AF phase tothe noncollinear spiral magnetic structure where ferroelectricityappears. In contrast, the electromagnon mode ω2 is almosttemperature independent and the energy associated with it is highcompared to ω1.Using Eq. (10), the magneto–polariton dispersion spectrum (ω2

vs. q) is analyzed. It yields a sixth-order algebraic equation whosesolutions are shown in Fig. 4. The role of ME coupling is to lowerthe magnetic frequency (ω0) to ω1, whereas it pushes up thelattice frequency (ωp) to ω2. From the dispersion it is obviousthat there are two frequency regions near ω1 and ω2 where theelectromagnons interact very strongly with the electromagneticradiation. The nature of the excitations near ω2 is dominated bythe electric polarization mode whereas near ω1 the prominentexcitations are of magnetic origin. Since ω2 ω1 and the low-energy physics in such a system is dictated by ω1-excitations, thewindow (see the inset in Fig. 4) where the resonant interactionof electromagnetic radiation with the ω1-electromagnon occurs iscalculated as

Ω1 − ω1 ≈η2k2M21ω0λµ

[1

ω2p − ω20−

1Ω2p − ω

20

], (12)

which is very small. Sinceω1 takes a minimum value at T = Tl, theabove interval is largest at Tl compared to any other temperature.Further, this window vanishes in the limit of either η → 0 ork→ 0.Since, in the present formulation, one can excite the electro-

magnons by applying an ac electric field, the ac magnetization mz

1560 S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560

Fig. 4. Magneto–polariton dispersion (ω2 vs. q) using the parameters from the fitin Fig. 1. The values shown along the y-axis are the corresponding ω2 ’s. The insetcorresponds to a zoomed version of the gap between ω21 andΩ

21 .

can be calculated using Eq. (8) as

mz =

∣∣∣∣∣ ω2 −Ω2p

(4π/λ)(2ηkM1)

∣∣∣∣∣ E0. (13)

Considering the frequency to be ω ∼ 20 cm−1, which correspondsto terahertz region, the highest electric field intensity is E0 ∼103 V/cm. Putting theses values along with the parametersestimated above, one can calculate the magnitude of the acmagnetization as mz = ( 0.5×10

−2√µλ

) Vcm . In our calculation above, we

have used the parameters 4πλ= 47 470 cm−2 and w

µ= 4.1 ×

103 cm−2. By choosing w = 0.001 (in proper units), the value ofmz is obtained as 0.4 × 10−2µB/Mn atom, which is comparablewith the experimental value [3].In conclusion, we summarize the main findings of the paper.

We consider a Landau free energy (from symmetry considerations)to address both the magnetic phases in multiferroic TbMnO3. Weshow that in ME materials, new hybrid spin–lattice excitationsexist which can be excited by ac electric fields. This is done by

coupling the excitations to electromagnetic radiation. By solvingthe dynamical equations, the energy of excitations as well as themagneto–polariton dispersion is obtained. In order to compare ourfinding with the available experiments, we derive an expressionfor the frequency and temperature dependent dielectric constant.The imaginary part of the dielectric function is computed andcompared with the observed spectrum. We believe that suchexcitations can be exploited for the design of magneto–opticaldevices.

References

[1] G.A. Smolenskii, I.E. Chupis, Sov. Phys. Usp. 25 (1982) 475.[2] D.I. Khomskii, Bull. Am. Phys. Soc. C 21 (2001) 002.[3] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, Y. Tokura, Nature 426(2003) 55.

[4] N. Hur, S. Park, P.A. Sharma, J.S. Ahn, S. Guha, C.-W. Cheong, Nature 429 (2004)392.

[5] Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, Y. Tokura, Phys. Rev.Lett. 96 (2006) 207204.

[6] K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, T. Arima, Phys. Rev. Lett. 97 (2006)097203.

[7] G. Lawes, A.B. Harris, T. Kimura, N. Rogado, R.J. Cava, A. Aharony, O. Entin-Wohlman, T. Yildrim,M. Kenzelmann, C. Broholm, A.P. Ramirez, Phys. Rev. Lett.95 (2005) 087205.

[8] V.G. Bar’yachtar, I.E. Chupis, Sov. Phys. Solid State 10 (1969) 2818.[9] V.G. Bar’yachtar, V.A. L’vov, D.A. Jablonskii, JETP Lett. 37 (1983) 673.[10] H. Katsura, N. Nagaosa, A.V. Balatsky, Phys. Rev. Lett. 95 (2005) 057205.[11] A.B. Harris, T. Yildrim, A. Aharony, O. Entin-Wohlman, Phys. Rev. B 73 (2006)

184433.[12] M. Mostovoy, Phys. Rev. Lett. 96 (2006) 067601.[13] I.A. Sergienko, E. Dagotto, Phys. Rev. B 73 (2006) 094434.[14] M. Kenzelmann, A.B. Harris, S. Jonas, C. Broholm, J. Schefer, S.B. Kim, C.L. Zhang,

S.-W. Cheong, O.P. Vajk, J.W. Lynn, Phys. Rev. Lett. 95 (2005) 087206.[15] T. Kimura, S. Ishihara, H. Shintani, T. Arima, K.T. Takahashi, K. Ishizaka,

Y. Tokura, Phys. Rev. B 68 (2003) 060403(R).[16] V.G. Bar’yachtar, I.E. Chupis, Sov. Phys. Solid State 11 (1970) 2628.[17] A. Pimenov, A.A. Mukhin, V.Yu. Ivanov, V.D. Travkin, A.M. Balbashov, A. Loidl,

Nat. Phys. 2 (2006) 97.[18] S.-W. Cheong, M. Mostovoy, Nat. Mater. 6 (2007) 13.[19] I. Dzyaloshinskii, J. Phys. Chem. Solids 4 (1958) 241.[20] T. Moriya, Phys. Rev. 120 (1960) 91.[21] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Harcourt Asia Pte Ltd.,

Singapore, 1976.[22] H. Katsura, A.V. Balatsky, N. Nagaosa, Phys. Rev. Lett. 98 (2007) 027203.