Eur. Phys. J. B (2013) 86: 201DOI: 10.1140/epjb/e2013-40071-x

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Phonon excitations and magnetoelectric couplingin multiferroic RMn2O5Safa Golrokh Bahoosh1, Julia M. Wesselinowa2, and Steffen Trimper3,a

1 Max Planck Institute of Microstructure Physics, 06120 Halle, Germany2 Department of Physics, University of Sofia, 1164 Sofia, Bulgaria3 Institute of Physics, Martin-Luther-University, 06120 Halle, Germany

Received 28 January 2013 / Received in final form 4 March 2013Published online 2nd May 2013 – c© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2013

Abstract. Multiferroic rare-earth manganites are theoretically studied by focusing on the coupling to thelattice degrees of freedom. We demonstrate analytically that the phonon excitations in the multiferroicphase are strongly affected by the magnetoelectric coupling, the spin-phonon interaction and the anhar-monic phonon-phonon interaction. Based on a microscopic model, the temperature dependence of thephonon dispersion relation is analyzed. It offers an anomaly at both the ferroelectric and the magnetictransition indicating the mutual coupling between multiferroic orders and lattice distortions. Dependingon the sign of the spin-phonon coupling the phonon modes become softer or harder in accordance withexperimental observations. We show that the phonon spectrum can be also controlled by an external mag-netic field. The phonon energy is enhanced by increasing that field. The applied Green’s function techniqueallows the calculation of the macroscopic magnetization depending on both the phonon-phonon and thespin-phonon couplings.

1 Introduction

Recently, a large class of manganese oxides as RMnO3and RMn2O5, with rare earth R = Y, Tb, Dy, etc. hasbeen discovered to reveal a strong magnetoelectric cou-pling (ME) [1]. The material offers multiferroic propertiescharacterized by the coexistence of magnetic and ferro-electric orders. The ME coupling leads to various novelphysical effects, such as the colossal magneto dielectric ef-fects [2] and magnetospin-flop effects [3]. As reported inreference [4] the ME coupling in TbMn2O5 is so strongthat the electric polarization can be reversed by apply-ing a magnetic field. The remarkable ME effects in thesematerials have attracted a great deal of interest due tothe fascinating physical properties and the potential ap-plications in novel multi-functional ME devices. Regard-less the innovative aspects, the underlying detailed mech-anism of the multiferroics is under a permanent debate,for a recent review see [5–7]. Since in the past phonons andtheir coupling with other degrees of freedom have playeda crucial role in understanding classic ferroelectrics, oneshould expect that they have a great impact on magneto-electric multiferroics. Recent investigations using Ramanand infrared (IR) spectroscopy, by transmittance and re-flectance measurements, have demonstrated the impor-tance of phonon effects in multiferroics. There is an ex-perimental evidence for a strong spin-phonon coupling

a e-mail: steffen.trimper@physik.uni-halle.de

in RMn2O5 [8,9]. The experimental results show pro-nounced phonon anomalies around the magnetic and elec-tric phase transition temperatures [10–15]. These anoma-lies are attributed to the multiferroic character of thematerials. Raman phonons in RMnO3 orthorhombic andhexagonal manganites have been studied in reference [16]as a function of the rare-earth ion and the tempera-ture. The sign and the magnitude of such anomalousphonon shifts seem to be correlated with the ionic ra-dius of the multiferroic RMn2O5 [11]. So the phononexcitation becomes a soft mode in case that R is Biand is hardening if the rare-earth part is Dy. An in-termediate behavior is observed for Eu. Based on thetemperature dependence of the far-IR transmission spec-tra of multiferroic YMn2O5 and TbMn2O5 single crys-tals, the occurrence of electromagnons was reported inRMn2O5 compounds in reference [17]. The phonon en-ergy and its damping are different for diverse compounds.So varying selection rules for electromagnons in RMn2O5and RMnO3 suggest different magneto-elastic couplingmechanisms in the multiferroic systems. Molecular-spindynamics study of electromagnons in RMn2O5 are dis-cussed in reference [18]. The so-called shell model lat-tice dynamic calculations are proposed in reference [19]for RMn2O5 (R = Ho, Dy) materials. Here spin-phononcouplings have been taken into account. Theoreticallythe influence of phonons is studied in reference [20].The exchange-striction induces a biquadratic interaction

http://www.epj.orghttp://dx.doi.org/10.1140/epjb/e2013-40071-xofftheoSchreibmaschinentextTH-2013-41

Page 2 of 10 Eur. Phys. J. B (2013) 86: 201

between spins and transverse phonons. Furthermore, theDzyaloshinskii-Moriya interaction is considered in refer-ences [20,21]. More recently, the spin-phonon coupling inmultiferroic Mn compounds were analyzed in a classi-cal spin-model [22]. Using a microscopic model in refer-ence [23] the properties of thin films composed of multi-ferroic material have been studied.

The new aspect focused on in the present paper isthe calculation of the phonon spectrum within a micro-scopic model for RMn2O5. Here the magnetic subsystemis described by a Heisenberg spin Hamiltonian with twocompeting coupling parameters which give rise to frus-tration. The magnetic subsystem offers a phase transitionat TN . In modeling the ferroelectric behavior we have totake into account the improper nature of ferroelectricity.To that aim we concentrate the analysis of the ferro-electric subsystem on a parameter range where the po-larization is induced by the magnetoelectric coupling ex-clusively. With other words the polarization is zero forT ≤ TN . Lowering the temperature further the mag-netoelectric (ME) coupling leads to a finite polarizationat TC , where TC < TN . Due to the complexity of themanganites the ferroelectric behavior is described withinour analytical approach by both order-disorder and dis-placive type aspects. The order-disorder properties arecharacterized by pseudo-spins in the frame of an Isingmodel in a transverse field [24], while the displacive be-havior is characterized by lattice distortions. Based on atractable analytical model we show that the magnetoelec-tric coupling, the spin-phonon interaction as well as theanharmonic phonon-phonon interaction contribute to thephonon excitations in a significant manner. So the temper-ature dependent phonon spectrum exhibits an anomaly atthe ferroelectric and the magnetic phase transitions. Therelated excitation is obtained applying temperature de-pendent Green’s function methods. The methods allowsalso the computation of the macroscopic magnetization.Furthermore, we demonstrate that the phonon mode canbe also controlled by an external magnetic field. The an-alytical finding is compared with experimental results.

2 The model

2.1 Hamiltonian

The known mechanisms of the spin-polarization couplingcan be categorized in two different types. One of themis based on the inverse Dyzaloshinskii-Moriya interaction,where the local dipole moment, denoted by uij , couplesto two canted spins Si and Sj in the form proportionalto uij · êij × (�Si × �Sj). The unit vector êij connects thecenters of the magnetic ions at lattice sites i and j. Themicroscopic origin of such coupling was investigated inseveral papers such as [25,26]. The properties of a wholeclass of multiferroic materials as RMnO3 can be under-stood on the basis of this coupling. The second type ofcoupling arises from exchange-striction, wherein the mag-netic ions are assumed to be influenced directly by the ex-change interaction giving rise to a coupling proportional

to êij · (ui − uj)(�Si · �Sj) [27–29]. Because such spin-latticecoupling is believed to be relevant for RMn2O5 we followthe line of the second kind of coupling.

As already emphasized [20] a more realistic modelshould include both aspects, namely a frustrated mag-netic system as well as symmetric exchange-striction. TheME coupling induces the polarization at temperature TCwhich is lower than the magnetic transition tempera-ture TN . Within an analytical approach one has to incor-porate the ferroelectric behavior in the total Hamiltonianof the system. In our model that subsystem is character-ized by a displacive type interaction and an order-disorderbehavior. Whereas the order-disorder type interactionis simply described by an Ising model in a transversefield [24] the displacive behavior is written in terms oflattice distortions. The Hamiltonian reads

H = Hm + Hf + Hmf . (1)

The magnetic part Hm is expressed by the Heisenberg spinHamiltonian

Hm = −12∑

ij

Jij �Si · �Sj + 12∑

〈ij〉J̃ij �Si · �Sj − h

∑

i

Szi . (2)

Here, h = gμBB is the external magnetic field. TheHamiltonian Hm consists of the isotropic Heisenberg ex-change interaction with nearest neighbor ferromagnetic in-teraction J > 0, denoted as ij, and next nearest neighborantiferromagnetic coupling J̃ > 0, indicated by 〈ij〉 inequation (2). Due to the mixed ferromagnetic and anti-ferromagnetic coupling the system exhibits frustration incase J̃ > J/4 [30]. Let us make some remarks concerningthe magnetic interaction. As pointed out in reference [31],compare also [32], the magnetic excitations in the mul-tiferroic material are more different. Within a numericalsimulation the authors have identified five different ex-change coupling parameters appearing in the Heisenbergspin Hamiltonian. Following [29], the exchange interac-tions between neighboring Mn3+ and Mn4+ ions are allantiferromagnetic. As a result, magnetic frustration mustexist in the smallest closed loop of Mn spins consisting offive magnetic ions. So, zigzag chains of frustrated spins ex-ist in a direction parallel to the a axis as it is illustrated inFigure 1a. In the framework of an analytical calculation,it would be to complicated to formulate the situation by aHamiltonian and solving the corresponding equations forsuch a sophisticated spin structure. Therefore, we havesimplified the model as depicted in Figure 1b. It com-prises the frustrated magnetic subsystem represented byequation (2). The two different couplings J and J̃ in thisequation reflect the competition between different mag-netic orders which gives reason to frustration.

The ferroelectric subsystem Hf is characterized by lat-tice distortions describing the displacive type behavior andpseudospins within the Ising model in a transverse field

http://www.epj.org

Eur. Phys. J. B (2013) 86: 201 Page 3 of 10

Fig. 1. (a) Mn-spin configuration in RMn2O5 structure show-ing frustration. (b) Typical sketch of a frustrated spin systemwith nearest neighbor ferromagnetic interaction and next near-est neighbor antiferromagnetic coupling.

reflecting the order-disorder behavior. Therefore Hf is

Hf = Hph + Htim

Hph =12

∑

k

(PkP−k + ω2(0)kQkQ−k

)

+13!

∑

kk1

B(k, k1)QkQ−k1Qk1−k

+14!

∑

k,k1k2

A(k, k1, k2)Qk1Qk2Q−k−k2Q−k1+k

Htim = −12∑

ij

J ′ijBzi B

zj − 2Ω

∑

i

Bxi . (3)

The first part of Hph represents the harmonic part ofthe lattice distortions in terms of the normal coordi-nate Qk, the momentum Pk and the harmonic phononfrequency ω(0)k of the lattice mode with wave vector k.The following terms describe the anharmonic interactions,where the third order coupling is given by B(k, k1) and thequartic one by A(k, k1, k2). In terms of phonon creationand annihilation operators the normal coordinate and the

momentum are expressed by:

Qk = (2ω(0)k)−1/2(ak + a

†−k

)

Pk = i(ω(0)k/2)1/2(a†k − a−k

). (4)

The order-disorder part is represented by Htim which isdue to [24] written in terms of pseudo-spin operators Bzand Bx. J ′ij denotes the nearest neighbor pseudo-spin in-teraction and Ω is a frequency which allows a flip processbetween the two orientations of the moment Bz . The twoorientations are given by the eigenvalues ±1 of the opera-tor Bz. The ferroelectric subsystem is considered typicallyin a rotated frame according to

Bxj =(

12− ρj

)sin ν +

12

(bj + b

†j

)cos ν;

Byj =i

2

(b†j − bj

);

Bzj =(

12− ρj

)cos ν −

(bj + b

†j

)sin ν; ρj = b

†jbj. (5)

Here, b and b† are Pauli-operators which fulfill the com-mutator relation [bi, b

†j] = (1 − 2ρi) δij . The angle ν is

determined in such a manner that the polarization is zeroprovided the system is within the range TC ≤ T ≤ TN andthe magnetoelectric coupling is not relevant. This techni-cal trick reflects the improper nature of the ferroelectric-ity. As demonstrated below the ferroelectricity is directlyinduced by the magnetoelectric coupling which will be in-troduced now.

The coupling between the ferroelectric and the mag-netic subsystems includes a quadratic coupling in the mag-netic order parameter and a linear one in the ferroelectricorder parameter. Because the magnetic subsystem is ex-pressed in terms of the spin-operator �S and the ferroelec-tric one in terms of pseudo-spin operators �B, the relevantmagnetoelectric coupling Hmf is given by

Hmf =∑

ijl

KijlBzi�Sj · �Sl. (6)

The underlying physical mechanism of the coupling is thatthe cooperative alignment of magnetic moments at a cer-tain temperature is the reason for the polarization. TheME coupling in equation (6) describes the direct influenceof the magnetic order parameter on the secondary polarorder parameter. However both subsystems are simulta-neously influenced by lattice distortions. Hence the modelhas to be completed by spin-phonon interactions. Such aspin-phonon coupling had been already discussed withinthe pseudo-spin approach in reference [24]. As pointed outthe main effect comes from the modulation of an internalcrystal field. Following that approach and former studiesfor ferroelectric thin films in [33] as well as for multifer-roic materials in reference [34], the pseudo-spin phononcoupling Hfph can be expressed in terms of normal dis-placement coordinates Q, compare equation (3). The total

http://www.epj.org

Page 4 of 10 Eur. Phys. J. B (2013) 86: 201

interaction is written in the form

Hfph = −∑

k

F (f)(k)QkBz−k

− 12

∑

kk1

R(f)(k, k1)QkQ−k1Bzk1−k

Hmph = −∑

k

F (m)(k)QkSz−k

− 12

∑

kk1

R(m)(k, k1)QkQ−k1Szk1−k. (7)

Here, the first term Hfph models the coupling between theferroelectric order parameter and the phonons, while thesecond term Hmph is responsible for the coupling betweenphonons and magnetic spins. We are aware that higherorder couplings, quadratic in spins or pseudo-spins, canbe also taken into account. Such terms seem to be irrele-vant in searching the occurrence of a polarization due tomagnetic ordering. The complete Hamiltonian will be an-alyzed in the subsequent section using Green’s functions.

2.2 Green’s functions and excitation spectrum

A powerful technique to get the dispersion relation andrelated macroscopic quantities is the Green’s functionmethod at finite temperatures. For the magnetic subsys-tem let us define the Green’s function due to [35]

Gmq (t − t′) ≡〈〈

S+q (t); S−q (t

′)〉〉

= −iΘ(t − t′) 〈[ S+q (t) , S−q (t′)]〉

. (8)

Using the commutation relation and making a simple de-coupling (RPA) it results

Gmq (E) =2〈σM 〉

E − Em(�q ) with M = 〈Sz〉 = 〈σM〉

Em(q) = 〈Sz〉[J0 − Jq + J̃q − J̃0 − K0 cos ν

(12− 〈ρ〉

)]

+ F (m)(0)〈Q〉 + R(m)(0) 〈Q2〉 . (9)

Here Em is the spin-wave energy of the magnetic sub-system modified by the magnetoelectric coupling K0 andthe magnetic spin-phonon interaction F (m) and R(m). Themagnetization M is given by M = 〈Sz〉 = 〈σM 〉.

Because the ferroelectric subsystem is expressed interms of pseudo-spin operators (see Eqs. (3), (5) and (7)),the related Green’s function can be defined in a similarmanner as for the magnetic system in equation (8). Differ-ent to the magnetic case the Green’s function is a matrixabbreviated as Gflm(t):

(〈〈bl; b†m〉〉 〈〈bl; bm〉〉〈〈b†l ; b†m〉〉 〈〈b†l ; bm〉〉

)≡

(Gf11lm G

f12lm

Gf21lm Gf22lm

).

Here the notation is the same as in equation (8). A typicalGreen’s function reads

Gf11q = 〈σP 〉(

qE − Ef −

′qE + Ef

)

Ef (q ) =√

(ε11q )2 − (ε12q )2. (10)

The coefficients q, ′q as well as ε11q and ε

11q can be found

in Appendix A.The phonon part is also described by a Green’s func-

tion matrix denoted as Gphq (t). They are defined by:(〈〈aq ; a†q〉〉 〈〈aq ; aq〉〉〈〈a†q ; a†q〉〉 〈〈a†q ; aq〉〉

)≡

(G11q G

12q

G21q G22q

). (11)

These set of Green’s functions fulfills the following equa-tion of motion:

(G11q G

12q

G21q G22q

)=

(1 0

0 −1

)+

(�q

11 �q12

�q21 �q

22

)(G11q G

12q

G21q G22q

),

(12)where the coefficients are also defined in Appendix B. Theharmonic phonon frequency in equation (3) is modified bythe different couplings. The renormalized phonon excita-tion energy is given by the poles of the related Green’sfunction and reads

ω2q = ω2(0)q + 2ω(0)q

(B̄q〈Q〉 + 12 Āq〈Q

2〉 − 12R̄(m)(q)〈σM 〉

−12R̄(f)(q)〈σP 〉 cos ν

). (13)

Let us stress that the renormalized phonon frequency inequation (13) is apart from the coupling parameters de-termined by the magnetization 〈σM 〉 and the polariza-tion 〈σP 〉. Those macroscopic quantities like magnetiza-tion and polarization can be also found using Green’sfunction. The analytical results will be discussed in thesubsequent section.

3 Results and discussion

In the previous section we found an analytical expres-sion for the phonon excitation of the multiferroic mate-rial equation (13). The spectrum is determined by themagnetization 〈σM 〉, the polarization 〈σP 〉, the anhar-monic phonon interactions B̄q, Āq as well as the vari-ous spin-phonon coupling parameters R̄(m) and R̄(f). Inthis section the results will be discussed and comparedwith experimental observations. Depending on the sign ofthe spin-phonon interaction constant R (see Eq. (7)), thephonon frequencies become either harder or softer. Whichsituation is actually realized depends on the interactionbetween the ferroic subsystems and hence it is a material-specific property. In BiMnO3 and YCrO3 one observesan interaction between the ferromagnetic and the ferro-electric subsystem, whereas YMnO3 and BiFeO3 offer a

http://www.epj.org

Eur. Phys. J. B (2013) 86: 201 Page 5 of 10

Fig. 2. Temperature dependence of the phonon energy

for TbMn2O5 with K0 = −7 K, R(m)0 = −0.35 cm−1,R

(f)0 = −0.45 cm−1, F (m)0 = F (f)0 = 4 cm−1. The bare phonon

frequency is ω0 = 96.3 cm−1. The mode shows a softening

behavior for negative spin-phonon coupling R.

coupling between an antiferromagnetic and ferroelectricsystems [1]. As demonstrated in reference [36] BiCrO3films again exhibit a multiferroic coupling between an-tiferroelectricity and antiferromagnetism (or weak ferro-magnetism). Hexagonal YMnO3 shows likewise a couplingbetween ferroelectric and magnetic ordering [37]. Themeasured temperature dependence of the phonon mode inTbMn2O5 reveals a softening of that mode [4] which sug-gests a negative coupling parameter R < 0. In the oppositecase R > 0 the phonon mode becomes harder. Followingthis line we have analyzed the influence of the magneticand the ferroelectric subsystem as well as the ME cou-pling on the phonon excitation energy. In particular, letus emphasize the anomalies around the ferroelectric andmagnetic phase transitions. The numerical calculations forthe optical phonon energy is based on the following modelparameters which are relevant for TbMn2O5: TC = 37 K,TN = 43 K, J ′0 = 75 K, K0 = −7 K, s = 1/2, S = 2,ω0 = 96.3 cm−1. The anharmonic interaction parame-ters are estimated to be A = −1 cm−1 , B = 0.5 cm−1,F

(m)0 = F

(f)0 = 4 cm

−1. Notice, that we have used theunit K (Kelvin) for all energy-related quantities, i.e. theBoltzmann constant kB is set to 1. As mentioned abovethe spin-phonon interaction constant R can be positive ornegative leading to hardening or softening of the phononmode, respectively. The experimental data presented forTbMn2O5 show a softening of the phonon mode which isconsistent with R < 0. A softening of the phonon modeat zero-wave vector ω(q = 0, T ) with increasing temper-ature is also observed for DyMn2O5, BiMn2O5 [12] andEuMn2O5 [13]. The frequency of the mode versus the tem-perature is shown in Figure 2. The harmonic phonon en-ergy ω0 introduced in equation (3) is renormalized due tothe anharmonic spin-phonon and phonon-phonon inter-action terms and becomes temperature dependent. Thephonon excitation is obtained in equation (13). The spin-phonon interactions are dominant at low temperatures,

Fig. 3. Temperature dependence of the phonon energy

for TbMn2O5 with K0 = −7 K, R(m)0 = 0.70 cm−1,R

(f)0 = 0.90 cm

−1, F (m)0 = F(f)0 = 4 cm

−1. The mode offershardening for positive spin-phonon coupling R.

whereas at higher temperatures, above TN , there remainsonly the anharmonic phonon-phonon coupling. Totally thephonon energy decreases slightly. The dispersion relationin Figure 2 shows two kinks at the ferroelectric phasetransition temperature TC = 37 K and at the magnetictransition TN = 43 K. The first anomaly occurs whenthe ferroelectric order disappears. As above TC the MEcoupling K0 is irrelevant, the kink in the dispersion re-lation is a direct consequence of the ME coupling. Thesecond kink reflects the influence of the magnetic orderon the phonon energy. The frequency shift below TC ismainly originated by the anharmonic spin-phonon inter-actions introduced in equation (7). Let us remind that thespin-phonon coupling comes from the exchange interactionJij = J(ri − rj) (and J̃ij , compare Eq. (2), as well as J ′ij ,see Eq. (3)). Assuming the interaction depends on the ac-tual lattice coordinates they can be expanded with respectto the phonon displacements ui and uj . Hence the spin-phonon coupling parameters are determined by the firstand second order derivatives of the related exchange cou-pling Jij . The results shown in Figure 2 are in agreementwith experimental observations in different RMn2O5 com-pounds [10–15]. In TbMn2O5 a small increase will be ob-served in the phonon curve between TC and TN [11]. Sucha behavior can be reproduced in our approach if a posi-tive coupling R(m) > 0 is assumed within the temperatureinterval TC ≤ T ≤ TN . A hardening of the phonon modeis reported in HoMn2O5 [10], which is consistent with ourstudies adopting positive spin-phonon couplings R(f) > 0and R(m) > 0 (see Fig. 3). The mentioned anomalies at TCand TN are also reproduced. Let us make a technical re-mark. Although the geometry of the lattice of RMn2O5 isquite complex, we have used a bcc structure for simplicity.The reason is that in our approach the geometrical struc-ture is primarily included by the wave vector dependenceof the coupling parameter. Moreover numerical simula-tions indicate the results will be not changed drasticallyusing an orthorhombic structure.

http://www.epj.org

Page 6 of 10 Eur. Phys. J. B (2013) 86: 201

Fig. 4. Temperature dependence of the phonon energy fordifferent magnetoelectric coupling K0 = −3 K (black/lowercurve); K0 = −7 K (red/middle curve); K0 = −10 K(blue/upper curve).

The ME coupling parameter is introduced in equa-tion (6). The effect of the magnetoelectric coupling con-stant K0 between the magnetic and electric subsystemsis shown in Figure 4. As shown, the phonon energy de-pends on the ME coupling K0 in a significant manner.The kink at TC is shifted to higher temperatures, i.e.TC is increased for stronger ME coupling. The magneticphase transition temperature TN remains unchanged bythe ME coupling K0 [38]. There is also an experimentalevidence for different coupling strengths and even for dif-ferent coupling mechanisms between the magnetic and theferroelectric subsystems. So the replacement of Y atomsby magnetic Ho atoms in YMnO3 leads to a stronger sup-pression of the thermal conductivity [39]. Another study inreference [40] predicts that the polarization in orthorhom-bic HoMnO3 with biquadratic ME coupling would be en-hanced by up to two orders of magnitude with respect tothat in orthorhombic TbMnO3. In that compound the MEinteraction term is linear in the electrical dipole moments.

The phonon energy is very sensitive to the anhar-monic spin-phonon interaction constants R(m) and R(f).The situation is depicted in Figure 5. The phonon en-ergy increases strongly with increasing R(m) below TN(see Fig. 5a), or with R(f) below TC , in the multiferroicphase, compare Figure 5b. Synchrotron X-ray studies givesome evidence of lattice modulation in the ferroelectricphase of YMn2O5 [41], however the atomic displacementsseem to be extremely small. Moreover, several phonons inTbMn2O5 exhibit explicit correlations to the ferroelectricproperties of these materials [11].

Until now the calculations are performed for zero ex-ternal fields. However the magnetization and polarizationin multiferroic substances are affected by such externalelectric or magnetic fields. The influence of an externalmagnetic field h = gμBB (see Eq. (2)), on the polariza-tion P is demonstrated experimentally in TmMn2O5 [42],in MnWO4 [43] and in orthorhombic YMnO3 [44]. Thestrong ME coupling in these materials can be illustratedby studying the influence of an applied magnetic field

(a) ω(q = 0, T, R(m))

(b) ω(q = 0, T, R(f))

Fig. 5. Temperature dependence of the phonon en-ergy for different magnetic spin-phonon couplings: mag-netoelectric couplings: (a): R(m) = −0.45 K (black/lowercurve); R(m) = −0.70 K (red/middle curve); R(m) = −0.95 K(blue/upper curve); (b): R(f) = −0.45 K (black/lowercurve); R(f) = −0.90 K (red/middle curve); R(f) = −1.95 K(blue/upper curve).

on the phonon spectrum. The result of our calculationis shown in Figure 6. The phonon energy ω increaseswith increasing magnetic field h. Significant changes of thephonon energy due to magnetic fields are reported for or-thorhombic TbMnO3 [45] and for hexagonal HoMnO3 [46].Unfortunately experimental data for RMn2O5 are stillmissing. Both phase transitions are indicated by the cor-responding kinks in the phonon energy. For higher tem-peratures beyond 45K we find no peculiarities. Figure 6shows that under the influence of a magnetic field thephase transition temperatures TC and TN are altered,where the effect is more pronounced for TC as a furtherindication of the magnetoelectric behavior. So the phasetransition temperature TC is enhanced by about 4–5 K.These changes are very sensitive to the model parame-ters. The increase of TC with the magnetic field h anda vanishing kink at TN is observed in reference [47]. Thedependence of the phonon frequency on the magnetic fieldis depicted in Figure 7. As expected the phonon excitation

http://www.epj.org

Eur. Phys. J. B (2013) 86: 201 Page 7 of 10

Fig. 6. Phonon energy versus temperature for different exter-nal magnetic field: h = 0 (black/lower curve), h = 1 K (readcurve), h = 5 K (blue curve), h = 10 K (green curve).

Fig. 7. Phonon frequency versus the magnetic field at fixedtemperature, T = 30 K.

energy for wave vector �q = is enhanced when the field is in-creased. As remarked in reference [48] the effect of a mag-netic field or hydrostatic pressure are strongly dependenton the R atom in the RMn2O5 systems. The compressionof these materials is supposed to be equivalent to applyinga magnetic field [49]. In references [9,50] it has been shownthat the coupling between the magnetic orders and the di-electric properties in multiferroic RMn2O5 (R = Ho, Dy,Tb) is mediated by the lattice distortion. The mentionedcompounds have different ionic sizes. Hence the thermalexpansion data display specific differences of the latticestrains. In case R stands for Ho and Dy then a contractionappears along the c-axis, while one observes an expansionalong the a- and b-axis at TC . The opposite behavior isobserved for Tb. The origin of this fact is due to the dif-ferent spin modulations along the c-axis. The spin-latticecoupling leads to different stress in the lattice.

As demonstrated in the previous section, the Green’sfunction technique allows also to calculate macroscopicquantities like the magnetization M = 〈σM 〉 and the

(a) M(T )

(b) ω(T )

Fig. 8. Temperature dependence of the (a) magnetizationand (b) the phonon energy for changed spin-phonon cou-pling: R(m) = 0.95 cm−1 (blue), R(m) = 0.70 cm−1 (red),R(m) = 0.70 cm−1 (green).

influence of the phonon degrees of freedom on M . Theresults are shown in Figure 8. In Figure 8a one finds theinfluence of the spin-phonon coupling on the magnetiza-tion M(T ). Increasing the spin-phonon coupling parame-ter R(m) the magnetization decreases. As expected thenthe magnetic transition temperature TN is lowered. InFigure 8b the phonon energy is represented as functionof the temperature with different spin-phonon couplings.The phonon excitation energy increases strongly when thespin-phonon coupling R(m) is enhanced. In the same man-ner the magnetization is slightly decreased. Obviously thelattice degrees of freedom affect the magnetization andthe phonon spectrum. Thus, for rare-earth R = Tb thesystem offers a tensile strain. The ordered Mn magneticmoments are suppressed at high pressure. The kink at TCis due to the ME coupling and can be observed again.Our results are in agreement with the experimental datafor orthorhombic BiMnO3 and LuFe2O4 [51,52]. There areno data available for the influence of pressure on the mag-netic properties of MF RMn2O5 systems. For the polariza-tion the influence of the coupling is not very pronounced.

http://www.epj.org

Page 8 of 10 Eur. Phys. J. B (2013) 86: 201

The effect is clearly stronger for the magnetization thanfor the polarization.

4 Conclusion

By using a realistic quantum model we have studiedthe multiferroic phases in RMn2O5 focusing on crucialroles of the phonon spectrum. The ferroelectric subsystemis characterized by both displacive-like couplings mani-fested by lattice distortions and order-disorder-like ele-ments expressed in terms of pseudo-spins. The magneticsubsystem comprises the nearest neighbor ferromagneticcoupling and next nearest neighbor antiferromagnetic in-teraction. The interplay between these two couplings leadsto collinear magnetic structures. As the consequence of thedetailed characterization of both subsystems the couplingbetween the ferroic phases is not only given by a directmagnetoelectric coupling but additionally by spin-phononcouplings. To be specific we have introduced two kinds ofspin-phonon couplings, namely the magnetic dominatedones, R(m), F (m) and the ferroelectric dominated onescharacterized by the parameters R(f), F (f). The systemundergoes a magnetic phase transition at the tempera-ture TN with a finite sublattice magnetization. A furtherlowering of the temperature offers a second phase tran-sition at TC < TN , below that the system develops aferroelectric phase with a nonzero polarization. The pa-per is focused on the influence of the various couplingson the phonon spectrum of the multiferroic system. Suchcollective properties can be elucidated by a Green’s func-tion technique. Due to the magnetoelectric coupling theharmonic phonon mode is renormalized and becomes tem-perature dependent. The effect is the stronger the higherthe magnetoelectric coupling is. Both phase transitions aremanifested by a kink in the phonon spectrum at the cor-responding transition temperatures. The behavior of themodified phonon mode is strongly influenced by the spin-phonon coupling. For positive couplings the mode becomesharder whereas for negative coupling the mode becomessofter. In the latter case we show that the phonon spec-trum is very sensitive to a change of the magnetic spin-phonon coupling, while for different couplings the kink inthe dispersion relation is maintained. In the same man-ner the ferroelectric pseudo-spin-phonon coupling, rele-vant for T < TC , alters the phonon spectrum. The phononmode can be also influenced by an external magnetic field,i.e. the phonon frequency can be triggered by a magneticfield. The Green’s function method allows to figure outthe behavior of macroscopic quantities as magnetizationor polarization. We show that the magnetization is in-fluenced by the spin-phonon coupling in such a mannerthat an increase of the coupling leads to a decrease of themagnetization.

One of us (S.G.B.) acknowledges support by the InternationalMax Planck Research School for Science and Technology ofNanostructures in Halle. J.M.W. is also indebted to the MaxPlanck Institute of Microstructure Physics in Halle for financialsupport.

Appendix A: Green’s functions for the Isingmodel in a transverse field

The Hamiltonian of the ferroelectric subsystem and mag-netoelectric coupling reads after rotation

Htim + Hmf = −Ω∑

i

[(1 − 2ρi) sin ν +

(b†i + bi

)cos ν)

]

− 18

∑

i,j

J ′i,j

( (b†ib

†j + b

†ibj + bib

†j + bibj

)

× sin2 ν −(b†j + bj − 2ρib†j − 2ρibj

)

× sin ν cos ν −(b†i + bi − 2b†iρj − biρj

)

× sin ν cos ν+(1 − 2ρj−2ρi+4ρiρj) cos2 ν)

+12

∑

i,j

KijSzi S

zj cosν−

∑

i,j,l

KijlρiSzj S

zl cosν

− 12

∑

i,j,l

Kijl

(b†i +bi

)Szj S

zl sin ν. (A.1)

The matrix Green’s function obeys(

Gf11q Gf12q

Gf21q Gf22q

)=

(2〈σP 〉 0

0 −2〈σP 〉

)

+

(εq

11 εq12

εq21 εq

22

)(Gf11q G

f12q

Gf21q Gf22q

),

with

εq11 = 2Ω sin ν + J ′0〈σP 〉 cos2 ν

− K0 cos ν〈σM 〉2 − 12J′q〈σP 〉 sin2 ν

+ F (f)〈Q〉 cos ν + 12R(f)〈Q2〉 cos ν

εq12 = −1

2J ′q〈σP 〉 sin2 ν

εq21 = −εq12 εq22 = −εq11. (A.2)

The solution for the Green’s functions are

Gf11q =〈σP 〉

[E + εq11

](E − εq11

)(E + εq11

)+

(εq12

)2

Gf12q =〈σP 〉εq12(

E − εq11)(

E + εq11) − (εq12

)2

Gf21q =〈σP 〉εq12(

E − εq11)(

E + εq11) − (εq12

)2

Gf22q = −〈σP 〉

[E − εq11

](E − εq11

)(E + εq11

) − (εq12)2 .

(A.3)

http://www.epj.org

Eur. Phys. J. B (2013) 86: 201 Page 9 of 10

The most relevant function Gf11 is given in the text asequation (10) where the coefficients are given by:

q =εq

11

2Ef(q)+

12, ′q =

εq11

2Ef(q)− 1

2. (A.4)

Here, we have defined Ef(q) =√

(εq11)2 − (εq12)2.

Appendix B: Phonon Green’s function

As mentioned in the text, the phonon Green’s functionsare also given by a matrix defined in equation (11)

(〈〈aq; a†q

〉〉 〈〈aq; aq〉〉〈〈

a†q; a†q〉〉 〈〈

a†q; aq〉〉

)≡

(G11q G

12q

G21q G22q

). (B.1)

These set fulfills the equation of motion given by equa-tion (12) with the following coefficients:

�q11 = ω0q + B̄q〈Q〉 +

12Āq〈Q2〉 − 12 R̄

(m)(q)〈σM 〉

− 12R̄(f)(q)〈σP 〉 cos ν

�q12 = B̄q〈Q〉 + 12 Āq〈Q

2〉 − 12R̄(m)(q)〈σM 〉

− 12R̄(f)(q)〈σP 〉 cos ν

�q11 − �q12 = ω0q , �q21 = −�q12, �q22 = −�q11.

(B.2)

Furthermore we have defined B̄(q) = B(q)/(2w0q) andĀ(q) = A(q)/(2w0q) and correspondingly R̄(m) and R̄(f).From here we conclude the phonon dispersion relation as:

ω(q) =√

(�q11)2 − (�q12)2. (B.3)The final result is given in the text as equation (13).

References

1. N.A. Hill, J. Phys. Chem. B 104, 6694 (2000)2. T. Kimura, S. Kawamoto, I. Yamada, M. Azuma,

M. Takano, Y. Tokura, Phys. Rev. B 67, 180401 (2003)3. T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,

Y. Tokura, Nature 426, 55 (2003)4. N. Hur, S. Park, P.A. Sharma, J. Ahn, S. Guha, S.-W.

Cheong, Nature 429, 392 (2004)5. S.-W. Cheong, M. Mostovoy, Nat. Mater. 6, 13 (2007)6. D. Khomskii, Physics 2, 20 (2009)7. K.F. Wang, J.M. Liu, Z.F. Ren, Adv. Phys. 58, 321 (2009)8. C. Wang, G.-C. Guo, L. He, Phys. Rev. B 77, 134113

(2008)9. C.R. dela Cruz, R. Yen, B. Lorenz, S. Park, S.-W. Cheong,

M.M. Gospodinov, W. Ratcliff, J.W. Lynn, C.W. Chu, J.Appl. Phys. 99, 08R103 (2006)

10. B. Mihailova, M.M. Gospodinov, B. Güttler, F. Yen, A.P.Litvinchuk, M.N. Iliev, Phys. Rev. B 71, 172301 (2005)

11. R. Valdés Aguilar, A.B. Sushkov, S. Park, S.-W. Cheong,H.D. Drew, Phys. Rev. B 74, 184404 (2006)

12. A.F. Garcia-Flores, E. Granado, H. Martinho, R.R.Urbano, C. Rettori, E.I. Golovenchits, V.A. Sanina, S.B.Oseroff, S. Park, S.-W. Cheong, Phys. Rev. B 73, 104411(2006)

13. E. Garcia-Flores, A.F. Granado, H. Martinho, C. Rettori,E.I. Golovenchits, V.A. Sanina, S.B. Oseroff, S. Park,S.-W. Cheong, J. Appl. Phys. 101, 09M106 (2007)

14. C.L. Lu, J. Fan, H.M. Liu, K. Xia, K.F. Wang, P.W. Wang,Q.Y. He, D.P. Yu, J.-M. Liu, Appl. Phys. A 96, 991 (2009)

15. J. Cao, L.I. Vergara, J.L. Musfeldt, A.P. Litvinchuk, Y.J.Wang, S. Park, S.-W. Cheong, Phys. Rev. B 78, 064307(2008)

16. L. Martin-Carron, A. de Andres, M.J. Martinez-Lopez,M.T. Casais, J.A. Alonso, J. Alloys Compd. 323–324, 494(2001)

17. A.B. Sushkov, R.V. Aguilar, S. Park, S.-W. Cheong, H.D.Drew, Phys. Rev. Lett. 98, 027202 (2007)

18. K. Cao, G.-C. Guo, L. He, J. Phys.: Condens. Matter 24,206001 (2012)

19. A.P. Litvinchuk, J. Magn. Magn. Mater. 321, 2373 (2009)20. C. Jia, J. Berakdar, Europhys. Lett. 85, 57004 (2009)21. C. Jia, J. Berakdar, Phys. Stat. Sol. B 247, 662 (2010)22. M. Mochizuki, N. Furukawa, N. Nagaosa, Phys. Rev. B 84,

144409 (2011)23. S. Kovachev, J.M. Wesselinowa, J. Phys.: Condens. Matter

22, 255901 (2010)24. R. Blinc, B. Žekš, Adv. Phys. 21, 693 (1972)25. H. Katsura, N. Nagaosa, A.V. Balatsky, Phys. Rev. Lett.

95, 057205 (2005)26. C. Jia, S. Onoda, N. Nagaosa, J.H. Han, Phys. Rev. B 74,

224444 (2006)27. L.C. Chapon, G.R. Blake, M.J. Gutmann, S. Park, N. Hur,

P.G. Radaelli, S.-W. Cheong, Phys. Rev. Lett. 93, 177402(2004)

28. L.C. Chapon, P.G. Radaelli, G.R. Blake, S. Park, S.-W.Cheong, Phys. Rev. Lett. 96, 097601 (2006)

29. G.R. Blake, L.C. Chapon, P.G. Radaelli, S. Park, N. Hur,S.-W. Cheong, J. Rodriguez-Carvajal, Phys. Rev. B 71,214402 (2005)

30. S.V. Tjablikov, Methods in the Quantum Theory of Magne-tism (Plenum Press, New York, 1967)

31. J.-H. Kim, M.A. van der Vegte, A. Scaramucci,S. Artyukhin, J.-H. Chung, S. Park, S.-W. Cheong,M. Mostovoy, S.-H. Lee, Phys. Rev. Lett. 107, 097401(2011)

32. M. Mostovoy, A. Scaramucci, N.A. Spaldin, K.T. Delaney,Phys. Rev. Lett. 105, 087202 (2010)

33. J.M. Wesselinowa, S. Kovachev, Phys. Rev. B 75, 045411(2007)

34. I. Apostolova, A.T. Apostolov, J.M. Wesselinowa, J. Phys.:Condens. Matter 21, 036002 (2009)

35. W. Nolting, Fundamentals of Many-body Physics(Springer-Verlag, Berlin, 2009)

36. N. Lee, M. Varela, H.M. Christen, Appl. Phys. Lett. 89,162904 (2006)

37. Z.J. Huang, Y. Cao, Y.Y. Sun, Y.Y. Xue, C.W. Chu, Phys.Rev. B 56, 2623 (1997)

38. S.G. Bahoosh, J.M. Wesselinowa, S. Trimper, submittedto J. Phys. D (2012)

39. P.A. Sharma, J.S. Ahn, N. Hur, S. Park, S.B. Kim, S. Lee,J.-G. Park, S. Guha, S.-W. Cheong, Phys. Rev. Lett. 93,177202 (2004)

http://www.epj.orghttp://dx.doi.org/10.1103/PhysRevB.67.180401http://dx.doi.org/10.1038/nature02018http://dx.doi.org/10.1038/nmat1804http://dx.doi.org/10.1103/Physics.2.20http://dx.doi.org/10.1080/00018730902920554http://dx.doi.org/10.1103/PhysRevB.77.134113http://dx.doi.org/10.1103/PhysRevB.71.172301http://dx.doi.org/10.1103/PhysRevB.74.184404http://dx.doi.org/10.1103/PhysRevB.73.104411http://dx.doi.org/10.1103/PhysRevB.78.064307http://dx.doi.org/10.1103/PhysRevLett.98.027202http://dx.doi.org/10.1209/0295-5075/85/57004http://dx.doi.org/10.1080/00018737200101348http://dx.doi.org/10.1103/PhysRevLett.95.057205http://dx.doi.org/10.1103/PhysRevB.74.224444http://dx.doi.org/10.1103/PhysRevLett.93.177402http://dx.doi.org/10.1103/PhysRevLett.96.097601http://dx.doi.org/10.1103/PhysRevB.71.214402http://dx.doi.org/10.1103/PhysRevLett.107.097401http://dx.doi.org/10.1103/PhysRevLett.105.087202http://dx.doi.org/10.1103/PhysRevB.56.2623http://dx.doi.org/10.1103/PhysRevLett.93.177202

Page 10 of 10 Eur. Phys. J. B (2013) 86: 201

40. I.A. Sergienko, C. Şen, E. Dagotto, Phys. Rev. Lett. 97,227204 (2006)

41. I. Kagomiya, S. Matsumoto, K. Kohn, Y. Fukuda,T. Shoubu, H. Kimura, Y. Noda, N. Ikeda, Ferroelectrics286, 167 (2003)

42. M. Fukunaga, Y. Sakamoto, H. Kimura, Y. Noda, N. Abe,K. Taniguchi, T. Arima, S. Wakimoto, M. Takeda,K. Kakurai, K. Kohn, Phys. Rev. Lett. 103, 077204 (2009)

43. R.P. Chaudhury, B. Lorenz, Y.Q. Wang, Y.Y. Sun, C.W.Chu, Phys. Rev. B 77, 104406 (2008)

44. I. Fina, L. Fabrega, X. Marti, F. Sanchez, J. Fontcuberta,Appl. Phys. Lett. 97, 232905 (2010)

45. H. Barath, M. Kim, S.L. Cooper, P. Abbamonte,E. Fradkin, I. Mahns, M. Rübhausen, N. Aliouane, D.N.Argyriou, Phys. Rev. B 78, 134407 (2008)

46. S.-W. Cheong, in Proceedings of the international con-ference on Low Energy Electrodynamics in Solids, Tallinn,2006, p. 62

47. S.M. Feng, Y.S. Chai, J.L. Zhu, N. Manivannan, Y.S. Oh,L.J. Wang, Y.S. Yang, C.Q. Jin, K.H. Kim, New J. Phys.12, 0730006 (2010)

48. Y. Noda, H. Kimura, M. Fukunaga, S. Kobayashi,I. Kagomiya, K. Kohn, J. Phys.: Condens. Matter 20,434206 (2008)

49. C.R. dela Cruz, B. Lorenz, Y.Y. Sun, Y. Wang, S. Park,S.-W. Cheong, M.M. Gospodinov, C.W. Chu, Phys. Rev.B 76, 174106 (2007)

50. C.R. dela Cruz, F. Yen, B. Lorenz, M.M. Gospodinov,C.W. Chu, W. Ratcliff, J.W. Lynn, S. Park, S.-W. Cheong,Phys. Rev. B 73, 100406 (2006)

51. C.C. Chou, S. Taran, J.L. Her, C.P. Sun, C.L. Huang,H. Sakurai, A.A. Belik, E. Takayama-Muromachi, H.D.Yang, Phys. Rev. B 78, 092404 (2008)

52. X. Shen, C.H. Xu, C.H. Li, Y. Zhang, Q. Zhao, H.X. Yang,Y. Sun, J.Q. Li, C.Q. Jin, R.C. Yu, Appl. Phys. Lett. 96,102909 (2010)

http://www.epj.orghttp://dx.doi.org/10.1103/PhysRevLett.97.227204http://dx.doi.org/10.1103/PhysRevLett.103.077204http://dx.doi.org/10.1103/PhysRevB.77.104406http://dx.doi.org/10.1103/PhysRevB.78.134407http://dx.doi.org/10.1103/PhysRevB.76.174106http://dx.doi.org/10.1103/PhysRevB.73.100406http://dx.doi.org/10.1103/PhysRevB.78.092404

IntroductionThe modelResults and discussionConclusionAppendix A: Green's functions for the Ising model in a transverse fieldAppendix B: Phonon Green's functionReferences