chenglong jia and jamal berakdar- coupled spin–phonon excitations in helical multiferroics

8/3/2019 Chenglong Jia and Jamal Berakdar- Coupled spinphonon excitations in helical multiferroics

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Coupled spinphonon excitations in

helical multiferroicsChenglong Jia* and Jamal Berakdar

Institut fur Physik, Martin Luther Universitat Halle-Wittenberg, Heinrich-Damerow-Strae 4, 06120 Halle, Germany

Received 23 June 2009, revised 24 September 2009, accepted 24 September 2009

Published online 9 February 2010

PACS 71.70.Ej, 71.70.Gm, 75.85.t, 77.80.e

* Corresponding author: e-mail [email protected], Phone: 49 345 5528528, Fax: 49 345 5527393

Both the DzyaloshiskiiMoriya interaction and the exchange

striction are shown to affect dynamically the magnetoelectric

excitations in the helical multiferroics. The exchange striction

results in a biquadratic interaction between the spins and the

transverse phonons, giving rise to quantum fluctuations of the

ferroelectric polarization P. This leads to low-lying phonon

modes that are perpendicular to P and to the helical spins at

small wave vector but are parallel to P at a wave vector close to

the magnetic modulation vector. For spin-1/2 helimagnet, the

local polarization can be completely reversed by the spin

fluctuation, and so does the direction of the on-site spin

chirality, which allows for a finite differential scattering

intensity of polarized neutrons from a cycloidal magnet.

2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Details of the coupling mechanismsof the magnetic and the ferroelectric (FE) order in multi-

ferroics is currently under active research. This is due to thefundamental physics involved and to promising technologi-

cal applications [1]. Our focus here is the helical multi-

ferroics such as perovskiteRMnO3 with R Tb, Dy, Gd, andEu1xYx [2]. The experimental finding is that RMnO3 has an

incommensurate spiral magnetic order and a finite FE

polarization. The driving mechanisms of this ordering is an

interplay between the exchange interaction and theDzyaloshiskiiMoriya (DM) interaction. Specifically, the

spinorbit coupling with a strength a related to the d(p)-

orbitals of the magnetic (oxygen) ions results in the FE

polarization [3, 4] P a^eij Si Sj. ^eij is a unit vectorconnecting the sites i andj. Generally, it is to beexpected thatthe magnetoelectric coupling will affect not only thematerial

static properties but also the dynamical response. Based on

the above spin-current model, the dynamical properties of

DM interaction were studied in Refs. [57]. A novel

magnonphonon excitation so-called electromagnon, was

theoretically predicted. When the spiral plane rotates with

respected to the axis of the helical wave vector, so does the

induced electric polarization, which couples the magneticexcitation to the electric field E of the radiation in the

direction perpendicular to the spin spiral plane [5].

Experimental observations in RMnO3 [8] and

Eu0.75Y0.25MnO3 [9] seem to be consistent with this finding.However, a detailed study of the terahertz spectrum of

Eu1xYxMnO3 [10] revealed that infrared absorption alongthe spontaneous polarization direction is also possible,

which is not explained by the theory mentioned above. This

violation suggests that the static and the dynamic magneto-

electric coupling may be different [11]. We carried out a

detailed investigations of the dynamical properties of the

multiferroics and find that both, the DM interaction and the

(super-)exchange striction play an essential role and need tobe taken into account.

2 Theoretical model We consider a one-dimen-

sional spin chain along the z-axis with a frustrated spininteraction. An effective model that captures the spinphonon coupling [5, 12] corresponds to the Hamiltonian

H Hs HDM HpHs

Phijinn

J1ri rjSi Sj

Phlminnn

J2rl rmSl Sm

HDM lP

i

ui ez Si Si1

Hp k2Piu

2i 12MPi

P2i

; (1)

Phys. Status Solidi B, 13 (2010) / DOI 10.1002/pssb.200983028 p s sbasic solid state physics

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where the notation hijinn indicates that i and j are nearest-neighbors (nn), and hlminnn corresponds to l and m beingnext-nearest-neighboring (nnn). The competition betweenthe nn ferromagnetic interaction (J1 < 0) and the nnn

antiferromagnetic interaction (J2 > 0) leads to magneticfrustration and realizes a spiral spin ordering with the wavevector cosQ J1=4J2 [1315]. Hp describes opticalphonons. The spinphonon interaction HDM originates

from a spinorbital coupling and breaks the inversion

symmetry along the chain. Minimizing the energy yields thecondition of the atomic displacement and the local spin

configuration

ui lk

ez Si Si1: (2)

Particularly, if the zx helical spins are aligned along thechain, i.e., Si Ssin iQ; 0; cos iQ, a uniform electricpolarization P along the x direction is induced by thecondensation of the transverse optical phonons, P eu0 elS2=ksin Qex with a Born charge e. Generally, uxcannot be softened through the hybridization between the

transverse optical phonons and the magnons because of

k=M ) JS. The spontaneous FE polarization Px

is frozenat eux0 in the FE phase. However, after accounting for thesuper-exchange striction, we have transverse acoustic

phonons, which induces the fluctuation of the polarization

hybridized with the spin bosons and soften thus the

transverse phonon behavior.Considering small atomic displacements perpendicular

to the chain, u?i ez 0, the exchange energy Jfalls off as apower law with the separation of the magnetic ions

J1;2jri rjj % J1;2 1 g1;2

2u?i u?j 2

h i(3)

where g is in the range of 614 [16]. The emerging

transverse acoustic phonon mode is coupled to the spinswith the bi-quadratic interaction $ u?i u?j 2Si Sj.This dynamical coupling does not contribute any additional

static electric polarization but induces the fluctuation of

the electric dipole moment due to the low-frequency

excitation modes of transverse acoustic phonon. We write

explicitly the atomic displacements into two parts: (i) the

statical part ui ux0; 0; 0 and (ii) the dynamical partdui du

xi ; du

y

i ; 0. Retaining terms up to the second orderin the quantum fluctuation, the spin-current model delivers

the following coupling terms:

~HDM lScosQP

i

duxi ~Sxi1 ~Sxi lSP

i

duyi ~Syi cosQi1 ~Syi1 cosQi

(4)

in the rotated spin frame: Sxi ~Sxi cos iQ ~Szi sin iQ,S

yi ~Syi , and Szi ~Sxi sin iQ ~Szi cos iQ.

3 Results and analysis In spin-1/2 helical multi-ferroics, such as LiCu2O2 [14], the spin fluctuations may

spontaneously reverse the local spin. Defining the vector of

spin chirality as the average of the outer product of two

adjacent spins ci si si1=jsi si1j, in accord with thespin current model the direction of local FE polarization is

determined by the on-site spin chirality [14]. The dynamical

DM interaction in Eq. (4) yields the coupling term betweenthe spin and the spin chirality in the spin-1/2 multiferroics,Pi c

xi sxi1 sxi

Pi s

xi cxi1 cxi , which indicates that

when the spin at site i is flipped, si ! si, the direction ofspin-chirality ci and ci1 are also reversed. Assuming allspins point along their corresponding classical directions in

the ground state of the spin-1/2 helical magnet as inNaCu2O2, where the J1 J2 spin model provides a gooddescription of the helix state [15]. So the spin interaction can

be ferromagnetically given as JsQsi sj where Q is takenas the pitch angle along the chain. An effective model that

describes the interplay between the helical spin and spin-

chirality has the form

Hsc Pi;j

Jssi sj Jcci cjgP

i

sxi cxi1 cxi :(5)

TheHilbert space canbe considered as thetensorproduct

space jii ! jszi is jczi ic: Now if the spin at site i is flipped,the spin and spin-chirality excitations are mixed due to the

spinphonon coupling. The expected value of spin-chiralityis given by

hci 1 hsi; (6)

which is less than one. The experimental data for a finitedifferential scattering intensity of polarized neutrons from

LiCu2O2 [14] suggests hci % 0:3 which is consistent withthe estimated value hci 0:44 based on the orderedmoment, 0.56mB per magnetic copper site [15].

For RMnO3, the helical spin ordering occurs, corre-

sponding to thecondensationof the spin bosons. By using thestandard linear-spin-wave approximation, a dynamical

magnonphonon interaction reads

~HDM lScosQP

q

duxq~Sxqcos q 1

lSX

q

duyq~SyqQeiQ eiqQ

2:

(7)

duyq is hybridized with the spin at q Q (optical magnons),but duxq is coupled to

~Sx at q (acoustical magnons). The

polarization correlation functions are given as

( duxqjduxq )v2 v2s

Mv4 v2v2p v2s v2pv2s v2sp;

( duyqjduyq )1

M v2 v2p l2S3

2M

Pq0qQ

Gsq0" # ;

(8)

2 C. Jia and J. Berakdar: Coupled spinphonon excitations in helical multiferroics

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where vp is the frequency for the transverse phonon,

vs(q) is the energy dispersion of the spin excitation,

v0spq 2Aq 2Bql2S3 cos2Q1 cos q=k1=2,and Gsq QAq Q2BqQ1 cosq 2Q=v

2

vsq Q withAq J2 cos 2Q 1

21 cos 2Qcos 2q

J2 cos 2Q 121 cos 2Qcos 2q

l2

S2 sin2 Q

2k2 cos q;

(9)

Bq J14

1 cosQcos q J24

1 cos 2Qcos 2q

l2

S2 sin2 Q

4kcos q:

(10)

At small wave vectors, q $ 0 and vp %ffiffiffiffiffiffiffiffiffi

k=Mp

, the TA

phonon is decoupled from the spins. The antisymmetric DM

interaction dominates over the spinphonon coupling. duy0 is

coupled via ~SyQ ~SyQ to the rotation of the spin plane andthe direction of the polarization along the chain. However, ata wave vector close to the magnetic modulation vector, i.e.,

q $ Q, both the symmetric and antisymmetric magneto-electric interaction respond to the fluctuations of the

polarization. Especially, in the direction parallel to the

FE polarization P, there is a low-frequency range around

vx

ffivs

Q

where ux couples resonantly to light. Introdu-

cing an easy-plane spin anisotropy DSy2

into the spinsystem, we observe nearly the same low-frequency behavior

of the polarization correlation functions vx %ffiffiffiffiffiffiffiffiffi

JSDp % vy.

These conclusions are also qualitatively consistent with

experiment observations for Eu1xYxMnO3 [10].

4 Summary In conclusion, we studied the origin of themagnetoelectric dynamics in the helical multiferroics. At a

small wave vector, the DM interaction determines the low-

frequency behavior of the phonons. For a wave vector close

to that of the magnetically modulated structure, the exchange

striction induces fluctuations in the FE polarization, and

additional low-lying mode parallel to the FE polarization

emerges. Due to the dynamical DM interaction, the spin-chirality is strongly coupled to the spin fluctuation which

implies a large quantum fluctuation of the spin-chirality in

the ordered spin-1/2 system and results in a finite scattering

intensity of polarized neutrons from a cycloidal helimagnet.

Acknowledgements This work is supported by the GermanScience Foundation DFG through SFB762-B7- functionality of

oxide interfaces.

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chenglong jia and jamal berakdar- coupled spin–phonon excitations in helical multiferroics

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