chenglong jia and jamal berakdar- coupled spin–phonon excitations in helical multiferroics
TRANSCRIPT
-
8/3/2019 Chenglong Jia and Jamal Berakdar- Coupled spinphonon excitations in helical multiferroics
1/3
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
b
status
solidi
www.pss-b.comphysica
2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
-
8/3/2019 Chenglong Jia and Jamal Berakdar- Coupled spinphonon excitations in helical multiferroics
2/3
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
physica
ssps
tatus
solidi b
2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com
-
8/3/2019 Chenglong Jia and Jamal Berakdar- Coupled spinphonon excitations in helical multiferroics
3/3
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.
References
[1] Y. Tokura, Science 312, 1481 (2006). S.-W. Cheong andM. Nostovoy, Nature Mater. 6, 13 (2007).
[2] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and
Y. Tokura, Nature (London) 426, 55 (2003).T. Goto, T. Kimura, G. Lawes, A. P. Ramirez, andY. Tokura, Phys. Rev. Lett. 92, 257201 (2004).M. Kenzelmann, A. B. Harris, S. Jonas, C. Broholm,
J. Schefer, S. B. Kim, C. L. Zhang, S.-W. Cheong,O. P. Vajk, and J. W. Lynn, Phys. Rev. Lett. 95,087206 (2005).Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He,
T. Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204(2006).
J. Hemberger, F. Schrettle, A. Pimenov, P. Lunkenheimer,V. Yu. Ivanov, A. A. Mukhin, A. M. Balbashov, andA. Loidl, Phys. Rev. B 75, 035118 (2007).
[3] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett.
95, 057205 (2005). I. A. Sergienko and E. Dagotto, Phys.Rev. B 73, 094434 (2006).
[4] C. Jia, S. Onoda, N. Nagaosa, and J.-H. Han, Phys. Rev. B74,224444 (2006).
C. Jia, S. Onoda, N. Nagaosa, and J.-H. Han, Phys. Rev.B 76, 144424 (2007).
[5] H. Katsura, A. V. Balatsky, and N. Nagaosa, Phys. Rev. Lett.
98, 027203 (2007).
[6] I. E. Chupis, Low Temp. Phys. 33, 715 (2007).A. Cano and E. I. Kats, Phys. Rev. B 78, 012104 (2008).
[7] A. Pimenov, T. Rudolf, F. Mayr, A. Loidl, A. A. Mukhin, and
A. M. Balbashov, Phys. Rev. B 74, 100403(R) (2006).[8] A. Pimenov, A. A. Mukhin, V. Yu. Ivanov, V. D. Travkin,
A. M. Balbashov, and A. Loidl, Nature Phys. 2, 97 (2006).[9] R. Valdes Aguilar, A. B. Suchkov, C. L. Zhang, Y. J. Choi,
S. W. Cheong, and H. D. Drew, Phys. Rev. B 76, 060404(R)(2007).
[10] A. Pimenov, A. Loidl, A. A. Mukhin, V. D. Travkin, V. Yu.
Ivanov, andA. M. Balbashov, Phys. Rev. B 77, 014438 (2008).[11] Y. Takahashi, N. Kida, Y. Yamasaki, J. Fujioka, T. Arima,
R. Shimano, S. Miyahara, M. Mochizuki, N. Furukawa, and
Y. Tokura, Phys. Rev. Lett. 101, 187201 (2008).R. Valde s Aguilar, M. Mostovoy, A. B. Sushkov, C. L.Zhang, Y. J. Choi, S.-W. Cheong, and H. D. Drew, Phys.Rev. Lett. 102, 047203 (2009).
[12] C. Jia and J. Berakdar, Eur. Phys. Lett. 88, 57004 (2009).[13] T. Kimura, S. Ishihara, H. Shintani, T. Arima, K. T. Takaha-
shi, K. Ishizaka, and Y. Tokura, Phys. Rev. B 68, 060403(R)
(2003).
[14] S. Park, Y. J. Choi, C. L. Zhang, and S.-W. Cheong, Phys.
Rev. Lett. 98, 057601 (2007).
S. Seki, Y. Yamasaki, M. Soda, M. Matsuura, K. Hirota,and Y. Tokura, Phys. Rev. Lett. 100, 127201 (2008).
[15] L. Capogna, M. Mayr, P. Horsch, M. Raichle, R. K. Kremer,
M. Sofin, A. Maljuk, M. Jansen, and B. Keimer, Phys. Rev. B
71, 140402(R) (2005).[16] W. A. Harrison, Electronic Structure and the Properties of
Solids (Dover, New York, 1980).
Phys. Status Solidi B (2010) 3
Original
Paper
www.pss-b.com 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim