Electrically active magnetic excitations in antiferromagnets (Review Article)
The magnetic resonance operation by electric field is highly nontrivial but the most demanding function in the future spin-electronics. Recently observed in a variety of multiferroics materials named the collective electrically active magnetic excitations, frequently referred to as “electromagnons...
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irk-123456789-1174672017-05-24T03:04:22Z Electrically active magnetic excitations in antiferromagnets (Review Article) Krivoruchko, V.N. К 80-летию Виктора Валентиновича Еременко The magnetic resonance operation by electric field is highly nontrivial but the most demanding function in the future spin-electronics. Recently observed in a variety of multiferroics materials named the collective electrically active magnetic excitations, frequently referred to as “electromagnons”, reveal a possible way to implement such a function. Experimental advances in terahertz spectroscopy of electromagnons in multiferroics as well as related theoretical models are reviewed. The earlier theoretical works, where the existence of electric-dipole active magnetic excitations in antiferro- and ferrimagnets with collinear spin structure has been predicted, are also discussed. Multi-sublattice magnets with electrically active magnetic excitations at room temperature give a direct possibility to transform one type of excitation into another in a terahertz time-domain. This is of crucial importance for the magnon-based spintronics as only the short-wavelength exchange magnons allow the signal processing on the nanoscale distance. 2012 Article Electrically active magnetic excitations in antiferromagnets (Review Article) / V.N. Krivoruchko // Физика низких температур. — 2012. — Т. 38, № 9. — С. 1018-1031. — Бібліогр.: 97 назв. — англ. 0132-6414 PACS: 75.80.+q, 75.40.Gb, 76.50.+g http://dspace.nbuv.gov.ua/handle/123456789/117467 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 80-летию Виктора Валентиновича Еременко К 80-летию Виктора Валентиновича Еременко |
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К 80-летию Виктора Валентиновича Еременко К 80-летию Виктора Валентиновича Еременко Krivoruchko, V.N. Electrically active magnetic excitations in antiferromagnets (Review Article) Физика низких температур |
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The magnetic resonance operation by electric field is highly nontrivial but the most demanding function in
the future spin-electronics. Recently observed in a variety of multiferroics materials named the collective electrically
active magnetic excitations, frequently referred to as “electromagnons”, reveal a possible way to implement
such a function. Experimental advances in terahertz spectroscopy of electromagnons in multiferroics as well as
related theoretical models are reviewed. The earlier theoretical works, where the existence of electric-dipole active
magnetic excitations in antiferro- and ferrimagnets with collinear spin structure has been predicted, are also
discussed. Multi-sublattice magnets with electrically active magnetic excitations at room temperature give a direct
possibility to transform one type of excitation into another in a terahertz time-domain. This is of crucial importance
for the magnon-based spintronics as only the short-wavelength exchange magnons allow the signal
processing on the nanoscale distance. |
| format |
Article |
| author |
Krivoruchko, V.N. |
| author_facet |
Krivoruchko, V.N. |
| author_sort |
Krivoruchko, V.N. |
| title |
Electrically active magnetic excitations in antiferromagnets (Review Article) |
| title_short |
Electrically active magnetic excitations in antiferromagnets (Review Article) |
| title_full |
Electrically active magnetic excitations in antiferromagnets (Review Article) |
| title_fullStr |
Electrically active magnetic excitations in antiferromagnets (Review Article) |
| title_full_unstemmed |
Electrically active magnetic excitations in antiferromagnets (Review Article) |
| title_sort |
electrically active magnetic excitations in antiferromagnets (review article) |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2012 |
| topic_facet |
К 80-летию Виктора Валентиновича Еременко |
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http://dspace.nbuv.gov.ua/handle/123456789/117467 |
| citation_txt |
Electrically active magnetic excitations in antiferromagnets (Review Article) / V.N. Krivoruchko // Физика низких температур. — 2012. — Т. 38, № 9. — С. 1018-1031. — Бібліогр.: 97 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT krivoruchkovn electricallyactivemagneticexcitationsinantiferromagnetsreviewarticle |
| first_indexed |
2025-07-08T12:18:46Z |
| last_indexed |
2025-07-08T12:18:46Z |
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1837081174586949632 |
| fulltext |
© V.N. Krivoruchko, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9, pp. 1018–1031
Electrically active magnetic excitations
in antiferromagnets
(Review Article)
V.N. Krivoruchko
O.O. Galkin Donetsk Institute for Physics and Technology, National Academy of Science of Ukraine
72 R. Luxemburg Str., Donetsk 83114, Ukraine
E-mail: krivoruc@krivoruc.fti.ac.donetsk.ua; krivoruc@gmail.com
Received March 20, 2012
The magnetic resonance operation by electric field is highly nontrivial but the most demanding function in
the future spin-electronics. Recently observed in a variety of multiferroics materials named the collective electri-
cally active magnetic excitations, frequently referred to as “electromagnons”, reveal a possible way to implement
such a function. Experimental advances in terahertz spectroscopy of electromagnons in multiferroics as well as
related theoretical models are reviewed. The earlier theoretical works, where the existence of electric-dipole ac-
tive magnetic excitations in antiferro- and ferrimagnets with collinear spin structure has been predicted, are also
discussed. Multi-sublattice magnets with electrically active magnetic excitations at room temperature give a di-
rect possibility to transform one type of excitation into another in a terahertz time-domain. This is of crucial im-
portance for the magnon-based spintronics as only the short-wavelength exchange magnons allow the signal
processing on the nanoscale distance.
PACS: 75.80.+q Magnetomechanical effects, magnetostriction;
75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.);
76.50.+g Ferromagnetic, antiferromagnetic, and ferrimagnetic resonances; spin-wave resonance.
Keywords: ferrimagnet, electromagnons, spintronics, electron paramagnetic resonance.
Соntents
1. Introduction ........................................................................................................................................ 1018
2. Electromagnons in multiferroics ........................................................................................................ 1019
2.1. Methods, samples, and ferroelectricity ....................................................................................... 1020
2.2. Evidences for electromagnons in multiferroics .......................................................................... 1021
2.3. Theoretical considerations .......................................................................................................... 1023
3. Electric-dipole active exchange modes in collinear magnets ............................................................. 1024
3.1. General conditions for the existence of AFER ........................................................................... 1025
3.1.1. The structure 1 (+) ........................................................................................................... 1025
3.1.2. The structure 1 (–) .......................................................................................................... 1025
3.2. AFER in α-Fe2O3 and Cr2O3 ..................................................................................................... 1026
4. Electric-dipole active magnetic resonances in ferrimagnets ............................................................... 1028
5. Conclusions ........................................................................................................................................ 1029
References .............................................................................................................................................. 1029
1. Introduction
In 1961 a new effect has been discovered in spectrosco-
py of solid state, namely, the applied electric field effect on
the electron spin resonance [1]. This effect is manifested
by a linear on electric field Е splitting of absorption lines
in the electron paramagnetic resonance spectrum and evi-
dences the external electric field effect on the energy levels
of magnetic ion. Another manifestation of the electric ef-
fect, discovered in 1962 [2], was the so-called paraelectric
resonance, i.e., an induction of transitions between magnet-
ic levels of paramagnetic ions by a high-frequency electric
field. Differently speaking, the magnetic resonance could
be observed by placing the sample into the ac electric field
antinodes instead of ac magnetic field ones. In this way a
possibility to excite transitions forbidden for a convention-
al magnetic resonance was demonstrated. In further inves-
tigations, electrically induced transitions between spin le-
vels were detected for different magnetic ions and explored
in a number of reports (see review [3] and textbooks [4,5]).
Electrically active magnetic excitations in antiferromagnets
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9 1019
It should be noted that the possibility to induce the mag-
netic resonance transitions by an alternating electric field
follows already from the spin Hamiltonian of the form
H = Ei (Rijk SjSk + Tijk Hj Sk); i, j, k = x, y, z,
which is used for description of linear electric field effects
(here S is the spin operator, H stands for external magnetic
field) [3–5]. From viewpoint of theory, there are no prin-
cipal differences between the static and alternating field; it
is only important that related magnetoelectric constants,
Rijk and Tijk, must be nonzero as well as the corresponding
matrix elements. Let us remind that linear electric effects
are feasible (the magnetoelectric constants are nonzero) for
paramagnetic ions in the non-center-symmetric positions
[3–5]. Obviously, in concentrated magnetic compounds the
existence of ions with electric-dipole active magnetic tran-
sitions yields collective magnetic excitations. But the li-
mited experimental possibilities, such as a hard frequency
range — terahertz and even optic frequency range — im-
pede progress in the field.
Terahertz (THz) frequency range (region of frequency
from 0.1 THz up to 10 THz, energy scale 1 THz ≈ 4 meV)
involves the short-wave part of millimeter range, all sub-
millimeter region and a part of far infrared range (wave
length from 3 mm up to 0.03 mm). This region of electro-
magnetic radiation remains unexplored since long ago. At
the same time, the radiation in this frequency range pos-
sesses a set of specific characteristics, which makes it rather
promising for applications in fundamental and applied phys-
ics, chemistry, biology, medicine and so on (for more details
see recent reviews [6,7] and references therein). At present,
a so-called “THz-gap” is actively mastered by traditional
vacuum electronics as well as by optoelectronics.
During the last years, the THz radiation was found to be
very sensitive and informative tool for investigation quasi-
particle excitations in solid state in this frequency/energy
range — polaritons, excitons, phonons and others. In fact,
nowadays the THz time-domain spectroscopy enters into
toolset of many physical laboratories [6,7]. A class of the
system where THz spectroscopy has already displayed its
unique performances is the so-called multiferroics — ma-
terials, where the magnetic order and ferroelectric polariza-
tion coexist. Exactly in multiferroics it was succeeded to
detect by the THz spectroscopy technique and identify a
new type of collective electrically active magnetic excita-
tion frequently termed as “electromagnons”.
The first part of this paper (Sec. 2) presents an overview
of recent experimental advances in THz spectroscopy of
electromagnons in multiferroics. The proposed theoretical
models of electromagnons origin in multiferroics are also
discussed. Next part (Sec. 3) summarizes the earlier theo-
retical works where existence of electric-dipole active
magnetic excitations in antiferromagnets has been pre-
dicted. The idea was put forward for the first time by
Ukrainian physicists in 1988 in series of publications, in
particular, with Victor Eremenko participation. Here, the
general conditions for appearance of electrically active mag-
netic oscillations in magnetic structures which ground state
is collinear in the exchange approximation are formulated.
As examples, the four-sublattices hematite (α-Fe2O3) and
chromium (III) oxide (Cr2O3), are considered. It is shown
that THz spectroscopy could be a fruitful, and sometimes
even sole, experimental tool for investigation magnetic
excitations in these antiferromagnets. In Sec. 4, the elec-
tric-dipole active magnetic resonances in compounds pos-
sessing both spontaneous magnetization and electric pola-
rization are briefly discussed.
Possibility to transform magnetic excitation into electric
signal and vice-versa is highly nontrivial but the most de-
manding function in the future spin-electronics. Due to
magnetic ordering well above room temperature, the multi-
sublattice magnets with electrically active magnetic excita-
tions give an opportunity for implementation of such func-
tion and thus are promising candidates for future applica-
tions in spin based electronics.
2. Electromagnons in multiferroics
Materials with the simultaneous occurrence of magnet-
ism and ferroelectricity, which are frequently referred to as
multiferroics [8] (in earlier Russian literature often named
as seignetto-magnets), are characterized by a strong coupl-
ing between magnetic order and electric polarization. Such
perovskite manganites as TbMnO3 and GdMnO3 are ex-
amles of typical family members of multiferroics with a
ferroelectricity induced by magnetic order. Exactly for
these compounds, apparently for the first time, the strong
experimental evidence was obtained of new type of mag-
netic excitations, namely the electric-dipole active mag-
nons or electromagnons [9,10]. These new elementary ex-
citations were manifested in the noncommensurate spin
ordered phase by additional absorption peaks in the dielec-
tric constant spectrum ε(ω) within the THz frequency re-
gion. These peaks vanished in applied magnetic field,
which produced a commensurate spin order. Such a beha-
vior points to the magnetic origin of excitations.
It should be noted here that the existence of electric-
dipole active magnons had been theoretically predicted
much earlier than the electromagnons were first observed.
In the end of 60th — beginning the 70th years, the theore-
ticians V.G. Bar’yakhtar and I.E. Chupis have demonstrat-
ed in a number of works [11,12] that in seignetto-magnets,
the liner oscillations of electric and magnetic dipoles are
coupled. Eventually, oscillations of magnetic moments
could be caused by electric field and, vice-versa, the elec-
tric-dipole excitations by magnetic field. This new elemen-
tary excitations, called by the authors the seignette-mag-
nons, appear as magnetic resonances in the dielectric
constant spectrum ε(ω) in a response to the electric field
component of light Eω. This is in contrast to the case of the
V.N. Krivoruchko
1020 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9
magnetic resonances appearing in the magnetic permeabili-
ty spectrum μ(ω) as a result of the spin excitations driven
by the magnetic component of light Hω. The concept pro-
posed in Refs. 11, 12 was developed in following publica-
tions, the review of which is given in Ref. 13. Unfortunate-
ly, as already was mentioned, the limited experimental
facilities postponed the full-scale investigations in the field
and a discovery of seignette-magnons.
Renewed attention to multiferroics has been inspired by
discovery of the electric polarization flop by applying
magnetic field in a perovskite manganite [14]. Obviously,
the dynamical interplay between magnetic and electric
excitations has been reasonably expected for these mate-
rials, as well. Indeed, soon after the work [14], the obser-
vation of electric-dipole active magnetic excitations were
reported for TbMnO3 and GdMnO3 as far-infrared absorp-
tion near 2.5 meV [9,10].
2.1. Methods, samples and ferroelectricity
Typically, in experiments with THz radiation, the value
of the complex optical constant ( ) ( ) ( )n ω = ε ω μ ω is de-
termined from the measured transmittance spectrum. In
this frequency region the contribution of magnetic permea-
bility ( )μ ω to ( )n ω in most cases is negligible and fre-
quently ( ) 1μ ω ≈ is assumed. The transmittance experi-
ments at THz frequencies can be carried out in a Mach–
Zehnder interferometer arrangement [15] which allows
measurements of amplitude and phase shift in a geometry
with controlled polarization of the radiation. The absolute
values of the complex dielectric permittivity ε = ε1 + iε2
are determined directly from the measured spectra using
the Fresnel optical formulas for transmittance T = |t|2 of a
plane-parallel sample:
2
1
2 2
1
(1 )
exp ( )
1
T
r t
t T i
r t
−
= ϕ =
−
,
where ( 1) / ( 1)r = ε − ε + and 1 exp ( 2 / ).t i d= − π ε λ
Here r is the reflection amplitude of a thick sample, t1 is
the “pure” transmission amplitude, d is the sample thick-
ness, λ is the radiation wavelength, and Tϕ is the phase
shift in the transmittance.
The transmittance experiment with different experimen-
tal geometries and varied polarization of the incident radia-
tion usually allows to separate unambiguously the magnet-
ic and dielectric contributions to the measured spectra.
However, in some cases both contributions are equally
strong and four independent experiments are necessary to
extract both, the dielectric permittivity ε(ω) and the mag-
netic permeability ( ).μ ω
A majority of existing experimental results are obtained
on single (or poly-) crystalline samples of RMnO3, and a
brief discussion of the magnetic phase diagram of these
compounds is not out of place.
In many of the manganite materials with perovskite
structure, RMnO3, the nearest-neighbor ferromagnetic ex-
change, J1, is weak relative to the next-neighbor antiferro-
magnetic interaction J2 (see Fig. 1). As the result, the mag-
netic order is sensitive to the Mn–O–Mn bond angle or,
equivalently, to the ionic radius of R. The phase diagram
of RMnO3 consists of the A-type (ferromagnetic ab-plane
antiferromagnetically ordered), ab-cycloidal (Fig. 2) and
E-type (↑↑↓↓) (Fig. 1) states. When J2/|J1| > 0.5, minimum
magnetic energy achieves for cycloidal structure with non-
commensurate period. In RMnO3 crystals with R = Tb, Dy,
Eu1–xYx a cycloidal structure is realized within ab plane.
Such magnetic order induces a nonzero static electric pola-
rization P || ab plane. The magnetically induced ferroelec-
tricity has been explained by the inverse Dzyaloshinskii–
Moriya (DM) interaction [16–18], in which the microscop-
ic polarization is expressed as PА ~ enm×(Sn × Sm) for a
pair of neighboring spins, Sn and Sm, with unit vector enm
connecting them.
In perovskites with rare earth ions of smaller radius
(R = Ho, Tm, Yb, Lu, Y1–xLux) minimum magnetic energy
achieves for E-type magnetic order in ab plane, hosting
also the ferroelectricity. It was shown that the ferroelectric-
ity in the E-type phase originates from the symmetric ex-
change term expressed as PS ~ πnm (Sn·Sm) which poten-
Fig. 1. The schematic spin configuration of the E-type state in ab
and bc planes.
J1
J2 a
J
c
c
2
b
a
b
c
b
Fig. 2. Cycloidal spin spiral structure; the spins rotate around an
axis normal to the spiral wave vector Q. The induced electric
polarization P is normal to both the wave vector and the spin
rotation axis: ⊥P Q .
Q
P
Electrically active magnetic excitations in antiferromagnets
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9 1021
tially produces larger spontaneous polarization than anti-
symmetric DM mechanism [19–21]. Indeed, it was found
[20,21] that the electric polarization of the E-type phase is
dominated by the symmetric spin exchange (Sn·Sm) and is
more than ten times larger that of the bc-cycloidal phase
dominated by the antisymmetric one ~ (Sn × Sm). The
magnetoelectric response is also enhanced near the bicriti-
cal phase boundary between these multiferroic phases of
different origins. (Note for reference, the largest magneti-
cally induced electrical polarization was documented re-
cently in CaMn7O12 [22]. The value of PS = 2870 μС/m2
for CaMn7O12 is four time large than a spontaneous pola-
rization in TbMnO3.)
2.2. Evidences for electromagnons in multiferroics
Pimenov et al. have measured the optical spectra of
TbMnO3 and GdMnO3 in the energy range 0.4–4.8 meV
[9,10]. At low temperature in zero magnetic field, they
found a single peak-structure in the imaginary part of ε(ω)
at 2.9 meV for TbMnO3 and at 2.5 meV for GdMnO3 (see
Fig. 3). These absorptions were shown to be permitted if
Eω || a axis and disappeared when the bc-spiral order is
transformed to the A-type antiferromagnetic order by ap-
plying the external magnetic field along c axis.
Pioneering works [9,10] stimulated a series of publica-
tions concerning the electric-dipole active spin dynamics in
crystals of different symmetry with noncommensurate
magnetic structure and multiferroic properties: RMnO3
[23–44], BiFeO3 [45,46], RMn2O5 [47–49], Cu(Fe,Ca)O2
[50], CuFe1–xGaxO2 [51,52], and Ba2CoGeO7 [53]. These
investigations revealed new excitations in THz time do-
main region and posed a set of questions.
Note here that different crystal imperfections (polycrys-
talline structure of a sample, ferroelectric domain walls, etc.)
and (especially) impurity of rare-earth ions also interact with
the electric component of light and give contributions to the
ε(ω) spectrum [54,55]. To avoid these problems, at least
partly, the systems RMnO3 where Mn3+ ion is the only
magnetic ion are found to be the most relevant for revealing
the physics of electromagnons in multiferroics.
Leaving behind the historical sequence of experimental
data obtaining, let us concentrate on the settled ones.
Most comprehensive data are obtained on the com-
pounds with R = Y, Lu, Y1–xLux, Eu, Eu1–xYx. Figures 4
and 5 illustrate the results on THz spectroscopy of elec-
tromagnons in perovskites manganites RMnO3 with R =
= Y1–xLux, Eu1–xYx [44]. In Fig. 4, the representative spec-
tra are displayed of the A-type, ab-cycloidal and E-type
Fig. 3. Spectra of the a-axis dielectric permittivity of GdMnO3 in
the THz frequency range at 0 and 2 T. Upper panel: real part of
the dielectric permittivity; lower panel: imaginary part. Triangles
represent the results from the THz transmittance. Circles — spec-
tra obtained via Kramers–Kronig analysis of the reflectance.
Electromagnon is detected as a broad peak at 25 cm–1. (Repro-
duced from Ref. 10.)
20
25
H c��
e a��~
2 T
GdMnO3
��
15 K
0 T
100
0
1
2
3
2 T
0 T
��
Frequency, cm
–1
Electro-
magnon
Fig. 4. The Im [εμ] spectra for: (a) A-type (collinear in ab plane),
(b) ab-cycloidal, and (c) E-type (↑↑↓↓ in ab plane) spin phases of
Eu0.8Y0.2MnO3, Eu0.4Y0.6MnO3, and Y0.5Lu0.5MnO3, respective-
ly. (Reproduced from Ref. 44.)
(a)
(b)
(c)
2.0
1.5
1.0
0.5
0
1.5
1.0
0.5
0
1.5
1.0
0.5
0
2 4 6 8 10
Photon energy, meV
Im
[
]
��
Im
[
]
��
Im
[
]
�� 4 K
100 K
4 K
100 K
4 K
100 K
30 K
30 K
30 K
V.N. Krivoruchko
1022 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9
states. The plateau-like absorption bands are observed in
the higher-T collinear sinusoidal phase for all the com-
pounds in addition to the tail of the higher-lying phonon
modes; this broadened band has been assigned to the pre-
cursory electromagnon-like excitation arising from the
cycloidal spin fluctuation [39]. The sharp antiferromagnet-
ic resonance (AFMR) with much smaller spectral intensity,
which is active for the magnetic field component of light,
is discerned at 2.5 meV in the A-type phase [Fig. 4(a)]. In
contrast, the development of two clear peaks, at 2.2 and
6.7 meV, is identified in the cycloidal-spin phase
[Fig. 4(b)]. For the E-type phase [Fig. 4(c)], an electro-
magnon-like feature is also revealed in this study: three
pronounced peaks appear at 2.9, 4.1 and 5.9 meV that are
in a frequency region typical of magnetic excitations.
Figure 5(a) shows the temperature dependence of
Im [εμ] spectra for LuMnO3. In paramagnetic phase
(90 K), only the contribution from the low energy tail of
the higher-lying phonon modes can be seen. The broad
absorption band develops from slightly above TN = 40 K
with decreasing temperature, and then it shows rapid en-
hancement below TN. This material undergoes the transi-
tion from the collinear sinusoidal to the E-type phase at
TC = 35 K, accompanied by the emergence of ferroelec-
tricity. In accordance with this phase transition, the broad
absorption band turns into peak structures at 2.9, 4.7 and
5.7 meV with slight hardening of the frequencies while
decreasing temperature. The spectra at the lowest tempera-
ture T = 4 K for the E-type phase are displayed for the
Y1−yLuyMnO3 samples in Fig. 5(b). All the spectra show
the three-peak feature in common.
Thus, the latest data in the THz time-domain spectros-
copy on electromagnons provide the strong evidence that
the full spectra of electro-active magnons in RMnO3 con-
sists of three perfectly resolved absorption peaks around
2.5–3, 4.5–5, and 5.5–6 meV, in contrast to a single broad
1–5 meV peak, as it was reported by earlier measurements
[9,10,26,27]. The low-lying electromagnon survives even
in the collinear spin ordered phase and thus this electro-
magnon can not be ascribed as arising due to the DM inte-
raction in multiferroics.
The models describing electric-dipole active excitations
in perovskites manganites will be considered in the next
subsection.
Among the multiferroic compounds, bismuth ferrite
BiFeO3 is obviously one of the most interesting systems
with both ferroelectric and magnetic transitions above
room temperature: with a Néel temperature at TN = 650 K
and a ferroelectric transition temperature at TC = 1100 K
[56], exhibiting one of the largest polarization values,
~ 100μC/m2 [57]. Another interesting point is that when it
undergoes an antiferromagnetic ordering at 650 K an in-
commensurate structure is formed with an extremely long
period of 620 Å.
Electromagnons in BiFeO3 were for the first time un-
raveled by Cazayous et al. [45] and Singh et al. [46] by
means of low-energy inelastic light scattering technique.
The two intense one-magnon branches are detected at 18.2
and 26.6 cm−1. The authors considered these modes as not
a pure spin wave but corresponding to magnetic excita-
tions with electric dipole moment in and out of the cyc-
loidal plane. Yet, a possible origin of the observed mag-
netic resonances is not uniquely identified. (There are
actually three reasonable models for the data, see Ref. 46
for more details.)
The observation of electric-dipole active magnons in
the RMn2O5 compounds including non-rare-earth metal
YMn2O5 [48] are reported [47–49]. The electromagnons
polarization selection rules provide evidence in favor of
symmetric exchange coupling mechanism in this system.
THz spectroscopy also has been used to reveal the elec-
tromagnon excitation in the paraelectric ↑↑↓↓ collinear
magnetic phase of triangular lattice antiferromagnet
CuFe1−xGaxO2 [50]. Surprisingly, this mode was found to
vanish in the ferroelectric helimagnetic phase. These facts
Fig. 5. (a) Temperature dependence of Im [εμ] spectra for
LuMnO3; (b) Im [εμ] spectra of the E-type phase at 4 K for
Y1–yLuyMnO3 (y = 0.25, 0.5, 0.75 and 1.0) with vertical offset for
clarity. Peak positions are indicated by the vertical dotted lines.
(Reproduced from Ref. 40.)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0 2 4 6 8 10
Photon energy, meV
Im
[
]
��
LuMnO3
27 K
33 K
39 K
50 K
90 K
15 K
18 K
21 K
4 K
(a)
3.0
2.5
2.0
1.5
1.0
0.5
0
2 4 6 8 10
R = Lu
R = Y Lu0.25 0.75
R = Y Lu0.5 0.5
4 K
(b)
R = Y Lu0.75 0.25
Photon energy, meV
Im
[
]
��
Electrically active magnetic excitations in antiferromagnets
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9 1023
prove that neither ferroelectricity nor noncollinear magnet-
ism are necessary condition for the appearance of electro-
magnon excitation, while the existing theories on electro-
magnon have focused on noncollinear magnets like heli-
magnets. The anticorrelation between the noncollinear mag-
netism and the emergence of electromagnon excludes the
exchange striction mechanism as the origin of dynamical
magnetoelectric coupling. The electric activity of the mag-
non in this compound was attributed by the authors to the
modulation of the p–d hybridization at the spin-twisted ex-
cited state via the spin-orbit interaction. This discovery sug-
gests that similar electromagnon modes should be observa-
ble in a broad variety of paraelectric collinear magnets.
Magnons and electromagnons in a four-sublattice
CuFeO2 have been investigated by both THz time-domain
spectroscopy [51] and by means of inelastic neutron scat-
tering [52]. The authors demonstrated that only the high-
energy mode generates the oscillating electric polarization,
which can be coupled with an ac electric field.
2.3. Theoretical considerations
A common opinion at present is that a whole series of
perovskite can show different multiferroic states, where
two representative mechanisms, the symmetric exchange
strictions and antisymmetric DM strictions, are effective.
For the latter case, the noncommensurate cycloidal mag-
netic structure produces the macroscopic (spontaneous)
polarization, ~ ( ) ~ [ ( ) ( ) ],А nm n m× × ∝ ∇ − ∇P e S S S S S S
parallel to the spiral plane (here spins Sn and Sm in points n
and m should not be coupled by inversion) [16–18].
In contrast, the ferroelectricity in the E-type phase origi-
nates from the symmetric exchange term expressed as
( )~ · .S nm n mP S Sπ It is naturally to expect that both me-
chanisms cause the dynamical interplay between magnetic
and electric excitations. However, the dynamical response
of multiferroic systems was found to be more complex and
there is no general consensus on this issue so far.
To analyze the results of theoretical investigations, one
may start by describing the perovskites manganites RMnO3
within a frustrated Heisenberg model with the Hamiltonian
of the form [35,58–60]
0 1 2
n.n. n.n.n.
· ·n m n mJ J= + +∑ ∑S S S SH
2
n.n.
· (S ) .c n m n
n
J D α
α+ +∑ ∑S S (1)
Here J1 (< 0) stands for the ferromagnetic interaction be-
tween the nearest-neighbor spins in the ab plane; J2 и Jc
(both > 0) is the next-nearest-neighbor in this plane and
interlayer interactions, respectively (see Fig. 1). As already
mentioned, in this class of materials, the exchange interac-
tion J1 is very small relative to that of other perovskites
compounds and the condition J2/|J1| > 0.5 is typically ful-
filled. Thus, the ground state is a cycloidal spin order
with the spiral angle θ being equal to cos θ = –J1/2J2.
A uniaxial anisotropy parameter Dα establishes the direc-
tions of the cycloidal plane. The evaluation of system pa-
rameters gives, for example, for DyMnO3, TbMnO3 and
Gd0.7Tb0.3MnO3 [35,58–60] that for all these compounds a
noncommensurate magnetic structure is realized with the
cb-cycloidal plane (i.e., α = a) for DyMnO3 and TbMnO3,
and ab-cycloidal plane (i.e., α = c) for Gd0.7Tb0.3MnO3.
When the external electric field E is applied, the spin
Hamiltonian of the system can be written as H = H0 – Е·Р,
where H0 is a Hamiltonian of the frustrated Heisenberg
model (1), and Р is an electric dipole moment which de-
pends on spin configuration. The electric dipole moment
associated with a pair of spins on the nearest-neighbor
bonds has been calculated within a microscopic theory
[58–60]. It was found that the dominated symmetric spin
depend component is given by
( )
n.n.
,S nm n m= ⋅∑P S Sπ n ≠ m. (2)
Here the vector πnm is nonzero if the inversion symmetry
is absent at the center of n and m positions. For the cyc-
loidal spin structure, the local symmetric spin polariza-
tion terms PS are aligned antiferroelectrically [58–60].
Thus, static part of (2) is zero PS = 0 and the symmetric
term does not violate the fact that the total static ferro-
electricity occurs solely due to the antisymmetric spin
dependent term of the form: ~ ( ).А nm n m× ×P e S S But the
dynamical fluctuations of PS contribute to the electro-
magnon excitation. Especially, in the noncollinear ground
state, the single-magnon processes at the zone edge origi-
nate from the term (2) (see Fig. 6).
The lower-lying magnetic excitations of the model (1),
(2) are described by the liner spin wave approach. Using a
standard procedure (see, e.g., [61]), the spin operator can
be approximated by the Bose ones and then, by well
known u-v-Bogolyubov transformation, diagonalized the
Hamiltonian. In a spin wave (magnon) creation k
+β and
annihilation kβ operators the polarization (2) is written as
one-, two-, etc. magnon processes [35,58–60]:
(1)
2 22
sin )( )(S kkiS SN +
π ππ
= θ× β − β −P kπ
(2)
22
sin ( /2) )( ) ...( k k k kk k
k
iS + +
− −− − ππ
− θ β β − β β +∑ kπ (3)
Here the vector π(1) has only a component parallel to the
anisotropy axis α. The second term in Eq. (3) corresponds
to the absorption due to simultaneous two-magnon absorp-
tion process. The optically active two-magnon excitations
are possible over the full magnon bandwidth with the con-
tribution of + k and – k magnons, and hence should spread
the spectral weight up to the energy twice of the magnon
bandwidth. A possible effect of electric-dipole active two-
magnon process on the observed characteristics is dis-
V.N. Krivoruchko
1024 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9
cussed in Ref. 30. In general, the effect of multi-magnon
absorptions are negligible except near the Néel ordered
state and, in the following, we focus on the first compo-
nent in Eq. (3).
In the one-magnon process, spin waves with k2π =
= (2π/а, 0, 0), k = (0, 2π/b, 0) and energy ω2π are induced.
Introducing the electric polarization tensor, Pα(ω) =
= ααα(ω)Eαω, for the imaginary part of the tensor ααα(ω)
the expression is obtained [35,60]
(1) 2 2 2
2 2 ,Im ( ) ( )sin ( ) .aNS kαα π π αα ω = π θδ ω − ω δ (4)
The performed estimations yield that one-magnon absorp-
tion (4) is rather strong in a wide range of parameters ex-
cept for near ferromagnetic and antiferromagnetic phases.
The approach (1), (2) perfectly reproduces the higher
energy peak at ~ 8 meV for TbMnO3 and peak at ~ 5 meV
for DyMnO3 [35,60]. However, within this scenario the
lower energy peak with substantial oscillator strength
around ~ 2 meV is not obtained. One should remember that
in general the full spectra of electro-active magnons in
RMnO3 consist of three well manifested peaks of absorp-
tion around 2.5–3, 4.5–5, and 5.5–6 meV [44]. The inclu-
sion of the antisymmetric term ~ ( )А nm n m× ×P e S S does
not improve the situation, because this interaction is nonze-
ro in cycloidal spin phase and may be responsible for the
one-magnon absorption only in geometry Eω || a. Thus,
within the framework of the model (1), (2), the question
about the origin of lower energy peak remains open.
Low-lying absorption can be obtained assuming that the
spin configuration is subjected to the significant deviation
from a pure cycloidal one with uniform spin rotation angle
and contains higher harmonics [43]. The proposed model is
based on the models (1), (2) but, in addition, contains the
single-ion spin anisotropy within cycloid plane [i.e., in
plane normal to axis α in Eq. (1)] and the biquadratic inte-
raction between two neighbor spins in the ab plane. The
latter is given by
2
biq biq
n.n.
( ) ;n mJ= − ∑ S SH , .⊥ αn m (5)
These two additional terms cause a nonuniform spin rota-
tion and effectively modulate a cycloid structure.
Within the approach [43], the calculated electromagnon
spectrum was in a good agreement with experiment. Par-
ticularly, it was found that irrespective of spiral-plane
orientation the absorption at low energy is enhanced
strongly by weak anisotropy within cycloid plane or by the
weak biquadratic interaction.
Yet, further investigations are necessary to examine the
model [43]. In particular, as pointed in Ref. 38, the effects
due to a magnetically induced incommensurate oscillatory
polarization can be also important. A further generalization
of the approach [43] which includes the spin-phonon
coupling has been considered in Ref. 62. Considering the
lattice degree of freedom the authors succeeded in repro-
ducing the entire phase diagram of RMnO3 for the first
time and explained several experimental puzzles. Howev-
er, the dynamical properties of the systems are not dis-
cussed in Ref. 62.
One of a key test for any theoretical model will be its
ability to reproduce a sensitivity of the electric-dipole ac-
tive magnon spectra to the rare-earth ionic radius observed
in RMnO3 multiferroics.
The physical picture proposed in Refs. 35, 43, 58–60,
62 reveals an important condition for a search of new class
of materials. Namely, in dynamics the effect of symmetric,
~ S·S, magnetostriction can be larger that the usually ex-
pected antisymmetric, ~ S×S, contribution. This enables
the researchers to search for the electric-dipole active
magnetic excitations not only in multiferroics but in com-
pounds without static electric polarization, as well. The
necessary condition is only the absence of coupling of
magnetic ions positions by inversion symmetry. A few
examples of such multi-sublattice magnets with collinear
magnetic order are discussed in the next section.
3. Electric-dipole active exchange modes in collinear
magnets
The above-considered systems are antiferromagnets
with noncommensurate magnetic structure. In such a case,
the appearance of electric dipole moment is due to disap-
pearance of a centre of inversion in the ordered phase and
direct coupling between magnetic and electric order para-
meters. There is a limited number of compounds where the
Fig. 6. The schematic zone-edge magnon in ab and bc planes.
(Reproduced from Ref. 44.)
�P
P
P ||a�
Zone- ||c)edge mode (E�
Zone- ||a)edge mode (E�
P
P ||c�
�P
c
2
Electrically active magnetic excitations in antiferromagnets
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9 1025
both conditions are satisfied. Overwhelming majority of
magnetic compounds contain a few magnetic ions in primi-
tive magnetic cell and are ordered into commensurate and
(in exchange approximation) collinear magnetic phase. In
this case, the spectrum of proper magnetic excitation con-
tains acoustic modes (АМs), with in-phase oscillation of
spins within a magnetic elementary cell, as well as modes
similar to optic phonons, i.e., with antiphase oscillation of
spins in an elementary cell. Frequency of such oscillations,
which are called the exchange modes (EМs), is of the order
of exchange interaction energy and, in typical cases, per-
tain to THz and even optic frequency range. Excitation of
EMs by a magnetic component of light is possible only if
the former is magnetically-active, i.e., is coupled to oscilla-
tions of the ferromagnetic vector М of the system. Howev-
er, the coupling of EM and М is due to relativistic and ex-
change-relativistic interactions, and therefore the intensity
of absorption by EM is weak compared to AM. For non-
magnetic-active EMs, i.e., for those of EMs which are not
coupled with oscillations of the ferromagnetic vector М,
the interaction with magnetic field is impossible by virtue
of selection rules, for example, in the case of EMs which
are odd under inversion. The natural way to resonantly
excite magnetic-system EM which is odd under inversion,
is to expose on them to a high-frequency electric field.
The presence of electrically-active EMs in a multi-
sublattice magnet with magnetic ions located in noncent-
rosymmetric positions and appearance of an antiferroelec-
tric resonance (AFER), were theoretically predicted in
Refs. 63–70, while some remarks concerning such a possi-
bility can be found even in earlier publications [71,72].
(We keep here the term «antiferroelectric resonance» be-
cause the electric-dipole active EM is oscillation of the
antiferromagnetic vectors L and to distinguish this type of
oscillations from electromagnons in multiferroics.) It is
significant that in AFER the electric component of light is
directly connected with the vectors L. This is a main dis-
tinction to the case of multiferroics.
3.1. General conditions for the existence of AFER
Let us formulate the general conditions for electric-
dipole active magnetic oscillations in magnetic structures
whose ground state in the exchange approximation is colli-
near [64]. As is known, for symmetry analysis it is conve-
nient to introduce linear combinations of sublattice magne-
tizations Mζ (ζ = 1, 2, …, n); that are the ferromagnetism
M and antiferromagnetism Lν (ν = 1, 2, … n – 1) vectors.
In the general case the EM and AM are of electric-dipole
type if the dynamic compounds of the antiferromagnetism
and ferromagnetism vectors which correspond to them
transform just like the compounds of the electric polariza-
tion vector P. More specifically, in centrosymmetric crys-
tals it is necessary to distinguish between structures which
are even and odd under inversion.
3.1.1. The structure
_
1 (+). If a system whose structure
is even under inversion contains magnetic ions which are
not located at inversion centers, then in addition to the ba-
sic even antiferromagnetism vector L0, there exists at least
one other antiferromagnetism vector ℓ which is odd under
inversion. In the thermodynamic potential of such a mag-
net it is possible to have invariants of the form
Kαβγ Eα L0β ℓγ, α, β, γ = x, y, z; Kαβγ = const. (6)
In a static electric field, Eq. (6) leads to the appearance of
ℓγ ~ Eα. In analogy with the magnetoelectric effect this
phenomenon can be called antiferroelectric effect (AFEE).
In an ac electric field E ~ Eωexp(iωt), the relation (6)
causes oscillation of the vector ℓ, and for ω = ω0 (ω0 is the
exchange frequency) excitation of EM — an antiferroelec-
tric resonance. Such type of homogeneous excitation is
always decoupled with even under inversion excitations.
This specific of dynamics does not depend on external
magnetic field orientation and is common for all magnetic
crystals with even spin structure.
A direct indication of the possibility of experimental
observation of AFER is provided by experiments in which
an ac electric field induces transitions between magnetic
levels of paramagnetic ions in noncentrosymmetric sites,
i.e., transitions from a state (l, m) to states (l, m + 1) where
l is the orbital and m the magnetic quantum number. Such
transitions were observed in [2] (see also Refs. 3–5). As
we have shown here, in centrosymmetric crystals which
have high concentrations of magnetic ions and magnetic
structures which are even under inversion (e.g., of α-Fe2O3
type) one consequence of these transitions will be magnet-
ic excitations of exchange type. Thus, the physics of AFER
combines the physical mechanisms of electric-dipole ac-
tive paramagnetic resonance associated with impurity
magnetic ions in noncentrosymmetric position [3,73], and
of absorption and scattering of light in magnetically-
ordered crystals [74–76]. In systems with magnetic ions
occupying centers of inversion, the electric dipole activity
of the magnetic modes can be due to, e.g., the additional
effect of a constant electric field.
3.1.2. The structure
_
1 (–). For structures odd under in-
version, it is necessary to distinguish between two cases
according to parity relative to a translation τ.
i) Systems with
_
1 (–), τ(+) pertain to antiferromagnets
whose thermodynamic potentials include an invariant of
the form
Kαβγ Eγ L0α mβ, (7)
where mβ is the magnetic moment of the system. If in this
case also exists an antiferromagnetic vector ℓ which is even
under inversion, then it is also possible to have an invariant
of the form (6). This invariant gives rise to the presence of
AFER and AFEE, while (7) makes possible excitation of
AM by an electric field, i.e., electromagnons. As far as an
V.N. Krivoruchko
1026 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9
invariant ~ ℓαМβ is also possible, AM and EM will be ex-
cited by both magnetic and electric fields of light.
ii) In systems with
_
1 (–), τ(–) (the magnetic unit cell is
larger than the crystallographic unit cell) the invariant (7)
is forbidden and there is no static magnetoelectric effect.
Nevertheless, an invariant of the form (6) is possible,
where now the antiferromagnetic vector ℓ is odd under
translation and even under inversion. In this system there
will be both AFEE and AFER. However, in this magnetic
phase the electric-dipole active and magnetic-dipole-active
spin excitations are uncoupled in dynamics.
The essential features of AFER in noncentrosymme-
tric crystals will be illustrated with two examples: hema-
tite (α-Fe2O3, TN = 948 K) and chromium oxide (Cr2O3,
TN = 307 K), where the former possesses even under in-
version magnetic order, while the magnetic structure of the
latter is odd.
3.2. AFER in α-Fe2O3 and Cr2O3
Hematite and chromium (III) oxide are well known
four-sublattice rhombohedral antiferromagnets with the
crystallographic group R
_
3 c ≡ 6
3dD . Magnetic ions are in
fourfold position 4c{3z}. We introduce the following linear
combinations of the sublattice magnetizations Mζ:
M = M1 + M2 + M3 + M4 = 4 M0 m,
L1 = M1 − M2 – M3 + M4 = 4 M0 ℓ1,
L2 = M1 – M2 + M3 – M4 = 4 M0 ℓ2, (9)
L3 = M1 + M2 – M3 – M4 = 4 M0 ℓ3,
where M0 is the magnitude of the sublattice magnetization.
i) For α-Fe2O3, the classification of the vectors (9) rela-
tive to the irreducible representations of the group 6
3dD is
shown in Table 1. The vectors m and ℓ1 generate even, and
ℓ2 and ℓ3, odd, representations of this group. Using the ta-
ble, it is easy to obtain the spin system Hamiltonian H0
(see, e.g., Refs. 63, 64, 77). In linear approximation, the
vectors (9) homogeneous oscillations correspond to two
AMs and two EMs. The АМs are due to oscillation of even
regarding to inversion vectors m and ℓ1, and are coupled
with magnetic light component, i.e., are magnetically-
active oscillations. In contrast, the EMs are due to odd re-
garding to inversion vectors, ℓ2 and ℓ3, oscillation of and
thus cannot be coupled with magnetic light component, as
well as with the AMs. On the other hand, electric field
could be coupled with the vectors ℓ2 and ℓ3 (see Table 1),
i.e., EMs are electric-dipole active excitations. These prop-
erties of the AMs and EMs preserve both in easy-axis and
easy-plane phases.
When electric field Е is applied the system Hamiltonian
is given by H = H0 – Е·Р, where the additional component
Е·Р is expressed as [63,64]
(2M0)–1 E·P = Ex {R3 (ℓ1z ℓ3y – ℓ1y ℓ3z) +
+ R2 (ℓ1z ℓ2x – ℓ1x ℓ2z) + r3 (mx ℓ3z – mz ℓ3x)} +
+ Ey {R3 (ℓ1x ℓ3z – ℓ1z ℓ3x) + R2 (ℓ1z ℓ2y – ℓ1y ℓ2z) +
+ r3 (my ℓ3z – mz ℓ3y)} + Ez {π1 m ℓ3 + π2 ℓ1 ℓ2 +
+ R3z (ℓ1x ℓ3y – ℓ1y ℓ3x) + r2z (mx ℓ2y – my ℓ2x)}.
Here the antiferroelectric constants πi have their origin in
exchange, while Ri and ri are due to relativistic-exchange
effects; terms of relativistic origin are neglected. The mi-
croscopy origin of the invariants ~ (π1 m·ℓ3 + π2 ℓ1·ℓ2) is
due to the symmetric exchange strictions, Eq. (2), i.e., the
electric-dipole originates from dynamic polarization of
electronic bonds of two magnetic ions not connected by
inversion. The generation of other terms is provided by the
antisymmetric DM interaction ~ enm×(Sn × Sm).
Thus, for the easy-plane phase (ℓ1y = 1, H || x) we have,
to linear approximation in the spin deviations, for the com-
ponents of the electric polarization vector [64]:
Px = – 4M0 (R3 – r3 m) ℓ3z, Py = 4M0 R2 ℓ2z,
Pz = 4M0 [(π2 + r2z m) + (π1 m – R3z) ℓ3z].
Introducing the electric polarization tensor, Pi(ω) =
= αij(ω)Ejω, we find that the spin part of αij(ω) has the
following nonzero components:
αxx(ω) = – 4M0 γ2 2
3R (He1 + He2) 2 2 1
01( – ) ,−ω ω
αyy(ω) = – 4M0 γ2 2
2R (He1 + He3) 2 2 1
02( – ) ,−ω ω
αyz(ω) = – αzy(ω) = i 4M0 γ2 R2 [R3z + (π2 – π1) m]ω ×
2 2 1
02( – ) ,−× ω ω
αzz(ω) = 4M0 γ2 [R3z + (π2 – π1) m](π1 m – R3z) ×
× (He1 + He2) 2 2 1
02( – ) .−ω ω
Here Heν is the effective fields of exchange interactions;
the explicit expressions for EMs ω01/γ and ω02/γ are given
in Refs. 63, 64 (γ stands for gyromagnetic ratio).
Table I. Magnetic configurations which are irreducible rela-
tive to the crystallographic group 6
3 .dD
Irreducible
representation
Irreducible
spin configurations
Polarization of ac
electric and mag-
netic fields
A1g ℓ1z —
A2g mz Hz
Eg (m+, m–), (ℓ1 + ℓ2–) (H+, H–)
A1u ℓ3z —
A2u ℓ2z Ez
Eu (ℓ2+, ℓ2–),(ℓ3+, – ℓ3–) (E+, E–)
Electrically active magnetic excitations in antiferromagnets
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9 1027
Similar calculations for the easy-axis phase (ℓ1z = 1,
H || z) yield: Px = 4M0 (R3 ℓ3z + R2 ℓ2x), Py = 4M0 R2 ℓ2y,
and Pz = 4M0 π2 ℓ2z. Calculating the oscillation amplitudes
of the antiferromagnetic vectors ℓ2(t) and ℓ3(t) under the
action of electric field of light, one obtains the following
nonzero spin contributions to the electric tensor αij(ω):
αxx(ω) = αyy(ω) = 4M0 γ2 2
3{R (He1 + He2) +
+ 2
2R (He1 + He3)} ×
2 2 2 1 2 2 1
01 02 01 02 ( – ) ( – ) ( – ) ,− −× ω ω ω ω ω ω ω
αxy(ω) = – αyx(ω) = i 4M0 γ2 2
3{R (He1 + He2) +
+ 2
2R (He2 + He3)} ×
2 2 1 2 2 1
01 02 2 ( – ) ( – ) .H − −× ω ω ω ω ω
The explicit expressions for EM ω01/γ and ω02/γ in this
phase are given in Refs. 63, 64. Thus in hematite the ab-
sorption of electric component of light on EMs is enhanced
by the exchange fields Heν.
The magnitude of antiferroelectric constants for α-Fe2O3
can be estimated from experiments on the shift of the pa-
ramagnetic resonance lines for Fe3+ ions [3]. The latter
give a value of ~10–2 for the single-ion spin-Hamiltonian
constant. In systems with a high concentration of magnetic
ions, contributions to the magnetoelectric effects come also
from ion–ion interactions (in particular from exchange and
relativistic-exchange interactions), which in individual
cases increase the value of the constant by an order of mag-
nitude [78]. Therefore in α-Fe2O3 (apparently) Ri ~ 10–1.
For the value of π it is reasonable to expect an order of
magnitude larger. It should also be noted that the contribu-
tion from the exchange origin invariants is proportional to
the magnetization; therefore, in the canted phases their
effect can significantly exceed those from the exchange-
relativistic invariants.
(ii) The crystal Cr2O3 possesses a magnetic structure
which is odd under inversion [77]. The magnetic class of
Cr2O3 includes inversion combines with the time-reversal
operation
_
,IR and this result in a linear magnetoelectric
effect [79–81]. The ground state is ℓ3 || z. In the exchange
approximation the acoustic type of oscillations correspond
to transverse oscillations of the vectors ℓ3 and m, while
those of exchange type correspond to transverse oscillation
of the vectors ℓ1 and ℓ2. Due to the Dzyaloshinksii interac-
tion, HD, the modes are dynamically coupled. Without
pausing for detailed calculations we will present the final
results [64]. In linear approximation, for the electric pola-
rization we have: Px = 4M0 (r3mx – R3ℓ1y) and Py = 4M0×
× (r3my + R3ℓ1x). The nonzero spin contribution to the
electric polarization tensor at АМs, ωA1/γ = εA + H and
ωA2/γ = εA – H, is described by the expressions
αxx(ω) = αyy(ω) = 4M0 γ2 2
3r {HA3 – 2
DH (He2 + He3)–1} ×
2 2 2 2 2 1 2 2 1
1 2( – + ) ( – ) ( – ) ,A A AH − −× ω ε ω ω ω ω
αxy(ω) = – αxy(ω) = i 4M0 γ2 2
3r {HA3 – 2
DH (He2 + He3)–1} ×
× ω H 2 2 1 2 2 1
1 2( – ) ( – ) .A A
− −ω ω ω ω
Within the same accuracy, the αij(ω) nonzero component
structure at EМs, ω01/γ = ε0 + H and ω02/γ = ε0 – H, has
the form
αxx(ω) = αyy(ω) = 4M0 γ2 2
3R (He2 + He3) ×
2 2 2 2 2 1 2 2 1
0 01 02( – ) ( – ) ( – ) ,H − −× ω ε + ω ω ω ω
αxy(ω) = – αyx(ω) = – i 4M0 γ2 2
3R (He2 + He3) ×
× ω H 2 2 1 2 2 1
01 02( – ) ( – ) .− −ω ω ω ω
The explicit expressions for EM and EM at zero magnetic
field, εА and ε0, are given in Ref. 64.
Since the vectors (ℓ1,m), which are even under inversion,
and the vectors (ℓ2,ℓ3), which are odd under inversion, par-
ticipate in the EM and AM oscillations, these modes can be
excited both by magnetic and electric components of light.
However, from an experimental point of view the impor-
tant is the magnitude of the absorption by the EM and AM.
The estimations of the magnetic χij(ω) and electric αij(ω)
susceptibility near the resonance frequency yield: α(ω0) ~
~ γ M0 R2
(ω – ω0)–1, χ(ω0) ~ γM0 (HA/He) (ω – ω0)–1 for
EМ, and α(ωA) ~ γ M0 R2
(HA/He)1/2 (ω – ωA)–1, χ(ωA) ~
~ γ M0 (HA/He)1/2 (ω – ωA)–1 for АМ. Thus, the intensity of
the absorption an electric field of light by EM is
α(ω0)/α(ωΑ) ~ (HA/He)1/2 times large that the absorption
by АМ. For magnetic field of light the situation is reverse:
χ(ω0)/χ(ωA) ~ (HA/He)1/2. It is also clear that for R2 >
> HA/He, the excitation of EM by electric field of light is
easier than by magnetic field.
For quantitative estimates of the antiferroelectric inte-
raction constants in Cr2O3 we make use of the results of
experimental and theoretical studies of the magnetoelectric
effect in this compound [78,82]. The latter give for the pa-
rameters of the spin Hamiltonian the value π ≈ 0.5, R ≈ 0.05.
Taking into account that HA/He ~ 10–4 in Cr2O3, the condi-
tions for detection of EM as peak in ε(ω) is more favorable
than those in μ(ω).
The EM frequencies of α-Fe2O3 and Cr2O3 are located
in the infrared wavelength band. Therefore the features of
experimental observation of AFER in these compounds are
close to those at THz frequencies [15] or optics [75,76]. If
the dimensions of the sample are comparable with or smaller
than the wavelength of the ac electric field, then the results
above is directly applicable. For bulk samples of α-Fe2O3
and Cr2O3, the equations of motion for the vectors (9) must
be considered jointly with the Maxwell equations. In this
case, the magnetooptic effects will be restructured. In partic-
ular, near the exchange resonances it is possible to resonant-
ly enhance such magnetooptic effects as Faraday rotation,
the Cotton–Mouton effect, and the Kerr effect. A detailed
analysis of these questions is outside the framework of the
present report. The reader is referred to Refs. 66–70.
V.N. Krivoruchko
1028 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9
4. Electric-dipole active magnetic resonances in
ferrimagnets
The above considered systems are antiferromagnets,
i.e., without spontaneous ferromagnetic moment. However,
the most interesting materials for both future applications
as well as the quest for fundamental principles are those
where ferromagnetic magnetization and ferroelectric pola-
rization are coexisting. A sole to date such type of com-
pound where electromagnons are experimentally revealed
is hexaferrite Ba2Mg2Fe12O22 [83,84].
As is known [85], at room temperature magnetic struc-
ture of Ba2Mg2Fe12O22 can be considered as two antifer-
romagnetically ordered sublattices L and S. Below 195 K
this structure transforms into the proper spiral with spiral
axis [001]. Then, below 50 K, the ordered conical-spin
phase appears with finite conical angle θ for each sublat-
tices. Kida et al. [83] observed a noticeable optical aniso-
tropy in the THz spectra for the ordered conical-spin
phase. The results obtained are presented in Figs. 7(a) and
7(b) where, respectively, the real (ε1) and imaginary (ε2)
parts of the complex dielectric permittivity at 5 K are
shown for different polarization configurations. A gigantic
sharp resonance around 2.8 meV in ε2(ω) and a dispersive
structure in ε1(ω) are clearly seen, when electric and mag-
netic fields polarizations of the light were set parallel to
Eω || [001] and Hω || [100]. Based on the complementary
measurements of the magnon dispersion (the inelastic
neutron scattering spectra and temperature dependence of
the complex dielectric constant) the authors [84] firmly
identified that the observed electric-dipole active mode
around 2.8 meV is magnetic in origin. In the configura-
tion Eω || [120] and Hω || [100] another resonance can be
discerned around 8 meV [Fig. 7(b)]. The magnitude of
this resonance is much smaller than that of 2.8 meV. The
mode at 8 meV disappears for Eω || [001]. Therefore, this
mode can be assigned also to an electric-dipole active
mode, while allowed only for the in-plane Eω.
It was demonstrated [83,84] that both longitudinal and
transverse conical-spin structures can give rise to electric-
dipole active magnetic resonance, irrespective of the pres-
ence or absence of spontaneous electric polarization.
Therefore, contrary to the origin of the ferroelectricity, i.e.,
the antisymmetric DM interaction (Sn × Sm), the symme-
tric exchange interaction (2) acting along [001] is likely the
dominant source of the observed electromagnons in this
hexaferrite.
Obviously, an interesting system is rare-earth garnets
with the lattice of Y3Fe5O12. For compound Dy3Fe5O12,
the attempt to find electromagnons was performed recently
in Ref. 86. Far-infrared spectra of magneto-dielectric
Dy3Fe5O12 garnet were studied between 13 and 100 cm–1
and at low temperatures between 5 and 80 K. A combina-
tion of transmission, reflectivity, and rotating analyzer el-
lipsometry was used to identify the type of the dipole ac-
tivity of the infrared modes. In addition to purely dielectric
and magnetic modes, the authors observed several hybrid
modes with a mixed magnetic and electric dipole activity.
Microscopically, these modes can originate from the super-
exchange interaction between magnetic moments of Fe and
Dy ions. It was speculated that the proximity of the Dy3+
exchange resonances to the frequency of the lowest optical
phonon, local electric polarization, and the noncollinear
spin structure for the Dy–Fe magnetic system are responsi-
ble for the mode hybridization. Further experimental and
theoretical investigations are needed to understand unam-
biguously these results.
Among the compounds with garnet lattice a special place
take those where magnetic ions occupy only rare-earth posi-
tions {c}. Examples of such garnets are Tb3Ga5O12 and
Dy3Al5O12. At low temperature (Т < 1 K) the magnetic
moments of rare-earth ions ordered forming noncollinear
magnetic structure called the “kagome structure”, when
local easy-axis (and the spins of ions) are directed along
three mutually orthogonal directions. Magnetic-strictions
properties of such garnet were considered in [87]. Investi-
gation the dynamical properties of these interesting sys-
tems is just in the beginning [88–90]. Recent advance in
Fig. 7. Electric-dipole active magnetic resonance in Ba2Mg2Fe12O22.
Real ε1 (a) and imaginary ε2 (b) parts of the complex dielectric
constant spectra ε(ω) measured in the ordered conical spin phase;
T = 5 K, H = 0. (Reproduced from Refs. 83, 84.)
16
20
0 0.5 1.0 1.5 2.0
Frequency, THz
0 2 4 6 8 10
1
2
3
4
5
6
1 E— || [001],� H || [100]�
(b)
(a)
Photon energy, meV
1
2
3
4
4 E— || [120], H || [100]� �
3 E— || [100], H || [120]� �
2 E— || [100], H || [001]� �
��
��
1
2
3
4
Electrically active magnetic excitations in antiferromagnets
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 9 1029
THz time-domain spectroscopy may stimulate the investi-
gations on a variety of kagome spin structure magnets to
uncover the electric-dipole active magnetic resonances in
these compounds.
Just recently a new ferromagnetic ferroelectric Mn2GeO4
has been revealed in experiments [91]. The compound is
shown to feature both a ferroelectric polarization and a
ferromagnetic magnetization that are directly coupled and
point along the same direction. The authors showed that a
spin spiral generates ferroelectricity, and a canted com-
mensurate order leads to weak ferromagnetism. Symmetry
suggests that the direct coupling between the ferromagnet-
ism and ferroelectricity is mediated by DM interactions
that exist only in the ferroelectric phase, controlling both
the spiral rotation and the canting of the commensurate
structure. The coupling between ferromagnetism and fer-
roelectricity permits to expect the existence of electric-
dipole active magnetic excitations in this system.
As for theoretical models, a detailed analysis of electro-
magnons in ferrimagnets was carried out only for the case
of tetragonal Mn2Sb [92,93]. Neutron diffraction experi-
ments revealed that magnetic structure of Mn2Sb can be
imagined as two ferromagnetically ordered (001) planes of
MnI and MnII ions with different magnetic moments. The
magnetic moments of MnI and MnII layers are antiparallel
and are not coupled by inversion (two sublattices a and c
subsystems). As it follows from the theoretical considera-
tions [92,93], in this ferrimagnet the electric-dipole active
excitations are vibrations of odd regarding to inversion
antiferromagnetic vectors (as in the case of α-Fe2O3).
There are two proper frequencies of electromagnons cor-
responding two types of the a and c sublattices magnetiza-
tion oscillation. As predicted, these elementary excitations
could be detected as magnetic resonances in the dielectric
susceptibility αij(ω) of the system. Undoubtedly, the expe-
rimental testing is needed to verify the existence of the
electric-dipole active magnons in Mn2Sb. Unfortunately,
the compound demonstrates a metallic-type conductivity,
which will cause additional difficulties for detection of
electromagnons.
5. Conclusions
In the above sections, only a little part of systems with
electric-dipole active magnetic excitations has been re-
viewed. The author omitted the discussion of multiferroic
properties and, accordingly, electromagnons modes de-
tected in systems where the coupling mechanism between
magnetic and electric properties is not uniquely identified
and is under discussion. Concerning theoretical investiga-
tions, electromagnons in rare-earth orthoferrites RFeO3
and orthochromites RCrO3 crystallizing into crystallo-
graphic group 16
2hD Pbnm= are also predicted. In these
compounds, a primitive magnetic cell contains four sublat-
tices of d-subsystem (Fe or Cr) and four rare-earth sublat-
tices, and thus, any consistent description of these mate-
rials would be rather cumbersome. The interested reader is
referred to Refs. 65, 89. The discussion of electric-dipole
active nuclear magnetic excitations can be found in the
original reports [93,94].
In conclusion, recent rediscoveries of the so-called mul-
tiferroic materials, where the magnetic order and ferroelec-
tric polarization coexist, have led to a surge of interest in
this rather unusual class of materials. Intense research ac-
tivities on these systems are partly motivated by the pure
desire to understand the fundamental questions. At the
same time, the immense potential for future applications,
e.g., in spin wave electronics [95,96], have made these
multiferroic compounds one of the most sought-after re-
cent topics in material science. The common studies of the
magnon excitations are based on coupling between mag-
netic system and magnetic components of microwave field.
However, this method is effective only for excitation of
long-wave (acoustic) vibrations. Only recently the possibil-
ity to generate the exchange magnetic excitations was ex-
perimentally demonstrated for Y3Fe5O12 by method of
parametric pumping [97]. In contrast, magnetic resonances
driven by the electric component of field give a direct
mode to effectively control of the magnetic sate by exter-
nal electric field and to transform one type of excitation
into another exactly in terahertz time-domain. Those are of
crucial importance for the magnon-based spintronics de-
vices as only short-wavelength exchange magnons allow
signal processing at nanoscales.
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