University of Groningen
Multiferroics
Cheong, Sang-Wook; Mostovoy, Maxim
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10.1038/nmat1804
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Cheong, S-W., & Mostovoy, M. (2007). Multiferroics: a magnetic twist for ferroelectricity.
Nature Materials
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(1), 13-20. https://doi.org/10.1038/nmat1804
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nature materials | VOL 6 | JANUARY 2007 | www.nature.com/naturematerials 13
REVIEW ARTICLE
SANG-WOOK CHEONG
1,2
AND MAXIM MOSTOVOY
3
1
Rutgers Center for Emergent Materials and Department of Physics &
Astronomy, 136 Frelinghuysen Road, Piscataway 08854, New Jersey, USA.
2
Laboratory of Pohang Emergent Materials and Department of Physics, Pohang
University of Science and Technology, Pohang 790-784, Korea.
3
Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG
Groningen, The Netherlands.
e-mail: sangc@physics.rutgers.edu; M.Mostovoy@rug.nl
In 1865, James Clerk Maxwell proposed four equations governing the
dynamics of electric elds, magnetic elds and electric charges, which
are now known as Maxwell’s equations
1
. ey show that magnetic
interactions and motion of electric charges, which were initially
thought to be two independent phenomena, are intrinsically coupled
to each other. In the covariant relativistic form, they reduce to just two
equations for the electromagnetic eld tensor, succinctly re ecting the
uni ed nature of magnetism and electricity
2
. A number of interesting
parallels exist between electric and magnetic phenomena, such as the
quantum scattering of charge o magnetic ux (Aharonov–Bohm
e ect
3
) and the scattering of magnetic dipoles o a charged wire
(Aharonov–Casher e ect
4
). e formal equivalence of the equations
of electrostatics and magnetostatics in polarizable media explains
numerous similarities in the thermodynamics of ferroelectrics and
ferromagnets, for example their behaviour in external elds, anomalies
at a critical temperature, and domain structures. ese similarities
are particularly striking in view of the seemingly di erent origins
of ferroelectricity and magnetism in solids: whereas magnetism is
related to ordering of spins of electrons in incomplete ionic shells,
ferroelectricity results from relative shi s of negative and positive
ions that induce surface charges.
Magnetism and ferroelectricity coexist in materials called
multiferroics. e search for these materials is driven by the prospect
of controlling charges by applied magnetic elds and spins by applied
voltages, and using this to construct new forms of multifunctional
devices. Much of the early work on multiferroics was directed towards
bringing ferroelectricity and magnetism together in one material
5
.
is proved to be a di cult problem, as these two contrasting order
parameters turned out to be mutually exclusive
6–10
. Furthermore, it
was found that the simultaneous presence of electric and magnetic
dipoles does not guarantee strong coupling between the two, as
microscopic mechanisms of ferroelectricity and magnetism are quite
di erent and do not strongly interfere with each other
11,12
.
e long-sought control of electric properties by magnetic
elds was recently achieved in a rather unexpected class of
materials known as ‘frustrated magnets’, for example the perovskites
RMnO
3
, RMn
2
O
5
(R: rare earths), Ni
3
V
2
O
8
, delafossite CuFeO
2
,
spinel CoCr
2
O
4
, MnWO
4
, and hexagonal ferrite (Ba,Sr)
2
Zn
2
Fe
12
O
22
(refs 13–20). Curiously, it is not the strength of the magnetoelectric
coupling or high magnitude of electric polarization that makes these
materials unique; in fact, the coupling is weak, as usual, and electric
polarization is two to three orders of magnitude smaller than
in typical ferroelectrics. e reason for the high sensitivity of the
dielectric properties to an applied magnetic eld lies in the magnetic
origin of their ferroelectricity, which is induced by complex spin
structures, characteristic of frustrated magnets
15,21–27
. Recent reviews
of this rapidly developing eld can be found in refs 28–30. Here, we
mainly focus on the relationship between magnetic frustration and
ferroelectricity, discuss di erent types of multiferroic materials and
mechanisms inducing electric polarization in magnetic states, and
outline the directions of the future research in this eld.
PROPER AND IMPROPER FERROELECTRICS
Why is it di cult to nd materials that are both ferroelectric and
magnetic
8,10,31
? Most ferroelectrics are transition metal oxides, in
which transition ions have empty d shells. ese positively charged
ions like to form ‘molecules’ with one (or several) of the neighbouring
negative oxygen ions. is collective shi of cations and anions inside
a periodic crystal induces bulk electric polarization. e mechanism
of the covalent bonding (electronic pairing) in such molecules is
the virtual hopping of electrons from the lled oxygen shell to the
Multiferroics: a magnetic twist for
ferroelectricity
Magnetism and ferroelectricity are essential to many forms of current technology, and the quest for
multiferroic materials, where these two phenomena are intimately coupled, is of great technological
and fundamental importance. Ferroelectricity and magnetism tend to be mutually exclusive and interact
weakly with each other when they coexist. The exciting new development is the discovery that even a
weak magnetoelectric interaction can lead to spectacular cross-coupling effects when it induces electric
polarization in a magnetically ordered state. Such magnetic ferroelectricity, showing an unprecedented
sensitivity to ap plied magnetic fi elds, occurs in ‘frustrated magnets’ with competing interactions between
spins and complex magnetic orders. We summarize key experimental fi ndings and the current theoretical
understanding of these phenomena, which have great potential for tuneable multifunctional devices.
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REVIEW ARTICLE
empty d shell of a transition metal ion. Magnetism, on the contrary,
requires transition metal ions with partially lled d shells, as the
spins of electrons occupying completely lled shells add to zero and
do not participate in magnetic ordering. e exchange interaction
between uncompensated spins of di erent ions, giving rise to long-
range magnetic ordering, also results from the virtual hopping of
electrons between the ions. In this respect the two mechanisms are
not so dissimilar, but the di erence in lling of the d shells required
for ferroelectricity and magnetism makes these two ordered states
mutually exclusive.
Still, some compounds, such as BiMnO
3
or BiFeO
3
with magnetic
Mn
3+
and Fe
3+
ions, are ferroelectric. Here, however, it is the Bi ion
with two electrons on the 6s orbital (lone pair) that moves away
from the centrosymmetric position in its oxygen surrounding
32
.
Because the ferroelectric and magnetic orders in these materials are
associated with di erent ions, the coupling between them is weak. For
example, BiMnO
3
shows a ferroelectric transition at T
FE
≈ 800 K and a
ferromagnetic transition at T
FM
≈ 110 K, below which the two orders
coexist
12
. BiMnO
3
is a unique material, in which both magnetization
and electric polarization are reasonably large
12,33,34
. is, however, does
not make it a useful multiferroic. Its dielectric constant ε shows only
a minute anomaly at T
FM
and is fairly insensitive to magnetic elds:
even very close to T
FM
, the change in ε produced by a 9-T eld does not
exceed 0.6%.
In the ‘proper’ ferroelectrics discussed so far, structural instability
towards the polar state, associated with the electronic pairing, is the
main driving force of the transition. If, on the other hand, polarization
is only a part of a more complex lattice distortion or if it appears as
an accidental by-product of some other ordering, the ferroelectricity
is called ‘improper’
35
(see Table 1). For example, the hexagonal
manganites RMnO
3
(R = Ho–Lu, Y) show a lattice transition which
enlarges their unit cell. An electric dipole moment, appearing below
this transition, is induced by a nonlinear coupling to nonpolar
lattice distortions, such as the buckling of R–O planes and tilts of
manganese–oxygen bipyramids (geometric ferroelectricity)
11,31,36
.
Another group of improper ferroelectrics, discussed recently,
are charge-ordered insulators. In many narrowband metals with
strong electronic correlations, charge carriers become localized at
low temperatures and form periodic superstructures. e celebrated
example is the magnetite Fe
3
O
4
, which undergoes a metal–insulator
y
z
P
P
a b
c d
Fe
3+
LuFe
2
O
4
Fe
2+
YNiO
3
Ni
3–δ
Ni
3+δ
x
P
P
Figure 1 Ferroelectricity in charge-ordered systems. Red/blue spheres correspond to cations with more/less positive charge. a, Ferroelectricity induced by simultaneous
presence of site-centred and bond-centred charge orders in a chain (site-centred charges and dimers formed on every second bond are marked with green dashed
lines). b, Polarization induced by coexisting site-centred charge and ↑↑↓↓ spin orders in a chain with the nearest-neighbour ferromagnetic and next-nearest-neighbour
antiferromagnetic couplings. Ions are shifted away from centrosymmetric positions by exchange striction. c, Charge ordering in bilayered Lu(Fe
2.5+
)
2
O
4
with a triangular lattice
of Fe ions in each layer. The charge transfer from the top to bottom layer gives rise to net electric polarization. d, Possible polarization induced by charge ordering and the
↑↑↓↓-type spin ordering in the a–b plane of perovskite YNiO
3
.
Table 1 Classifi cation of ferroelectrics
Mechanism of inversion symmetry breaking Materials
Proper Covalent bonding between 3d
0
transition metal
(Ti) and oxygen
BaTiO
3
Polarization of 6s
2
lone pair of Bi or Pb BiMnO
3
, BiFeO
3
,
Pb(Fe
2/3
W
1/3
)O
3
Improper Structural transition
‘Geometric ferroelectrics’
K
2
SeO
4
, Cs
2
CdI
4
hexagonal RMnO
3
Charge ordering
‘Electronic ferroelectrics’
LuFe
2
O
4
Magnetic ordering
‘Magnetic ferroelectrics’
Orthorhombic RMnO
3
,
RMn
2
O
5
, CoCr
2
O
4
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REVIEW ARTICLE
nature materials | VOL 6 | JANUARY 2007 | www.nature.com/naturematerials 15
transition at ~125 K (the Verwey transition) with a rather complex
pattern of ordered charges of iron ions
37
. When charges order in
a non-symmetric fashion, they induce electric polarization. It has
been suggested that the coexistence of bond-centred and site-centred
charge orders in Pr
1–x
Ca
x
MnO
3
leads to a non-centrosymmetric
charge distribution and a net electric polarization
38
(see Fig. 1a).
A polar lattice distortion induced by charge ordering has been
reported in the bilayer manganite Pr(Sr
0.1
Ca
0.9
)
2
Mn
2
O
7
, which also
shows an interesting reorientation transition of orbital stripes
39
.
Charge ordering in LuFe
2
O
4
, crystallizing in a bilayer structure,
also appears to induce electric polarization. e average valence
of Fe ions in LuFe
2
O
4
is 2.5+, and in each layer these ions form a
triangular lattice. As suggested in ref. 40, the charge ordering below
~350 K creates alternating layers with Fe
2+
:Fe
3+
ratios of 2:1 and 1:2,
inducing net polarization (see Fig. 1c).
FERROELECTRICITY IN FRUSTRATED MAGNETS
Naturally, improper ferroelectricity puts lower constraints on the
coexistence with magnetism. In fact, materials with electric dipoles
induced by magnetic ordering are the best candidates for useful
multiferroics, because such dipoles are highly tuneable by applied
magnetic elds. e current revolution in the eld of multiferroics
began with the discovery of the high magnetic tuneability of
electric polarization and dielectric constant in the orthorhombic
rare-earth manganites Tb(Dy)MnO
3
and Tb(Dy)Mn
2
O
5
(refs
13,14,41,42). e onset of ferroelectricity in TbMnO
3
clearly
correlates with the appearance of spiral magnetic ordering at
~28 K (ref. 43). In applied magnetic elds, Tb(Dy)MnO
3
shows a
spin- op transition, at which the polarization vector rotates by 90°
(see Fig. 2a) and the dielectric constant ε (in DyMnO
3
) increases
a
b
c
d
P
n
P
n
H
TbMn
2
O
5
Time (s)
2,000 1,000 0
0
1
2
H (T)
P (nC cm
2
)
–40
–20
20
20
–20
–40
02468
0
40
40
0
H (T)
40
7 T
8 T
9 T
5 T
3 T
0 T
35
30
25
ε
b
20
0
0
2 K
4 K
9 K
12 K
15 K
18 K
21 K
0
150
300
450
600
DyMnO
3
H
b
(T)
1234567
10 20 30 40 50
T (K)
H//a
DyMn
2
O
5
600
400
1
2
2
3
1
200
0 0
200
400
600
0 3 6
H
b
(T)
ΔP
c
(μC m
–2
)
Δε
a
(%)
ΔP
a
(μC m
–2
)
9
T = 9 K
TbMnO
3
P
c
P
a
b
a
c
Figure 2 High magnetic tunability of magnetic ferroelectrics. a, Electric polarization in perovskite TbMnO
3
versus magnetic fi eld along the b axis
13
. The magnetic fi eld of ~5 T
fl ips the direction of electric polarization from the c axis to the a axis. Numbers show the sequence of magnetic fi eld variation. b, Dielectric constant ε along the a axis versus
magnetic fi eld along the b axis at various temperatures in perovskite DyMnO
3
. The sharp peak in ε(H) accompanies the fl ipping of electric polarization from the c axis to a axis.
c, The highly reversible 180° fl ipping of electric polarization along the b axis in TbMn
2
O
5
can be activated by applying magnetic fi elds along the a axis
14
. d, The temperature
dependence of ε along the b axis in DyMn
2
O
5
in various magnetic fi elds. The magnitude of the step-like increase of ε below ~25 K is strongly fi eld-dependent. Parts b and d
are reprinted with permission from refs 42 and 41, respectively. Copyright (2004) APS.
nmat1804 Cheong Review.indd 15nmat1804 Cheong Review.indd 15 11/12/06 10:29:2011/12/06 10:29:20
16 nature materials | VOL 6 | JANUARY 2007 | www.nature.com/naturematerials
REVIEW ARTICLE
by ~500% in a narrow eld range. is colossal magneto-dielectric
e ect is shown in Fig. 2b.
Many of the rare-earth manganites RMn
2
O
5
, where R denotes
rare earths from Pr to Lu, Bi and Y, show four sequential magnetic
transitions: incommensurate sinusoidal ordering of Mn spins
at T
1
= 42–45 K, commensurate antiferromagnetic ordering at
T
2
= 38–41 K, re-entrance transition into the incommensurate
sinusoidal state at T
3
= 20–25 K, and nally, an ordering of rare-
earth spins below T
4
≤ 10 K (refs 44–49). Ferroelectricity sets in at T
2
and gives rise to a peak in ε at this magnetic transition (see Fig. 3d).
Furthermore, ε in DyMn
2
O
5
shows a remarkably strong dependence
on magnetic elds below T
3
(see Fig. 2d)
41
. e magnetic eld rotates
the electric polarization of TbMn
2
O
5
by 180° in a highly reversible
way
14
, as shown in Fig. 2c.
Complex magnetic structures and phase diagrams are observed
in all multiferroics showing strong interplay between magnetic and
dielectric phenomena
13–20
. All these materials are ‘frustrated’ magnets,
in which competing interactions between spins preclude simple
magnetic orders. e disordered paramagnetic phase in frustrated
magnets extends to unusually low temperatures. For example, the
Curie–Weiss temperature, T
CW
, of YMn
2
O
5
, obtained by tting its
magnetic susceptibility χ with the high-temperature asymptotic,
χ ≈ (C/(T+T
CW
), is ~250 K (see Fig. 3b). e temperature T
CW
re ects
the strength of interaction between spins, and in usual magnets it
gives a good estimate of the spin ordering temperature. e fact that
the long-range magnetic order sets in at T
1
≈ 45 K, which is about
ve times smaller than T
CW
, is clear indication for the presence of
signi cant magnetic frustration in YMn
2
O
5
.
HOW MAGNETIC SPIRALS INDUCE FERROELECTRICITY
e key questions are how it is possible that magnetic ordering
can induce ferroelectricity and what the role of frustration is.
e coupling between electric polarization and magnetization is
governed by the symmetries of these two order parameters, which
are very di erent. e electric polarization P and electric eld E
change sign on the inversion of all coordinates, r → –r, but remain
invariant on time reversal, t → –t. e magnetization M and
magnetic eld H transform in precisely the opposite way: spatial
inversion leaves them unchanged, whereas the time reversal changes
sign. Because of this di erence in transformation properties, the
linear coupling between (P, E) and (M, H) described by Maxwell’s
equations is only possible when these vectors vary both in space
and in time: for example, spatial derivatives of E are proportional
to the time derivative of H and vice versa.
e coupling between static P and M can only be nonlinear.
Nonlinear coupling results from the interplay of charge, spin, orbital
and lattice degrees of freedom. It is always present in solids, although it
is usually weak. Whether it can induce polarization in a magnetically
ordered state crucially depends on its form. A small energy gain
proportional to –P
2
M
2
does not induce ferroelectricity, because it
is overcompensated by the energy cost of a polar lattice distortion
proportional to +P
2
. is fourth-order term accounts for small
changes in dielectric constant below magnetic transition, observed
for example in YMnO
3
and BiMnO
3
(refs 11,12). If magnetic ordering
is inhomogeneous (that is, M varies over the crystal), symmetries
also allow for the third-order coupling of PM∂M. Being linear
in P, arbitrarily weak interaction of this type gives rise to electric
polarization, as soon as magnetic ordering of a proper kind sets in.
For cubic crystals, the allowed form of the magnetically induced
electric polarization is
22,23,26
P ∝ [(M · ∂)M – M(∂ · M)] . (1)
is is where frustration comes into play. Its role is to induce spatial
variations of magnetization. e period of magnetic states in
frustrated systems depends on strengths of competing interactions
and is o en incommensurate with the period of crystal lattice. For
example, a spin chain with a ferromagnetic interaction J < 0 between
neighbouring spins has a uniform ground state with all parallel spins.
An antiferromagnetic next-nearest-neighbour interaction J´ > 0
frustrates this simple ordering, and when su ciently strong, stabilizes
a spiral magnetic state (see Fig. 4a):
S
n
= S [e
1
cosQx
n
+ e
2
sinQx
n
] , (2)
where e
1
and e
2
are two orthogonal unit vectors and the wavevector
Q is given by
cos(Q/2) = –J´/(4J).
Like any other magnetic ordering, the magnetic spiral
spontaneously breaks time-reversal symmetry. In addition it
breaks inversion symmetry, because the change of the sign of all
coordinates inverts the direction of the rotation of spins in the spiral.
us, the symmetry of the spiral state allows for a simultaneous
T
N
T
3
T
FE
H//a
H//c
1.2
1.4
1.6
1.8
2.0
2.2
χ (10
–2
e.m.u.
per mole)
T
N
T
FE
H//b
H//a
H//c
1.5
ε
a
27
30
33
36
39
42
45
12
14
16
18
20
22
ε
b
010 3020 40 50
0204060
T (K)T (K)
T (K)
0 100 200 300
T (K)
0 100 200 300
0
30
60
90
1/χ (mole per e.m.u.)
0
30
60
90
120
1/χ (mole per e.m.u.)
ba
dc
2.5
3.0
2.0
χ (10
–2
e.m.u.
per mole)
Figure 3 Temperature dependence of the inverse magnetic susceptibility of
magnetically frustrated materials. a, (Eu
0.75
Y
0.25
)MnO
3
and b, YMn
2
O
5
both have
magnetic transition temperatures (T
N
) signifi cantly lower than the corresponding
Curie–Weiss temperatures. c,d, Anisotropic magnetic susceptibility as well as the
dielectric constant along the ferroelectric polarization direction for (Eu
0.75
Y
0.25
)MnO
3
(c)
and YMn
2
O
5
(d). The sharp peak of the dielectric constant curve indicates the onset of
a ferroelectric transition. The data show that in (Eu
0.75
Y
0.25
)MnO
3
, the collinear magnetic
state with the magnetic easy b axis is paraelectric, whereas the one with the easy
a–b plane (magnetic spiral) is ferroelectric with electric polarization along the a axis. In
ferroelectric YMn
2
O
5
, spins have tendency to orient along the a axis, but rotate slightly
from the a axis to the b axis on cooling at the temperature at which the dielectric
constant shows a step-like increase. (Data are from Y. J. Choi, C. L. Zhang, S. Park and
S.-W. Cheong, manuscript in preparation.)
nmat1804 Cheong Review.indd 16nmat1804 Cheong Review.indd 16 11/12/06 10:29:2011/12/06 10:29:20
![Figure 4 Frustrated spin chains with the nearest-neighbour FM and nextnearest-neighbour AFM interactions J and J ´ . a, The spin chain with isotropic (Heisenberg) HH = Σn [J Sn · Sn+1 + J ´Sn · Sn+2]. For J ´ /|J | > 1/4 its classical ground state is a magnetic spiral. b, The chain of Ising spins σn = ±1, with energy HI = Σn [J σnσn+1 + J ´σnσn+2] has the up–up–down–down ground state for J ´ /|J | > 1/2.](/figures/figure-4-frustrated-spin-chains-with-the-nearest-neighbour-2jp89q0i.webp)
![Figure 5 Effects of the antisymmetric Dzyaloshinskii–Moriya interaction. The interaction HDM = D12 · [S1 × S2]. The Dzyaloshinskii vector D12 is proportional to spin-orbit coupling constant λ, and depends on the position of the oxygen ion (open circle) between two](/figures/figure-5-effects-of-the-antisymmetric-dzyaloshinskii-moriya-uj537f4x.webp)
