Recent Progress in the Thermodynamics of Ferrotoroidic Materials

Abstract

Recent theoretical and experimental progress on the study of ferrotoroidic
materials is reviewed. The basic field equations are first described and then
the expressions for magnetic toroidal moment and toroidization are derived.
Relevant materials and experimental observation of magnetic toroidal moment and
toroidal domains are summarized next. The thermodynamics of such magnetic
materials is discussed in detail with examples of ferrorotoidic phase
transition studied using Landau modelling. Specifically, an example of
application of Landau modelling to the study of toroidocaloric effect is also
provided. Recent results of polar nanostructures with electrical toroidal
moment are finally reviewed.

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Multiferr. Mater. 2014; 1:9–22
Research Article Open Access
Antoni Planes, Teresa Castán, and Avadh Saxena
Recent progress in the thermodynamics of
ferrotoroidic materials
Abstract: Recent theoretical and experimental progress on
the study of ferrotoroidic materials is reviewed. The basic
eld equations are rst described and then the expressions
for magnetic toroidal moment and toroidization are de-
rived. Relevant materials and experimental observation of
magnetic toroidal moment and toroidal domains are sum-
marized next. The thermodynamics of such magnetic ma-
terials is discussed in detail with examples of ferrorotoidic
phase transition studied using Landau modelling. Specif-
ically, an example of application of Landau modelling to
the study of toroidocaloric eect is also provided. Recent
results of polar nanostructures with electrical toroidal mo-
ment are nally reviewed.
DOI 10.2478/muma-2014-0002
Received ; accepted
1 Introduction
According to the common point of view, ferroelastic, fer-
roelectric and ferromagnetic materials constitute the fam-
ily of ferroic materials [1]. More recently ferrotoroidic ma-
terials have also been included in this family [2, 3]. Ferro-
toroidics describe materials where toroidal moments show
cooperative long range order. Ferrotoroidic materials in-
trinsically belong to the class of multiferroic materials [4].
Ferrotoroidicity spontaneously emerges at a phase transi-
tion from a paratoroidic to a ferrotoroidic phase in which
both time and spatial inversion symmetries are simultane-
ously broken [2–5]. The order parameter for this transition
is toroidization. Note that in ferroelectrics only the spa-
tial inversion symmetry is broken whereas in ferromagnets
only the time reversal symmetry is broken. In ferroelastics
neither symmetry is broken; only the rotational symmetry
is broken [5].
Antoni Planes, Teresa Castán: Departament d’Estructura i Con-
stituents de la Matèria, Facultat de Física, Universitat de Barcelona,
Diagonal 647, 08028 Barcelona, Catalonia
Avadh Saxena: Theoretical Division, Los Alamos National Labora-
tory, Los Alamos, New Mexico 87545, USA
Ferrotoroidal order can be understood in terms of an
ordering of magnetic-vortex like structures characterized
by a toroidal (dipolar) moment. This order is also related
to (asymmetric) magnetoelectricity, i.e. α
ij
≠ α
ji
[3]. In
the present article we are mainly concerned with mag-
netic toroidal moments [2, 6–9] as observed for instance,
in LiCo(PO
4
)
3
[10]. Electrical toroidal moments can also ex-
ist in nanostructures such as polar dots [11–13] but not as
a long range ordered state in bulk materials as no symme-
try is broken. In Section 7 we will summarize recent results
on nanoscale ferroelectric materials which exhibit electric
toroidization as a consequence of dipolar vortex forma-
tion.
Figure 1 shows the symmetry properties of the four fer-
roic vector order parameters. Polarization is a polar vec-
tor, magnetization is an axial vector (it contains a sense
of time), and toroidization is an axio-polar vector. Note
that strain is a (second rank polar) tensor order parame-
ter. In addition, there are physical properties described by
a second rank axial tensor such as magnetogyration [14,
15] likely present in the spin-half antiferromagnet potas-
sium hyperoxide, KO
2
, as well as in CdS, (Ga
x
In
1−x
)
2
Se
3
,
Pb
5
Ge
3
O
11
and Bi
12
GeO
20
.
Therefore, the table can be conceivably generalized to
include tensor ferroics. The two entries in the left column
would be strain (second rank polar tensor) and magneto-
gyration (second rank axial tensor) but at present the ten-
sor analogs of polarization and toroidization that would
complete the table have not been properly identied. Note
that in principle this idea could be generalized further to
third (and higher) rank tensor ferroic properties. We intend
to present these results elsewhere in the near future.
Any ferroic order is usually accompanied by domain
walls [1]. Indeed, ferrotoroidic domain walls have been
observed in LiCo(PO
4
)
3
using nonlinear optics, i.e. sec-
ond harmonic generation. Litvin has provided a sym-
metry based classication of such domains [16] as well
as ferrotoroidal crystals [17, 18]. Another material, BCG
(Ba
2
CoGe
2
O
7
), also exhibits spontaneous toroidal mo-
ments [19]. Similarly MnTiO
3
thin lms show ferrotoroidic
ordering [20]. Multiple ferrotoroidic phase transitions
have been studied in Ni-Br and Ni-I boracites [9, 21]. In-
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10 | Antoni Planes, Teresa Castán, and Avadh Saxena
terestingly, some quasi-one dimensional materials such as
pyroxenes [22] also exhibit ferrotoroidic behaviour.
Figure 1: Symmetry properties of the four vectorial ferroic orders.
2 Basic eld equations
We will introduce here the toroidic moment and toroidiza-
tion following ideas published by Dubovik et al. in Ref. [8].
Assume distributions of charges ρ(r) and currents j(r) lo-
calized in a given region of space. These distributions cre-
ate electric and magnetic elds that satify Maxwell equa-
tions. In the presence of matter, within the continuum
dipolar approximation, these equations are usually ex-
pressed as [23]:
∇ × H −
∂D
∂t
= j , (1)
∇ · D = ρ, (2)
∇ × E +
∂B
∂t
= 0 , (3)
∇ · B = 0. (4)
The electric displacement, D, and the magnetic eld, H,
are dened as:
D = ε
0
E + P, (5)
H =
1
µ
0
B − M, (6)
where P and M are the electric and magnetic polarizations
of the medium, and ε
0
and µ
0
the electric permittivity and
the magnetic permeability of free space. Electric polariza-
tion (or simply polarization) and magnetic polarization (or
magnetization) are introduced as volume densities of the
electric and magnetic moments which are dened from a
multipole expansion far from charge and current distribu-
tion of the electric scalar potential, φ, and the magnetic
vector potential, A, at dipolar order respectively [24]. These
potentials are dened from eqs. (1) and (4) as:
E = −∇φ −
˙
A , (7)
B = ∇ × A. (8)
We will see that the dipolar approximation is not sucient
for some peculiar (electric and magnetic) moment cong-
urations. In this case, higher order terms in the expansion
must be taken into account.
Now let us assume a situation where the congura-
tion of magnetic or electric moments is spiral-like. This is
illustrated in Fig. 2. For this conguration, the magneti-
zation (or polarization) is along the z-axis, while the pro-
jection on the xy-plane, which has a circle-like congura-
tion, is zero. This circle-like conguration can be under-
stood as being originated by a toroidal conguration of
loop-currents or electric dipoles in the magnetic and elec-
tric cases respectively. For this kind of congurations M
or P alone do not provide enough information about the
ordering of magnetic/electric moments. Irene A. Beards-
ley [25] noticed that in this case an arbitrary amount of
a divergence-free magnetization/polarization distribution
can be added to M or P without aecting the external
eld created by the distribution of magnetic/electric mo-
ments. This divergence-free term can be written as the curl
of some vector that characterizes the circle-like congura-
tion of magnetic/electric moments in the xy-plane. In the
magnetic case, this vector is the magnetic toroidization,
∇ ×T
M
, while in the electric case it is the electric toroidiza-
tion, ∇ × T
E
. Note that the existence of magnetic toroidiza-
tion implies that both, time-reversal and spatial-inversion
symmetries are broken. Therefore, magnetic toroidization
is represented by an axiopolar (or time-odd polar) vector.
In contrast, no broken symmetry is associated with electric
toroidization. As we will discuss later in Section 7, these
toroidizations are related to the moment of the distribu-
tions of magnetic and electric moments respectively. In
practice, this means that the magnetization M must be re-
placed by M+∇×T
M
, while in the electric case, the electric
polarization P must be replaced by P + ∇ × T
E
.
When electric and magnetic toroidizations are taken
into account, the new macroscopic Maxwell equations
take the same formal expressions as the standard ones af-
ter the following redenition of the elds.
D →
b
D = ε
0
E + P + ∇ × T
E
= ε
0
E +
b
P , (9)
H →
b
H =
1
µ
0
B − M − ∇ × T
M
=
1
µ
0
B −
b
M . (10)
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Recent progress in the thermodynamics of ferrotoroidic materials | 11
Figure 2: Spiral-like conguration of moments along the surface of a
cylinder.
Therefore, the new macroscopic Maxwell equations read:
∇ ×
b
H −
∂
b
D
∂t
= j , (11)
∇ ·
b
D = ρ, (12)
∇ × E +
∂B
∂t
= 0 , (13)
∇ · B = 0. (14)
The energy density accounting for the interaction of
the generalized polarization and magnetization with ex-
ternal elds are given by E·
b
P and B·
b
M respectively. Hence,
the coupling energies of the eld with electric and mag-
netic toroidizations are
R
E · [∇ × T
E
] d
3
r and
R
B · [∇ ×
T
M
] d
3
r respectively. These terms can be written in the
form:
Z
E · [∇ × T
E
] d
3
r =
Z
∇ · (T
E
× E)d
3
r +
Z
(∇ × E) · T
E
d
3
r,
(15)
Z
B · [∇ × T
M
] d
3
r =
Z
∇ · (T
M
× B)d
3
r +
Z
(∇ × B) · T
M
d
3
r,
(16)
where the rst terms on the right-hand sides of both eq.
(15) and eq. (16) are zero taking into account the Gauss the-
orem. In the electric case, ∇ × E = 0, and thus this energy
vanishes. In the magnetic case, it is given by
R
(∇ × B) ·
T
M
d
3
r. Therefore, this shows that the conjugate eld of the
magnetic toroidization is ∇ × B.
We can now generalize the ideas discussed above and
assume systems with spiral-like congurations of toroidal
moments (either electric or magnetic) arising from toroidal
congurations of electric or magnetic moments. This kind
of double-vortex conguration should be characterized by
divergence-free vectors expressed as the curl of higher or-
der toroidal moments. These moments are denoted as hy-
pertoroidal moments [26]. In the case of magnetism, for in-
stance, this means that in the macroscopic Maxwell equa-
tions the toroidization T
M
, should be replaced by:
T
M
→ T
M
+ ∇ × T
M
(2)
, (17)
where T
M
= T
M
(1)
is the magnetic rst-order toroidization
and T
M
(2)
is the magnetic second-order toroidization (or
hypertoroidization). Hence, magnetization should be re-
placed by:
M →
b
M
(2)
= M + ∇ × T
M
(1)
+ ∇ × (∇ × T
M
(2)
) . (18)
Of course this idea can be formally generalized to any or-
der by dening higher order hypertoroidal moments (see
Fig. 3). At order n we will have:
b
M
(n)
= M + ∇ × T
M
(1)
+ ∇ × ∇ × T
M
(2)
+ ... + ∇ × .... × ∇ × T
M
(n)
.
(19)
Indeed, one can proceed similarly in the electric case.
Therefore, Maxwell equations at the nth order, similar to
equations (11-14), can be established by dening the fol-
lowing elds:
b
D
(n)
= ε
0
E +
b
P
(n)
, (20)
b
H
(n)
=
1
µ
0
B −
b
M
(n)
. (21)
With similar arguments as those given above, it is easy
to show that the eld conjugated to the nth order magnetic
hypertoroidization is, ∇ × .... × ∇ × B.
Figure 3: Magnetic moment, toroidal moment and the generation of
successive hypertoroidal moments.
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12 | Antoni Planes, Teresa Castán, and Avadh Saxena
3 The magnetic toroidal moment
and toroidization
Similar to polarization and magnetization, electric and
magnetic toroidizations are dened in the continuum ap-
proximation as volume densities of electric and magnetic
toroidal moments respectively. Since the symmetries as-
sociated with electric toroidization are trivial (no change
of sign is expected either under spatial inversion or under
time reversal) no phase transition to an electric toroidal
phase should be envisaged. Actually, this is consistent
with the fact that toroidal moment associated with elec-
tric moment vorticity is not expected to occur in the ther-
modynamic limit [27]. From the discussion in the preced-
ing section, it seems intuitively reasonable to foresee that
the toroidal moment should be related to the moment of
the distribution of magnetic moments. This is what we
will discuss in the rest of this section where we will in-
troduce the toroidal moment based on the multipolar ex-
pansion beyond the dipolar approximation. We will also
introduce a second denition based on symmetry consid-
erations which is of interest from a more macroscopic ther-
modynamic point of view.
In the former case we consider a nite distribution of
steady currents J(r). The vector potential of this distribu-
tion is given by
A(R) =
µ
0
4 π
Z
V
J(r)
|R − r|
dv, (22)
where R is the vector position of a point P, r the vector po-
sition of the volume element dv and V the volume of the
distribution, and the Coulomb gauge (∇ · A = 0) has been
assumed. The multipole expansion of A(R) (about r = 0)
takes the form [24],
A(R) =
µ
0
4 π
∞
X
n=0
(−1)
n
n!
Z
V
J(r)[r · ∇]
n

1
R

dv. (23)
It is easy to see that the zeroth-order term vanishes
for a steady current distribution for which the continuity
equation yields ∇ · J = 0. The rst order term in the expan-
sion is the dipolar term that can be expressed as,
A
(1)
= −m × ∇
1
R
=
m × R
R
3
, (24)
where m is the magnetic dipolar moment dened as,
m =
1
2
Z
V
(r × J)dv. (25)
The next term can be expressed as the sum of magnetic
quadrupolar and toroidal contributions. The quadrupolar
part is given by,
( A
quad
)
(2)
i
= −ε
ijk
q
kl
∇
i
∇
l
1
R
(26)
where ε
ijk
is the Levi-Civita symbol and q
kl
is the magnetic
quadrupolar moment given by,
q
ij
=
2
3
Z
V
(r × J)
i
r
j
dv, (27)
which is a traceless symmetric tensor. The toroidal term is
given by,
A
(2)
tor
= ∇(t · ∇)
1
R
+ tδ(R), (28)
where t is the toroidal moment that can be expressed as
t =
1
4
Z
v
r × [r × J(r)] dv. (29)
This pseudovector represents the dual antisymmetric part
of the complete tensor which appears in the second order
term of the multipole expansion. Dening m(r ) =
1
2
[r×J(r)]
as the distribution of magnetic moments, the toroidal mo-
ment can be written as,
t =
1
2
Z
v
[r × m(r)]dv, (30)
which indicates that the toroidal moment can be under-
stood as the moment of the distribution of magnetic mo-
ments (see Fig. 3b).
For a discrete distribution of N point charges q
α
of
mass m
α
localized at positions r
α
with velocities u
α
, the
current density can be written as
J(r) =
N
X
α=1
q
α
u
α
δ(r − r
α
) . (31)
The magnetic moment then takes the form:
m =
1
2
N
X
α=1
q
α
r
α
× u
α
, =
N
X
α=1
m
α
(32)
where m
α
is the magnetic moment of charge α. Similarly,
the toroidal moment of interest here can be written as,
t =
1
4
N
X
α=1
q
α
(r
α
× [r
α
× u
α
]) =
1
2
N
X
α=1
[r
α
× m
α
] . (33)
For a system consisting of a distribution of N spins s
α
localized at positions r
α
, m
α
= gµ
0
s
α
, where g is the gyro-
magnetic ratio and µ
B
the Bohr magneton. Therefore, the
corresponding toroidal moment is given by
t
α
=
1
2
gµ
0
N
X
α=1
(r
α
× s
α
) . (34)
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Recent progress in the thermodynamics of ferrotoroidic materials | 13
It is worth noting that treatment similar to the one dis-
cussed above can be developed from the multipolar expan-
sion of the electric scalar potential ϕ. In particular, from
the rst order term the electric dipolar moment can be de-
ned as
p =
Z
V
rρ(r)dv. (35)
Once the toroidal moment is introduced, magnetic
toroidization, hereafter simply denoted as toroidization,
T, is dened as the volume density of toroidal moments.
That is T = dt/dv. We have already seen that its corre-
sponding conjugated eld is ∇ × B and hence, the energy
density of a distribution of toroidal moments character-
ized by a toroidization T is given by E = −T· (∇ × B). There-
fore, a net toroidization might be induced by means of a
current density J = ∇×B. Schmid [2] noticed that reversing
toroidal dipoles by means of such a eld in order to modify
toroidization appears unfeasible since this would require
the action of coherent circular currents of very small size
(comparable to the unit cell of the crystal).
The observation of toroidization in the absence of ap-
plied elds indicates the existence of long range order as-
sociated with toroidal moments. This long range order is
usually denoted as ferrotoroidic order and should be re-
lated to some kind of coupling between toroidal moments.
Taking into account the basic symmetries of the toroidal
moment, the occurrence of ferrotoroidic order supposes
the simultaneous breaking of spatial inversion and time
reversal symmetries.
Materials with toroidal moments are expected to in-
trinsically display magnetoelectric coupling. In these sys-
tems, an applied magnetic eld breaks inversion symme-
try and thus induces polarization. On its turn, an applied
electric eld breaks time reversal symmetry and induces
magnetization. Therefore, we expect that these materials
respond to applied electric and magnetic elds according
to the following equations,
P =
χ
e
E + α
T
B , (36)
M = α
T
E + χ
m
B , (37)
where χ
e
and χ
m
are respectively, electric and magnetic
susceptibility tensors, and α
T
is the magnetoelectric ten-
sor (all are rank-2 tensors). From a thermodynamic point of
view, if the free energy of the system is F, these polariza-
tions and magnetizations should be expressed as −∂F/∂E
and −∂F/∂B, respectively. Therefore, we expect that the
free energy of a magnetoelectric term is of the type F
m−e
=
−E α
T
B (or −α
T
ij
E
i
B
j
, in coordinate notation). The decom-
position of this magnetoelectric term into pseudoscalar,
vector, and symmetric traceless terms enables one to ex-
press F
m−e
in the form (see Ref. [3]),
F
m−e
∼ −E · B − T
′
· [E × B] − Q
ij
[ E
i
B
j
+ E
j
B
i
] , (38)
where T
′
is a vector with the same symmetry properties as
toroidal moment and toroidization. Identifying this vector
with toroidization T supposes that its conjugated eld is
G = E × B. This assumption is in agreement with recent
experiments that showed that toroiodal moments can be
controlled by this eld [28]. Therefore, polarization, P
t
,
and magnetization, M
t
, intrinsically associated with the
energy term −G · T, induced, respectively, under applica-
tion of electric and magnetic elds, can be expressed as
P
t
= −
∂T · [E × B]
∂E
= B × T, (39)
M
t
= −
∂T · [E × B]
∂B
= T × E, (40)
where we have taken into account that T·[E×B] = B·[T×E] =
E · [B × T].
In the far-eld approximation, from multipolar expan-
sions of electric (scalar) and vector potentials, the toroidal
eld G can be expressed as,
G = E × B = A(p × m) + B(p × r) + C(r × m), (41)
where, p and m are electric and magnetic dipole moments,
and the coecients A, B, and C are coecients that decay
with distance r as r
−6
, r
−7
and r
−7
respectively. Taking into
account eq. (41), it is worth pointing out that if p and m are
parallel, then G = 0 as expected. On the other hand, G has
maximum strength when p and m are perpendicular.
Taking G as the eld conjugated to toroidization, sug-
gests the following alternative denition of the toroidal
moment:
t =
µ
0
4 π
(p × m). (42)
This denition neglects residual terms in eq. (41) associ-
ated with the magnetic and electric moments. The choice
is supported by the fact that these terms decay (with dis-
tance) faster than the magneto-toroidal one.
It is worth noticing that this denition of the toroidal
moment is not strictly equivalent to the denition in eq.
(29) resulting from the second order term in the multipole
expansion of the vector potential. Nevertheless, it is, in
fact, expected to provide a good measure of the toroidal
moment in systems which are simultaneously ferroelectric
and ferromagnetic [29]. In these systems the coupling of
t (or the toroidization obtained as the volume density of
this toroidal moment) to G leads to a magnetoelectric re-
sponse similar to that of magnetic ferrotoroidics [3]. In gen-
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14 | Antoni Planes, Teresa Castán, and Avadh Saxena
eral, however, a non-zero toroidal moment should be pos-
sible even in antiferroelectric and antiferromagnetic sys-
tems. Indeed, resonant x-ray diraction observations of or-
bital currents in CuO provide direct evidence of antiferro-
toroidic ordering [30]. Actually, these situations can only
be considered when the standard denition (arising from
the multipole expansion) of the toroidal moment is taken
into account. Note that a multipole expansion including
the toroidal moment has been considered in [31].
4 Materials and relevant
experimental results
At present direct measurements of toroidization or toroidal
moment seem very dicult. Present experimental tech-
niques can only detect magnetization and magnetic mo-
ment, for instance from polarized neutron scattering or
Lorentz microscopy. In principle, these techniques should
be able to detect specic arrangements of magnetic mo-
ments characterized by toroidal moments that might order
to yield net toroidization and thus, ferrotoroidal order. In
practice, this appears to be unfeasible.
Indirectly ferrotoroidal order can be inferred from
an asymmetric magnetoelectric response. Therefore, this
needs measurement and analysis of the appropiate mag-
netoelectric tensor components. Sannikov [32] has argued
that observation of α
ij
≠ α
ji
is an indication of possible
ferrotoroidic order. It is however important to take into
account that this condition is not sucient to justify the
occurrence of toroidization. Asymmetric behaviour of the
magnetoelectric tensor has been reported for some bo-
racites (G
2
phase of Co-I and Ni-Cl boracites). It has been
reported also for some oxides such as Ga
2−x
Fe
x
O
3
and
Cr
2
O
3
[33, 34].
Visualization of ferrotoroidal order requires an exper-
imental technique that is sensitive to both space inversion
and time reversal broken symmetries which is the inher-
ent feature associated with ferrotoroidal order. As shown
by Van Aken and co-authors [10] non-linear optics oers
this possibility. These authors used optical second har-
monic generation (SHG) to resolve ferrotoroidal domains
in LiCoPO
4
. Similar experiments were already carried out
some years before [35, 36] but were much less conclusive in
relation to the existence of ferrotoroidal domains. In this
technique, electromagnetic light eld E(ω) of given fre-
quency is incident on a crystal and induces a polarization
at double the frequency which acts as a wave source. The
symmetry aects the corresponding susceptibility. This
means that the second harmonic generation light from do-
mains with opposite order should have a phase shift of
180
◦
.
LiCoPO
4
crystallizes in the orthorhombic Pnma
olivine structure [37, 38]. It displays unique properties
including large linear magnetoelectric eect and large
Li-ionic conductivity. Co
2+
ions belonging to (100) Co-
O layers carry the magnetic moments that are strongly
coupled by superexchange Co-O-Co interactions. Layers,
however, are only weakly coupled by higher order interac-
tions. Thus, the system behaves as a magnetic 2-d system
to a very good approximation.
Antiferromagnetic order in the material occurs be-
low a Néel temperature T
N
= 21.4 K. Due to large magne-
tocrystalline anisotropy the magnetic moments are con-
ned to directions lying within b-c planes, approximately
4.6
◦
away from the b axis. Actually, the Co magnetic mo-
ments are not completely compensated, and the system
shows a small net magnetic moment which, in fact, is not
consistent with the orthorhombic symmetry and can only
be understood assuming a small monoclinic distortion.
Interestingly, the monoclinic symmetry allows for a non-
zero dielectric polarization and a non-zero toroidal mo-
ment (along the pseudo-orthorhombic a-axis) to occur.
Van Aken et al. SHG experiments [10] detected four dif-
ferent domain states. It was shown later from symmetry
considerations [39] that the four domains are equivalent
with dierent orientations of the net magnetic moment.
Thus they carry toroidal moments with signs and direc-
tions mutually coupled. The fact that the SHG signal inten-
sity is observed to disappear precisely at the Néel temper-
ature, corroborates the coupling between magnetic and
toroidal order parameters.
In addition to Ba
2
CoGe
2
O
7
[19] and MnTiO
3
thin lms
[20] toroidal moments have been considered in BiFeO
3
and related multiferroics [40–42]. Another candidate ma-
terial is the magnetoelectric MnPS
3
as indicated by neu-
tron polarimetry [43]. Based on an initio calculations the
olivine Li
4
MnFeCoNiP
4
O
16
is possibly a ferrotoroidic ma-
terial [44]. Note that apart from quasi-one dimensional
materials called pyroxenes [22] the toroidal moment in the
molecular context is also of interest, e.g. in dysprosium tri-
angle based systems [45]. In a related context a physical re-
alization of toroidal order is an interacting system of disks
with a triangle of spins on each disk [46].
It is worth noting that the existence of a true long
range ordered ferrotoroidic phase has only been estab-
lished in a suciently reliable way in very few cases and,
perhaps the clearer evidence has been provided by Van
Aken et al. SHG results [10]. In other cases, only indirect
results suggest the existence of such a phase. The fact that
interaction between toroidal moments is very weak as indi-
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Recent progress in the thermodynamics of ferrotoroidic materials | 15
cated by the short range dipolar interaction (see eq. (41)),
suggests that any small amount of disorder in the mate-
rial is enough to yield a toroidal glassy state, which repre-
sents a frozen state with local order only [47]. The existence
of toroidal glass in Ni
0.4
Mn
0.6
TiO
3
has been foreseen from
the behaviour of the magnetoelectric response which was
observed to strongly depend on cooling history [48]. This
is indeed a very interesting result suggesting that materials
which are candidates to display ferrotoroidal order should
also be analysed within this point of view. In fact, possible
observation of toroidal glass completes the quartet of fer-
roic glasses, namely spin glass, relaxor ferroelectrics and
strain glass [49].
5 Thermodynamics
We consider a macroscopic body where the vector ferroic
properties, namely polarization, P, magnetization, M, and
toroidization, T, coexist. Only part of the polarization and
magnetization will be assumed to be intrinsic, thus orig-
inating from preexisting electric and magnetic moments.
The remaining part will arise from the toroidization orig-
inating from magnetic toroidal momemts in the presence
of external magnetic and electric elds respectively. That
is,
P = P
i
+ P
t
, (43)
M = M
i
+ M
t
. (44)
For this kind of closed systems, the fundamental thermo-
dynamic equation reads
dU = τdS + E · dP + B · dM, (45)
where U is the internal energy density, S the entropy den-
sity and τ the temperature.
Taking into account eq. (39), the term E · dP can be
expressed as,
E · dP = E · dP
i
+ E · dP
t
= E · dP
i
+ G · dT + T · [E × dB].
(46)
Similarly, using eq. (40), the term B · dM can be expressed
as,
B · dM = B · dM
i
+ B · dM
t
= B · dM
i
+ G · dT + T · [dE × B].
(47)
Therefore,
dU = τdS + E · dP
i
+ B · dM
i
+ G · dT + d(G · T). (48)
Helmholtz, F, and Gibbs, G, free energies are dened
as follows,
F = U − τS (49)
G = F − E · P − B · M = F − E · P
i
− B · M
i
− 2G · T. (50)
Their dierential expressions are,
dF = −Sdτ + E · dP + B · dM, (51)
dG = −Sdτ − P · dE − M · dB, (52)
which can be alternatively expressed as,
dF = −Sdτ + E · dP
i
+ B · dM
i
+ G · dT + d(G · T), (53)
dG = −Sdτ − P
i
· dE − M
i
· dB − T · dG. (54)
Note that this expression suggests that we can assume that
the three ferroic properties, polarization, magnetization,
and toroidization can be assumed as independent (vector)
quantities thermodynamically conjugated to the electric,
magnetic and toroidal elds, respectively.
The response of the system to applied electric and
magnetic elds is given by the generalized susceptibility,
ξ =
∂
2
G
∂E
2
∂
2
G
∂B∂E
∂
2
G
∂E∂B
∂
2
G
∂B
2
!
=−
∂P
∂E
∂P
∂B
∂M
∂E
∂M
∂B
!
= −
χ
e
α
α
T
χ
m
!
.
(55)
Diagonal terms, χ
e
and χ
m
, dene electric and magnetic
susceptibilities. These susceptibilities have two contribu-
tions. The intrinsic contribution, given by χ
e
i
= ∂P
i
/∂E
(= −∂
2
G /∂E
2
), and χ
m
i
= ∂M
i
/∂B (= −∂
2
G /∂B
2
), and the
toroidal contributions arising from toroidization. These
last contributions are given by,
χ
e
t
=
∂P
t
∂E
=
∂B × T
∂E
= B ×
∂T
∂E
, (56)
χ
m
t
=
∂M
t
∂B
=
∂T × E
∂B
=
∂T
∂B
× E. (57)
A toroidal susceptibility can also be dened as,
χ
T
= ∂T /∂G = −∂
2
G /∂G
2
. (58)
Neglecting the intrinsic contributions to the polarization
and magnetization, this toroidal susceptibility can be ex-
pressed as,
χ
T
=
"
E ×
1
∂T
∂B
× E
−
1
∂T
∂E
× B
× B
#
−1
=
"
E ×
1
χ
m
t
+
1
χ
e
t
× B
#
−1
,
(59)
which shows that it is related to χ
e
t
and χ
m
t
.
Cross terms dene the magnetoelectric coecients.
Maxwell relations require that second derivatives of G are
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16 | Antoni Planes, Teresa Castán, and Avadh Saxena
independent of the order in which they are performed.
Therefore, this yields
α
T
= α. (60)
Assuming that the whole magnetoelectric interplay arises
from the existence of toroidization, the magnetoelectric
coecient is given by,
α =
∂P
t
∂B
=
∂T × B
∂B
=
∂T
∂B
× B + T × I, (61)
and
α
T
=
∂M
t
∂E
=
∂E × T
∂E
= E ×
∂T
∂E
+ I × T, (62)
where I is the identity tensor. Notice that thermodynamic
stability requires that ξ is positive-denite. This implies
that both χ
e
and χ
m
must be positive-denite and, χ
e
χ
m
≥
α
T
α.
Thermal response is determined by the second order
derivatives of G involving temperature. On the one hand,
the heat capacity C is given by
∂
2
G
∂τ
2
=
C
τ
. (63)
Taking into account Maxwell relations, the derivatives in-
volving temperature and elds satisfy,
∂
2
G
∂τ∂E
=
∂
2
G
∂E∂τ
⇒
∂S
∂E
=
∂P
∂τ
, (64)
and
∂
2
G
∂τ∂B
=
∂
2
G
∂B∂τ
⇒
∂S
∂B
=
∂M
∂τ
. (65)
It can also be obtained that
∂
2
G
∂τ∂G
=
∂
2
G
∂G∂τ
⇒
∂S
∂G
=
∂T
∂τ
. (66)
These expressions determine the cross-response to electric
or magnetic eld and temperature. They are adequate for
the study of thermal response of the materials to applied
external elds which are commonly denoted as caloric ef-
fects. These eects are quantied by the entropy change
that occurs by isothermally applying or removing a given
eld, and the temperature change that results when the
same eld is applied or removed adiabatically. From a
practical point of view, materials displaying large caloric
eects are nowadays of great interest thanks to their poten-
tial use in energy harvesting, and particularly, in refrigera-
tion applications [50]. In general in ferroic and multiferroic
materials, large caloric eects are expected in the vicinity
of phase transitions to ferroic and multiferroic phases due
to the expected strong temperature dependence of thermo-
dynamic properties [51]. In the case of toroidal materials,
caloric eects have been analyzed from a theoretical per-
spective in Ref. [52]. The entropy change induced by appli-
cation of an electric eld, (0 → E), which quanties the
electrocaloric eect, can be obtained from integration of
eq. (64) as,
∆S(τ, 0 → E) =
E
Z
0
∂P
∂τ
· dE. (67)
Similarly the entropy change induced by application of
a magnetic eld, (0 → B), which quanties the magne-
tocaloric eect, can be obtained from integration of eq.
(65) as,
∆S(τ, 0 → B) =
B
Z
0
∂M
∂τ
· dB. (68)
Taking into account eqs. (43) and (44), the preceding en-
tropy changes characterizing electrocaloric and magne-
tocaloric eects, can be, respectively, decomposed into
two terms associated with intrinsic contributions and con-
tributions arising from the toroidal moment. The intrinsic
electro- and magnetocaloric terms are respectively,
∆S
i
( τ, 0 → E) =
E
Z
0
∂P
i
∂τ
· dE, (69)
∆S
i
( τ, 0 → B) =
B
Z
0
∂M
i
∂τ
· dB. (70)
The contributions arising from the toroidal moment can be
written in the form,
∆S
t
( τ, 0 → E) =
E
Z
0
∂P
t
∂τ
· dE =
E
Z
0

B ×
∂T
∂τ

· dE, (71)
∆S
t
( τ, 0 → B) =
B
Z
0
∂M
t
∂τ
· dB =
B
Z
0

∂T
∂τ
× E

· dB. (72)
A change of entropy can be isothermally induced by appli-
cation of a toroidal eld G = E × B. This entropy change
characterizes the toroidocaloric eect, and when taken
into account with eq. (66), it can simply be expressed as,
∆S(τ, 0 → G) =
G
Z
0
∂T
∂τ
· dG =
E
Z
0

B ×
∂T
∂τ

· dE
+
B
Z
0

∂T
∂τ
× E

· dB
= ∆S
t
( τ, 0 → E) + ∆S
t
( τ, 0 → B), (73)
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Recent progress in the thermodynamics of ferrotoroidic materials | 17
which shows that the toroidocaloric entropy change is
simply the sum of the electrocaloric and magnetocaloric
contributions associated with the toroidal moment as ex-
pected.
Similar expressions can be written for electrically and
magnetically induced adiabatic temperature changes. The
corresponding total change can be obtained by taking into
account that from eqs. (63), (64) and (65), the constant en-
tropy condition (adiabaticity in thermodynamic equilib-
rium) can be expressed as,
C
τ
dτ +
∂P
∂τ
· dE +
∂M
∂τ
· dB = 0. (74)
Therefore, the adiabatic temperature change induced by
application of an electric eld is given as,
∆τ(S, 0 → E) =
E
Z
0
τ
C
∂P
∂τ
· dE, (75)
and the adiabatic temperature change induced by applica-
tion of a magnetic eld as,
∆τ(S, 0 → B) =
B
Z
0
τ
C
∂M
∂τ
· dB. (76)
On its turn, the adiabatic temperature change induced by
application of a toroidal eld is given by,
∆τ(S, 0 → G) =
G
Z
0
τ
C
∂T
∂τ
· dT =
E
Z
0
τ
C

B ×
∂T
∂τ

· dE
+
B
Z
0
τ
C

∂T
∂τ
× E

· dB,
(77)
where the last two terms in the right-hand side correspond
to the sum of electrocaloric and magnetocaloric contribu-
tions associated with the toroidal moment. That is,
∆τ(S, 0 → G) = ∆τ
t
( S, 0 → E) + ∆τ
t
( S, 0 → B). (78)
It is worth noting that both ∆S(τ, 0 → G) and ∆τ(S, 0 → G)
vanish if either E or B is zero or if they are parallel. For the
sake of simplicity let us consider that E = (E, 0, 0) and B =
(0 , B, 0). In this case, G = (0, 0, EB) and assuming electric
and magnetic isotropy, P = (P, 0, 0), M = (0, M, 0) and
T = (0, 0, T), and eqs. (73) and (77) are simply expressed
as
∆S(τ, 0 → EB) =
EB
Z
0
∂T
∂τ
d(EB) = B
E
Z
0
∂T
∂τ
dE + E
B
Z
0
∂T
∂τ
dB,
(79)
and
∆τ(S, 0 → EB) =
EB
Z
0
τ
C
∂T
∂τ
d(EB)
= B
E
Z
0
τ
C
∂T
∂τ
dE + E
B
Z
0
τ
C
∂T
∂τ
dB, (80)
respectively.
6 Landau and Ginzburg-Landau
modelling and domains
Phase transitons in ferroic and multiferroic materials are
associated, as we have discussed above, with some sym-
metry change. This change is captured by an order pa-
rameter which is zero at temperatures above the transi-
tion and non-zero below it. Landau and Ginzburg-Landau
theories provide reliable expressions of the free energy of
the materials in the region of the transition in homoge-
neous and non-homogeneous cases, respectively. The ap-
proach is phenomenological in nature and its combination
with the thermodynamics formalism provides a powerful
method to study macroscopic and mesoscopic behaviour
of ferroic and multiferroic materials. This approach en-
ables one to relate measurable quantities to the input pa-
rameters of the theories that can be determined either from
experiments or from rst-principle calculations. In Landau
theory the free energy is expressed as a series expansion of
the order parameter. As this free energy must be invariant
under the symmetry operations of the system, only those
terms allowed by symmetry are included in the series ex-
pansion.
In ferrotoroidic materials toroidization is the primary
order parameter but magnetization and polarization must
also be included in the free energy. So far, few Landau
models have been proposed to account for ferrotoroidal
transition in specic materials. Sannikov [21] already pro-
posed a model to account for the anomalous behaviour of
the component α
32
of the magnetoelectric tensor near the
cubic (43m1
′
) to the orthorrombic (m
′
m2
′
) phase transi-
tion in boracites.
Based on a group theoretic analysis Sannikov has pro-
vided a free energy for ferrotoroidic phase transitions in
boracites [21, 32]. It consists of a usual double well F(T) =
aT
2
+ bT
4
in the toroidization T and harmonic terms cP
2
and dM
2
in polarization and magnetization. In addition,
it has symmetry allowed coupling terms between various
components of the three dipolar vectors, namely of the
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18 | Antoni Planes, Teresa Castán, and Avadh Saxena
form P
i
T
2
j
, M
i
T
3
j
and the trilinear coupling P
i
M
j
T
k
. The
free energy is then analysed to obtain a phase diagram
in terms of the free energy coecients indicating the vari-
ous ferrotoroidic phase transitions. Note that Litvin has ex-
tended the symmetry analysis for ferroics to ferrotoroidic
materials including the possible toroidic domains and do-
main walls [16–18]. This analysis is very helpful in obtain-
ing the free energy for ferrotoroidic phase transitions for
any crystal symmetry.
Ederer and Spaldin, taking into account that the sym-
metries which allow for a macroscopic toroidal moment
are the same that give rise to an antisymmetric compo-
nent of the linear magnetoelectric eect tensor, proposed
the simplest possible Landau free energy that describes
a phase transition between a paratoroidic and a ferro-
toroidic phase that includes the energies associated with
the eect of electric and magnetic eld on polarization and
magnetization, and their coupling to toroidization [53]. It
has the following form,
F(T, T, P, M) =
1
2
A
0
( τ − τ
0
c
) T
2
+
1
4
CT
4
+
1
2
χ
−1
p
P
2
+
1
2
χ
−1
m
M
2
− B · M − E · P + κT · (P × M), (81)
where χ
p
and χ
m
are electric and magnetic susceptibilities
respectively, A
0
is the toroidic stiness, and C > 0 is the
nonlinear toroidic coecient. κ measures the strength of
the magnetoelectric coupling. The last term in the previous
free energy represents the lowest possible order coupling
term between the three order parameters consistent with
the required space and time reversal symmetries.
Minimization of the free energy (81) with respect to po-
larization and magnetization provides the equilibrium val-
ues of polarization and magnetization. They are given by
the following equations,
P = χ
p
E − χ
p
κ(M × T), (82)
and
M = χ
m
B − χ
m
κ(T × P). (83)
We can now solve these two equations assuming (for sim-
plicity) that E = (E, 0, 0) and B = (0, B, 0) and, assuming
isotropy, P = (P, 0, 0), M = (0, M, 0) and T = (0, 0, T). We
obtain:
P = χ
p
E − κχ
p
χ
m
BT + O(T
2
) ' χ
p
E − αB (84)
and
M = χ
m
H − κχ
p
χ
m
ET + O(T
2
) ' χ
m
B − αE, (85)
where we have neglected the nonlinear magnetoelectric ef-
fects in the above two equations. The magnetoelectric co-
ecient α = κχ
p
χ
m
T is a quadrilinear product of electric
susceptibility, magnetic susceptibility, the coupling con-
stant κ and the toroidization. Thus, either for κ = 0 or τ = 0
there is no magnetoelectric eect.
Substitution of P (84) and M (85) in the free energy (81)
gives the following general type of eective free energy:
F
e
= F
0
( E, H) +
1
2
A
0
( τ − τ
c
) T
2
+
1
3
βT
3
+
1
4
CT
4
+ λT,
(86)
where,
F
0
= −
1
2

χ
p
E
2
+ χ
m
B
2

, (87)
τ
c
= τ
0
c
+
κ
2
A
0
χ
p
χ
m
[ χ
m
B
2
+ χ
p
E
2
] , (88)
β = 3κ
3
χ
2
m
χ
2
p
EB = 3κ
2
χ
m
χ
p
λ, (89)
λ = κχ
m
χ
p
EB. (90)
The eective free energy (86) corresponds to the free en-
ergy of a toroidal system subjected to an eective applied
toroidal eld λ (proportional to G). Interestingly, the coef-
cient of the third order term, β, depends also on λ (and
thus, on G). When G = 0, β = 0 then (86) describes
a paratoroidal-to-ferrotoroidal second-order phase tran-
sition. Under the application of a toroidal eld G ≠ 0
(and β ≠ 0), the transition becomes rst-order for |λ| >
|β|
3
C
2
/27. It is worth pointing out that the addition of non-
linear terms in eqs. (84) and (85) would lead to higher order
terms in the expansion (86) that go beyond the minimal
model. However, within the spirit of the Landau Theory, it
is expected that such terms are not essential.
It is worth noting that the free energy (86) does not in-
clude a term directly coupling G and T, which is in agree-
ment with the thermodynamic formulation developed in
section 55. In Ref. [53], however, P, M and T were assumed
to be independent order parameters and a term −G · T was
included in the free energy. The same eective free energy
(86) is obtained, but in this case the parameter λ is given
by, (κχ
m
χ
p
− 1)EB. Similar results are obtained with this
model which leads to a richer physics due to the fact that
the λ and β terms can have dierent sign. The model has
been used in [52] in order to study toroidocaloric eects in
ferrotoroidic materials. The entropy of the system can be
obtained as,
S(τ, T, G) = −
∂F
e
∂τ
= −
1
2
A
0
T
2
( τ, G), (91)
where T(τ, G) is the equilibrium value of the toroidal order
parameter which is a solution of ∂F
e
/∂T = 0. Then, the
change of entropy isothermally induced by application of
a toroidal eld is obtained as,
S(T, G = EH) − S(T, G = 0) = −
1
2
A
0
[ T
2
( T, G = EH)
− T
2
( T, G = 0)]. (92)
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Recent progress in the thermodynamics of ferrotoroidic materials | 19
Figure 4: Toroidal order parameter as a function of temperature (in
arbitrary units) for three dierent values of the coupling parameter
κ = 0.90, κ = 1 and κ = 1.05 and selected values of the applied
toroidal eld G. The arrow in the lower panel indicates the occur-
rence of the rst order transition. Taken from [52].
It is easy to show (see [51]) that this expression coincides
with the general thermodynamic expression (73). The vari-
ation of toroidization as a function of temperature for three
dierent values of the coupling parameter (κ) is shown
in Figure 4. The corresponding isothermal entropy change
(∆S) or the toroidocaloric eect is depicted in Figure 5.
The present Landau approach to ferrotoroidic materi-
als can be generalized by including symmetry allowed gra-
dient term (∇T)
2
[9], i.e. the Ginzburg term. This should
allow one to study domains and domain walls in ferro-
toroidic materials as observed in LiCo(PO
4
)
3
using optical
second harmonic generation techniques [10]. With doping
induced disorder in such materials we expect that novel
phases such as toroidic tweed and toroidic glass should
also exist and remain to be observed experimentally with
certainty [48]. With symmetry allowed coupling of strain to
toroidization, if we apply stress to such a crystal we expect
toroidoelastic eects, i.e. a change in toroidization with
hydrostatic pressure or shear. We expect that these impor-
tant topics will be explored in near future.
Figure 5: Toroidocaloric eect, i.e. the isothermal change in en-
tropy, as a function of transition temperature (in arbitrary units) and
selected values of the applied toroidal eld G for three dierent val-
ues of the coupling parameter κ = 0.90, κ = 1 and κ = 1.05. Taken
from [52].
7 Electric ferrotoroidics at the
nanoscale
We have already discussed in Section 33 that no broken
symmetry is associated with electric toroidization which
is consistent with the fact the formation of electric mo-
ment vortex is forbidden in the thermodynamic limit.
However, the situation can drastically change when the
scale of the material decreases towards the nanoscale. It
has been predicted that vortex structures can be stabi-
lized below a certain critical temperature in both ferro-
electric and ferromagnetic nanodots [54, 55]. These zero-
dimensional structures have been studied in detail from
ab initio simulations of an eective hamiltonian under ap-
propriate boundary conditions [11]. They numerically sim-
ulated Pb(Zr,Ti)O
3
ferroelectric nanodisks and nanorods
under open circuit-like electric boundary conditions and
found vortex states comprising electric dipoles that form
a closed structure yielding spontaneous electric toroidal
moment. Indeed, in recent years electrotoroidic behavior
has been found experimentally [56–60]. Atomistic simula-
tions of KTaO
3
seem to indicate an incipient ferrotoroidic
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20 | Antoni Planes, Teresa Castán, and Avadh Saxena
response wherein quantum vibrations suppress the forma-
tion of polar vortices [61]. In addition, electrogyration or
electric eld induced optical rotation has been predicted
in materials exhibiting electrotoroidic behavior [13]. They
suggest that these nanostructures are potentially interest-
ing for data storage applications. Note that the eect of
long-range elastic interactions on the electric toroidization
in a ferroelectric nanoparticle has been considered in [62].
From a practical point of view, using these nanostruc-
tures based on electric toroidal moment as writing mem-
ory nanodevices is not straightforward since, in princi-
ple, these moments cannot be switched by standard meth-
ods as they are unaected by applied electric elds. How-
ever, several solutions have been envisaged. In [63] it has
been shown that vortices, both electric and magnetic, can
be manipulated by inhomogeneous static elds. The cou-
pling with elasticity enables controlling vortices by me-
chanical load, which gives rise to a rich temperature-stress
phase diagram [64]. Another possibility, considers nan-
odots with the shape of nanorings with an o-central hole.
The interest in these objects relies on the fact that they
are characterized by a transverse hypertoroidal moment,
which is a polar vector and thus sensitive to an homoge-
neous applied electric eld [65]. Following similar ideas,
it has been recently proposed [66] and numerically pre-
dicted that ferroelectric nanotori can possess an homoge-
neous hypertoroidal moment as well as exhibit the coexis-
tence of axial toroidal moment and hypertoroidal moment
phases. In these nanoscale objects the hypertoroidal mo-
ment could be manipulated by an homogeneous applied
electric eld. Note that as a dierent application, metama-
terials based on the electric toroidal moment have been
proposed [67].
8 Conclusions
With the recent emergence of magnetoelectric and mul-
tiferoic materials the fourth primary ferroic property,
namely ferrotoroidicity, has gained special attention. In
the present article we have developed a general thermo-
dynamic framework for the study of phase transitions, do-
main walls and caloric eects within the context of Lan-
dau theory in ferrotoroidic materials such as LiCo(PO
4
)
3
.
Both magnetic and electric toroidal moments were con-
sidered although the latter can only exist in polar nanos-
tructures. The generalization to hypertoroidal moments
was also presented. We discussed a variety of materials
where toroidic order has been observed. In the presence
of sucient disorder the other three primary ferroics ex-
hibit glassy behaviour, namely as spin glass, relaxor ferro-
electrics and strain glass [49]. Similarly, toroidal glass has
been potentially observed as well in the study of dynamics
of a linear magnetoelectric Ni
0.4
Mn
0.6
TiO
3
[48]. Presence
of toroidal moments is also an indicator of magnetoelectric
coupling in the material [68]. Similarly, toroidal magnon
excitations in multiferroics relate to magneto-optical ef-
fects [69]. In this article we did not consider the elds and
radiation from moving toroidal dipoles which is also an
important area of research [70, 71]. Clearly, exploration
of toroidal phenomena is a fertile area of research with a
great potential for both fundamental science and device
applications. Indeed, toroidal metamaterials have been
proposed, studied and experimentally realized [72–74] in-
cluding in double-ring [75] and double-disk [76] structures.
Acknowledgement: This work received nancial support
from CICyT (Spain), Project No. MAT2013-40590-P and was
partially supported by the U.S. Department of Energy.
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