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Proc. R. Soc. A (2011) 467, 2369–2386
doi:10.1098/rspa.2010.0521
Published online 2 March 2011
The number and types of all possible rotational
symmetries for flexoelectric tensors
BY H. LE QUANG*
AND
Q.-C. HE
Université Paris-Est, Laboratoire de Modélisation et Simulation Multi Echelle,
UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France
Flexoelectricity is due to the electric polarization generated by a non-zero strain
gradient in a dielectric material without or with centrosymmetric microstructure. It is
characterized by a fourth-order tensor, referred to as flexoelectric tensor, which relates the
electric polarization vector to the gradient of the second-order strain tensor. This paper
solves the fundamental problem of determining the number and types of all possible
rotational symmetries for flexoelectric tensors and specifies the number of independent
material parameters contained in a flexoelectric tensor belonging to a given symmetry
class. These results are useful and even indispensable for experimentally identifying or
theoretically/numerically estimating the flexoelectric coefficients of a dielectric material.
Keywords: flexoelectricity; symmetry class; dielectric material; flexoelectric tensor
1. Introduction
A necessary condition for piezoelectricity to be generated in a dielectric material
undergoing a uniform strain field is that the microstructure of this material is
devoid of centrosymmetry. Physically, this necessary condition comes from the
fact that, in a dielectric material subjected to a uniform strain field, non-zero
electric polarization owing to a relative displacement between the centres of
positive and negative charges can be induced only if its microstructure has no
centrosymmetry. Mathematically, the necessary condition in question is owing to
the fact that the third-order tensor characterizing piezoelectricity reduces to 0
when it is invariant under the central inversion transformation.
However, when a non-uniform strain field is applied to a dielectric material even
with a centrosymmetric microstructure, electric polarization can be produced,
since the corresponding non-zero strain gradient can destroy centrosymmetry and
thus leads to a displacement of the centre of positive charge with respect to the
one of negative charge. This physical phenomenon, called flexoelectricity, was first
predicted by Mashkevich & Tolpygo (1957) and later described by Kogan (1963)
and Tagantsev (1986, 1991).
Flexoelectricity has been experimentally observed in a variety of systems such
as crystal plates (Bursian & Trunov 1974), isotropic elastomers (Marvan &
Havranek 1988), thin films (Catalan et al. 2004), liquid crystals (Meyer 1969) and
*Author for correspondence ([email protected]).
Received 7 October 2010
Accepted 17 January 2011
2369
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H. Le Quang and Q.-C. He
biological membranes. The experimental identification of flexoelectric constants
was made by Ma & Cross (2001, 2002, 2006) and Zubko et al. (2007) for
perovskites, which exhibit unusually high flexoelectricity. Some authors also
reported large flexoelectric effects in low systems like nanographitic systems
(Kalinin & Meunier 2008) and two-dimensional boron-nitride sheets (Naumov
et al. 2009). The theoretical and numerical estimates for flexoelectric constants
were provided, for example, by Sahin & Dost (1988), Tagantsev (1986, 1991)
and Maranganti & Sharma (2009). For a comprehensive list of references about
flexoelectricity and an interesting discussion on possible important applications
of flexoelectricity, we refer to the papers of Sharma et al. (2007) and Tagantsev
et al. (2009).
Flexoelectricity is characterized by a fourth-order tensor, called flexoelectric
tensor, which relates the electric polarization vector to the gradient of the secondorder strain tensor. Two fundamental questions, which seem not to have been
addressed and a fortiori not to have been answered in studying flexoelectricity,
are the following ones:
— What are the number and types of all the rotational symmetries that
flexoelectric tensors can have?
— How many independent material parameters have a flexoelectric tensor
exhibiting a given type of symmetry?
These two questions and the responses to them are of both theoretical and
practical importance. This can be recognized with no doubt while recalling how
crucial their elastic counterparts are in studying linear elasticity. A flexoelectric
tensor F is of the same order as an elastic tensor C. However, the former has
the index symmetry Fijkl = Fikjl , while the latter possesses the index symmetries
Cijkl = Cjikl = Cklij . Consequently, the space formed by all flexoelectric tensors is
much more complex than the space consisting of all elastic tenors. This can be
seen by noting that, in the most general case, F contains 54 independent material
parameters, whereas C comprises 21 ones.
The problem of determining the number and types of all rotational symmetries
for elastic tensors was, for the first time, explicitly and rigorously formulated and
treated in a seminal paper of Huo & Del Piero (1991) about 20 years ago. The
fundamental problem posed by Huo & Del Piero in the context of elasticity has
then been approached in different ways and captured the attention of scientists
in the fields of mechanics, physics, applied mathematics and engineering (e.g.
Zheng & Boehler 1994, He & Zheng 1996, Forte & Vianello 1996, 1997, Xiao
1997, Chadwick et al. 2001, Bóna et al. 2004, 2007, Moakher & Norris 2006).
The objective of the present work is to answer the aforementioned two questions
posed for flexoelectric tensors. The methods used to achieve this objective are
the techniques of harmonic decomposition and Cartan decomposition, which
have been shown to be physically meaningful and mathematically efficient (e.g.
Forte & Vianello 1997). The results obtained in the present work are useful
and even indispensable for experimentally identifying or theoretically/numerically
estimating the flexoelectric coefficients of a dielectric material.
The paper is organized as follows. In §2, we introduce the basic notions
used throughout the paper and define what is meant by a symmetry class
for flexoelectric tensors. In §3, the harmonic irreducible decomposition of a
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Symmetries for flexoelectric tensors
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flexoelectric tensor is derived in the most general case. In §4, the Cartan
decomposition is applied to a generic flexoelectric tensor. With the help of
the results obtained in §§3 and 4 and by applying some relevant results from
the three-dimensional rotation group theory, in §5, we deduce and specify the
number and types of all possible rotational symmetries for flexoelectric tensors
and determine the number of independent material parameters and the matrix
form for a flexoelectric tensor with a given rotational symmetry. In §6, a few
concluding remarks are provided.
2. Formulation of the problem
In this paper, as a general rule, light-face Greek or Latin letters, for example,
a and a, denote scalars; bold-face minuscules, such as a and b, and boldface majuscules, like A and B, designate vectors and second-order tensors,
respectively; blackboard bold fonts such as D, E, H and F are reserved for thirdorder tensors while outline letters such as D, E, H and F are used to symbolize
fourth-order tensors.
First, let us introduce a Cartesian coordinate system {x, y, z} associated
to a right-handed orthonormal basis {e1 , e2 , e3 }. We denote by V the
three-dimensional inner-product space over the reals R and by Lin
the space of all linear transformations (second-order tensors) on V.
The inner product of two vectors a and b of V is noted as a · b.
The subspace of symmetric tensors Sym is given by Sym = {A ∈ Lin|Aa ·
b = a · Ab, ∀a, b ∈ V}. The three-dimensional orthogonal tensor group O(3)
is defined as O(3) = {Q ∈ Lin|Qa · Qb = a · b, ∀a, b ∈ V}. The three-dimensional
rotation tensor group SO(3) is given by SO(3) = {Q ∈ O(3)| det Q = 1}. Next,
Q(a, q) stands for the rotation about a ∈ V through an√angle q ∈ [0,√
2p); in
particular, the rotations Q(e1 + e2 + e3 , 2p/3) and Q(2( 5 − 1)e2 + ( 5 + 1)
e3 , 2p/3) are in short noted by Q̃ and Q̂, respectively.
In the §3, use will be made of the following standard group notations: the
identity subgroup being formed by the second-order identity tensor I, SO(2)
standing for the group of all rotations Q about e3 such that Qe3 = e3 , O(2)
denoting the group consisting of all orthogonal tensors Q such that Qe3 = ±e3 , Zr
(r ≥ 2) corresponding to a cyclic group with r elements generated by Q(e3 , 2p/r),
Dr (r ≥ 2) designating a dihedral group with 2r elements generated by Q(e3 , 2p/r)
and Q(e1 , p), T representing a tetrahedral group with 12 elements generated by
D2 and Q̃, O being an octahedral group containing 24 elements generated by D4
and Q̃ and I symbolizing the dodecahedral group having 60 elements generated
by D5 and Q̂. Recall that the subgroups T , O and I map a tetrahedron, a cube
and a dodecahedron onto themselves, respectively.
Now, we consider a deformable dielectric material in which the electric
polarization p is generated by the infinitesimal strain tensor 3 and the gradient
of the latter, namely E = V3. More precisely, p is related to 3 and E through the
linear relation
pi = Dijk 3jk + Fijkl Ejkl ,
Proc. R. Soc. A (2011)
(2.1)
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2372
H. Le Quang and Q.-C. He
where Dijk are the components of the third-order piezoelectric tensor D and Fijkl
the components of the fourth-order flexoelectric tensor F. In the case where the
microstructure of a medium exhibits centrosymmetry, the third-order tensor D
must be invariant under the central inversion transformation and is necessarily
null, so that equation (2.1) reduces to
pi = Fijkl Ejkl ,
(2.2)
where the flexoelectric polarization p depends only on the strain gradient E.
Remark that E has the property Eijk = Ejik owing to the symmetry 3ij = 3ji
of 3. Thus, the matrix components of F verify the following index permutation
symmetry:
Fijkl = Fikjl .
(2.3)
For later use, it is convenient to introduce the space of flexoelectricity tensors as
F = {F = Fijkl ei ⊗ ej ⊗ ek ⊗ el |Fijkl = Fikjl }.
(2.4)
The symmetry group of a flexoelectricity tensor F ∈ F is defined as the subgroup
G(F) of SO(3) such that F is invariant with respect to each element of G(F):
G(F) = {Q ∈ SO(3)|Q ∗ F = F},
(2.5)
where Q ∗ F = Qir Qjm Qkn Qls Frmns ei ⊗ ej ⊗ ek ⊗ el . It can then be shown that: (i)
the symmetry group G(F) is a closed subgroup of SO(3); and (ii) G(Q ∗ F) =
QG(F)QT for any orthogonal tensor Q ∈ O(3). When G is a subgroup of SO(3)
such that Q ∗ F = F for all Q ∈ G, then G ⊆ G(F). In this case, we say that F
exhibits G-symmetry.
Physically, it is natural and meaningful to say that the flexoelectric properties
of two materials exhibit the same type of symmetry if the symmetry groups of the
corresponding flexoelectric tensors coincide to within a rotation. Consequently,
two flexoelectric tensors, F1 ∈ F and F2 ∈ F , are said to be equivalent, noted as
F1 ∼ F2 , when their symmetry groups are conjugate to each other. More precisely,
we write
F1 ∼ F2 ⇔ G(F1 ) ∼ G(F2 )
⇔ ∃Q ∈ SO(3) such that G(F1 ) = QG(F2 )QT .
(2.6)
This equivalence relation results in a partition of the flexoelectric tensor space F ,
namely a family of non-empty subsets, (Fi )1≤i≤N , of F such that no two elements
of (Fi )1≤i≤N overlap and the union of (Fi )1≤i≤N is equal to F . An element of this
partition, say Fi , is referred to as a symmetry class for flexoelectric tensors.
Let Fi ∈ Fi and consider the symmetry group G(Fi ) of Fi . Then, the collection
of all the conjugates of G(Fi ) in the set of subgroups of SO(3), i.e.
{Gi } = {G(Fi )} = {G̃ ⊆ SO(3)|G̃ = QG(Fi )QT , Q ∈ SO(3)},
Proc. R. Soc. A (2011)
(2.7)
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Symmetries for flexoelectric tensors
2373
constitutes an intrinsic characterization of the type of rotational symmetries
exhibited by the elements of Fi . For this reason, it is often more convenient
to define Fi through {Gi }. In the following, for a subgroup G of SO(3), we denote
by {G} the collection of all the conjugates of G in the set of subgroups of SO(3)
and define F (G) as the set
F (G) = {F ∈ F |G(F) ∈ {G}}.
(2.8)
With the notions and definitions presented above, the answer to the first one
of the two fundamental questions addressed in §1 consists in determining the
number N of the elements contained in the partition (Fi )1≤i≤N and specifying
{Gi } for each element of this partition. The response to the second question is to
find the number of independent material parameters comprised in Fi ∈ Fi , and
the matrix form of Fi ∈ Fi with 1 ≤ i ≤ N .
3. Harmonic decomposition
By definition, a tensor of order n ≥ 2 is totally symmetric if and only if its
components remain unchanged under any permutation of the indexes. A tensor
of order n ≥ 2 is said to be harmonic if and only if it is simultaneously traceless
and totally symmetric. In particular, a scalar and a vector are considered in this
work as harmonic tensors of orders 0 and 1, respectively. For simplicity, a general
harmonic tensor of order n will be symbolized by H (n) and the space of nth-order
harmonic tensors by H(n) .
Next, we proceed to make the irreducible harmonic decomposition of the space
(n )
F of flexoelectricity tensors via the one of a tensor product space. Letting E1 1
(n )
and E2 2 be the spaces of tensors of orders n1 and n2 , respectively, the tensor
(n )
(n )
(n )
(n )
product space of E1 1 and E2 2 , denoted by E = E1 1 ⊗ E2 2 , is defined as a linear
(n )
(n )
combination of the tensor products of the elements of E1 1 by the elements of E2 2 .
(n )
(n )
The harmonic irreducible decomposition of E = E1 1 ⊗ E2 2 into a basis space of
harmonic tensors, designated by the index a, is known to be given by (e.g. Zuber
2006, Auffray 2008)
(n1 )
E = E1
(n2 )
⊗ E2
=
n
1 +n2
Ea(k) .
(3.1)
k=|n1 −n2 |
(n )
(n )
In particular, if the product tensor space E = E1 1 ⊗ E2 2 is composed of totally
symmetric tensors of even (or odd) order, then the harmonic decomposition of
(n )
(n )
(n )
(n )
E = E1 1 ⊗ E2 2 , designated by Sp (E1 1 ⊗ E2 2 ), contains only the even- (or odd-)
order harmonic tensors (e.g. Jerphagnon et al. 1978). For example, the harmonic
decomposition of a second-order tensor space corresponding to the tensor product
space of two first-order tensor spaces is given by
(1)
(1)
(1)
(2)
Sp (H1 ⊗ H2 ) = H(0)
a ⊕ Ha ⊕ Ha .
Proc. R. Soc. A (2011)
(3.2)
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2374
H. Le Quang and Q.-C. He
For a space of symmetric second-order tensors, its harmonic decomposition
contains single even-order harmonic tensor space. Consequently, its harmonic
decomposition reduces to
(1)
(0)
(2)
Sp (H(1)
a ⊗ Ha ) = Ha ⊕ Ha .
(3.3)
The space F consisting of all fourth-order flexoelectric tensors F satisfying the
index permutation symmetry Fijkl = Fikjl can be considered as the tensor product
of a space of first-order tensor, a space of symmetric second-order tensor and a
space of first-order tensor. Applying the formulae (3.1) and (3.3), we obtain the
harmonic decomposition of F as follows:
(0)
(2)
(1)
F = Sp (H(1)
a ⊗ Hc ⊗ (Hb ⊕ Hb ))
(0)
(2)
(1)
(1)
(1)
= Sp (H(1)
a ⊗ Hc ⊗ Hb ) ⊕ Sp (Ha ⊗ Hc ⊗ Hb )
(1)
(1)
(1)
(2)
(3)
= Sp (H(1)
a ⊗ Hc ) ⊕ Sp (Ha ⊗ (Ha ⊕ Ha ⊕ Ha ))
(1)
(1)
(1)
(1)
(2)
(1)
(3)
= Sp (H(1)
a ⊗ Hc ) ⊕ Sp (Ha ⊗ Ha ) ⊕ Sp (Ha ⊗ Ha ) ⊕ Sp (Ha ⊗ Ha )
(0)
(1)
(2)
(1)
(2)
(1)
(2)
(3)
= Hb ⊕ Hb ⊕ Hb ⊕ H(0)
g ⊕ Hg ⊕ Hg ⊕ Hh ⊕ Hh ⊕ Hh
(2)
(3)
(4)
⊕ Hx ⊕ Hx ⊕ Hx .
Consequently, the space F of fourth-order flexoelectric tensors admits the
harmonic decomposition
2
F=
i=1
(0)
Hi
⊕
3
i=1
(1)
Hi
⊕
4
i=1
(2)
Hi
⊕
2
i=1
(3)
Hi
⊕
1
(4)
Hi .
(3.4)
i=1
Recalling that the dimension of an nth-order harmonic tensor is 2n + 1, we see
(0)
(1)
(2)
(3)
(4)
that the dimensions of Hi , Hi , Hi , Hi and Hi are 1, 3, 5, 7 and 9,
respectively, so that the dimension of the space F characterized by equation (3.4)
is 54.
Thus, every fourth-order flexoelectric tensor F can formally be written in terms
of its harmonic decomposition components ai , ai , Hi , Hi and Hi :
F = (a1 , a2 , a1 , a2 , a3 , H1 , H2 , H3 , H4 , H1 , H2 , H1 ).
(3.5)
The expression of ai , ai , Hi , Hi and Hi can be specified by using a general
method of Spenser (1970), which consists in first decomposing F into totally
symmetric tensors and then splitting each totally symmetry tensor into harmonic
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2375
Symmetries for flexoelectric tensors
tensors. Applying this method, we obtain the following expressions:
1
1
a1 = (Fpqqp + 2Fppqq ), a2 = epqk emnk F(mp)nq ,
3
3
1
[a1 ]k = epqk (Fpmmq + 2Fmmpq ), [a2 ]k = emnp Fmnkp ,
9
1
[a3 ]k = (epqk Fpqmm + epqn Fpqkn ),
6
[H1 ]km = Fppkm ,
and
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
1
[H2 ]km = (epqk elnm + epqm elnk )F(lp)nq − a2 dkm ,
2
1
[H3 ]km = elnq (epqm F(lp)nk + epqk F(lp)nm ),
2
⎪
1
⎪
⎪
[H4 ]km = elnq (epqm F(lpk)n + epqk F(lpm)n ),
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
1
⎪
⎪
[H1 ]kmn = [S1 ](kmn) − (dmn [a3 ]k + dkm [a3 ]n + dkn [a3 ]m ),
⎪
⎪
⎪
5
⎪
⎪
⎪
⎪
1
⎪
⎪
[H2 ]kmn = [S2 ](kmn) − (dmn [a1 ]k + dkm [a1 ]n + dkn [a1 ]m )
⎪
⎪
5
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
[H1 ]klmn = F(klmn) − (dkl [H1 ]mn + dkm [H1 ]ln
⎪
⎪
7
⎪
⎪
⎪
⎪
+ dkn [H1 ]lm + dlm [H1 ]kn + dln [H1 ]km + dmn [H1 ]kl )⎪
⎪
⎪
⎪
⎪
a1
⎪
⎭
− (dkl dmn + dkm dln + dkn dlm ).
15
(3.6)
In these equations, dij and eijk are the Kronecker delta and permutation symbol,
respectively; either •i1 i2 ...in or [•]i1 i2 ...in stands for the components of an nth-order
tensor •; •(i1 i2 ...ir )ir+1 ...in denotes the tensor components by adding together the
r! components of tensor • obtained by permuting the indices i1 , i2 , . . . , ir in all
possible ways and by dividing the resulting sum by r!; the third-order tensors S1
and S2 are given by
[S1 ]kmn = epqn F(pk)qm
and
[S2 ]kmn = epqn F(pkm)q .
(3.7)
Bearing in mind the expressions (3.6) and (3.7), the flexoelectric tensor F has
the following explicit harmonic decomposition:
1
1
1
Fijkl = [H1 ]ijkl + eijm [H1 ]klm + ejkm [H1 ]ilm + eikm [H1 ]jlm + eilm [H2 ]jkm
3
3
4
1
1
1
+ ejlm [H2 ]ikm + eklm [H2 ]ijm + (dij [H1 ]kl + dik [H1 ]jl + dil [H1 ]jk
4
4
7
1
+ djk [H1 ]il + djl [H1 ]ik + dkl [H1 ]ij ) + (3eijm ekln + eikm ejln + ejkm eiln )[H2 ]mn
6
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2376
H. Le Quang and Q.-C. He
2
+ (2djk [H3 ]il − dik [H3 ]jl + djl [H3 ]ik − 2dil [H3 ]jk + dkl [H3 ]lj − dij [H3 ]kl )
9
1
+ (djl [H4 ]ik + dkl [H4 ]ij + dil [H4 ]jk − dij [H4 ]kl − dik [H4 ]jl − djk [H4 ]il )
6
3
1
+ (dij eklm + djk eilm + dik ejlm )[a1 ]m + (2ekli [a2 ]j − eklj [a2 ]i )
10
12
1
1
+ (3eijk [a3 ]l + 3eijl [a3 ]k + eikl [a3 ]j + ejkl [a3 ]i ) + (11eikm djl
15
15
− 4ejkm dil − 5ejlm dik + 10eilm djk − 5eklm dij + 3eijm dkl )[a3 ]m
1
(2ekim djl + 2elim djk − eljm dik − ekjm dil − elkm dij )[a2 ]m
12
a1
a2
+ (dij dkl + dik djl + dil djk ) + (djl dik + dij dkl − 2dil djk ).
(3.8)
3
15
The decomposition (3.5) implies that: (i) Q ∗ F = (a1 , a2 , Q ∗ a1 , Q ∗ a2 , Q ∗ a3 ,
Q ∗ H1 , Q ∗ H2 , Q ∗ H3 , Q ∗ H4 , Q ∗ H1 , Q ∗ H2 , Q ∗ H1 ) for any rotation Q;
and (ii) G(F) = G(a1 ) ∩ G(a2 ) ∩ G(a3 ) ∩ G(H1 ) ∩ G(H1 ) ∩ G(H2 ) ∩ G(H3 ) ∩ G(H4 ) ∩
G(H1 ) ∩ G(H2 ) ∩ G(H1 ), where the symmetry groups of a vector, a second-order
tensor, a third-order tensor and a fourth-order tensor are defined as
⎫
G(a) := {Q ∈ SO(3)|Q ∗ a = a}, ⎪
⎪
⎪
G(H) := {Q ∈ SO(3)|Q ∗ H = H}, ⎬
(3.9)
G(H) := {Q ∈ SO(3)|Q ∗ H = H} ⎪
⎪
⎪
⎭
and
G(H) := {Q ∈ SO(3) | Q ∗ H = H},
+
with Q ∗ a := Qij aj ei , Q ∗ H := Hlp Qil Qjp ei ⊗ ej , Q ∗ H := Hlpr Qil Qjp Qkr ei ⊗ ej ⊗
ek and Q ∗ H := Hlprw Qil Qjp Qkr Qmw ei ⊗ ej ⊗ ek ⊗ em .
4. Cartan decomposition
Let P (n) be the space of homogeneous polynomials of degree n (0 ≤ n ≤ 4) in
terms of three variables x, y and z. It is well known that an isomorphism 4 exists
between the space S (n) of nth-order totally symmetry tensors and P (n) :
(n)
S (n) S(n) → 4(S(n) ) := Si1 i2 ...in ri1 ri2 . . . rin ∈ P (n) ,
(4.1)
where r := xe1 + ye2 + ze3 . Analogously, the space of harmonic tensors H(n) ⊂S (n)
is also isomorphic under 4 to the space of homogeneous harmonic polynomials
(n)
of degree n, designated by Ph ⊂P (n) . For p = 4(S(n) ) ∈ P (n) and Q ∈ SO(3), we
define the action of Q on P (n) as
(n)
(Q ∗ p)(x, y, z) := Si1 i2 ...in rj1 rj2 . . . rjn Qj1 i1 Qj2 i2 . . . Qjn in ,
(4.2)
or equivalently
(n)
(Q ∗ 4)(S(n) ) := 4(Si1 i2 ...in Qj1 i1 Qj2 i2 . . . Qjn in ej1 ⊗ ej2 ⊗ · · · ⊗ ejn ).
Proc. R. Soc. A (2011)
(4.3)
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Symmetries for flexoelectric tensors
2377
According to this definition and for later use, the actions of Q(e3 , q), Q(e1 , p), Q̃
and Q̂ on p(x, y, z) ∈ P (n) are specified by
Q(e3 , q) ∗ p = p(x cos q + y sin q, −x sin q + y cos q, z),
Q(e1 , p) ∗ p = p(x, −y, −z),
Q̃ ∗ p = p(y, z, x)
and
Q̂ ∗ p = p(x , y , z ),
(4.4)
where x , y and z are the components of r such that r = x e1 + y e2 + z e3 = Q̂T r.
It is a classical result that the space of homogeneous polynomials P (n) is the
(n)
direct sum of the space of homogeneous harmonic polynomials Ph and the space
(n)
of polynomials, denoted as Pq , which are multiples of r := x 2 + y 2 + z 2 . It can
(n)
be also shown that for each p ∈ P (n) , there exists a unique q ∈ Pq such that
(n)
h := (p + rq) ∈ Ph , and h is called the harmonic part of p.
(n)
Cartan decomposition. The space of homogeneous harmonic polynomials Ph
can be expressed as
n
(n)
(n)
(n)
Ph = K0 ⊕
Km
.
(4.5)
m=1
(n)
Here, Kl
n−l
= span(ul , tl ), where ul and tl are the harmonic parts of z n−l (hl )
√
(n)
and z (hl ) with h := x + −1y. In particular, we have t0 = 0, so that K0 =
span(u0 ).
(n)
Using equation (4.4), it can be shown that a rotation Q(e3 , q) acts on K0 as
(n)
the identity and on Km (1 ≤ m ≤ n) as a rotation through an angle mq. More
precisely,
⎫
Q(e3 , q) ∗ u0 = u0 ,
⎪
⎬
Q(e3 , q) ∗ um = um cos(mq) + tm sin(mq) .
(4.6)
⎪
⎭
and
Q(e3 , q) ∗ tm = tm cos(mq) − um sin(mq)
(n)
(n)
Next, a reflection Q(e1 , p) acts on K0
and on Km (1 ≤ m ≤ n) as follows:
Q(e1 , p) ∗ u0 = (−1)n u0 , Q(e1 , p) ∗ um = (−1)n−m um
.
(4.7)
and
Q(e1 , p) ∗ tm = (−1)n−m+1 tm
Thus, for each H (n) ∈ H(n) , by setting 4(H (n) ) = h = a0 u0 + nm=1 am um +
n
(n)
(n)
(n)
−1
−1
−1
m=1 bm tm , U0 = 4 (u0 ), Um = 4 (um ) and Tm = 4 (tm ), the nth-order
(n)
harmonic tensor H can be written as
n
(n)
(n)
Hm(n) ,
(4.8)
H = H0 +
m=1
(n)
H0
(n)
= a0 U0
(n)
Hm
(n)
= am Um
(n)
(n)
and
+ bm Tm with 1 ≤ m ≤ n. Remark also
where
(n)
(n)
that each component Hm of H has, in general, a ‘horizontal’ part am Um and
(n)
(n)
a ‘vertical’ part bm Tm . Thus, Hm is said to be horizontal if it is a multiple of
(n)
(n)
(n)
Um and to be vertical if it is a multiple of Tm . Clearly, H0 is horizontal.
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H. Le Quang and Q.-C. He
For later use, we recall below two results:
Lemma 4.1. Let A, B be two second-order harmonic tensors then (i) {G(A)}
is either {D2 }, {O(2)} or {SO(3)}; and (ii) {G(A) ∩ G(B)} is either {I}, {Z2 }, {D2 },
{O(2)} or {SO(3)}.
Lemma 4.2. Let a, b be two vectors then (i) {G(a)} is either {SO(3)} or {SO(2)};
and (ii) {G(a) ∩ G(b)} is either {I}, {SO(2)} or {SO(3)}.
The proofs of lemmas 4.1 and 4.2 can be found, for example, in Forte & Vianello
(1996) or Le Quang et al. (submitted).
Finally, the Cartan decompositions and the symmetry groups of a second-,
third- and fourth-order harmonic tensors are provided, e.g. by Forte & Vianello
(1996, 1997) and Le Quang et al. (submitted).
— For n = 2, simple computations of the harmonic polynomials u0 , um and
tm show that
1
u0 = r − z 2 ,
3
(u1 , t1 ) = (zx, zy)
and
(u2 , t2 ) = (x 2 − y 2 , 2xy).
(4.9)
Proposition 4.3. For H = H0 + H1 + H2 ∈ H(2) , the following implications hold:
and
(i)
G(H) ⊇ SO(2) ⇔ H1 = H2 = 0 ⇔ G(H) ⊇ O(2);
(ii)
G(H) ⊇ Zk ⇔ H1 = H2 = 0 ⇔ G(H) ⊇ O(2),
(k ≥ 3);
(iii)
G(H) ⊇ Dk ⇔ H1 = H2 = 0 ⇔ G(H) ⊇ O(2),
(k ≥ 3);
(iv)
G(H) ⊇ Z2 ⇔ H1 = 0;
(v)
G(H) ⊇ D2 ⇔ H1 = 0, and H2 is horizontal in K2
(vi)
G(H) ⊇ T ⇔ G(H) ⊇ O ⇔ G(H) ⊇ I ⇔ H = 0.
(2)
— For n = 3, simple computations of the harmonic polynomials u0 , um and
tm imply that
3
1
1
3
2
2
u0 = z − rz, (u1 , t1 ) = z x − rx, z y − ry ,
5
5
5
(u2 , t2 ) = (z(x 2 − y 2 ), 2xyz)
and
(u3 , t3 ) = (x 3 − 3xy 2 , 3x 2 y − y 3 ).
(4.10)
Proposition 4.4. For H = H0 + H1 + H2 + H3 ∈ H(3) , the following results hold:
(i)
G(H) ⊇ SO(2) ⇔ H1 = H2 = H3 = 0;
(ii)
G(H) ⊇ O(2) ⇔ H = 0;
(iii)
G(H) ⊇ Zk ⇔ H1 = H2 = H3 = 0 ⇔ G(H) ⊇ SO(2),
(iv)
G(H) ⊇ Dk ⇔ H = 0 ⇔ G(H) ⊇ O(2),
(v)
G(H) ⊇ Z2 ⇔ H1 = H3 = 0;
(vi)
G(H) ⊇ D2 ⇔ H0 = H1 = H3 = 0, and H2 is vertical in K2 ;
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(k ≥ 4);
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and
(vii)
G(H) ⊇ Z3 ⇔ H1 = H2 = 0;
(viii)
G(H) ⊇ D3 ⇔ H0 = H1 = H2 = 0, and H3 is horizontal in K3
(ix)
G(H) ⊇ T ⇔ H0 = H1 = H3 = 0, and H2 is vertical in K2 .
(3)
(3)
— For n = 4, the corresponding harmonic polynomials u0 , um and tm have the
following expressions:
⎫
⎪
u0 = 8z 4 + 3(x 2 + y 2 )(r − 9z 2 ),
⎪
⎪
⎪
⎪
2
2
⎪
⎪
(u1 , t1 ) = (zx(7z − 3r), zy(7z − 3r)),
⎪
⎬
2
2
2
2
(4.11)
(u2 , t2 ) = ((x − y )(7z − r), 2xy(7z − r)), .
⎪
⎪
⎪
⎪
(u3 , t3 ) = (zx(x 2 − 3y 2 ), zy(3x 2 − y 2 ))
⎪
⎪
⎪
⎪
⎭
4
2 2
4
2
2
and
(u4 , t4 ) = (x − 6x y + y , 4xy(x − y ))
Proposition 4.5. For H = H0 + H1 + H2 + H3 + H4 ∈ H(4) , the following results
hold:
(i)
G(H) ⊇ SO(2) ⇔ H1 = H2 = H3 = H4 = 0 ⇔ G(H) ⊇ O(2);
(ii)
G(H) ⊇ Zk ⇔ H1 = H2 = H3 = H4 = 0 ⇔ G(H) ⊇ O(2),
(k ≥ 5);
(iii)
G(H) ⊇ Dk ⇔ H1 = H2 = H3 = H4 = 0 ⇔ G(H) ⊇ O(2),
(k ≥ 5);
(iv)
G(H) ⊇ Z2 ⇔ H1 = H3 = 0;
(v)
G(H) ⊇ D2 ⇔ H1 = H3 = 0,
(4)
and both H2 and H4 are horizontal in
(4)
K2 and K4 ;
and
(vi)
G(H) ⊇ Z3 ⇔ H1 = H2 = H4 = 0;
(vii)
G(H) ⊇ D3 ⇔ H1 = H2 = H4 = 0, and H3 is vertical in K3 ;
(viii)
G(H) ⊇ Z4 ⇔ H1 = H2 = H3 = 0;
(ix)
G(H) ⊇ D4 ⇔ H1 = H2 = H3 = 0,
(x)
G(H) ⊇ O ⇔ G(H) ⊇ T ⇔ {∃l ∈ R|H = l(U0 + 5U4 )}
(xi)
G(H) ⊇ I ⇔ H = 0.
(4)
(4)
and H4 is horizontal in K4 ;
(4.12)
5. Symmetry groups and symmetry classes
It is a classical result (e.g. Golubitsky et al. 1985, Forte & Vianello 1996, 1997)
that any closed subgroup of SO(3) is conjugate to one of the groups in the
collection S defined by
S := {I, Zr , Dr , T , O, I , SO(2), O(2), SO(3)},
(5.1)
where r ≥ 2.
Owing to the fact that all the symmetry groups are closed subgroups of SO(3),
the following result follows directly from the above classical result.
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H. Le Quang and Q.-C. He
Proposition 5.1. For each F ∈ F there is exactly one G ∈ S such that G is
conjugate to G(F).
With the definition (2.8), it can be shown from proposition 5.1 (Forte &
Vianello 1996, 1997) that F (G1 ) and F (G2 ) are disjoint when S G1 = G2 ∈ S
and
F=
F (G).
(5.2)
G∈S
Thus, the determination of the symmetry classes of flexoelectric tensors is now
reduced to answering the following question: for each subgroup G ∈ S, is the set
F (G) empty or not? The response to this question is provided by the following
two theorems:
Theorem 5.2. For r ≥ 5,
F (Zr ) = F (Dr ) = F (I ) = ∅.
(5.3)
Theorem 5.3. For 2 ≤ r ≤ 4,
F (I), F (Zr ), F (Dr ), F (SO(2)), F (O(2)),
F (O), F (T ) and F (SO(3)) are not empty.
(5.4)
The following lemma is needed to prove theorems 5.2 and 5.3.
Lemma 5.4. For F ∈ F , the following equivalences hold:
and
(i)
G(F) ⊇ Zk ⇔ G(F) ⊇ SO(2)(k ≥ 5)
(ii)
G(F) ⊇ Dk ⇔ G(F) ⊇ O(2)(k ≥ 5).
Proof. Assuming that F ∈ F and G(F) ⊇ Zk (or Dk ) with k ≥ 5, it has
been previously established that G(F) = G(a1 ) ∩ G(a2 ) ∩ G(a3 ) ∩ G(H1 ) ∩ G(H2 ) ∩
G(H3 ) ∩ G(H4 ) ∩ G(H1 ) ∩ G(H2 ) ∩ G(H1 ) ⊇ Zk (or Dk ). Using the results stated
in propositions 4.3–4.5 and lemmas 4.1 and 4.2, it follows that G(ai ) ⊇ Zk (or
Dk ) with k ≥ 5 ⇔ G(ai ) ⊇ SO(2), G(Hi ) ⊇ Zk (or Dk ) ⇔ G(Hi ) ⊇ O(2), G(Hi ) ⊇ Zk
(or Dk ) ⇔ G(Hi ) ⊇ SO(2) (or O(2)), G(Hi ) ⊇ Zk (or Dk ) ⇔ G(Hi ) ⊇ O(2). Thus,
G(F) ⊇ Zk (or Dk ) with k ≥ 5 implies that G(F) ⊇ SO(2) (or O(2)). The inverse
implication is trivial.
Theorem 5.2, which states that some symmetry classes are empty, is now proven
by contradiction as follows:
(a) Assume that F ∈ F (Zr ) for r ≥ 5. This means that there exists a rotation
Q such that QG(F)QT = Zr or, equivalently, G(Q ∗ F) = Zr . We have shown
in part (i) of lemma 5.4 that if G(Q ∗ F) = Zr then G(Q ∗ F) ⊇ SO(2).
This is equivalent to G(F) ⊇ QT SO(2)Q. This is in contradiction with the
assumption, because it is not possible for a conjugate of Zr to contain a
conjugate of SO(2).
(b) Suppose that F ∈ F (Dr ) for r ≥ 5. For some rotation Q, QG(F)QT = Dr
or, equivalently, G(Q ∗ F) = Dr . It has been shown in part (ii) of lemma
5.4 that, if G(Q ∗ F) = Dr , then G(Q ∗ F) ⊇ O(2) or, equivalently, G(F) ⊇
QT O(2)Q. As before, a conjugate of Dr cannot be a conjugate of O(2).
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Symmetries for flexoelectric tensors
(c) Consider F ∈ F (I ). Therefore, for some rotation Q, QG(F)QT = I or,
equivalently, G(Q ∗ F) = I . Owing to the fact that G(Q ∗ F) = I ⊃ D5 , part
(ii) of lemma 5.4 implies that G(Q ∗ F) ⊇ O(2). Then, we have G(F) ⊇
QT O(2)Q = O(2). This is in contradiction with the assumption, since a
conjugate of I cannot contain a conjugate of O(2), which is a group of
higher order than I .
To prove theorem 5.3 stating that some symmetry classes are not empty, we
show how to construct tensors exhibiting a given symmetry.
(i) To construct a tensor F ∈ F with G(F) = I, we first choose two secondorder harmonic tensors H1 and H2 such that G(H1 ) ∩ G(H2 ) = I and then
all remaining harmonic decomposition components of F in equation (3.5)
are arbitrary. It is then immediate that G(F) = I.
(ii) Consider a tensor F ∈ F with the harmonic decomposition components of
F in equation (3.5) chosen as
ai arbitrary, ai spanned by the vector e3 ,
(2)
(2)
(2)
Hi = a0i U0 + a2i U2 + b2i T2 ,
(3)
(3)
(4)
(4)
(3)
Hi = a0i U0 + a2i U2 + b2i T2
and
(4)
(4)
(4)
Hi = a0i U0 + a2i U2 + b2i T2 + a4i U4 + b4i T4 ,
(n)
(n)
where the coefficients ali and bli are arbitrary and different from zero.
By using propositions 4.3–4.5 and lemmas 4.1 and 4.2, we can show that
G(F) = Z2 .
(iii) Choose F ∈ F with the non-zero components of the harmonic
decomposition of F in equation (3.5) as follows:
(2)
(2)
(3)
ai arbitrary, Hi = a0i U0 + a2i U2 , Hi = b2i T2
and
(4)
(4)
(4)
Hi = a0i U0 + a2i U2 + a4i U4 ,
(n)
(n)
where the coefficients ali and bli are arbitrary and different from zero.
It implies from propositions 4.3–4.5 that G(F) = D2 .
(iv) Set F ∈ F with the harmonic decomposition components of F in
equation (3.5) as
ai arbitrary, ai spanned by the vector e3 ,
(2)
(3)
(3)
(3)
Hi = a0i U0 , Hi = a0i U0 + a3i U3 + b3i T3
and
(4)
(4)
(n)
(n)
(4)
Hi = a0i U0 + a3i U3 + b3i T3 ,
where the coefficients ali and bli are arbitrary and different from zero.
In view of propositions 4.3–4.5 and lemmas 4.1 and 4.2, it follows that
G(F) = Z3 .
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(v) Take F ∈ F with the non-zero components of the harmonic decomposition
of F in equation (3.5) as follows:
(2)
(3)
ai arbitrary, Hi = a0i U0 , Hi = a3i U3
(4)
(4)
Hi = a0i U0 + b3i T3 ,
and
(n)
(n)
with ali and bli being arbitrary and different from zero. Then, with the
help of propositions 4.3–4.5, we can show that G(F) = D3 .
(vi) Let F ∈ F with the harmonic decomposition components of F in
equation (3.5) verifying the following conditions:
ai arbitrary, ai spanned by the vector e3 ,
(2)
(3)
Hi = a0i U0 ,
Hi = a0i U0
(4)
(4)
(4)
Hi = a0i U0 + a4i U4 + b4i T4 ,
and
(n)
(n)
where the coefficients ali and bli are arbitrary and different from zero. It
follows from propositions 4.3–4.5 and lemmas 4.1 and 4.2 that G(F) = Z4 .
(vii) Choose F ∈ F with the non-zero components of the harmonic
decomposition of F in equation (3.5) given by
(2)
ai arbitrary, Hi = a0i U0
and
(4)
(4)
Hi = a0i U0 + a4i U4 ,
(n)
where the coefficients ali are arbitrary and different from zero. Accounting
for propositions 4.3–4.5 and lemma 4.1, we can write G(F) = D4 .
(viii) Consider F ∈ F with the non-zero harmonic decomposition components of
F in equation (3.5) given by
(3)
ai arbitrary, Hi = b2i T2
(n)
and
(4)
Hi = a0i (U0 + 5U4 ),
(n)
where the coefficients ali and bli are arbitrary and different from zero.
With the aid of propositions 4.3–4.5, we derive that G(C) = T .
(ix) Choose F ∈ F with the non-zero components of the harmonic
decomposition of F in equation (3.5) such that
ai arbitrary
and
(4)
Hi = a0i (U0 + 5U4 ),
(n)
where the coefficients ali are arbitrary and different from zero. Then, it
is inferred from propositions 4.3–4.5 that G(C) = O.
(x) For F ∈ F with its non-zero harmonic decomposition components in
equation (3.5) such that
(2)
ai arbitrary, Hi = a0i U0
(n)
and
(4)
Hi = a0i U0 ,
where the coefficients ali are arbitrary and different from zero, it follows
from propositions 4.3–4.5 and lemma 4.1 that G(F) = O(2).
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Symmetries for flexoelectric tensors
Table 1. Number of independent material parameters contained in a flexoelectric tensor belonging
to a given symmetry class.
symmetry class
number of independent components
I
{Z2 }
{D2 }
{Z3 }
{D3 }
{Z4 }
{D4 }
{T }
{O}
{O(2)}
{SO(2)}
SO(3)
54
28
15
18
10
14
8
5
3
7
12
2
(xi) Consider F ∈ F with the non-zero harmonic decomposition components of
F in equation (3.5) given by
ai arbitrary, ai spanned by the vector e3 ,
(2)
Hi = a0i U0 ,
(3)
Hi = a0i U0
and
(4)
Hi = a0i U0 ,
(n)
where the coefficients ali are arbitrary and different from zero.
Propositions 4.3–4.5 and lemmas 4.1 and 4.2 imply that G(F) = SO(2).
(xii) Choosing F ∈ F such that ai are the only non-zero components of the
harmonic decomposition of F in equation (3.5), it can be shown with the
help of propositions 4.3–4.5 that G(F) = SO(3).
Above, we have proven theorems 5.2 and 5.3. Owing to the fact that the
determination of the symmetry classes for all flexoelectric tensors amounts to
identifying what are the groups in the collection S of equation (5.1) for which the
sets defined by equation (2.8) are not empty, we can deduce from theorems 5.2
and 5.3, one of the main results of this paper:
Theorem 5.5. The number of all possible rotational symmetry classes for all
flexoelectric tensors is 12. These 12 symmetry classes are characterized by the 12
sets formed by the conjugates of 12 subgroups of SO(3):
I, {Zr }, {Dr }, {O}, {T }, {SO(2)}, {O(2)}, SO(3)
(5.5)
where 2 ≤ r ≤ 4.
By using the foregoing procedure elaborated to construct a flexoelectric
tensor F ∈ F exhibiting a required symmetry and by exploiting the harmonic
decomposition presented in equation (3.4), we can exactly calculate the number
of independent components of F belonging to a given symmetry class. More
precisely, the number of independent components of F is determined as the sum
of the numbers of the independent components involved in all the elements of the
harmonic decomposition (3.4). The result is provided in table 1. Note that in the
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H. Le Quang and Q.-C. He
tetragonal D4
(8)
trans. iso. O(2)
(7)
Orthotropic D2
(15)
tetrahedral T
(5)
trigonal D3
(10)
triclinic I
(54)
isotropic
(2)
trigonal Z3
(18)
cubic O
(3)
trans. iso. SO(2)
(12)
monoclinic Z2
(28)
tetragonal Z4
(14)
Figure 1. Geometrical illustration of the 12 symmetry classes for flexoelectric tensors and the
number of independent material parameters for each symmetry class.
isotropic case, relative to any orthonormal basis, the flexoelectric tensor F has
the components given directly by equation (3.4)
a1
a2
(5.6)
Fijkl = (dij dkl + dik djl + dil djk ) + (dij dkl + dik djl − 2dil djk ),
15
3
where a1 and a2 are two material parameters. In the 11 anisotropic cases, we can
also specify the corresponding matrix forms, but this is not done here to avoid
rendering the present paper lengthy.
In figure 1, a geometrical picture and a name are associated to each symmetry
class and the relations between the symmetry classes are shown.
6. Concluding remarks
In this work, we have solved the fundamental problem of determining the number
and types of all possible rotational symmetries for flexoelectric tensors. The
methods used to obtain this solution are quite similar to those employed by
Forte & Vianello (1996, 1997). The results derived in this work are helpful
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Symmetries for flexoelectric tensors
2385
and even necessary for getting a better understanding of flexoelectricity and for
identifying or estimating the flexoelectric coefficients of a dielectric material in
the anisotropic cases.
A flexoelectric tensor is algebraically more complex than an elastic tensor,
even though they are both of fourth order. Consequently, the number of symmetry
classes for flexoelectric tensors is 12 while the one for elastic tensors is 8. Moreover,
the largest number of independent flexoelectric constants is 54 whereas that
of independent elastic constants is 21. These facts imply that flexoelectricity
is a much more complex phenomenon even at the macroscopic level, and the
identification or estimation of flexoelectric constants is a tough problem in the
highly anisotropic cases.
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