## Abstract

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.

## 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 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 second-order 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 is of the same order as an elastic tensor . However, the former has the index symmetry *F*_{ijkl}=*F*_{ikjl}, while the latter possesses the index symmetries *C*_{ijkl}=*C*_{jikl}=*C*_{klij}. 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, contains 54 independent material parameters, whereas 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 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, *α* and *a*, denote scalars; bold-face minuscules, such as **a** and **b**, and bold-face majuscules, like **A** and **B**, designate vectors and second-order tensors, respectively; blackboard bold fonts such as and are reserved for third-order tensors while outline letters such as , , and 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 {**e**_{1},**e**_{2},**e**_{3}}. We denote by the three-dimensional inner-product space over the reals and by Lin the space of all linear transformations (second-order tensors) on . The inner product of two vectors **a** and **b** of is noted as **a**·**b**. The subspace of symmetric tensors Sym is given by . The three-dimensional orthogonal tensor group *O*(3) is defined as . The three-dimensional rotation tensor group *SO*(3) is given by . Next, ** Q**(

**a**,

*θ*) stands for the rotation about through an angle

*θ*∈[0,2

*π*); in particular, the rotations

**(**

*Q***e**

_{1}+

**e**

_{2}+

**e**

_{3},2

*π*/3) and are in short noted by and , 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 **e**_{3} such that **Q****e**_{3}=**e**_{3}, *O*(2) denoting the group consisting of all orthogonal tensors **Q** such that **Q****e**_{3}=±**e**_{3}, *Z*_{r} (*r*≥2) corresponding to a cyclic group with *r* elements generated by **Q**(**e**_{3},2*π*/*r*), *D*_{r} (*r*≥2) designating a dihedral group with 2*r* elements generated by **Q**(**e**_{3},2*π*/*r*) and **Q**(**e**_{1},*π*), representing a tetrahedral group with 12 elements generated by *D*_{2} and , being an octahedral group containing 24 elements generated by *D*_{4} and and symbolizing the dodecahedral group having 60 elements generated by *D*_{5} and . Recall that the subgroups , and 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 ** ε** and the gradient of the latter, namely

*E*=∇

**. More precisely,**

*ε***p**is related to

**and**

*ε**E*through the linear relation 2.1where

*D*

_{ijk}are the components of the third-order piezoelectric tensor and

*F*

_{ijkl}the components of the fourth-order flexoelectric tensor . In the case where the microstructure of a medium exhibits centrosymmetry, the third-order tensor must be invariant under the central inversion transformation and is necessarily null, so that equation (2.1) reduces to 2.2where the flexoelectric polarization

**p**depends only on the strain gradient .

Remark that has the property *E*_{ijk}=*E*_{jik} owing to the symmetry *ε*_{ij}=*ε*_{ji} of ** ε**. Thus, the matrix components of verify the following index permutation symmetry:
2.3For later use, it is convenient to introduce the space of flexoelectricity tensors as
2.4The symmetry group of a flexoelectricity tensor is defined as the subgroup of

*SO*(3) such that is invariant with respect to each element of : 2.5where . It can then be shown that: (i) the symmetry group is a closed subgroup of

*SO*(3); and (ii) for any orthogonal tensor

**Q**∈

*O*(3). When

*G*is a subgroup of

*SO*(3) such that for all

**Q**∈

*G*, then In this case, we say that 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, and , are said to be equivalent, noted as , when their symmetry groups are conjugate to each other. More precisely, we write
2.6This equivalence relation results in a partition of the flexoelectric tensor space , namely a family of non-empty subsets, , of such that no two elements of overlap and the union of is equal to . An element of this partition, say , is referred to as a *symmetry class* for flexoelectric tensors.

Let and consider the symmetry group of . Then, the collection of all the conjugates of in the set of subgroups of *SO*(3), i.e.
2.7constitutes an intrinsic characterization of the type of rotational symmetries exhibited by the elements of . For this reason, it is often more convenient to define through . 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 as the set
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 and specifying for each element of this partition. The response to the second question is to find the number of independent material parameters comprised in , and the matrix form of 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 *n*th-order harmonic tensors by .

Next, we proceed to make the irreducible harmonic decomposition of the space of flexoelectricity tensors via the one of a tensor product space. Letting and be the spaces of tensors of orders *n*_{1} and *n*_{2}, respectively, the tensor product space of and , denoted by , is defined as a linear combination of the tensor products of the elements of by the elements of . The harmonic irreducible decomposition of into a basis space of harmonic tensors, designated by the index *α*, is known to be given by (e.g. Zuber 2006, Auffray 2008)
3.1In particular, if the product tensor space is composed of totally symmetric tensors of even (or odd) order, then the harmonic decomposition of , designated by , 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
3.2For a space of symmetric second-order tensors, its harmonic decomposition contains single even-order harmonic tensor space. Consequently, its harmonic decomposition reduces to
3.3

The space consisting of all fourth-order flexoelectric tensors satisfying the index permutation symmetry *F*_{ijkl}=*F*_{ikjl} 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 as follows:
Consequently, the space of fourth-order flexoelectric tensors admits the harmonic decomposition
3.4Recalling that the dimension of an *n*th-order harmonic tensor is 2*n*+1, we see that the dimensions of , , , and are 1, 3, 5, 7 and 9, respectively, so that the dimension of the space characterized by equation (3.4) is 54.

Thus, every fourth-order flexoelectric tensor can formally be written in terms of its harmonic decomposition components *α*_{i}, **a**_{i}, **H**_{i}, and :
3.5

The expression of *α*_{i}, **a**_{i}, **H**_{i}, and can be specified by using a general method of Spenser (1970), which consists in first decomposing into totally symmetric tensors and then splitting each totally symmetry tensor into harmonic tensors. Applying this method, we obtain the following expressions:
3.6In these equations, *δ*_{ij} and *ε*_{ijk} are the Kronecker delta and permutation symbol, respectively; either •_{i1i2…in} or [•]_{i1i2…in} stands for the components of an *n*th-order tensor •; •_{(i1i2…ir)ir+1…in} denotes the tensor components by adding together the *r*! components of tensor • obtained by permuting the indices *i*_{1},*i*_{2},…,*i*_{r} in all possible ways and by dividing the resulting sum by *r*!; the third-order tensors and are given by
3.7

Bearing in mind the expressions (3.6) and (3.7), the flexoelectric tensor has the following explicit harmonic decomposition: 3.8

The decomposition (3.5) implies that: (i) for any rotation **Q**; and (ii) , where the symmetry groups of a vector, a second-order tensor, a third-order tensor and a fourth-order tensor are defined as
3.9with **Q*****a**:=*Q*_{ij}*a*_{j}**e**_{i}, **Q*****H**:=*H*_{lp}*Q*_{il}*Q*_{jp}**e**_{i}⊗**e**_{j}, and .

## 4. Cartan decomposition

Let 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 *φ* exists between the space of *n*th-order totally symmetry tensors and :
4.1where ** r**:=

*x*

**e**

_{1}+

*y*

**e**

_{2}+

*z*

**e**

_{3}. Analogously, the space of harmonic tensors is also isomorphic under

*φ*to the space of homogeneous harmonic polynomials of degree

*n*, designated by . For and

**Q**∈

*SO*(3), we define the action of

**Q**on as 4.2or equivalently 4.3According to this definition and for later use, the actions of

**Q**(

**e**

_{3},

*θ*),

**Q**(

**e**

_{1},

*π*), and on are specified by 4.4where

*x*

^{′},

*y*

^{′}and

*z*

^{′}are the components of

*r*^{′}such that .

It is a classical result that the space of homogeneous polynomials is the direct sum of the space of homogeneous harmonic polynomials and the space of polynomials, denoted as , which are multiples of *ρ*:=*x*^{2}+*y*^{2}+*z*^{2}. It can be also shown that for each , there exists a unique such that , and *h* is called the harmonic part of *p*.

*Cartan decomposition*. The space of homogeneous harmonic polynomials can be expressed as
4.5Here, , where *u*_{l} and *t*_{l} are the harmonic parts of and with . In particular, we have *t*_{0}=0, so that .

Using equation (4.4), it can be shown that a rotation **Q**(**e**_{3},*θ*) acts on as the identity and on (1≤*m*≤*n*) as a rotation through an angle *mθ*. More precisely,
4.6Next, a reflection **Q**(**e**_{1},*π*) acts on and on (1≤*m*≤*n*) as follows:
4.7Thus, for each , by setting , , and , the *n*th-order harmonic tensor *H*^{(n)} can be written as
4.8where and with 1≤*m*≤*n*. Remark also that each component of *H*^{(n)} has, in general, a ‘horizontal’ part and a ‘vertical’ part . Thus, is said to be horizontal if it is a multiple of and to be vertical if it is a multiple of . Clearly, is horizontal.

For later use, we recall below two results:

### Lemma 4.1

*Let* **A***,* **B** *be two second-order harmonic tensors then (i)* *is either* {D_{2}},{*O*(2)} *or* {*SO*(3)}; *and* (*ii*) *is either* {**I**}, {*Z*_{2}}, {*D*_{2}}, {*O*(2)} *or* {*SO*(3)}.

### Lemma 4.2

*Let* **a***,* **b** *be two vectors then* (*i*) *is either* {*SO*(3)} *or* {*SO*(2)}; *and* (ii) *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*u*_{0},*u*_{m}and*t*_{m}show that 4.9

### Proposition 4.3

For *the following implications hold*:

— For

*n*=3, simple computations of the harmonic polynomials*u*_{0},*u*_{m}and*t*_{m}imply that 4.10

### Proposition 4.4

For , the following results hold:

— For

*n*=4, the corresponding harmonic polynomials*u*_{0},*u*_{m}and*t*_{m}have the following expressions: 4.11

### Proposition 4.5

For the following results hold: 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 *Σ* defined by
5.1where *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.

### Proposition 5.1

*For each* *there is exactly one G*∈*Σ such that G is conjugate to* .

With the definition (2.8), it can be shown from proposition 5.1 (Forte & Vianello 1996, 1997) that and are disjoint when *Σ*∋*G*_{1}≠*G*_{2}∈*Σ* and
5.2Thus, the determination of the symmetry classes of flexoelectric tensors is now reduced to answering the following question: for each subgroup *G*∈*Σ*, is the set empty or not? The response to this question is provided by the following two theorems:

### Theorem 5.2

*For r*≥5,
5.3

### Theorem 5.3

*For 2≤r*≤4,
5.4*The following lemma is needed to prove theorems* 5.2 *and* 5.3.

### Lemma 5.4

*For* *the following equivalences hold:
*

### Proof.

Assuming that and (or *D*_{k}) with *k*≥5, it has been previously established that (or *D*_{k}). Using the results stated in propositions 4.3–4.5 and lemmas 4.1 and 4.2, it follows that (or *D*_{k}) with *k*≥5 , (or *D*_{k}) , (or *D*_{k}) (or *O*(2)), (or *D*_{k}) . Thus, (or *D*_{k}) with *k*≥5 implies that (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:

Assume that for

*r*≥5. This means that there exists a rotation**Q**such that or, equivalently, . We have shown in part (i) of lemma 5.4 that if then . This is equivalent to . This is in contradiction with the assumption, because it is not possible for a conjugate of*Z*_{r}to contain a conjugate of*SO*(2).Suppose that for

*r*≥5. For some rotation**Q**, or, equivalently, . It has been shown in part (ii) of lemma 5.4 that, if , then or, equivalently, . As before, a conjugate of*D*_{r}cannot be a conjugate of*O*(2).Consider . Therefore, for some rotation

**Q**, or, equivalently, . Owing to the fact that , part (ii) of lemma 5.4 implies that . Then, we have . This is in contradiction with the assumption, since a conjugate of cannot contain a conjugate of*O*(2), which is a group of higher order than .

To prove theorem 5.3 stating that some symmetry classes are not empty, we show how to construct tensors exhibiting a given symmetry.

To construct a tensor with , we first choose two second-order harmonic tensors

**H**_{1}and**H**_{2}such that and then all remaining harmonic decomposition components of in equation (3.5) are arbitrary. It is then immediate that .Consider a tensor with the harmonic decomposition components of in equation (3.5) chosen as where the coefficients and are arbitrary and different from zero. By using propositions 4.3–4.5 and lemmas 4.1 and 4.2, we can show that .

Choose with the non-zero components of the harmonic decomposition of in equation (3.5) as follows: where the coefficients and are arbitrary and different from zero. It implies from propositions 4.3–4.5 that .

Set with the harmonic decomposition components of in equation (3.5) as where the coefficients and are arbitrary and different from zero. In view of propositions 4.3–4.5 and lemmas 4.1 and 4.2, it follows that .

Take with the non-zero components of the harmonic decomposition of in equation (3.5) as follows: with and being arbitrary and different from zero. Then, with the help of propositions 4.3–4.5, we can show that .

Let with the harmonic decomposition components of in equation (3.5) verifying the following conditions: where the coefficients and are arbitrary and different from zero. It follows from propositions 4.3–4.5 and lemmas 4.1 and 4.2 that .

Choose with the non-zero components of the harmonic decomposition of in equation (3.5) given by where the coefficients are arbitrary and different from zero. Accounting for propositions 4.3–4.5 and lemma 4.1, we can write .

Consider with the non-zero harmonic decomposition components of in equation (3.5) given by where the coefficients and are arbitrary and different from zero. With the aid of propositions 4.3–4.5, we derive that .

Choose with the non-zero components of the harmonic decomposition of in equation (3.5) such that where the coefficients are arbitrary and different from zero. Then, it is inferred from propositions 4.3–4.5 that .

For with its non-zero harmonic decomposition components in equation (3.5) such that where the coefficients are arbitrary and different from zero, it follows from propositions 4.3–4.5 and lemma 4.1 that .

Consider with the non-zero harmonic decomposition components of in equation (3.5) given by where the coefficients are arbitrary and different from zero. Propositions 4.3–4.5 and lemmas 4.1 and 4.2 imply that .

Choosing such that

*α*_{i}are the only non-zero components of the harmonic decomposition of in equation (3.5), it can be shown with the help of propositions 4.3–4.5 that .

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 *Σ* 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):
5.5*where* 2≤*r*≤4.

By using the foregoing procedure elaborated to construct a flexoelectric tensor 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 belonging to a given symmetry class. More precisely, the number of independent components of 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 isotropic case, relative to any orthonormal basis, the flexoelectric tensor has the components given directly by equation (3.4)
5.6where *α*_{1} and *α*_{2} 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 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.

- Received October 7, 2010.
- Accepted January 17, 2011.

- This journal is © 2011 The Royal Society