Compact Explicit Matrix Representations of the Flexoelectric Tensor and a Graphic Method for Identifying All of Its Rotation and Reflection Symmetries H

Compact explicit matrix representations of the flexoelectric tensor and a graphic method for identifying all of its rotation and reflection symmetries H. Le Quang, Q.-C. He

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H. Le Quang, Q.-C. He. Compact explicit matrix representations of the flexoelectric tensor and a graphic method for identifying all of its rotation and reflection symmetries. Journal of Applied Physics, American Institute of Physics, 2021, 129 (24), pp.244103. 10.1063/5.0048386. hal-03267829

HAL Id: hal-03267829 https://hal.archives-ouvertes.fr/hal-03267829 Submitted on 22 Jun 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Compact explicit matrix representations of the flexoelectric tensor and a graphic method for identifying all of its rotation and reflection symmetries H. Le Quang1, a) and Q.-C. He1, 2 1)UniversitéGustave Eiffel, CNRS, MSME UMR 8208, F-77454 Marne-la-Vallée, France. 2)Southwest Jiaotong University, School of Mechanical Engineering, Chengdu 610031, PR China. (Dated: 17 May 2021) Flexoelectricity is an electromechanical phenomenon produced in a dielectric material, with or without centrosymmetric microstructure, undergoing a non-uniform strain. It is characterized by the fourth-order flexoelectric tensor which links the electric polarization vector with the gradient of the second-order strain tensor. Our previous work [Le Quang and He, Roy. Soc. London, Ser. A 467, 2369-2386, 2011] solved the fundamen- tal theoretical problem of determining the number and types of all rotational symmetries that the flexoelectric tensor can exhibit. In the present one, compact explicit matrix representations of the flexoelectric tensor are provided so as to facilitate the use of it with any possible rotational symmetry. The number and types of all reflection symmetries that the flexoelectric tensor can have are also determined. To identify the rotational symmetry and reflection symmetry of a given flexoelectric tensor, a simple and efficient graphic method based on the concept of pole figures is presented and illustrated.

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I. INTRODUCTION such as TiO2 ceramics and the polyvinylidene fluoride (PVDF). In parallel with these experimental works, the- Flexoelectricity is a coupled electromechanical phe- oretical studies were also conducted to demonstrate the nomenon appearing a dielectric material subjected to a size-dependent flexoelectric properties and surface effect non-uniform strain. In contrast to piezoelectricity, it of dielectric materials/structures in nanoscale, for exam- 18 19,20 can be generated even in a dielectric material whose ple, by Sahin and Dost , Tagantsev , Yurkov and 21 22 23 24 microstructure is centrosymmetric. Indeed, in the case Tagantsev , He et al. , Qi et al. and Bai et al. . Nu- of a dielectric material undergoing a uniform strain, merical approaches were elaborated according either to the electric polarization is produced if and only if the the first-principles method, for example, by Maranganti 25 26,27 microstructure of this material is non-centrosymmetric. and Sharma , Hong et al. or to the other theoreti- However, when a dielectric material with (or without) cal calculation methods such as finite element method by 28 29 a centrosymmetric microstructure is subjected to a non- Deng et al. and Yvonnet et al. , phase-field method by 30 31 uniform strain whose gradient is non null, a relative dis- Li et al. and Wang et al. to estimate the flexoelectric placement of the centers of the positive and negative properties of some dielectric materials/structures. For charges is resulted in and gives rise to an electric po- more references about flexoelectricity and for discussions larization. on potential important applications of flexoelectricity, 20 The flexoelectric effects can be produced in a multitude the reader is referred to Tagantsev et al. , Sharma et 32 33 34 35 of situations, for example, in bending crystal plates1, al. , Zubko et al. , Wang et al. , Narvaez et al. , Ab- 36 37,38 nanobeams and nanowires2–4 or when stretching thin dollahi et al. and Shu et al. . films5 on liquid crystals6 and on elastomers7. The flex- When a dielectric material is subjected to small de- oelectric constants describing the flexoelectric effects of formations and when the piezoelectric and flexoelectric some dielectric materials were observed and measured phenomena produced in it are linear, the electric polar- in a direct or indirect way in a few experimental works ization vector p is related to the infinitesimal strain ten- such as those made by Ma and Cross8–10 and Zubko sor ε and the gradient of the latter, namely E = ∇ε, by et al.11 for various perovskites which exhibit unusually a linear relation: high flexoelectricity, the ones of Kalinin and Meunier12 pi = Dijkεjk + FijklEjkl. (1) and Naumov et al.13 for low-dimensional structures like nanographitic systems and two-dimensional boron- Above, Dijk are the matrix components of the third- nitride sheets or by Zhang et al.14,15, Chu and Salem16 order piezoelectric tensor D and Fijkl stand for the ma- and Zhou et al.17 for dielectric materials and polymers trix components of the fourth-order flexoelectric tensor F. Due to the symmetry εij = εji of ε, the strain gradient E possesses the property Eijk = Ejik and the matrix components of have the following index permutation a) F Corresponding author symmetry: Tel: 33 (0) 160 957 797; Fax: 33 (0) 160 957 799; Email: hung.le-quang@univ-eiffel.fr Fijkl = Fikjl. (2) 2

Note that, if the microstructure of the dielectric mate- some concluding remarks are given. rial in question exhibits centrosymmetry, the requirement II. NOTATIONS AND DEFINITIONS that the third-order tensor D be invariant under the cen- D tral inversion transformation implies that is null, so Let V be a three-dimensional (3D) inner-product space that the constitutive law (1) reduces to over the reals R and Lin be the space of all linear transformations (second-order tensors) on V. The in- pi = FijklEjkl. (3) ner product of two vectors a and b of V is symbolized by a · b. The 3D orthogonal group O(3) is defined as In other words, when the dielectric material has a cen- O(3) = {Q ∈ Lin | Qa · Qb = a · b, ∀a, b ∈ V}. The trosymmetric microstructure, the piezoelectric effect dis- 3D rotation group SO(3) is given by SO(3) = {Q ∈ appears and the electric polarization vector p depends O(3) | det Q = 1}. In what follows, Q(a, θ) stands for only on the strain gradient E. the rotation about a ∈ V through an angle θ ∈ [0, 2π). In the linear constitutive law (1), the classical third- ˜ ˆ ˇ order piezoelectric tensor D has been completely inves- In particular, Q, Q and Q denote, respectively, the rotations Q (e + e + e , 2π/3), Q 2e + ϕ2e , 2π/3 and tigated and understood; however, the fourth-order flexo- 1 2 3 2 3 √ electric tensor F, which is much more complicated than Q (ϕe2 + e3, π) with the golden ratio ϕ = (1 + 5)/2. the usual fourth-order elastic tensor, is far from being For later use, it is convenient to introduce the following thoroughly studied and understood. In our previous standard group notations: one39, the number and types of all possible rotational (i) the identity group, denoted by I, is formed by the symmetries for the flexoelectric tensor F were specified second-order identity tensor I; and the number of independent material parameters of (ii) the cyclic group Zr (r ≥ 2) contains r elements F belonging to each possible symmetry class was deter- generated by Q(e3, 2π/r). In particular, when r → ∞, 40 mined. Later, Shu et al. gave the matrix representa- Zr becomes the group SO(2) consisting of all rotations tions of F for various symmetries. Q about e3 such that Qe3 = e3; The present work can be regarded as a continuation of (iii) the dihedral group Dr (r ≥ 2) comprises 2r ele- 39 our previous one . Precisely, novel 3 × 18 matrix rep- ments generated by Q(e3, 2π/r) and Q(e1, π). The cor- resentations of the flexoelectric tensor F are provided for responding form of Dr when r → ∞ is O(2) consisting all 12 rotational symmetries determined in our previous of all orthogonal tensors Q such that Qe3 = ±e3; study39. The matrix representations of F given in the (iv) the spatial groups T , O and I with T represent- present work are well-structured and much more com- ing the tetrahedral group of 12 elements generated by D2 pact than those of Shu et al.40. This should facilitate and Q˜ , O being the octahedral group of 24 elements gen- ˜ the practical use of F in various anisotropic cases. In ad- erated by D4 and Q, and I symbolizing the dodecahedral ˆ ˇ dition, the flexoelectric tensor F is further investigated group of 60 elements generated by D5, Q and Q. Recall in the present work by determining all of its reflection that the spatial groups T , O and I map a tetrahedron, a symmetries. It is proved that the 12 rotational symme- cube and a dodecahedron onto themselves, respectively. try classes of the flexoelectric tensor F are reduced to 8 Next, we define the space of flexoelectricity tensors as reflection symmetries and these 8 reflection symmetries follows: are identical to those of the fourth-order elastic tensor. Finally, a simple but efficient graphic method based on F = {F = Fijklei ⊗ ej ⊗ ek ⊗ el | Fijkl = Fikjl}. the notion of pole figures is suggested and illustrated for identifying the reflection symmetry and rotational sym- The symmetry group of a flexoelectricity tensor F ∈ F, metry that a given flexoelectric tensor may have. is denoted by G(F) and characterized as The paper is structured as follows. In Section II, some notations and definitions used throughout the paper are G(F) = {Q ∈ SO(3) | Q ∗ F = F} (4) presented. In Section III, the main results obtained in 39 our previous work on the symmetry groups and sym- where Q∗F = QirQjmQknQlsFrmnsei ⊗ej ⊗ek ⊗el. The metry classes of the flexoelectric tensor are recalled for definition of G(F) implies that G(F) is a closed subgroup the paper to be self-contained. In Section IV, compact of SO(3) and G(Q ∗ F) = QG(F)QT for any orthogonal explicit matrix representations of the flexoelectric tensor tensor Q ∈O(3). On the other hand, F is said to exhibit for all possible rotational symmetries are provided. Sec- G-symmetry when G ⊆ G(F) and Q ∗ F = F for all Q ∈ tion V is dedicated to finding out the reflection symmetry G. Consequently, two materials characterized by their classes of the flexoelectric tensor. In Section VI, a sim- respective flexoelectric tensors F1 ∈ F and F2 ∈ F are ple and efficient graphic method is elaborated to identify said to have the same type of symmetry if and only if not only the reflection symmetry but also the rotational the symmetry groups of F1 and F2 are conjugate to each symmetry of a given flexoelectric tensor. In Section VII, other, i.e.

T F1 ∼ F2 ⇔ G(F1) ∼ G(F2) ⇔ ∃Q ∈ SO(3) such that G(F1) = QG(F2)Q . (5) 3

TABLE I. List of notations Notations Descriptions p, pi Electric polarization vector and its components F, Fijkl Fourth-order (type-II) flexoelectric tensor and its components F˜, F˜iα Matrix representation of the flexoelectric tensor and its components I I F , Fijkl Fourth-order type-I flexoelectric tensor and its components D, Dijk Third-order piezoelectric tensor and its components ε, εij Infinitesimal strain tensor and its components E, Eijk Strain gradient tensor and its components E˜ , E˜α Vector representation of the strain-gradient tensor and its components H, H, H Second-, third-, fourth-order harmonic tensors F Space of flexoelectricity tensors Q Generic orthogonal tensor Q(a, θ) Rotation about a through an angle θ G(F) Rotation symmetry group of F {Gi} Rotation symmetry class Zn (n ≥ 2) Cyclic group of order n, generated by the n-fold rotation Q (e3, θ = 2π/n) Dn (n ≥ 2) Dihedral group of order 2n generated by Zn and Q(e1, π) SO(3) Tridimensional rotational group O (3) Tridimensional orthogonal group √ T Q e , π Q e , π Q v, π/ v 3 e e e Tetrahedral group of 12 elements generated by ( 3 ), ( 1 ), ( 2 3) with = 3√( 1 + 2 + 3) 3 O Octahedral group of 24 elements generated by Q(e3, π/2), Q(e1, π), Q(v, 2π/3) with v = 3 (e1 + e2 + e3) I Q e , π/ , Q e , π , Q w, π/ Dodecahedron group of 60 elements generated√ by ( 3 2 5) ( 1 ) ( 2 3) 1 5+1 with w = √ (2e2 + φe3) and φ = 4+φ2 2 SO(2) Subgroup of rotations Q(e3, θ) with θ ∈ [0; 2π) O(2) Subgroup generated by SO(2) and Q(e1, π) P(n) Reflection transformation through the plane of normal n

PF Set of reflection symmetry elements of F {PF} Reflection symmetry class of F Ph Set containing one reflection P(e3) 2pπ Pvk (k ≥ 1) Set containing k elements Pvk = {P(r3( k )}1≤p≤k with r3(θ) = sin θe1 + cos θe2 Phvk (k ≥ 1) Set containing k elements of Pvk completed by P(e3) PO Cubic set consisting of 9 reflections with respect to the nine planes of which the normals of 6 pass through the center of each edge of a regular cube and the normals of 3 through the center of each face of the latter PI Icosahedral set of 15 reflections with respect to the fifteen planes whose normals pass through the center of each edge of a regular icosahedron PO(3) Set composed of all reflections P(n)

With the above notion of conjugacy, a family of non- characterizes a symmetry class for flexoelectric tensors. empty subsets, (Fi)1≤i≤N , of F acts as a partition of the On the other hand, for a given flexoelectric tensor Fi ∈ flexoelectric tensor space F in the sense that no two ele- Fi with the symmetry group G(Fi), the collection of all ments of (Fi)1≤i≤N overlap and the union of (Fi)1≤i≤N the conjugates of G(Fi) in the set of subgroups of SO(3), is equal to F. Thus, each element Fi of this partition i.e.

˜ ˜ T {Gi} = {G(Fi)} = {G ⊆ SO(3) | G = QG(Fi)Q , Q ∈SO(3)}, (6)

constitutes an intrinsic characterization of the type of For the convenience of the reader, the notations used in rotational symmetries exhibited by the elements of Fi. this paper is summarized in Table I. Clearly, the deﬁnition of Fi through {Gi} is more convenient. Finally, we denote by {G} the collection of all the conjugates of G ∈ SO(3) in the set of subgroups of SO(3) and deﬁne F(G) as the set

F(G) = {F ∈ F | G(F) ∈ {G}}. (7) 4

III. ROTATIONAL SYMMETRY CLASSES OF THE tensor. First, using a general method due to Spencer41, FLEXOELECTRIC TENSOR any fourth-order flexoelectric tensor F ∈ F can be first decomposed into totally symmetric tensors and then split For the paper to be self-contained and for later use, the into harmonic tensors. The following explicit harmonic present section recalls the main results of our previous decomposition is established for the flexoelectric tensor study39 concerning the determination of the number and F: types of all the rotational symmetries for the flexoelectric

Fijkl = [H]ijkl 1 1 + (3 [H(1)] + [H(1)] + [H(1)] ) + ( [H(2)] + [H(2)] + [H(2)] ) 3 ijm klm jkm ilm ikm jlm 4 ilm jkm jlm ikm klm ijm 1 + (δ [H(1)] + δ [H(1)] + δ [H(1)] + δ [H(1)] + δ [H(1)] + δ [H(1)] ) 7 ij kl ik jl il jk jk il jl ik kl ij 1 + (3 + + )[H(2)] 6 ijm kln ikm jln jkm iln mn 2 + (2δ [H(3)] − δ [H(3)] + δ [H(3)] − 2δ [H(3)] + δ [H(3)] − δ [H(3)] ) 9 jk il ik jl jl ik il jk lk ji ij kl 1 + (δ [H(4)] + δ [H(4)] + δ [H(4)] − δ [H(4)] − δ [H(4)] − δ [H(4)] ) 6 jl ik kl ij il jk ij kl ik jl jk il 3 + (δ + δ + δ )[a(1)] 10 ij klm jk ilm ik jlm m 1 1 + (2 [a(2)] − [a(2)] ) + (2 δ + 2 δ − δ − δ − δ )[a(2)] 12 kli j klj i 12 kim jl lim jk ljm ik kjm il lkm ij m 1 + (11 δ − 4 δ − 5 δ + 10 δ − 5 δ + 3 δ )[a(3)] 15 ikm jl jkm il jlm ik ilm jk klm ij ijm kl m 1 + (3 [a(3)] + 3 [a(3)] + [a(3)] + [a(3)] ) 15 ijk l ijl k ikl j jkl i α α + 1 (δ δ + δ δ + δ δ ) + 2 (δ δ + δ δ − 2δ δ ). (8) 15 ij kl ik jl il jk 3 jl ik ij kl il jk

It can be seen from (8) that the harmonic decomposition H(1) and H(2) and a fourth-order harmonic tensor H. (1) of F contains : two scalars α1 and α2; three vectors a , The components of these harmonic tensors are explicitly (2) (3) (1) a and a ; four second-order harmonic tensors H , expressed in terms of Fijkl as H(2), H(3) and H(4); two third-order harmonic tensors

1 1 α = (F + 2F ), α = F , (9) 1 3 pqqp ppqq 2 3 pqk mnk (mp)nq

1 1 [a(1)] = (F + 2F ), [a(2)] = F , [a(3)] = ( F + F ), (10) k 9 pqk pmmq mmpq k mnp mnkp k 6 pqk pqmm pqn pqkn

1 1 [H(1)] = F − α δ , [H(2)] = ( + )F − α δ , km (ppkm) 3 1 km km 2 pqk lnm pqm lnk (lp)nq 2 km 1 1 [H(3)] = ( F + F ), [H(4)] = ( F + F ), (11) km 2 lnq pqm (lp)nk pqk (lp)nm km 2 lnq pqm (lpk)n pqk (lpm)n 5

1 [H(1)] = [S(1)] − (δ [a(3)] + δ [a(3)] + δ [a(3)] ), kmn (kmn) 5 mn k km n kn m 1 [H(2)] = [S(2)] − (δ [a(1)] + δ [a(1)] + δ [a(1)] ), (12) kmn (kmn) 5 mn k km n kn m

1 [ ] = F − (δ [H(1)] + δ [H(1)] + δ [H(1)] + δ [H(1)] + δ [H(1)] + δ [H(1)] ) H klmn (klmn) 7 kl mn km ln kn lm lm kn ln km mn kl α − 1 (δ δ + δ δ + δ δ ). (13) 15 kl mn km ln kn lm

Above, δij and ijk are the Kronecker delta and permuta- third-order tensors S1 and S2 are deﬁned as tion symbol, respectively; either •i1i2...in or [•]i1i2...in is [S(1)] = F , [S(2)] = F . the component of an nth-order tensor •; •(i1i2...ir )ir+1...in kmn pqn (pk)qm kmn pqn (pkm)q denotes the average of r! components obtained by per- (14) muting the indices i1, i2, . . . , ir in all possible ways; the Next, by using the Cartan method, the second-, third- and fourth-order harmonic tensors H(i), H(i) and H in Eq. (8) can be rewritten as follows:

(i) (2) (2) (2) (2) (2) H = α0i U0 + α1i U1 + β1i T1 + α2i U2 + β2i T2 with i = 1, 2, 3, 4, H(i) (3)U (3)U (3)T (3)U (3)T (3)U (3)T = α0i 0 + α1i 1 + β1i 1 + α2i 2 + β2i 2 + α3i 3 + β3i 3 with i = 1, 2, (4) (4) (4) (4) (4) (4) (4) (4) (4) H = α0 U0 + α1 U1 + β1 T1 + α2 U2 + β2 T2 + α3 U3 + β3 T3 + α4 U4 + β4 T4, (15)

where Ui, Ti, Ui, Ti, Ui and Ti are the tensors involved groups of SO(3): in the Cartan decomposition, whose explicit expressions 42,43 can be found in Forte and Vianello or Le Quang and I, {Z }, {D }, {O}, {T }, {SO(2)}, {O(2)},SO(3) (16) He39. r r With the help of the harmonic and Cartan decompositions presented above, it can be shown that the number where 2 ≤ r ≤ 4. of all possible rotational symmetry classes for all ﬂex- Alternatively, the 12 sets F(I), F(Zr), F(Dr) with oelectric tensors is 12. These 12 symmetry classes are 2 ≤ r ≤ 4, F(O), F(T ), F(SO(2)), F(O(2)), F(SO(3)) characterized by the conjugates of the following 12 sub- form a partition of the ﬂexoelectric tensor space F:

4 4 F = F(I) ∪r=2 F(Zr) ∪r=2 F(Dr) ∪ F(O) ∪ F(T ) ∪ F(SO(2)) ∪ F(O(2)) ∪ F(SO(3)). (17)

For more detail about the derivation of these results, the symmetry classes, we first adopt the following reduced reader can refer to our previous work39. suffix notations for the gradient E of the infinitesimal strain tensor ε and for the fourth-order flexoelectric tensor F ∈ F: IV. COMPACT EXPLICIT MATRIX h√ i REPRESENTATIONS OF THE FLEXOELECTRIC E˜γ = 2(1 − δjk) + δjk Ejkl, (18) TENSOR

In order to obtain explicit matrix expressions of the √ ˜ h i fourth-order ﬂexoelectric tensor for all the 12 rotational Fiγ = 2(1 − δjk) + δjk Fijkl, (19) 6

Above, γ and ζ are defined in Table II and the summation TABLE II. Suffix notation correspondences between (j, k, l) convention does not hold for j, k, m and n. and γ. By using the procedure elaborated in Le Quang and (j, k, l) or (m, n, s) γ or ζ He39 to construct a flexoelectric tensor F exhibiting a (1,1,1) 1 required symmetry and by exploiting the harmonic and (2,2,1) 2 Cartan decompositions in Eqs. (8) and (15) together (1,2,2) or (2,1,2) 3 (•) (•) with the Cartan decomposition parameters α and β (3,3,1) 4 • • (1,3,3) or (3,1,3) 5 provided in Table III, we can exactly calculate the num- (2,2,2) 6 ber of independent components contained in a flexoelec- (1,1,2) 7 tric tensor F ∈ F belonging to a given symmetry class. (1,2,1) or (2,1,1) 8 The corresponding results are shown in Table IV. (3,3,2) 9 Next, by adopting the above reduced suffix notations (2,3,3) or (3,2,3) 10 for the fourth-order flexoelectric tensor F, we obtain the (3,3,3) 11 explicit matrix forms of the elementary block matrices of (1,1,3) 12 (D,C,J)-type as well as the explicit matrix form of F˜ for (1,3,1) or (3,1,1) 13 each symmetry class. These explicit block matrix forms (2,2,3) 14 of F˜ are provided as follows: (2,3,2) or (3,2,2) 15 (1,2,3) or (2,1,3) 16 Identity class: I (1,3,2) or (3,1,2) 17  (5) (5) (5) (3)  (2,3,1) or (3,2,1) 18 Dx Cxy Cxz Jx ˜  (5) (5) (5) (3)  FI =  Cyx Dy Cyz Jy  , (24) (5) (5) (5) (3) Czx Czy Dz Jz where γ defined in Table II and the summation convention does not apply on j and k. Cyclic classes: Zr According to the previously reduced notation rules, the  (5) (5)  constitutive relation (3) between the electric polarization Dx Cxy 0 0 and the gradient of the infinitesimal strain tensor can now F˜ =  (5) (5)  , (25) Z2  Cyx Dy 0 0  be written in the following matrix form: (5) (3) 0 0 Dz Jz p = F˜ · E˜ (20)  (4) (4) (2) (2)  D Cxy Cxz Cyz · P1 ˜ F˜ =  (4) (4) (2) (2)  , (26) where the vectors p and E are specified by Z3  −Cxy D Cyz −Cxz · P1  (1) (1) (3) (1) T C C D J p = [p1 p2 p3] , zx zy z z T  (5) (5)  E˜ = [E˜1 E˜2 ... E˜18] , (21) D Cxy 0 0 F˜ =  (5) (5)  , (27) and F˜ is the 3 × 18 flexoelectric matrix whose block ma- Z4  −Cxy D 0 0  (3) (1) trix representation has the general expression 0 0 Dz Jz    (4) (4)  Dx Cxy Cxz Jx D Cxy 0 0 ˜ ˜  (4) (4)  F =  Cyx Dy Cyz Jy  . FSO(2) =  −Cxy D 0 0  , (28) (3) (1) Czx Czy Dz Jz 0 0 Dz Jz

Here, the diagonal block matrices Dx, Dy and Dz of D- Dihedral classes: Dr type and extra-diagonal block matrices Cxy, Cxz, Cyx, C C and C of C-type have the same size 1 × 5  (5)  yz zx zy D 0 0 0 while the extra-diagonal block matrices J , J and J of x x y z F˜ =  (5)  , (29) J-type are of the size 1 × 3. D2  0 Dy 0 0  (5) With the previously reduced notation rules, the action 0 0 Dz 0 of an orthogonal tensor Q ∈ O(3) on a ﬂexoelectric tensor  (2)  D(4) 0 0 C · P can be expressed in the following simple and explicit yz 1 F F˜ =  (4) (2)  , (30) matrix form: D3  0 D Cyz 0  (1) (3) 0 Czy Dz 0 QirQjmQknQlsFrmns = QirF˜rζ Q˜ζγ (22)  D(5) 0 0 0  (5) where Q˜ζγ , standing for the components of the 18 × 18 ˜ FD4 =  0 D 0 0  , (31) ˜ (3) matrix Q, are given by 0 0 Dz 0 √ h i  (4)  Q˜ζγ = 2(1 − δjk) + δjk D 0 0 0 ˜ (4) h√ i FO(2) =  0 D 0 0  , (32) × 2(1 − δ ) + δ Q Q Q . (23) (3) mn mn jm kn ls 0 0 Dz 0 7

TABLE III. Zero components · and non-zero independent material parameters • contained in a ﬂexoelectric tensor belonging to a given symmetry class

a b Parameters I {Z2}{D2}{Z3}{D3}{Z4}{D4} {T } {O} {O(2)}{SO(2)}{SO(3)} α1 •••••••••••• α2 •••••••••••• (i) a1 •··········· (i) a2 •··········· (i) a3 •·••·•····•· (2) α0i •••••••··••· (2) α1i •··········· (2) β1i •··········· (2) α2i •••········· (2) β2i ••·········· (3) α0i ••·•·•····•· (3) α1i •··········· (3) β1i •··········· (3) α2i ••·········· (3) β2i •••····•···· (3) α3i •··••······· (3) β3i •··•········ (4) α0 •••••••••••· (4) α1 •··········· (4) β1 •··········· (4) α2 •••········· (4) β2 ••·········· (4) α3 •··•········ (4) β3 •··••······· (4) α4 •••··••••··· (4) β4 ••···•······

a (4) (4) α4 = 5α0 b (4) (4) α4 = 5α0

TABLE IV. Number of independent material parameters contained in a ﬂexoelectric tensor belonging to a given symmetry class

Symmetry class I {Z2}{D2}{Z3}{D3}{Z4}{D4} {T } {O} {O(2)}{SO(2)}{SO(3)} Number of independent components 54 28 15 18 10 14 8 5 3 7 12 2

Spatial classes: rection matrices. By introducing two material parameters deﬁned as  D(5) 0 0 0  F˜ = (5) , (33) 1 1 T  0 D · P2 0 0  η = √ (F˜11 − F˜12), θ = √ (F˜22 − F˜21), (36) 0 0 D(5) 0 2 2  (3)  D 0 0 0 the elementary block matrices of (D,C,J)-types as well F˜ = (3) , (34) O  0 D 0 0  as of the correction matrices P1 and P2 are explicitly 0 0 D(3) 0 expressed as follows:  D(2) 0 0 0  D-type elements: ˜ (2) FSO(3) =  0 D 0 0  . (35) D(5) = d d d d d , (37) 0 0 D(2) 0 1 2 3 4 5 (4) D = d1 d2 η d4 d5 , (38) In Eqs. (25)-(35), the superscript of each elementary D(3) = d d d d d , (39) block matrix of D, C or J-type denotes the number of 1 2 3 2 3 (2) independent material parameters; P1 and P2 are two cor- D = d1 d2 η d2 η , (40) 8

C-type elements: •P O designating the cubic set of nine reflections with (5) respect to the nine planes of which the normals of C = c1 c2 c3 c4 c5 , (41) 6 pass through the center of each edge of a regular (4) C = c1 c2 θ c4 c5 , (42) cube and the normals of 3 through the center of (2) each face of the latter. C = 0 c2 c3 −c2 −c3 , (43) (1) •P denoting the icosahedral set consisting of fifteen C = c1 −c1 −c1 0 0 , (44) I reflections with respect to the fifteen planes whose J-type elements: normals pass through the center of each edge of a (3) (1) regular icosahedron; J = j1 j2 j3 ,J = 0 j2 −j2 , (45) Correction elements: •P O(3) representing the set composed of all reflections P(n) with n ∈ S2 where S2 is the unit sphere  0 0 0   1 0 0 0 0  defined by S2 = {x ∈ R3 | kxk = 1}.  1 0 0   0 0 0 1 0      P1 =  0 1 1  ,P2 =  0 0 0 0 1  . (46) We can show that the space F of flexoelectric tensors is  0 0 0   0 1 0 0 0  divided into the 8 reflection symmetry classes which are 0 0 1 0 0 1 0 0 characterized by the following 8 sets of reflection trans- Finally, the compact explicit matrix expressions for the formations: 12 symmetry classes of flexoelectric tensors are shown in ∅, {P }, {P }, {P }, {P }, {P }, {P }, {P }. Table V. h hv2 v3 hv4 hv∞ O O(3) (50) The characteristics of each reflection symmetry class, and V. REFLECTION SYMMETRY CLASSES OF THE especially its link with the rotational ones, are detailed in FLEXOELECTRIC TENSOR Table VI. It is important and interesting to remark that: (i) unlike the results as obtained by Chadwick et al.44 First, let us introduce P(n) ∈ O(3)\SO(3), the re- for the space of fourth-order elasticity tensors, according to which the classifications by rotational symmetry flection through the plane P(n) = {x ∈R3 | x · n = 0} perpendicular to a unit vector n, by groups and by reflection symmetry planes give the same response, the number of rotational symmetry classes for P(n) = I − 2n ⊗ n. (47) the space of fourth-order flexoelectric tensors is 12 while the one of reflection symmetry classes is only 8; (ii) these It can be seen from (47) that P is an even function of n 8 reflection symmetry classes for the space of flexoelectric in the sense that P(−n) = P(n). Thus, two unit normal tensors are exactly identical to the ones for the space of vectors are associated to each reflection. elasticity tensors, even through a fourth-order flexoelec- Next, we denote by PF the set of reflection symmetry tric tensor is algebraically more complex than a fourth- elements of F ∈ F. The reflection symmetry class of order elasticity tensor. F ∈ F, symbolized by {PF}, is defined as the collection of all the conjugates of PF, namely VI. A GRAPHIC METHOD FOR IDENTIFYING {P } = {P0 = RP RT | R ∈ SO(3)}. (48) F F F REFLECTION AND ROTATIONAL SYMMETRIES Note that {P } represents the reflection symmetry class F The important question now arises as to how to iden- and not the symmetry group of F ∈ F. We introduce the unit vector tify the reflection symmetry and rotational symmetry and of a given flexoelectric tensor from the knowledge of ri(θ) = sin θej + cos θek (49) its matrix relative to a basis. The present section aims at elaborating a simple but efficient graphic method to with {i, j, k} being a cycle permutation of {1, 2, 3}, and answer this question. the following sets of reflection transformations: Francois et al.45 initiated a graphic approach to identifying the reflection symmetry planes that a given fourth- •P h being the set which contains only the reflection order elastic tensor has. This approach is based on the P(e3). The label h means “horizontal”. notion of “pole figures”. Owing to the fact that the reflection symmetry classes of the fourth-order elastic ten- •P vk being the set defined by Pvk = 2pπ sor are identical to its rotational symmetry classes, the {P(r3( k )}1≤p≤k and consisting of k elements. The label v means “vertical”. identification of the formers leads also the one of the lat- ters. However, when the fourth-order elastic tensor is

•P hvk (k ≥ 1) being the set deﬁned by the k elements concerned, the situation is much more complicated, since

of Pvk completed by P(e3): this set comprises k+1 its reﬂection symmetry classes are 8 while its reﬂection elements. symmetry classes are 12. Thus, in this section we extend 9

TABLE V. Number of independent material parameters and compact explicit matrix expression for each of the 12 symmetry classesclasses of of ﬂexoelectric ﬂexoelectric tensors tensors

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TABLE VI. Symmetry plane stratification of the space F of fourth-order flexoelectric tensors System Reflection Number of reflection Rotational symmetry class symmetry planes symmetry class

Triclinic ∅. 0 I, {Z3} Monoclinic {Ph} 1 {Z2}, {Z4}, {SO(2)}

Orthotropic {Phv2 } 3 {D2}, {T }

Trigonal {Pv3 } 3 {D3}

Tetragonal {Phv4 } 5 {D4}

Trans. isotropic {Phv∞ } ∞ + 1 {O(2)} Cubic {PO} 9 {O} 3 Isotropic {PO(3)} ∞ {SO(3)} 10 the graphic approach of Francois et al.45 to being able The foregoing graphical approach is now applied to to identify not only the reflection symmetry of a given a given flexoelectric tensor F ∈ F with a given angle flexoelectric tensor but also its rotational symmetry. ψ ∈ [0; 2π[ for identifying all invariant directions defined 2 Let n ∈ S be the unit vector relative to the reflection by n = sin θ cos φe1 + sin θ sin φe2 + cos θe3 in the sense transformation P(n) = I − 2n ⊗ n through the plane that F is unchangeable under the rotational transforma- P(n). With no loss of generality, n is expressed by tion action Q(n, ψ). As before, by using the well-known Rodrigues expression of Q(n, ψ), i.e. n = sin θ cos φe1 + sin θ sin φe2 + cos θe3 (51) Q(n, ψ) = cos(ψ)I − sin(ψ) · n + [1 − cos(ψ)]n ⊗ n (53) where (φ, θ) ∈ [0, 2π[×[0, π[ denote, respectively, the longitude and colatitude angles relative to a system of spher- in which denotes the Levi-Civita third-order tensor, and ical coordinates. Then, by considering, for a given flexo- by introducing the following function electric tensor ∈ F, the function: F J(θ, φ, ψ) = kQ(θ, φ, ψ) ∗ F − Fk ˜ ˜ ˜ L(θ, φ) = kP(θ, φ) ∗ F − Fk = kP(θ, φ) · F˜ · P˜ (θ, φ) − F˜k = kQ(θ, φ, ψ) · F · Q(θ, φ, ψ) − Fk (54) (52) where Q(θ, φ, ψ) is a rotation operator parametrized with in which k · k is the Frobenius norm inherited from the the longitude, colatitude and rotation angles; Q˜ (θ, φ, ψ) scalar product on F, P(θ, φ) is a reflection operator is a 18 × 18 matrix whose components are defined by parametrized with the longitude and colatitude angles (23), the vanishing loci of J(θ, φ, ψ) allow us to obtain and P˜ (θ, φ) is a 18 × 18 matrix whose components are the invariant directions n that possesses. obtained by replacing Q with P in (23). Finally, the F We illustrate, in Figure 2, the vanishing loci of the vanishing loci of L(θ, φ) give the unit normals to the sym- function J(θ, φ, 2π/3) plotted on the θ − φ plane for a metry planes that has. F flexoelectric tensor belonging to the rotational symme- Concretely, the function (52) is numerically evaluated F try class {Z } whose matrix representation is provided in in a discrete way. Precisely, we introduce 3 Section IV. It can be seen from Figure 2 that there exists 2π 2π only one invariant axis spanned by n = e3 or n = −e3. Mij = L(θi, φj) with θi = i and φj = j This is in agreement with the fact that the rotational N N symmetry class {Z3} contains Q(e3, 2π/3). where the number N depends on the degree of numer- By combining the knowledge of the matrix representa- ical accuracy required. In our computations, N is set tions of all rotational symmetry classes with pole figures, to be equal to 160. To evaluate the function L(θ, φ), we can finally identify the rotational symmetry class to we first use the matrix representations of flexoelectric which a given flexoelectric tensor belongs. The corre- tensors presented in Section IV. In addition, the numeri- sponding identification procedure is summarized in Fig- cal values of the components of a flexoelectric tensor are ure 3. determined as random integers picked-up in the range {−10, 10}. We show, in Figure 1, the loci of the zeros of L(θ, φ) plotted on the θ −φ plane for all rotation symme- VII. CONCLUDING REMARKS try classes and all reflection symmetry classes. It can be seen from Figure 1 that the number of symmetry planes Flexoelectricity is an electromechanical phenomenon for a given rotation symmetry class or reflection sym- which has a great number of potential applications metry class coincides exactly with the one provided in including energy harvesting, sensors, actuators and Table VI. This also constitutes a validity verification of biotechnology. A full understanding of the fourth-order our theoretical results. flexoelectric tensor is essential not only to the fundamen- In addition, it can be observed from Figure 1 that, tal theory of flexoelectricity but also to all possible ap- even through the flexoelectric tensors belonging to both plications of flexoelectricity. In the present work, which 39 rotational symmetry classes {Z3} and I do not exhibit may be viewed as a continuation of our previous one , any reflection symmetry plane, we can differentiate them compact explicit matrix representations of the flexoelec- since the loci of the function L(θ, φ) for the flexoelectric tric tensor have been provided for all the 12 possible ro- tensors belonging to {Z3} are periodic in the φ-direction tational symmetry classes, so as to facilitate its use in with period 2π/3 while the ones for the flexoelectric ten- various situations; the reflection symmetry classes of the sors appertaining to I are not periodic. Similarly, even if flexoelectric tensor have been also determined and shown the flexoelectric tensors belonging to the rotational sym- to be identical to those of the fourth-order elastic ten- metry classes {D2} and T possess the same number of re- sor; a simple and efficient graphic method for identifying flection symmetry planes, the loci of the function L(θ, φ) the rotational symmetry and reflection symmetry of a for flexoelectric tensors belonging to {T } are periodic in given flexoelectric tensor has been elaborated and illus- the φ-direction with period π/2 but the counterpart of trated. These results contribute to developing the con- the flexoelectric tensors appertaining to {D2} are peri- tinuum theory of flexoelectricity and rendering the use of odic in φ-direction with period π. this theory easier in various anisotropic cases. 11

FIG. 1. Loci of the zeros of the function L(θ, φ) plotted on the θ − φ plane for all rotation symmetry classes and all reﬂection symmetry classes. 12

As a simple example of application of our results, we consider Pervoskites with general chemical formula ABO3, which are known as ferroelectric materials exhibiting higher permittivity ferroelectrics like flexoelectricity, piezoelectricity and pyroelectricity than the ones of usual dielectric materials, and which are now widely used for the production of electronic components and mi- crotransducers. Since the oxygen octahedral structure of Pervoskites, the symmetry behavior of the corresponding flexoelectricity tensor is cubic and characterized by the rotation symmetry class {O} or by the reflection symmetry class {PO}. The matrix of the flexoelectricity tensor ˜ FIG. 2. Loci of the function J(θ, φ, 2π/3) plotted on the θ −φ relative to an appropriate basis, denoted by FABO3 , is plane for a flexoelectric tensor F belonging to the rotational given by symmetry class {Z3}

  F1111 F2112 F2121 F2112 F2121 0000000000000 ˜ FABO3 =  0 0 0 0 0 F1111 F2112 F2121 F2112 F2121 0 0 0 0 0 0 0 0  . 0 0 0 0 0 0 0 0 0 0 F1111 F2112 F2121 F2112 F2121 0 0 0

˜ The matrix FABO3 has 3 independent components, i.e., the longitudinal flexoelectric coefficient F , transverse TABLE VII. Suffix notation correspondences between (j, k, l) 1111 and γ. flexoelectric coefficient F2112 and shear flexoelectric coefficient F2121. The values of F1111, F2112 and F2121 deter- (j, k, l) γ mined experimentally and computationally can be found (1,1,1) 1 in Wang et al.34 and Shu et al.37 for some Pervoskites. (1,2,2) 2 In parallel with the formulation used in the present (2,1,2) or (2,,2,1) 3 work in which the electric polarization vector p is linearly (1,3,3) 4 related to the strain gradient E through the fourth-order (3,1,3) or (3,3,1) 5 (2,2,2) 6 flexoelectric tensor , there is another formulation of the F (2,1,1) 7 electric polarization vector p linearly connected to the (1,1,2) or (1,2,1) 8 U second-order derivative of the displacement vector, = (2,3,3) 9 27 ∇∇u, as follows (see e.g. Hong et al. ): (3,2,3) or (3,3,2) 10 (3,3,3) 11 I I pi = FijklUjkl or pi = Fijkluj,kl. (55) (3,1,1) 12 (1,1,3) or (1,3,1) 13 I (3,2,2) 14 Here Fijkl, the tensor component of the type-I flexoelectric tensor I , possesses the index permutation symmetry (2,2,3) or (2,3,2) 15 F (3,1,2) or (3,2,1) 16 F I = F I . Compared with , called also type-II flex- ijkl ijlk F (2,1,3) or (2,3,1) 17 oelectric tensor, more suitable not only for formulating (1,2,3) or (1,3,2) 18 the thermodynamic theory of ferroelectric materials but also for making comparisons with experimental measurements, the definition of the type-I flexoelectric tensor FI is complicated for mathematical derivations of the micro- the reduced suffix notations described in Table VII the scopic theory of flexoelectricity. It can be shown that the fourth-order flexoelectric tenor I with component F I I F ijkl connections between F and F are given by can be expressed in a 3 × 18 matrix form F˜ I with com- ˜I ponents Fiγ . Consequently, the number of independent I 1 I I I F = (Fijkl + Fijlk),Fijkl = F + F − F . material parameters and compact explicit matrix expres- ijkl 2 ijkl ikjl iljk (56) sion for each of the 12 symmetry classes of flexoelectric I In addition, due to the fact that both flexoelectric tensors tensors F are exactly identical to the ones of F. FI and F exhibit mathematically the same permutation Finally, it is interesting and important to remark that symmetry with respect to two indexes, the flexoelectric the methods and results presented in the present work tensors FI and F will possess the same rotation and re- are directly applicable to the fourth-order flexomagnetic flection symmetry classifications. Moreover, by adopting tensor46. 13

FIG. 3. Procedure for identifying the rotational symmetry class to which a ﬂexoelectric tensor belongs with respect to a given basis

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