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

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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 To cite this version: 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)Universit´eGustave Eiffel, CNRS, MSME UMR 8208, F-77454 Marne-la-Vall´ee, 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 cen- trosymmetric microstructure, undergoing a non-uniform strain. It is characterized by the fourth-order flexo- electric 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. PACS numbers: Valid PACS appear here 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 " and the gradient of the latter, namely E = r", by et al.11 for various perovskites which exhibit unusually a linear relation: high flexoelectricity, the ones of Kalinin and Meunier12 pi = Dijk"jk + 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 "ij = "ji of ", the strain gradi- ent 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) = fQ 2 Lin j Qa · Qb = a · b; 8a; b 2 Vg. The trosymmetric microstructure, the piezoelectric effect dis- 3D rotation group SO(3) is given by SO(3) = fQ 2 appears and the electric polarization vector p depends O(3) j det Q = 1g. In what follows, Q(a; θ) stands for only on the strain gradient E. the rotation about a 2 V through an angle θ 2 [0; 2π). In the linear constitutive law (1), the classical third- ~ ^ ˇ order piezoelectric tensor D has been completely inves- In particular, Q, Q and Q denote, respectively, the ro- tations Q (e + e + e ; 2π=3), Q 2e + '2e ; 2π=3 and tigated and understood; however, the fourth-order flexo- 1 2 3 2 3 p electric tensor F, which is much more complicated than Q ('e2 + e3; π) with the golden ratio ' = (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 ! 1, 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 ! 1 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 ^ 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.
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