Orientation Dependence of Elastic and Piezoelectric Properties in Rhombohedral BiFeO3

Through a coordinate transformation approach, crystal orientation dependences of elastic and piezoelectric properties at room temperature have been investigated in a three-dimensional space for rhombohedral bismuth ferrite (BiFeO3). Elastic constants (stiffnesses) c11′, c12′, c13′ and piezoelectric constants d15′, d31′, d33′ along arbitrary orientations were obtained based on crystalline asymmetry characteristics of 3m point group BiFeO3. Parameters along specific orientations obtaining the largest values were presented. The max c11′ = 213 × 109 N/m2 could be achieved in planes with ϕ = 0° and 90°. The max c12′ = c13′ = 132.2 × 109 N/m2 could be achieved along directions at θ = 13° and θ = 77° inside three mirror planes, respectively. The max d15′ = 27.6 × 10−12 C/N and the max d31′ = 12.67 × 10−12 C/N could be both obtained along directions at θ = 69° inside mirror planes. The max d33′ = 18 × 10−12 C/N could be obtained at θ = 0°, along the spontaneous polarization axis. By adopting optimal directions, the elastic and piezoelectric parameters of BiFeO3 could be significantly enhanced which shows applications for the growth of BeFeO3 films with preferred orientations and enhanced properties.


Introduction
Bismuth ferrite (BiFeO 3 ) was first actively pursued as a room temperature single-phase multiferroic material for its coexistence of magnetic order and electric order, high Curie point, and G-type antiferromagnetic Neel point [1][2][3][4]. Then, as BiFeO 3 exhibits good ferroelectric and piezoelectric properties, it was also studied as a piezoelectric phase to combine other magnetostrictive phase to form composite multiferric with tailored properties, such as BiFeO 3 -CoFe 2 O 4 [5], BiFeO 3 -CuFe 2 O 4 [6], etc., which shows potential technical applications in multi-state magnetoelectric memories [7], weak magnetic fields detectors [8], and other novel sensors [9,10]. As a lead-free piezoelectric material, BiFeO 3 could be highly applied in piezoelectric MEMS devices because of the spontaneous polarization of as high as 100 µC/cm 2 [11] and the high Curie temperature [3]. Besides, BiFeO 3 thin films are reported to have lower dielectric constants than Pb-based piezoelectric materials which could generate high electromechanical coupling abilities [12,13]. On raising piezoelectric constants of BiFeO 3 , previous studies were focused on doping transition metals such as Sm, Yb, Ho in A site and Sc, In in B site of BiFeO 3 , piezoelectric constants d 33 of up to~20-~50 pC/N were achieved [14,15]. Most piezoelectrics have asymmetrical crystal structures, which result high anisotropy of parameters of the materials. Piezoelectric thin films with preferred orientations can obtaion enhanced or decreased properties compared with grain random distributed materials [16,17]. This also applies to doped films with preferred orientations can obtaion enhanced or decreased properties compared with grain random distributed materials [16,17]. This also applies to doped BiFeO3 with unchanged crystal structures. Whether the properties will be enhanced or decreased is strongly determined by asymmetry characteristics of crystal, i.e., the point groups. The crystal structure of BiFeO3 below 1100 K belongs to a rhombohedral system with point group of 3m [18]. It is meaningful to have a precise description of properties-orientation relations to guide further experimental works. In this research, using the coordinate transformation method, we investigated the orientation dependences of elastic and piezoelectric parameters of rhombohedral BiFeO3 with the 3m point group. Precise relations between values of parameters and arbitrary orientations were given, and orientations along which the maximum and minimum of these parameters could be obtained were specified. The result shows applications for the growth of BiFeO3 films with an oriented structure and enhanced properties.

Methods
The Curie point and the antiferromagnetic Neel point of BiFeO3 are 1100 [3] and 640 K [4], respectively. Below the Curie point, BiFeO3 is a member of rhombohedral crystal system with point group of 3m [18]. For this type of crystal, the asymmetrical characteristics lie in such a way that it has a threefold rotation axis along the c-axis (also the spontaneous polarization axis, displacements of Bi relative to O in rhombohedral BiFeO3 [19,20], also can be denoted as [111]-axis using coordinate with oxygen octahedra in perovskite Bi framework) and three mirror planes which are 120° apart (two of them are parallel to a-and b-axes, respectively). Because of α = β = 90° and γ = 120° for rhombohedral BiFeO3 crystal, the physical orthogonal coordinate system does not coincide rightly with a-, b-, and c-axes. The chosen coordinate system in this study is as follows: X-axis and Z-axis are chosen along a-axis and c-axis (i.e., X||a, Z||c), respectively; Y-axis is determined by using the right-hand rule, which is perpendicular to a-axis and at a 30° angle with b-axis ( Figure 1). To determine the anisotropy of properties originating from asymmetry of crystals, we use the coordinate system transformation method that is related to transforming the property tensors to obtain values in any arbitrary direction. It contains two-step rotations to reach a desired direction from the original coordinate system. Firstly, rotation through a clockwise angle  about the To determine the anisotropy of properties originating from asymmetry of crystals, we use the coordinate system transformation method that is related to transforming the property tensors to obtain values in any arbitrary direction. It contains two-step rotations to reach a desired direction from the original coordinate system. Firstly, rotation through a clockwise angle φ about the Z-axis, Materials 2018, 11, 2441 3 of 10 the original orthogonal coordinates XYZ can be changed into a set of new coordinates X'Y'Z' (Z'||Z); secondly, rotation through a clockwise angle θ about the X'-axis; the coordinate system X'Y'Z' can be changed into another set of coordinates X"Y"Z" (X"||X'). The transformation matrices A Z and A X corresponding to the first and the second rotations are N and M are [6 × 6] bond strain transformation matrices. a ij represents the element in row i and column j in transformation matrices A, the matrices N and M are  The elastic constants, including the compliance s ij (i, j = 1-6) (m 2 /N) and the stiffness c ij (i, j = 1-6) (N/m 2 ), are defined by Hooke's Law containing the relations between the strain x and the stress X, (x) = (s) (X) and (X) = (c) (x). The piezoelectric coefficient d ij (i = 1-3, j = 1-6) (C/N), relates polarization P, to stress X in the relation of (P) = (d) (X). For a three-dimensional space coordinate system, all physical quantities are presented in matrix forms. Matrices of elastic stiffnesses c ij and piezoelectric constants d ij are presented in [6 × 6] and [3 × 6] rank tensors, respectively. According to Euler's rotation laws, physical quantity matrices in a new orientation are obtained by two-step product operations with transformation matrices A, N, and M [21,22].
The symbols ' ' and 't' represent parameters in the new coordinate system and transpose matrix, respectively. The key to calculate Equations (5) and (6) is using the right original matrices (at θ = φ = 0 • ) which are obtained from Neumann's principle changing with different point groups. The original matrices of (c) and (d) for 3m point group can be obtained from Ref. [20]. The elastic stiffnesses c ij of BiFeO 3 at room temperature were obtained from Refs. [23,24]. The piezoelectric constants d ij of BiFeO 3 at room temperature were obtained from Refs. [24,25]. Three-dimensional and two-dimensional representation graphs of parameters were drawn using Maple 18, a mathematical software program.

Results and Discussion
In both three-dimensional and two-dimensional graphs, the distance between a point and the original point represents the absolute value of the relative physical quantity in the relative direction. Orientational dependences of elastic constants c ij were obtained by calculating (c ) in Equation (5) using data of five independent elastic constants (c 11 , c 12 , c 13 , c 33 , and c 44 ) for rhombohedral BiFeO 3 . Figure 2a shows the three-dimensional representation of the first term elastic constant c 11 . In the geometry in Figure 2a, moderate changes of c 11 could be observed with rhombohedral BiFeO 3 crystal rotating to different (θ, φ). The max c 11 = 213 × 10 9 N/m 2 is along φ = 0 • and 90 • (also see Figure 2b); while the min c 11 = 188 × 10 9 N/m 2 is along φ = 45 • and 135 • with a 11.7% decrease from the maximum. It also indicates that along principle axes of rhombohedral BiFeO 3 , the max c 11 could be obtained.

Results and Discussion
In both three-dimensional and two-dimensional graphs, the distance between a point and the original point represents the absolute value of the relative physical quantity in the relative direction. Orientational dependences of elastic constants cij′ were obtained by calculating (c′) in Equation (5) using data of five independent elastic constants (c11, c12, c13, c33, and c44) for rhombohedral BiFeO3. Figure 2a shows the three-dimensional representation of the first term elastic constant c11′. In the geometry in Figure 2a, moderate changes of c11′ could be observed with rhombohedral BiFeO3 crystal rotating to different (θ, ϕ). The max c11′ = 213 × 10 9 N/m 2 is along  = 0° and 90° (also see Figure 2b); while the min c11′ = 188 × 10 9 N/m 2 is along  = 45° and 135° with a 11.7% decrease from the maximum. It also indicates that along principle axes of rhombohedral BiFeO3, the max c11′ could be obtained.  Figure 3a shows the three-dimensional representation of the second term elastic stiffness c12′ of rhombohedral BiFeO3. The geometry exhibits rotation symmetry in that by rotating through the Z-axis about 120° it coincides with each other, and by rotating through the Z-axis about 60° the rotated and original shapes are axisymmetric about the Z-axis. This is because to the symmetry elements of rhombohedral BiFeO3 are three mirror-planes 120° apart from each other. Figure 3b shows the cross-section plots of c12′ in XOZ plane at  = 0° or 120°. Plot in Figure 3c at  = 60° is axisymmetric with the plot of Figure 3b about the Z-axis. The max c12′ = 132.3 × 10 9 N/m 2 can be obtained at θ = 13° when  = 0° and 120°, θ = −13° when  = 60°. This indicates the three maximums exist in the mirror plane of rhombohedral BiFeO3, but they are at an angle to the spontaneous polarization axis. The min c12′ = 49 × 10 9 N/m 2 can be obtained at θ = −77°. Figure 4a shows the three-dimensional representation of the second term elastic stiffness c13′ of rhombohedral BiFeO3. Unlike c12′, large c13′ is found along the transverse directions, rather than the longitudinal directions. Symmetrical lobes of c13′ also are along mirror planes, as shown in Figure  4b,c. The max c13′ = 132 × 10 9 N/m 2 can be obtained at θ of 77° inside mirror planes. The min c13′ = 44.6 × 10 9 N/m 2 can be obtained at θ of -13°. Values of c11′, c12′, and c13′ of rhombohedral BiFeO3 along several chosen orientations are shown in Table 1.  Figure 3a shows the three-dimensional representation of the second term elastic stiffness c 12 of rhombohedral BiFeO 3 . The geometry exhibits rotation symmetry in that by rotating through the Z-axis about 120 • it coincides with each other, and by rotating through the Z-axis about 60 • the rotated and original shapes are axisymmetric about the Z-axis. This is because to the symmetry elements of rhombohedral BiFeO 3 are three mirror-planes 120 • apart from each other. Figure 3b shows the cross-section plots of c 12 in XOZ plane at φ = 0 • or 120 • . Plot in Figure 3c at φ = 60 • is axisymmetric with the plot of Figure 3b about the Z-axis. The max c 12 = 132.3 × 10 9 N/m 2 can be obtained at θ = 13 • when φ = 0 • and 120 • , θ = −13 • when φ = 60 • . This indicates the three maximums exist in the mirror plane of rhombohedral BiFeO 3 , but they are at an angle to the spontaneous polarization axis. The min c 12 = 49 × 10 9 N/m 2 can be obtained at θ = −77 • . Figure 4a shows the three-dimensional representation of the second term elastic stiffness c 13 of rhombohedral BiFeO 3 . Unlike c 12 , large c 13 is found along the transverse directions, rather than the longitudinal directions. Symmetrical lobes of c 13 also are along mirror planes, as shown in Figure 4b Table 1.  Orientational dependences of piezoelectric constants dij′ were obtained by calculating (d′) in Equation (6) using data of four independent elastic constants (d15, d22, d31, and d33) for rhombohedral BiFeO3. Figure 5a shows the three-dimensional representation of shear piezoelectric constant d15′. There are three pairs of lobes in the geometry shape of d15′. Each pairs are along mirror planes of 3m BiFeO3, and two components of each pairs are centrosymmetric to the origin point. d15′ with large values could be obtained along orientations inside there lobes and it tends to be zero outside these lobes showing very evident spatial anisotropy. Figure 5b shows a cross-sectional plot of piezoelectric constant d15′ at  = 0°, 120°, and 240° ( Figure 5c is the relative plot at  = 60°). The max d15′ = 27.6 × 10 −12 C/N could be obtained at θ = 69° inside mirror planes.  Orientational dependences of piezoelectric constants dij′ were obtained by calculating (d′) in Equation (6) using data of four independent elastic constants (d15, d22, d31, and d33) for rhombohedral BiFeO3. Figure 5a shows the three-dimensional representation of shear piezoelectric constant d15′. There are three pairs of lobes in the geometry shape of d15′. Each pairs are along mirror planes of 3m BiFeO3, and two components of each pairs are centrosymmetric to the origin point. d15′ with large values could be obtained along orientations inside there lobes and it tends to be zero outside these lobes showing very evident spatial anisotropy. Figure 5b shows a cross-sectional plot of piezoelectric constant d15′ at  = 0°, 120°, and 240° ( Figure 5c is the relative plot at  = 60°). The max d15′ = 27.6 × 10 −12 C/N could be obtained at θ = 69° inside mirror planes. The transverse piezoelectric constant d31′ of rhombohedral BiFeO3 exhibits characteristics quite similar to that of d31′, shown in Figure 6. The max d31′ = 12.67 × 10 −12 C/N could also be obtained at θ = 69° inside mirror planes, which is about three times larger than d31′ = 4.5 × 10 −12 C/N along the [001]-axis. Notably, the transverse piezoelectric constant d31′ tends to be small along the spontaneous polarization axis ([001]-axis in this study). The result is in accordance with transverse piezoelectric stress constant values e31 (unit C/m 2 , e = d·c, where d and c are piezoelectric constant and elastic stiffness, respectively) of epitaxial BiFeO3 films with various preferred orientations, e31 along the spontaneous polarization axis and at a tilt 54.7° were −1.3 and −3.5 C/m 2 , respectively, which is also about three times larger [26]. Figure 7a shows the three-dimensional representation of longitudinal piezoelectric constant d33′ of rhombohedral BiFeO3. Symmetrical lobes of d33′ also can be found along mirror planes. The max d33′ = 18 × 10 −12 C/N could be obtained at θ = 0° along the spontaneous polarization axis. The cross-sectional two-dimensional plots in X'OZ planes are shown in Figure 7b,c. The value of d33′ along [100] and [010] is 13 × 10 −12 C/N. The result in this study is quite in accordance with d33 value of BiFeO3 crystal measured by Raman scattering, which is d33 = 16 × 10 −12 C/N [27]. For epitaxial BiFeO3 films, because of another measurement method applied (i.e., piezoelectric force microscopy, PFM) and strain from the substrates, values of d33 of epitaxial BiFeO3 films tend to be larger, in a range of 20-60 × 10 −12 C/N. However, experimental results showed epitaxial BiFeO3 films with oriented structure has about four times the enhancement of longitudinal piezoelectric constant compared with polycrystalline BiFeO3 films [28,29], which are generally in line with our results. Values of d15′, d31′, and d33′ of rhombohedral BiFeO3 along several chosen orientations are summarized in Table 1.     of BiFeO 3 crystal measured by Raman scattering, which is d 33 = 16 × 10 −12 C/N [27]. For epitaxial BiFeO 3 films, because of another measurement method applied (i.e., piezoelectric force microscopy, PFM) and strain from the substrates, values of d 33 of epitaxial BiFeO 3 films tend to be larger, in a range of 20-60 × 10 −12 C/N. However, experimental results showed epitaxial BiFeO 3 films with oriented structure has about four times the enhancement of longitudinal piezoelectric constant compared with polycrystalline BiFeO 3 films [28,29], which are generally in line with our results. Values of d 15 , d 31 , and d 33 of rhombohedral BiFeO 3 along several chosen orientations are summarized in Table 1.  From the orientation dependences of elastic and piezoelectric parameters investigated in this study, it can be found that the spatial anisotropy of cij and dij are related closely with the asymmetrical characteristics of rhombohedral BiFeO3 crystal with the 3m point group. As there is a rotation axis that is also the c-axis, several parameters tend to produce extreme values along this direction: c11 and d33 have the max values while d15 and d31 have the min values. Also, as there are  From the orientation dependences of elastic and piezoelectric parameters investigated in this study, it can be found that the spatial anisotropy of c ij and d ij are related closely with the asymmetrical characteristics of rhombohedral BiFeO 3 crystal with the 3m point group. As there is a rotation axis that is also the c-axis, several parameters tend to produce extreme values along this direction: c 11 and d 33 have the max values while d 15 and d 31 have the min values. Also, as there are three mirror planes locate at φ = 0 • , 60 • (240 • ), and 120 • , respectively (see Figure 8a-c, also denoting them with Miller index of (010), (110), and (100), respectively), the main lobes of several parameters are along the mirror planes: c 12 , c 13 , and all piezoelectric constants investigated. It also should be noted that Maxima directions are at a certain angle with the main crystalline axes, which may be attributed to the intrinsic lattice effects, which also appear in other piezoelectrics like BaTiO 3 [30] and LiNbO 3 [31]. The values of these parameters along arbitrary orientations were given in this study (some are listed in Table 1 Table 1 also indicates that as a lead-free piezoelectric material, BiFeO 3 has smaller elastic stiffnesses (larger elastic compliances) than tetragonal BaTiO 3 , and it has piezoelectric constants in the same level with those of KNbO 3 [32,33]. Furthermore, the largest values of these parameters were given in this study, which shows applications for the growth of BiFeO 3 films with oriented structure and enhanced properties.
denoting them with Miller index of (010), (110 ), and (100), respectively), the main lobes of several parameters are along the mirror planes: c12, c13, and all piezoelectric constants investigated. It also should be noted that Maxima directions are at a certain angle with the main crystalline axes, which may be attributed to the intrinsic lattice effects, which also appear in other piezoelectrics like BaTiO3 [30] and LiNbO3 [31]. The values of these parameters along arbitrary orientations were given in this study (some are listed in Table 1), which provides precise predictions for values of parameters with changing orientations. Table 1 also indicates that as a lead-free piezoelectric material, BiFeO3 has smaller elastic stiffnesses (larger elastic compliances) than tetragonal BaTiO3, and it has piezoelectric constants in the same level with those of KNbO3 [32,33]. Furthermore, the largest values of these parameters were given in this study, which shows applications for the growth of BiFeO3 films with oriented structure and enhanced properties.

Conclusions
Using the coordinate transformation method, the elastic and piezoelectric parameters at room temperature of rhombohedral BiFeO3 with the 3m point group along arbitrary orientations have been investigated. Several conclusions were obtained through this study: (1) The max elastic stiffness c11′ = 213 × 10 9 N/m 2 lies in planes with  = 0° and 90°. The max elastic stiffness c12′ = c13′ = 132.2 × 10 9 N/m 2 lie in directions at θ = 13° and θ = 77° inside three mirror planes, respectively.
(3) The elastic and piezoelectric parameters of BiFeO3 along arbitrary orientations were presented, and by adopting optimal directions these parameters could be significantly enhanced, which shows applications for the growth of BeFeO3 films with oriented structures and enhanced properties.

Conclusions
Using the coordinate transformation method, the elastic and piezoelectric parameters at room temperature of rhombohedral BiFeO 3 with the 3m point group along arbitrary orientations have been investigated. Several conclusions were obtained through this study: (1) The max elastic stiffness c 11 = 213 × 10 9 N/m 2 lies in planes with φ = 0 • and 90 • . The max elastic stiffness c 12 = c 13 = 132.2 × 10 9 N/m 2 lie in directions at θ = 13 • and θ = 77 • inside three mirror planes, respectively.
(3) The elastic and piezoelectric parameters of BiFeO 3 along arbitrary orientations were presented, and by adopting optimal directions these parameters could be significantly enhanced, which shows applications for the growth of BeFeO 3 films with oriented structures and enhanced properties.