First-principles study of the phonon, dielectric, and piezoelectric response in Bi2ZnTiO6 supercell

Computational Materials Science 101 (2015) 227–232

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First-principles study of the phonon, dielectric, and piezoelectric response in Bi2ZnTiO6 supercell Jian-Qing Dai ⇑, Yu-Min Song, Hu Zhang School of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, PR China

a r t i c l e

i n f o

Article history: Received 21 July 2014 Received in revised form 7 January 2015 Accepted 27 January 2015

Keywords: Lattice dynamics Dielectric response Piezoelectric property First-principles

a b s t r a c t First-principles calculations were used to investigate the lattice dynamics, electronic and lattice contributions to the dielectric and piezoelectric responses of the Bi2ZnTiO6 10-atom supercell with Cm symmetry. We explore the lattice contributions coming from individual atoms and individual modes. Detailed analysis shows that the large magnitudes of dielectric element e22 and piezoelectric element e26 are mainly due to two low frequency phonons (xk = 72.5 and 25.1 cm1) with A00 irreducible representation. On the other hand, the O3 (2a Wyckoff position in the Ti–O3–Zn linkage) atom provides extraordinary large contribution to the lattice dielectric and piezoelectric responses. Ó 2015 Published by Elsevier B.V.

1. Introduction Bi-containing perovskite is considered as promising alternative to the Pb-based piezoelectric materials due to their similar 6s2 electronic configuration. Much experimental effort has focused on the solid solutions of PbTiO3-Bi2ZnTiO6, which should be regarded as ‘‘low-lead’’ piezoelectric materials [1]. More recently, the entirely lead-free piezoelectric system of (Bi0.5Na0.5)TiO3-Bi(Zn0.5Ti0.5)O3 solid solution has been demonstrated by the observation of interesting piezoelectric property with high Curie temperature [2]. At the same time, theoretical investigations [3,4] have also predicted that remarkable ferroelectric and piezoelectric properties may be the intrinsic character for single phase Bi-based perovskites. However, synthesis of these Bi-based perovskite compounds via conventional methods is very difficult because of the smaller size of Bi as A-site cation. Thanks to the advances in high pressure technique, several single phase Bi2BTiO6 (B = Mg, Zn, Co, Ni) compounds in the perovskite form have been successfully synthesized [5–7] and the Bi2ZnTiO6 material was once expected to possess the highrecord spontaneous polarization [8]. According to the experimental characterization, crystal structure of Bi2ZnTiO6 belongs to the tetragonal P4mm space group regardless of the ultrahigh tetragonality (c/a ratio is as large as 1.211) and the unreasonably large thermal factors [8]. The subsequent first-principles investigations [9–11] preserved the tetragonal symmetry without further examining whether or not ⇑ Corresponding author. Fax: +86 871 65107922. E-mail address: [email protected] (J.-Q. Dai). http://dx.doi.org/10.1016/j.commatsci.2015.01.040 0927-0256/Ó 2015 Published by Elsevier B.V.

the P4mm phase is stable. Through carefully lattice dynamics investigation, we have pointed out that the P4mm structure is unstable because of the presence of a strongly soft polar mode [12]. Condensation of this unstable phonon leads to the Cm phase, which is the most probable ground structure with very similar primitive cell parameters to that of the P4mm symmetry. More important is that the spontaneous polarization of Bi2ZnTiO6 is not as significant as previous expected since the common adopted centrosymmetric reference structure shows metallic behavior [12]. In this paper, we present direct first-principles calculations of the lattice dynamics, dielectric and piezoelectric response of the Cm phase of Bi2ZnTiO6 supercell with Zn/Ti [0 0 1] ordering. The rest of the paper is organized as follows. After a brief discussion for the crystal structure of the Cm phase and the first-principles methods in Section 2, we provide a detailed analysis of the zone-center optical phonons and their contributions to dielectric response in Section 3 A and B. In Section 3 C, we calculate the ‘‘clamped-ion’’ and ‘‘internal-strain’’ contributions to piezoelectric tensor. In order to obtain further insight into the major piezoelectric contribution, we decomposed the internal-strain piezoelectric tensor into contributions from individual ions, as well as contributions from normal-mode contributions. While the discussion and summary are given in Sections 4 and 5, respectively. 2. Structure and calculation The calculated structural parameters [12] for the Cm phase Bi2(ZnTi)O6 supercell with Zn/Ti [0 0 1] ordering is shown in Table 1.

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For ZnO6 and TiO6 octahedron, the octahedron volume V, bond  bond angle along the c axis, and length, average bond length d, 2 2 bond angle variance r (° ) are also listed. [13] Clearly, the volume of ZnO6 octahedron are larger than that of TiO6 octahedron by about 15%, and the distortion of ZnO6 octahedron is more dominant than the TiO6 octahedron. At the B site, Zn and Ti cations move off center by 0.62 Å and 0.49 Å, respectively, creating short Zn-O3 and Ti-O4 bonds, which make an important contribution to the overall polarization. On the other hand, the distances of Zn-O4 and Ti-O3 bonds are 3.20 Å and 2.68 Å, respectively. In fact, the so large distance between Zn and O4 atom makes it more suitable to characterize the Zinc-Oxygen coordination polyhedron as ZnO5 pyramid instead of ZnO6 octahedron, as illustrated in Fig. 1. First principles calculations are performed by using density functional theory (DFT) and density functional perturbation theory (DFPT), which have been implemented in the VASP package [14– 16]. For the exchange–correlation functional, we adopt the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof form revised for solids (PBEsol) [17] since it is remarkable for the accuracy in predicting the volume of solids [18]. The valenceelectron configurations for Bi, Zn, Ti, and O are 5d106s26p3, 3d104s2, 2s22p63d24s2, and 2s22p4, respectively. The projector-augmented wave (PAW) pseudo potentials [19] are employed and the energy cutoff of 500 eV is used for the plane-wave expansion. We use the Gaussian broadening technique [20] of 0.05 eV to relax the structural parameters until the remaining Hellmann–Feynman forces are no more than 1 meV/Å. A 6  6  4 Monkhorst–Pack k-point mesh [21] centered at C is used for the Brillouin-zone sampling. The lattice dynamics, dielectric and piezoelectric tensors are determined via the DFPT [22] of the linear response of strain type perturbations [23], which is based on the systematic expansion of the variation expression of the DFT total energy in powers of different parameters [24].

3. Results 3.1. Zone-center phonons Group theoretical analysis [25] was undertaken to decompose the zone-center phonon modes into their irreducible representations (irreps) for the Bi2ZnTiO6 supercell with 1:1 chemical ordering along the [0 0 1] direction. As demonstrated in Ref. 12, the P4mm structure presents a doubly degenerate unstable E mode,

Fig. 1. Structure of Bi2ZnTiO6 unit cell with Cm symmetry, showing TiO6 octahedra and ZnO5 pyramids.

condensation of this unstable polar mode leads to the Cm phase, which is the most probable ground structure. For the P4mm symmetry, the optical modes at the C point can be classified as

Copt ¼ 7A1  2B1  9E: The singlet A1 and doubly degenerate E modes are both infrared (IR) and Raman active, while the singlet B1 modes are only Raman active. Due to the lower symmetry of the monoclinic Cm structure, there are only two irreps of A0 and A00 at the zone-center, and the optical phonon modes can be decomposed into

Copt ¼ 16A0  11A00 : All the modes in the Cm structure are both IR and Raman active. Inspecting the character tables for these irreducible representations [26], one can write the compatibility relations between the phonon modes in the P4mm and Cm structures. This gives the following correspondences:

16A0 ðCmÞ ! 7A1  9E 11A00 ðCmÞ ! 2B1  9E: For example, the doubly degenerate E modes in the P4mm phase split into the A0 and A00 irreps in the Cm structure. The A1 and B1 irreps are transformed into A0 and A00 irreps, respectively, as the symmetry changed from the P4mm to the Cm phase. The calculated optical phonon frequencies of the A0 and A00 modes for the Cm structure are presented in Table 2. For comparison, the frequencies of optical modes in the P4mm phase are also listed (the detailed structural parameters used in calculation are the same as in Ref. 12). First of all, the phonon frequencies with values above 300 cm1 for the two structures are quite similar, although the Cm phase is significantly more stable than the P4mm structure. Second, though condensation of the unstable E mode in the P4mm structure eliminates unstable modes in the resulting Cm phase, there are still four ‘‘soft’’ modes (two A0 and two A00 modes) with frequency lower than 100 cm1. The frequencies of the two A0 modes are 89.5 and 40.8 cm1, while the frequencies of the two A00 modes are 72.5 and 25.1 cm1, respectively. In order to clarify which ions are dominant in these modes, the involvement of each atom in the eigendisplacements [27] is reported in Table 3. Inspection of the results points out that the dominant contributions to the eigendisplacements of the two A0 modes are mainly from Bi2 and Bi1 atoms. The situation for the two A00 modes is different. Though the main contribution in the lowest 25.1 cm1 A00 mode is also due to the Bi2 and Bi1 atom, for the 72.5 cm1 A00 mode, however, the involvement of the O3 atom is more dominant than other atoms. It is interesting to see how the modes harden going from the P4mm structure to the more stable Cm phase. We focus on the behavior of the two E modes with frequencies of 45.9 and 170.3i cm1. By projecting the eigenvectors of the two E modes of the P4mm structure onto the normal mode eigenvectors of the Cm phase, the magnitude of the inner products indicates the coupling extent between the corresponding phonon modes. Detailed analysis shows that both eigenvectors of the doubly degenerate 45.9 cm1 E mode in the P4mm structure are closely associated with the 40.8 cm1 A0 and the 25.1 cm1 A00 modes, the inner products onto which are 0.64 and 0.57, respectively. For the unstable 170.3i cm1 E mode, however, the correspondence is rather complex. We find that one of the doubly degenerate orthogonal eigenvectors mainly turns into the A0 irrep, while the other eigenvector is (almost) only associated with the A00 modes. The projections of the eigenvector onto each mode in the Cm phase are shown in Table 2. Roughly speaking, the leading A0 modes corresponding to the unstable E mode are 355.0, 344.2, and 112.8 cm1, while the

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J.-Q. Dai et al. / Computational Materials Science 101 (2015) 227–232 Table 1 Calculated structural parameters for Bi2ZnTiO6 with Cm symmetry. The lattice parameters are a = 5.379 Å, b = 5.325 Å, c = 9.536 Å, and b = 88.785°. Atom

Site

x

y

z

ZnO6

TiO6

Bi1 Bi2 Zn Ti O1 O2 O3

2a 2a 2a 2a 4b 4b 2a

0.053 0.011 0.483 0.488 0.233 0.222 0.447

0 0 0 0 0.250 0.242 0

0.457 0.074 0.279 0.236 0.199 0.293 0.484

Zn–O1 Zn–O1 Zn–O3 Zn–O4 V (Å3) r2 (°2)  (Å) d

O4

2a

0.487

0

0.057

O3–Zn–O4 (°)

(Å) (Å) (Å) (Å)

2.030 2.051 1.960 3.204 12.31 377.3 2.221

Ti–O2 (Å) Ti–O2 (Å) Ti–O3 (Å) Ti–O4 (Å) V (Å3) r2 (°2)  (Å) d

1.933 2.009 2.684 1.707 10.45 206.7 2.046

174.7

O3–Ti–O4 (°)

175.1

Table 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 0 00 Calculated mode frequency xk (cm1), oscillator strength Sk;ij ¼ X20 Z k;ij Z k;ij =x2k , and mode effective charge jZ k j ¼ i Z k;i (in e) for the A and A modes. Inner products (IP) of the eigenvectors in the Cm phase with that of the unstable E modes in the P4mm phase are also listed. Note that the phonon frequency of the P4mm phase is taken from Ref. [12].

xk 889.9 617.8 538.1 520.3 488.8 412.8 355.0 344.2 300.2 223.9 198.5 172.8 139.9 112.8 89.5 40.8

A0

A00

P4mm xk

Sk;11

Sk;33

Sk;31

jZ k j

IP

xk

Sk;22

jZ k j

IP

0.00 0.37 0.01 0.23 0.47 0.01 0.09 3.66 5.35 0.19 0.00 0.03 0.02 7.30 0.01 5.05

0.42 0.02 0.00 0.17 0.00 2.43 0.37 0.42 0.17 0.19 0.08 0.28 0.07 0.49 0.05 0.00

0.00 0.08 0.00 0.20 0.03 0.14 0.19 1.24 0.95 0.19 0.02 0.10 0.03 1.91 0.02 0.03

4.23 2.65 0.32 2.24 2.34 4.50 1.70 4.90 5.46 1.23 0.55 0.82 0.45 2.96 0.32 1.37

0.01 0.09 0.03 0.29 0.01 0.29 0.40 0.59 0.18 0.20 0.00 0.06 0.05 0.39 0.02 0.15

565.3 523.2 386.3 346.6 321.6 257.6 185.1 145.8 136.4 72.5 25.1

1.06 0.00 1.15 2.78 0.08 6.03 0.01 1.38 0.00 60.33 14.54

3.79 0.21 2.63 3.82 0.59 4.61 0.14 1.29 0.02 4.22 1.40

0.01 0.00 0.05 0.11 0.00 0.07 0.12 0.19 0.15 0.83 0.40

Table 3 Involvement of each atom in some A0 and A00 modes with frequency xk (cm1) for the Cm structure. Atom

112.8 (A0 )

89.5 (A0 )

72.5 (A00 )

40.8 (A0 )

25.1 (A00 )

Bi1 Bi2 Zn Ti O1 O2 O3 O4

0.501 0.190 0.267 0.519 0.095 0.270 0.012 0.456

0.623 0.635 0.243 0.209 0.156 0.139 0.092 0.101

0.328 0.310 0.024 0.390 0.031 0.213 0.689 0.276

0.573 0.675 0.218 0.248 0.112 0.151 0.127 0.145

0.450 0.765 0.331 0.076 0.170 0.057 0.180 0.015

most dominant A00 modes having large inner products with this E mode are 72.5 and 25.1 cm1, respectively. It is important to note that projecting the unstable E mode onto the 72.5 cm1 A00 phonon results in the large inner product with the value of 0.83, suggesting that their mode patterns are very similar. 3.2. Static dielectric response We now turn our attention to the static dielectric tensor and the lattice dielectric response in the Cm structure. The static dielectric tensor can be written as sum of electronic e1,ij and lattice contribution eph,ij, [28]

eij ¼ e1;ij þ eph;ij ¼ e1;ij þ X20

X Z k;i Z k;j k

x2k

phonon frequencies and mode effective charge vector in i direction, while X20 ¼ 4pe2 =m0 V 0 is effective plasma frequency with mass m0 = 1 amu, charge e, and density 1/V0 (V0 is the 10-atom primitive cell volume). The calculated electronic and phonon dielectric contributions, as well as the total static dielectric tensor, are shown in Table 4. For the Cm structure, the dielectric tensor has four independent elements of e11, e22, e33, and e31. The average dielectric constant, P e ¼ 13 3a¼1 eii , is also listed in Table 4 to facilitate comparison with future experimental results for the ceramic sample. It is clearly seen that the electronic dielectric tensor has diagonal values of 5.3–6.8, and that the average electronic dielectric constant e1 is about 5.9. Though there are no available experiment results for the Bi2ZnTiO6 compound, we find that our computed electronic dielectric tensor is in good agreement with previous theoretical and experimental results of other perovskite materials such as BaTiO3 [24], CaTiO3 [28], and PbMg1/3Nb2/3O3 [31]. Particular focus now has been directed toward revealing behavior of the lattice dielectric response. The most important thing should be pay attention to is the anomalously large anisotropy of Table 4 Calculated electronic (e1,ij) and phonon contributions (eph,ij) to, and total static   dielectric tensor e ¼ e1;ij þ eph;ij for Bi2(ZnTi)O6 supercell. The average dielectric  P ij constants e ¼ 13 3a¼1 eii are also listed. ij

:

The static lattice dielectric response can be further expressed as the zero-frequency response of a system of classical Lorentz oscillators [29,30]. The xk and Z k;i are, respectively, the IR-active

890.6 (A1) 613.1 (E) 540.6 (E) 527.1 (A1) 494.5 (A1) 403.4 (A1) 355.5 (E) 325.4 (B1) 316.3 (B1) 285.0 (E) 214.0 (A1) 210.8 (E) 193.2 (A1) 153.6 (E) 134.4 (E) 95.9 (A1) 45.9 (E) 170.3i (E)

e1,ij eph,ij eij

11

22

33

31

6.01 22.79 28.79

6.32 87.37 93.69

5.28 5.15 10.43

0.14 2.48 2.62

e1 eph e

5.87 38.43 44.30

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the lattice dielectric contribution. For example, the magnitude of eph,22 is 87.4, which is near 17 times larger than eph,33. To further identify the origin of the large anisotropy, the oscillator strength Sk;ij and mode effective charge jZ k j for each phonon are also collected in Table 2. According to the symmetry, the oscillator strength of the A0 modes has three components of Sk;11 , Sk;33 , and Sk;31 , while the A00 modes only contributes to the Sk;22 component. It is evident that the extremely large eph,22 is mainly due to the large oscillator strengths of two A00 phonon modes with frequencies of 72.5 and 25.1 cm1, respectively. Especially for the 72.5 cm1 A00 phonon, the oscillator strength is as large as 60.3 and contributes about 69% to the total eph,22. Inspecting the eigendisplacements shows that this 72.5 cm1 A00 mode consists of the Bi atoms moving against other atoms but the O3 displacement is dominant. The nature of the phonon eigenvector for the 25.1 cm1 A00 mode, however, involves significant motion of the Bi1, Ti, O2, O4 atoms in opposition to that of the Bi2, Zn, O1, and O3 atoms. It is worth noting that the 72.5 cm1 A00 mode has very similar mode patterns as that of the unstable E mode in the P4mm structure, while the 25.1 cm1 A00 mode couples with both the 45.9 and the 170.3i cm1 E modes of the P4mm phase as previously mentioned. Finally, the A0 mode with frequency of 112.8 cm1, which consists mainly of the Bi1 atom moving against both the Ti and O4 atoms, has the largest contribution to eph,11, and is also considerably coupled with the unstable E phonon mode in the P4mm structure. Another informative way to analyze the lattice dielectric response is to decompose the lattice contribution eph,ij into contributions of individual ions ek,ij (shown in Table 5) [32]. First, the contributions to the dielectric response can be positive or negative for different atoms and different dielectric tensor elements. The negative contribution indicates the stronger inner electric field with opposite direction compared to the applied external field. Except for the negligible ek,22 of the O1 atom, all other oxygen atoms offer positive contributions. In contrast, the dielectric contributions from the Ti atom are negative. On the other hand, the Zn and Bi atoms can provide positive or negative contributions to dielectric response for different dielectric elements. Second, the two Bi atoms afford opposite contributions to eph,11 and eph,22. The Bi atoms in the Cm structure are grouped into two types of Bi1 and Bi2 as illustrated in Fig. 1. Surrounding the Bi atom there are seven oxygen atoms. The nearest and next-nearest neighbors of Bi1 atom are O3 and O2 atom, respectively, and O1 and O4 atom for the Bi2 atom. The large but opposite contributions from Bi1 and Bi2 atom to eph,11 and eph,22 reveal the different influences of chemical and coordination environments. Finally, we note that, for lattice dielectric response of eph,22, the dominant contribution is provided from the O3 atom, while the Bi2, O1, and O2 atom also afford relatively large contributions. For the dielectric element of eph,11, however, the Bi2 and O2 atom contribute the primary component.

3.3. Piezoelectricity The elements of macroscopic piezoelectric tensor can be separated into two parts: a homogeneous or clamped-ion strain Table 5 Contributions of individual ions (ek,ii) to the lattice dielectric tensor. Atom

WP

ek,11

ek,22

ek,33

Bi1 Bi2 Zn Ti O1 O2 O3 O4

2a 2a 2a 2a 4b 4b 2a 2a

3.61 8.42 1.00 7.78 0.03 7.83 4.58 4.57

20.96 17.26 8.35 9.04 13.61 11.14 44.78 6.77

0.59 0.00 0.29 0.67 0.70 0.60 0.95 1.40

contribution eim,hom evaluated at vanishing internal strain u [33], and an internal strain term eim,int that is due to the relative displacements of different charged sublattices:

eim ¼ eim;hom þ eim;int ¼ eim;hom þ

X eaj kj

V0

Z k;ji

@uk;j ; @fm

where V0 is the volume, aj is the lattice parameter, Z k;ji is the Born effective charge of atom k, fm is the strain tensor element. On the other hand, the internal strain term eim,int for a given ground state can be decomposed into normal-mode contributions [34]:

eim;int ¼

X Z k;i @nk;i ; V 0 @fm k

where Z k;i is the mode effective charge along i direction, and

@nk;i @fm

the

change in the amplitude of mode k with strain. Monoclinic Cm Bi2ZnTiO6 supercell has ten independent piezoelectric tensor components, all of which were calculated. The diagonal element of e11 (e33) describes the polarization induced along the a1 (a3) axis when the crystal is uniformly strained along the same axis. The off-diagonal components, for example, e13 and e15 describe the polarization induced along the a1 direction by a strain along the a3 axis and a shear strain in the a1a3 plane (the a1, a2, and a3 directions correspond to the a, b, and c axes, respectively, as illustrated in Fig. 1). The calculated piezoelectric tensor elements are listed in Table 6. The calculated value of e26 is 6.55 C/m2, which is almost the same as that of e15 in PbTiO3 [35]. Except for e26 and e24, the calculated magnitudes of all other piezoelectric elements are relatively small. From contributions of individual ions to the internal-strain terms eim,int (shown in Table 7), the largest ionic contribution of 4.01 C/m2 to e26 comes from the O3 atom, and the two types of Bi atoms also provide considerable contributions in despite of the large negative contribution of the Bi2 atom. For e24,int, however, both the O and Bi atoms have similar positive contributions while the Zn and Ti atoms give negative contributions. We next look at the phonon dependence of the internal-strain terms eim,int by analyzing the contributions of normal modes. According to the symmetry, the piezoelectric components of e26,int and e24,int are contributed from the A00 modes, while the A0 modes account for all the other terms. The results are collected in Table 8. For e26,int, we find that the dominating contributions originate from the 72.5 and 25.1 cm1 A00 modes. In fact, these two modes account for 88% of the magnitude of e26,int. On the other hand, the value of e24,int are almost entirely contributed from the only 72.5 cm1 A00 mode. The situation is somehow different for the A0 modes. For the off-diagonal element of e15,int, the main contributions are provided by the two low frequency A0 modes of 112.8 and 40.8 cm1. For the two diagonal components, three A0 modes with frequencies of 344.2, 300.2, and 112.8 cm1, respectively, are the main contributors to e11,int, while the 889.9 and 412.8 cm1 A0 modes account for the leading contributions of e33,int. Finally, it should be mentioned that the contributions from low frequency phonons to the off-diagonal piezoelectric components of e26,int, e24,int, and e15,int are very similar to that of the dielectric constants. 4. Discussion It is well known that the dielectric and piezoelectric responses belong to the polarization-related properties, which can be expressed as the derivatives of polarization with respect to applied electric field and macroscopic strain, respectively. Both can be decomposed into two terms: the clamped-ion response and a contribution produced by the induced relative displacement of ionic

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Table 6 Homogeneous-strain contribution (eim,hom) and internal-strain contribution (eim,int) to the total piezoelectric stress tensor element eim,tot (in C/m2). We use the Voigt notation, the Latin index i runs from 1 to 3, and the Greek index m from 1 to 6.

eim,hom eim,int eim,tot

11

12

13

15

24

26

31

32

33

35

0.25 1.33 1.09

0.13 0.71 0.84

0.02 0.29 0.31

0.14 1.67 1.53

0.18 3.08 2.90

0.02 6.53 6.55

0.20 0.02 0.22

0.15 0.35 0.50

0.51 2.30 1.79

0.08 0.01 0.07

Table 7 Contributions of individual ions to the internal-strain piezoelectric tensor eim,int (C/ m2). Atom

e11,int

e15,int

e24,int

e26,int

e33,int

Bi1 Bi2 Zn Ti O1 O2 O3 O4

0.82 0.48 0.18 0.15 0.27 0.16 0.37 0.10

0.36 1.09 0.10 1.26 0.04 0.75 0.11 0.60

0.56 0.47 0.33 1.27 0.35 0.73 0.84 0.65

2.35 3.11 1.13 0.09 1.59 0.47 4.01 0.19

0.13 0.15 0.10 0.31 0.19 0.22 0.58 1.02

sublattices. The latter can be directly traced to the presence of low frequency polar phonons, which dominate the properties producing large lattice responses with high sensitivity to mechanical and electrical boundary conditions [36]. The doubly degenerate E mode eigenvectors of the unstable phonon in the P4mm structure involve displacements of all atoms, of which point to the same direction except for the Bi1 atom with opposite displacement, along the tetragonal a-axis (or along b, having degenerate frequency). The Cm structure is resulted from condensation of the unstable doubly degenerate E mode in the P4mm phase, and has superposed on these E type displacements, additional displacements along the tetragonal c-axis of all the atoms having A1 and/or B1 phonon mode character. The condensation and combination of these modes gives the spontaneous polarization of 62.5 lC/cm2 along [vv1] (v = 0.245) direction in the primitive cell [12]. In this manuscript, we use the conventional unit cell, however, in which the spontaneous polarization lies in a1a3 plane (ac plane illustrated in Fig. 1). As mentioned in previous section, the resultant Cm structure has several low frequency modes. In particular, there exists a close correspondence between the unstable E mode in the P4mm structure and the 72.5 cm1 A00 mode in the Cm phase, of which the involvement of the atom O3 is dominant. For the Bi2ZnTiO6 supercell with Cm symmetry, we predict an extremely large anisotropy in the dielectric constant along different axes that is of order about 900% (!), an effect has yet to be investigated experimentally. The axis of largest dielectric constant does depend on the details of the structure, be in the b direction in our case, along which the spontaneous polarization is zero. On the other hand, the axis of smallest dielectric constant is in the c direction along which the spontaneous polarization has the maximum value. The result indicates the trend of the larger the spontaneous polarization, the smaller the dielectric response. Normal mode Table 8 Contributions of A0 and A00 modes to the internal-strain piezoelectric tensor eim,int (C/m2). A0

A00

xk

e11

e15

e33

xk

e24

e26

889.9 412.8 344.2 300.2 112.8 40.8

0.00 0.01 0.54 0.66 0.68 0.36

0.00 0.01 0.01 0.02 1.11 0.56

0.56 0.79 0.21 0.02 0.05 0.00

386.3 346.6 145.8 72.5 25.1

0.22 0.15 0.30 3.04 0.04

0.22 0.19 0.29 3.65 2.07

analysis shows that about 69% of the dielectric constant magnitude for eph,22 is due to the single A00 phonon with xk = 72.5 cm1 and large mode effective charge jZk j = 4.22. The extremely large dielectric contribution of this mode is due to the large mode effective charge, the relatively low frequency, and the nature of the phonon eigenvector, which involves significant motion of the Bi atoms moving against other atoms but the O3 displacement is dominant. The situation is similar for the piezoelectric response. The large lattice piezoelectric components of e26,int and e24,int also indicate large induced polarization along the b-axis under a macroscopic shear strain. In other words, induction of relative displacement between cationic and anionic sublattices is more easer along the b-axis than other directions along which the spontaneous polarization exists. In addition, the magnitude of e26,int, which is the b-axis piezoelectric response under shear strain in the ab plane, is observably larger than that of the e24,int. Obviously, the large values of e26,int and e24,int are attributed to the presence of the two low frequency A00 phonons, especially the phonon with xk = 72.5 cm1. Finally, the most valuable information indicated in contributions from individual ions to the dielectric and piezoelectric responses (shown in Tables 5 and 7) is that the large values of eph,22 and e26,int is mainly due to the very large displacements of the internal coordinates of the O3 atom to the applied electric field and macroscopic strain, respectively. As discussed in Ref. 12, because Zn cation is the nearest neighbor of the O3 atom, the effect of Zn-substitution/alloying with other divalent cation may be especially interesting. 5. Conclusions In summary, we have used first-principles methods to compute the zone-center phonons, electronic and lattice contributions to the dielectric and piezoelectric responses of the Bi2ZnTiO6 supercell with Cm symmetry. We provide a detailed analysis of the lattice contributions coming from individual atoms and individual modes, and roughly characterize the dominating factors. The large values of e22 and e26 are due to the presence of low frequency A00 phonons with xk = 72.5 and 25.1 cm1. From the viewpoint of contributions from individual ions, the O3 atom provides extraordinary large contribution to the dielectric and piezoelectric responses. This material may exhibit fantastic behavior under high pressure, epitaxial-strain constraint, or even the Zn-substitution/ alloying. We hope that the work described here can serve as a theoretical foundation for future investigations of this intriguing material. Acknowledgments This work was supported by the National Science Foundation of China (Grant Nos. 51162019 and 51462019). References [1] M.R. Suchomel, P.K. Davies, Appl. Phys. Lett. 86 (2005) 262905; I. Grinberg, M.R. Suchomel, W. Dmowski, S.E. Mason, H. Wu, P.K. Davies, A.M. Rappe, Phys. Rev. Lett. 98 (2007) 107601; J. Chen, P.H. Hu, X.Y. Sun, C. Sun, X.R. Xing, Appl. Phys. Lett. 91 (2007) 171907; X.D. Zhang, D. Kwon, B.G. Kim, Appl. Phys. Lett. 92 (2008) 082906. [2] S.-T. Zhang, F. Yan, B. Yang, J. Appl. Phys. 107 (2010) 114110.

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[3] H. Wang, B. Wang, Q.K. Li, Z.Y. Zhu, R. Wang, C.H. Woo, Phys. Rev. B 75 (2007) 245209. [4] P. Baettig, C.F. Schelle, R. LeSar, U.V. Waghmare, N.A. Spaldin, Chem. Mater. 17 (2005) 1376. [5] Y. Inaguma, T. Katsumata, Ferroelectrics 286 (2003) 833. [6] M. Azuma, K. Takata, T. Saito, S. Ishiwata, Y. Shimakawa, M. Takano, J. Am. Chem. Soc. 127 (2005) 8889. [7] D.D. Khalyavin, A.N. Salak, N.P. Vyshatko, A.B. Lopes, N.M. Olekhnovich, A.V. Pushkarev, I.I. Maroz, Y.V. Radyush, Chem. Mater. 8 (2006) 5104. [8] M.R. Suchomel, A.M. Fogg, M. Allix, H. Niu, J.B. Claridge, M.J. Rosseinsky, Chem. Mater. 18 (2006) 4987. Chem. Mater. 18 (2006) 5810. [9] T. Qi, I. Grinberg, A.M. Rappe, Phys. Rev. B 79 (2009) 094114. [10] H. Wang, H. Huang, W. Lu, H.L.W. Chan, B. Wang, C.H. Woo, J. Appl. Phys. 105 (2009) 053713. [11] S. Ju, G.-U. Guo, J. Chem. Phys. 129 (2009) 194704. [12] J.-Q. Dai, Z. Fang, J. Appl. Phys. 111 (2012) 114101. Pm 2 1 [13] The bond angle variance is defined as r2 ¼ m1 i¼1 ð/i  /0 Þ , where m is number of bond angles, /i is the ith bond angle, and /0 is the ideal bond angle for a regular polyhedron (for example, 90° for an octahedron). [14] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) R558. [15] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [16] M. Gajdoš, K. Hummer, G. Kresse, Phys. Rev. B 73 (2006) 045112. [17] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008) 136406. [18] R. Wahl, D. Vogtenhuber, G. Kresse, Phys. Rev. B 78 (2008) 104116. [19] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953.

[20] C. Elsässer, M. Fähnle, C.T. Chan, K.M. Ho, Phys. Rev. B 49 (1994) 13975. [21] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [22] S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Rev. Mod. Phys. 73 (2001) 515. [23] D.R. Hamann, X. Wu, K.M. Rabe, D. Vanderbilt, Phys. Rev. B 71 (2005) 035117. [24] X. Wu, D. Vanderbilt, D.R. Hamanm, Phys. Rev. B 72 (2005) 035105. [25] H.T. Stokes, D.M. Hatch, B.J. Campbell, ISOTROPY, 2007, . [26] G. Venkataraman, L.A. Feldcamp, V.C. Sahni, Dynamics of Perfect Crystals, MIT Press, Cambridge, 1975. [27] The involvement of each atom in eigendisplacements is quantified from P 2 individual contributions to the norm: hgjgi ¼ 10 i¼1 gi ¼ 1. [28] E. Cockayne, B.P. Burton, Phys. Rev. B 62 (2000) 3735. [29] L. He, J.B. Neaton, M.H. Cohen, D. Vanderbilt, C.C. Homes, Phys. Rev. B 65 (2002) 214112. [30] C. Wang, G.C. Guo, L. He, Phys. Rev. B 77 (2008) 134113. [31] N. Choudhury, Z. Wu, E.J. Walter, R.E. Cohen, Phys. Rev. B 71 (2005) 125134. [32] According to equation (52) in Ref. [Phys. Rev. B 55, 10335 (1997) 27], lattice dielectric tensor is the summation over atom and eigenmode m. By first summing on all modes m and using the orthonormalization of eigenvectors, we can get the dielectric contribution from individual ions. [33] S. de Gironcoli, S. Baroni, R. Resta, Phys. Rev. Lett. 62 (1989) 2853. [34] E. Cockayne, K.M. Rabe, Phys. Rev. B 57 (1998) 13973. [35] Z. Wu, R.E. Cohen, Phys. Rev. Lett. 95 (2005) 037601. [36] K.M. Rabe, C.H. Ahn, J.-M. Triscone, Physics of Ferroelectric: A Modern Perspective, Springer-Verlag, Berlin Heidelberg, 2007.