Hierarchical dielectric orders in layered ferroelectrics Bi₂SiO₅

Electric dipole engineering is now an emerging technology for high electron-mobility transistors, ferroelectric random access memory and multiferroic devices etc. Although various studies to provide insight into dipole moment behaviour, such as phase transition, order and disorder states, have been reported, macroscopic spontaneous polarization has been mainly discussed so far. Here, visualization of the electric dipole arrangement in layered ferroelectrics Bi₂SiO₅ by means of combined analysis of maximum entropy charge density and electrostatic potential distribution analysis based on synchrotron radiation X-ray powder diffraction data is reported. It was found that the hierarchical dipole orders, the weak-ferroelectric and ferroelectric configurations, were observed in the Bi₂O₂ and the SiO₃ layers, respectively, and the ferrielectric configuration was realised by the interlayer interaction. This discovery provides a new method to visualize the local polarization in ferroelectric materials.

1. Introduction

Designing and controlling the intense local electric field and/or polarization in solids is vital for emerging electronics, such as high-performance field-effect transistors, ferroelectric random access memory and multiferroic devices in the nano-scale (de Araujo et al., 1995; Auciello et al., 1998; Haertling, 1999; Scott, 2000; Dawber et al., 2005; Schilling et al., 2007; Chung et al., 2011; Yamada et al., 2012; Keeney et al., 2012a,b, 2013; Maity et al., 2012; Zhang et al., 2012). Dielectric properties have been mainly discussed in terms of macroscopic properties based on measurements of dielectric permittivity (∊) and electric polarization (P) under electric fields (E) for bulk samples so far. Recently, electrostatic potential (EP) analysis based on electron charge density (ECD) using the maximum entropy method (MEM) has been developed for the characterization of specific features originating from the electrostatic field/force on the microscopic scale (Sakata & Sato, 1990; Takata & Sakata, 1996; Takata, 2008; Tanaka et al., 2006; Kim et al., 2011). Using ECD/EP analysis, we succeeded in visualizing the relationship between internal electric fields and physical properties, such as thermal conductivity affected by rattling (Fujiwara et al., 2012) and superconductivity related to the bi-polaron (Kim et al., 2014).

Bi₂SiO₅ (BSO) has attracted much attention as an alternative to the traditional lead-based ferroelectric materials with a phase-transition temperature (T_c) of 673 K. BSO has an Aurivillius-like structure consisting of the [Bi₂O₂]²⁺ layer and [SiO₃]²⁻ layer (Fig. 1a) (Pirovano et al., 2001; Georges et al., 2006; Taniguchi et al., 2013). A relatively large spontaneous polarization (P_c) of 14.5 µC cm⁻² along the SiO₃ chain (c-axis) was predicted by first-principles calculations, while those along the a- and b-axes, P_a and P_b, are estimated to be small, 0.1 µC cm⁻² and 0 µC cm⁻², respectively (Taniguchi et al., 2013). From experimental P versus E measurements (Taniguchi et al., 2013), only the P_a value of 0.8 µC cm⁻² was detected, because the BSO crystals have a thin-plate shape, and the electrode for the P versus E measurements can only be formed on a large area of the b–c plane of the crystals. In addition, the polarization of BSO is suggested to originate from the SiO₃ layer and not from the Bi₂O₂ layer by the first-principles calculations (Taniguchi et al., 2013). Clarification of the origin and mechanism of the ferroelectricity in BSO is therefore crucial for further development of lead-free ferroelectric materials.

Figure 1 Schematic drawing of the crystal structure (a) and ECD using MEM distribution of the Bi2O2 (b, c) and the SiO3 layer (d, e) in the ferroelectric (300 K) and paraelectric (773 K) phases. The isosurface of ECD is 0.85 e Å^−3 and 1.50 e Å^−3 for the Bi2O2 and the SiO3 layer, respectively.

Here, we report the visualization of the electric dipole arrangement in layered ferroelectrics Bi₂SiO₅ by means of combined analysis of the ECD using MEM and EP distribution analysis based on high-precision synchrotron radiation X-ray powder diffraction data.

2. Experimental

Synchrotron radiation X-ray powder diffraction measurements of BSO were performed at BL02B2 beamline at SPring-8 with a large Debye–Scherrer camera to obtain high counting statics for accurate structure analysis (Nishibori et al., 2001; Takata et al., 2002). The BSO sample was sufficiently ground for a homogeneous distribution of intensity and sealed in a glass capillary with a diameter of 0.1 mm. The diffraction pattern was measured at 300 K and 773 K with a N₂ gas flowing temperature control system. The measurement wavelength was 0.35206 (1) Å to reduce absorption effects caused by the heavy atom (Bi) in the sample. The diffraction data were collected for 45 min on an image plate installed in the large Debye–Scherrer camera.

Determination of the precise structure was carried out by Rietveld refinement. Details of the process and the results are described in the supporting information . The total number of observed structure factors was 3921 and 2930 at 300 K and 773 K, respectively. The ECD was calculated by MEM using the ENIGMA program (Sakata et al., 1990; Tanaka et al., 2002). The electrostatic potential was calculated with a method developed by Tanaka et al. using the MEM electron charge density (Tanaka et al., 2006). The electrostatic potential [U(r)] is composed of the nucleus charge [U_nuc(r)] and the electron charge [U_ele(r)] components. In this study, ECD and EP were visualized using the OpenDx program provided by IBM Visualization Data Explorer. The procedure for the polarization calculations is described in the supporting information .

3. Results

The ECD/EP analysis is one of the best ways to understand the microscopic behaviour of polarizations in BSO. The ECD distributions directly obtained from integrated intensities of the X-ray diffraction pattern by MEM analysis reveal the deformation of both the BiO₄ square pyramids in the Bi₂O₂ layer and the SiO₃ tetrahedra in the SiO₃ layer [Figs. 1(b)–1(e)]. In the ferroelectric phase (300 K), the Bi atoms form a stronger covalent bond with one of the four equivalent first-neighbour O atoms in the paraelectric phase (773 K) (see Fig. S4 and Table S4 of the supporting information). The Bi(b)–O(b) and Bi(a)–O(c) pairs form electric dipole moments, and the two neighbouring electric dipoles form an almost antiparallel configuration in the Bi₂O₂ layer (Fig. 1b). On the other hand, the Si atoms in the ferroelectric phase form a stronger covalent bond (Fig. 1d) with three of the four equivalent first-neighbour O atoms in the paraelectric phase (Fig. 1e), showing that the SiO₃ cluster has an electric dipole moment. The electric dipoles of SiO₃ align in the ferroelectric configuration. From the ECD analysis using MEM, the results visualized the antiferroelectric order in the Bi₂O₂ layer and the ferroelectric order in the SiO₃ layer in the ferroelectric phase, as shown in Figs. 2(a) and 2(b). This is the reason why the large dipole moment of BSO originates from the SiO₃ layer instead of the Bi₂O₂ layer.

Figure 2 Schematic of the electric dipole configuration of Bi2SiO5 based on the polarization obtained by the ECD/EP method. (a) Three-dimensional electric dipole configuration with atom positions. (b) Schematics of dipole ordering for SiO3, Bi(b)–O(b) and Bi(a)–O(c). (c) Schematic dipole ordering showing a weak-ferroelectric configuration of the Bi2O2 layer. The residual dipole in the Bi2O2 layer is aligned antiparallel to the large dipole in the SiO3 layer showing the ferrielectric configuration.

Electric dipole moments in the crystal can be calculated from the electron charge using MEM and the nuclear charge using Ewald's method. It is, however, well known that the value of the polarization calculated from the charge distribution strongly depends on the method of selection of the crystallographic unit cell: the determination of the boundary of the crystallographic unit cell is critical for the calculation (Resta & Vanderbilt, 2007; Spaldin, 2012). This issue can be resolved by the Berry-phase theory (King-Smith & Vanderbilt, 1993; Resta, 1994; Neaton et al., 2005). In the current case, we introduce the concept of fragments for extracting experimentally individual dipole units originating from BiO and SiO₃ clusters. The boundary of fragments can be determined by the local minimum value of EP around the fragments (ECD/EP method) so that each fragment satisfies charge neutrality. An extracted fragment unit of SiO₃ is shown in Fig. 3 as an example. The partial electric polarization in the fragments can be estimated by (Spaldin, 2012; Gohda et al., 2000)

where V is the volume of the unit cell; e is the elementary charge (1.602 × 10⁻¹⁹ C); A_i is atomic number; (X_i, Y_i, Z_i) is the position of ith atom; (x_o, y_o, z_o) is the position of the center of mass in the fragment unit; ρ_i(x_i, y_i, z_i) is the electron density located at the ith pixel; (x_i, y_i, z_i) is the position of ith pixel for electron charge contribution; is the unit vector. Integration is carried out over the fragment unit; the value of the electron density is assigned to pixels in a unit cell divided into 256 × 128 × 128 pixels. The total and projected values of polarization are summarized in Table 1.

Table 1
Total and projected polarization estimated by the ECD/EP method, point charge model (PC model), P versus E measurement and first-principles calculation

P versus E measurements and the first-principles calculation are reported by Taniguchi et al. (2013). All values are shown in units of µC cm⁻².

		Total polarization	Projected polarization
		|P|	Pa	Pb	Pc
ECD/EP method	Bi2O2 layer	4.2 (1)	−1.8 (1)	0	−3.8 (1)
	Bi(b)–O(b) sublayer (upper square-pyramid component)	26.8 (3)	24.0 (3)	0	−11.9 (1)
	Bi(a)–O(c) sublayer (lower square-pyramid component)	27.0 (4)	−25.8 (4)	0	8.0 (1)
	SiO3 layer	27.3 (1)	1.4 (1)	0	27.3 (1)
	Bi2SiO5	23.5 (1)	0.3 (2)	0	23.5 (1)
PC model	Bi2SiO5	9.4	4.9	0	8.1
P versus E measurement	Bi2SiO5	–	0.8	–	–
First-principles calculation	Bi2SiO5	14.5	0.1	0	14.5

Figure 3 Extracted SiO3 fragment. (a) Two-dimensional EP map on the (004) plane with electric fields. The boundary defining the dipole unit can be determined by the local minimum value of the EP around the fragments (white dashed line). (b) Extracted three-dimensional perspective of the SiO3 fragment area (yellow) with the shape of the SiO3 molecule with an isosurface of 0.8 e Å^3 (grey).

The SiO₃ layer shows a large polarization along the c-axis originating from a large dipole moment of the SiO₃ fragment [27.3 (1) µC cm⁻²]. The projected values of the polarization along the a- and c-axis, P_a and P_c, in the SiO₃ layer are 1.4 (1) µC cm⁻² and 27.3 (1) µC cm⁻², respectively. It should be noted that P_b is zero due to the inversion symmetry operation along the b-axis. On the other hand, the Bi₂O₂ layer has a small but distinct polarization value in spite of the antiferroelectric order: the projections of the polarization in the Bi₂O₂ layers were −1.8 (1) µC cm⁻² for P_a and −3.8 (1) µC cm⁻² for P_c. This originates from the asymmetric distortion of the Bi₂O₂ pyramids, meaning that the Bi–O dipoles with antiparallel configuration do not fully cancel out the polarization in the layer: this is regarded as the weak-ferroelectric configuration, as shown in Fig. 2(c). Since the residual polarization in the Bi₂O₂ layer aligns in the antiparallel direction to the polarization in the SiO₃ layer, the distortion in the Bi₂O₂ layer is suggested to be induced by the large polarization of the SiO₃ layer for reducing the electrostatic energy in the crystal. The individual dipole moment [26.8 (3) µC cm⁻² and 27.0 (4) µC cm⁻²] of Bi–O is comparably large with that of the SiO₃ fragment [27.3 (1) µC cm⁻²], although the net value is as small as 4.2 (1) µC cm⁻² owing to the antiparallel configuration of the Bi–O dipoles.

The total P_a and P_c of BSO were estimated to be 0.3 (2) µC cm⁻² and 23.5 (1) µC cm⁻², respectively. P_b was zero because of the symmetry operation of the crystal structure. The value of P_a is roughly consistent with that predicted by theoretical calculation (0.1 µC cm⁻²) and that determined by P versus E measurements (0.8 µC cm⁻²) (Taniguchi et al., 2013). In addition, the large P_c value predicted by theoretical calculation [14.5 µC cm⁻²] was experimentally determined by a microscopic approach using ECD/EP analysis [23.5 (1) µC cm⁻²]. The result shows that the ECD/EP analysis using precise X-ray diffraction data can derive the local electric dipole moment in the crystal as well as in the polarization values from small amounts (less than 0.1 mg) of powder samples, and values are consistent with those predicted by the complete picture based on the Berry-phase theory. It should be noted that the values of polarization based on the point charge model, where the electron charge of atoms was assigned to each atomic position obtained by Rietveld analysis, largely deviated from any other results of theoretical prediction, P versus E measurements and the ECD/EP analysis; this result shows that the use of the ECD distribution is essential for estimation of accurate values of polarization. The method of ECD/EP analysis is, therefore, useful for characterization and design of newly synthesized dielectric materials, and thus for the development of emerging dielectric materials.

The hierarchical dipole ordering with structural distortion can be understood in terms of electrostatic energy. Firstly, the antiferroelectric configuration with the residual dipole moments (weak ferroelectricity) and the ferroelectric configuration are realised in the Bi₂O₂ and the SiO₃ layer, respectively. Next, the antiparallel arrangement between the net small dipoles in the Bi₂O₂ layer and the large dipoles in the SiO₃ layers reduces the interlayer electrostatic interaction; this configuration is regarded as an interlayer ferrielectric ordering. In addition, the neighbouring Bi₂O₂ and SiO₃ layers form a pair, namely, the dimerization of the layers (Figs. 2c and S3a). In this distortion the dipoles in the neighbouring layer being aligned in the antiparallel configuration become close to reduce the electrostatic energy. In actual fact, the cohesive energy of the low-temperature `ferroelectric' phase is lower than that of the high-temperature paraelectric phase by 23.1 meV in Bi<sub>2</sub>SiO<sub>5</sub> (Taniguchi et al., 2013), showing an energetic advantage of the `ferroelectric' phase with hierarchical dipole ordering.

The ECD/EP analysis can visualize the properties of covalency and polarization in the crystal using a tiny amount of powder sample. In addition, the method demonstrates its ability to visualize local polarization which cannot be detected by the macroscopic measurements in principle. In the case of BSO, for instance, a large dipole moment should be induced at the SiO₃ layer, and large polarization is stabilized by the antiparallel configuration with the Bi₂O₂ layer.

4. Summary

In summary, we have discovered a new approach to the visualization of the local electric dipole moments and their orders in a crystal by means of the combined analysis of ECD using MEM and EP distribution analysis based on synchrotron radiation X-ray powder diffraction data. Application of this method revealed hierarchical dipole ordering in Bi₂SiO₅: the weak ferroelectricity in the Bi₂O₂ layer, the ferroelectric order in the SiO₃ layer, and the ferrielectric order between the Bi₂O₂ and SiO₃ layers. The results suggest that ECD/EP analysis is a useful method to visualize the local polarization based on X-ray powder diffraction experiment.