#
Origin of Ferroelectricity in BiFeO_{3}-Based Solid Solutions

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}–LaFeO

_{3}system in rhombohedral R3c and tetragonal P4mm symmetries by ab initio density functional theory calculations and compare their electronic features with paraelectric orthorhombic Pnma symmetry. We show that a coherent accommodation of stereo-active lone pair electrons of Bi is the detrimental factor of ferroelectricity. A Bloch function arising from an indirect Bi_6p–Fe_3d hybridization mediated through O_2p is the primary origin of spontaneous polarization (P

_{s}) in the rhombohedral system. In the orthorhombic system, a similar Bloch function was found, whereas a staggered accommodation of stereo-active lone pair electrons of Bi exclusively results in paraelectricity. A giant P

_{s}reported in the tetragonal system originates from an orbital hybridization of Bi_6p and O_2p, where Fe-3d plays a minor role. The P

_{s}in the rhombohedral system decreases with increasing La content, while that in the tetragonal system displays a discontinuous drop at a certain La content. We discuss the electronic factors affecting the P

_{s}evolutions with La content.

## 1. Introduction

_{s}) by external stimuli. Lead titanate (PbTiO

_{3}) has tetragonal P4mm symmetry and features a high Curie temperature (T

_{C}) of 495 °C and a robust P

_{s}[1]. Ferroelectric tetragonal PbTiO

_{3}forms a solid solution with antiferroelectric rhombohedral PbZrO

_{3}[2,3]. Their dielectric and piezoelectric properties are maximal at around the morphotropic phase boundary (MPB) [4], and the properties around the MPB have been widely utilized for sensors, ultrasonic motors, and medical transducers [5,6,7]. Single crystals of Pb(Mg, Nb)O

_{3}–PbTiO

_{3}and Pb(Zn, Nb)O

_{3}–PbTiO

_{3}exhibit extraordinary high electromechanical coupling factors because a bridging phase in monoclinic symmetry [4,8,9,10] is capable of a rotation of P

_{s}under electric fields [8,11,12,13,14].

_{3}has been intensively studied from the viewpoints not only of multiferroic nature [15,16,17,18] but also of a Pb-free piezoelectric material [19,20,21,22]. In BiFeO

_{3}[18,23], P

_{s}coexists with an incommensurate spin structure, which can be approximated as an antiferromagnetic spin configuration [15,17]. A large P

_{s}of 80–100 μC/cm

^{2}along with a high T

_{C}of 830 °C [24,25] is present along [111]

_{c}in rhombohedral R3c symmetry [24,25] (the subscript ‘c’ indicates pseudo-cubic), while strained tetragonal films display a giant P

_{s}of over 130 μC/cm

^{2}[26].

_{3}–REFeO

_{3}[27,28,29,30,31,32,33,34,35] (RE: rare-earth elements) and BiFeO

_{3}–BaTiO

_{3}[20,21,36,37]. The simplest solid solution is the BiFeO

_{3}–LaFeO

_{3}system with a composition of Bi

_{1−x}La

_{x}FeO

_{3}, the detail of which has been comprehensively summarized in the review paper [27]. Karimi et al. [28] reported that a phase boundary between the ferroelectric R3c and the paraelectric orthorhombic Pnma phases exists at x ~ 0.23 at room temperature. Rusakov et al. reported that single-phase materials with R3c symmetry can be prepared after annealing for composition 0 ≤ x ≤ 0.1, and the Pnma phase is stable at 0.50 ≤ x ≤ 1; these results were verified by synchrotron radiation X-ray diffraction, electron diffraction, and high-resolution transmission electron microscopy [38]. They also found that an incommensurate phase in orthorhombic Imma symmetry is formed at 0.19 ≤ x ≤ 0.30. Karpinsky et al. [29] proposed a temperature-composition phase diagram, in which the ferroelectric R3c (x < 0.15) and the paraelectric Pnma (x > 0.4) phases are mediated through a bridging anti-polar phase in orthorhombic Pbam symmetry.

_{3}by experiments in conjunction with DFT calculations [30]. They reported that both an Fe–O bond anisotropy and off-center cation displacements are suppressed by the La doping. As a result, the degree of Fe 3d–4p orbital mixing decreases in the solid solution samples. An impact of the La content on the polarization and the electronic band structure was also reported by You et al. [40]. They reported that the La doping induces a chemically driven rotational instability. It modifies the local crystal field along with the electronic structure, which gives rise to a direct-to-indirect transition of the bandgap and provides an enhancement in ferroelectric photovoltaic effect. In contrast, Tan et al. reported that the La doping has little influence on P

_{s}in tetragonal BiFeO

_{3}[34]. In spite of extensive research by DFT studies [30,31,32,33,34,35], the P

_{s}’s evolution with the La content and its electronic origin still remain unclear.

_{3}-LaFeO

_{3}system (Bi

_{1−x}La

_{x}FeO

_{3}). The electronic feature and structural distortion in the paraelectric orthorhombic Pnma symmetry are also investigated because Bi

_{1−x}La

_{x}FeO

_{3}with x ≥ 0.5 is of the orthorhombic phase [29]. We show DFT energy evolutions with x but focus on the relation between the orbital hybridizations and the ferroelectric (paraelectric) distortions. We show that the Bloch function arising from a Bi_6p-Fe_3d hybridization mediated through O_2p is the primary origin of P

_{s}in the rhombohedral system. In the orthorhombic system, a similar Bloch function and the resultant structural distortion are constructed, whereas a staggered accommodation of stereo-active lone pair electrons of Bi never allow the presence of P

_{s}. We discuss a large P

_{s}and its dependence on x in the tetragonal system.

## 2. DFT Calculations

^{−6}eV, and a criterion for ionic relaxations of 1 meV/nm. The Γ-centered k-point mesh was set to 3 × 3 × 3 for the structural optimizations and 5 × 5 × 5 for density of states (DOS) and band structure calculations.

_{3}[34,45,46,47]. The bandgap value is enlarged when U–J is increased, while the essential feature, such as P

_{s}, and the valence-band electronic structure remain unchanged. One main reason why we adopted U–J = 6 eV is as follows: the bandgap becomes narrow for a specific Bi–La arrangement on the A site and eventually vanishes when the arrangement of Bi and La is an ordered configuration along the polar c axis, as will be described later. In order to maintain the bandgap above ca. 2 eV, we set U–J to 6 eV for Fe-3d throughout the calculations.

_{3}can be approximated as the G-type antiferromagnet [46], we set the spin arrangement in which the adjacent Fe ions have an antiparallel spin configuration as much as possible irrespective of the La content (x) on the A site.

_{1}/m for the orthorhombic ones, while the tetragonal P4mm remains unchanged, the details of which are summarized in Tables S1–S3. Although the orthorhombic cells were optimized in monoclinic P2

_{1}/m symmetry, we restricted the monoclinic angle of β to 90.0 degree, and then the crystal system is regarded as orthorhombic.

^{−4}, e.g., see Figure 4. The Fermi energy is set to zero throughout our manuscript.

_{3}–LaFeO

_{3}solid solution cell, there exists several choices of the arrangement of Bi and La. We adopted the rock-salt arrangement of Bi and La, especially along the polar c axis, as much as possible, as can be seen in Figures S1 and S2. This is because the electronic structure in the rhombohedral cell with a layered Bi and La ordering along the polar

**c**axis has a non-realistic metallic feature (Figure S10), which is not consistent with the experimental fact of an insulating nature of BiFeO

_{3}–LaFeO

_{3}solid solutions [47,48]. We therefore avoid such a Bi–La ordering along the specific crystallographic axis and adopt the rock-salt-like orderings of Bi and La. The details of the structure parameters are listed in Tables S4–S10 for the rhombohedral cells, Tables S11–S18 for the tetragonal cells, and Tables S19–S21 for the orthorhombic cells.

_{s}) using the following equation

## 3. Results and Discussion

#### 3.1. Ground-State Structures

_{total}) per the ABO

_{3}formula unit as a function of the La content (x) expressed by x = [La]/([Bi] + [La]). For comparing E

_{total}in different symmetries, x should be the same. Indeed, x has a discrete value depending on the total number of the A-site atoms, i.e., six in the rhombohedral cells, eight in the tetragonal ones, and four in the orthorhombic ones (Tables S1–S3). We, therefore, set the energies obtained by fitting the DFT energies of the rhombohedral cells by a quadratic function to zero of E

_{total}. The E

_{total}of the rhombohedral cells are lower than those of the tetragonal and orthorhombic cells irrespective of x. These results are different from those reported in the literature [31,50,51]. The orthorhombic cells are slightly higher in E

_{total}by ~0.1 eV while the tetragonal cells have a higher E

_{total}by ~0.3 eV. In reality, the tetragonal BiFeO

_{3}is stabilized under a compressive strain in epitaxial films [26].

_{s}of the rhombohedral cells as a function of x. The P

_{s}of the rhombohedral BiFeO

_{3}is 83.2 μC/cm

^{2}, which is in good agreement with the previous DFT calculations [26,31,52] and experiments [53]. With increasing x, the rhombohedral P

_{s}exhibits a monotonic decrease, which is consistent with other calculations [31]. The tetragonal BiFeO

_{3}(Figure 2c) possesses a giant P

_{s}of 133.4 μC/cm

^{2}, which is close to the values obtained for epitaxially strained films and from DFT calculations [26]. Zhang et al. [26] have reported that this large P

_{s}is associated with a high tetragonality c/a of 1.24 and also with a coherent displacement of Fe (∆

_{Fe}’) by 0.033 nm with respect to the Bi sublattice. These values accord with our calculations of c/a = 1.25 and ∆

_{Fe}’ = 0.032. The P

_{s}and c/a decrease with increasing x followed by discontinuous drops at 2/8 < x < 3/8. At x = 3/8, the c/a is almost in unity whereas the apparent P

_{s}of 52.0 μC/cm

^{2}exists. With a further increase in x, the P

_{s}again shows a monotonic decrease while the c/a remains constant (ca. ~1.0).

#### 3.2. Electronic Structures

#### 3.2.1. Ferroelectric Rhombohedral System

_{3}(x = 0) along with (c–g) the total and partial DOS results and (h,i) the electronic band structures in the valence band. The wavefunction of the band shown in the blue circle in i is depicted in Figure 4. The off-center displacements of Fe and Bi are stabilized by the following two orbital hybridizations, respectively: Fe_3d–O_2p (Figure 5a) and Bi_6p–O_2p (Figure 5b). Because Fe atoms have either a positive (↑) or a negative (↓) magnetic moment, the majority spin (↑) and minority spin (↓) bands have to be taken into account in a distinct manner. Here, we consider the orbital interaction in the ↑ band that leads to DOS components in the valence band in the range of −7 to 0 eV (in the right panel in Figure 3c–g). Although O1 and O2 have a slightly different magnetic moment, the DOS characters are almost identical both in the ↑ and ↓ bands. When the magnetic moments are ignored, O1 and O2 have the same site symmetry, and therefore, we do not distinguish O1 and O2. Due to the same reason, we regard Bi1 and Bi2 as identical Bi atoms.

^{2}6p

^{0}for Bi

^{3+}leads to zero DOS of Bi_6p in the valence band, because the states of the isolated Bi_6p are unoccupied. The Bi_6p–O_2p interaction (Figure 5b) results in a low-lying bonding state and a high-lying antibonding state. The low-lying states are occupied by electrons and thereby the Bi_6p states have apparent DOS components in the valence band (Figure 3e).

^{5}↑

^{0}. These five ↓ electrons of Fe1 are present at deep levels in the ↓ band [54]. We focus our attention on the interaction between the empty Fe1_3d (↑) and the occupied O_2p states (Figure 5a). The hybridization of these orbitals delivers an occupied bonding state and an empty antibonding state. Therefore, Fe1 has not only the major DOS in the band but also the minor DOS in the ↑ band, as displayed in Figure 3d.

_{6}octahedra and BiO polyhedrons. We note that the interaction between the bonding states of Fe1_3d–O_2p and Bi_6p–O_2p (Figure 5c) forms a coherent wavefunction that is spread throughout the crystal (Figure 4). Namely, the interaction between the Fe1_3d–O_2p- and the Bi_6p–O_2p-derived bonding states yields the low-lying bonding state, termed Bloch function, arising from the –Fe1–O–Bi–O– network (Figure 4). It results also in the high-lying antibonding state in the valence band, as shown in Figure 5c. We conclude that the Bloch function stemming from the indirect Bi_6p–Fe_3d hybridization mediated through O_2p is the primary origin of P

_{s}in the rhombohedral system.

_{3}cell. The similar electronic feature was found for the rhombohedral cell (x = 4/6) in Figure S4. It is interesting to note that the Bloch function is formed through the Bi–O–Fe bond avoiding La.

#### 3.2.2. Ferroelectric Tetragonal System

_{s}of the tetragonal BiFeO

_{3}is derived from the cooperative off-center displacements of Bi and Fe along the polar c axis with respect to the oxygen sublattice (Figure S5 for the BiFeO

_{3}cell). This off-center feature is maintained in the tetragonal cell with x = 2/8. Figure S6 shows (a) the crystal structure of the tetragonal cell (x = 2/8) along with (b–f) the total and partial DOS results and (g, h) the electronic band structures in the valence band. The wavefunction of the band shown in the blue circle in h is depicted in i. The Bloch function where a Bi_6p–O2_p interaction plays a central role (Figure S6i) is present at ca. −4.2 eV at the Γ point and spreads also along the c axis through the Bi_6p

_{z}orbital. Note that the Fe-3d state indeed does not participate in the Bloch function, which is in contrast to the rhombohedral system (Figure 4, Figures S3 and S4). Additionally, in the BiFeO

_{3}cell (Figure S5), the similar Bloch function with a small contribution of Fe-3d appears.

_{s}and c/a at 2/8 < x < 3/8 stems from the in-plane feature of the Bloch function formed by the Bi_6p–O2_p hybridization.

#### 3.2.3. Paraelectric Orthorhombic System

_{s}and c/a can be qualitatively understood from a decrease in the number of the Bi pillar along the

**c**axis. Figure S7 shows the arrangement of Bi and La in the tetragonal cells at (a) x = 2/8 and (b) x = 3/8. The tetragonal BiFeO

_{3}(x = 0) has a large P

_{s}, which is ascribed to the full set of the Bi pillars. The substitution of La on the A site decreases the number of the Bi pillar and two Bi pillars are maintained until x ≤ 2/8. These Bi pillars contribute to the formation of the Bloch function spreading along the c axis through the Bi_6p

_{z}orbital. With increasing x above 3/8, the number of the pillar becomes only one, and thereby the Bloch function has an in-plane feature, which is accompanied by a marked decrease in P

_{s}.

#### 3.3. Factors Affecting Ferroelectricity

_{s}in the rhombohedral system. The hybridized orbital is accompanied by the formation of the covalent bonds not only in the direction of P

_{s}but also in the opposite direction of P

_{s}, e.g., see Figure 4. The Bloch function is closely related to the bond lengths of Bi–Fe (see Tables S1–S3); this is the first (structural) factor leading to a robust P

_{s}.

_{s}, while the second shortest Bi–Fe bond is almost normal to P

_{s}. For the tetragonal system (Figure S9b), the first and second shortest Bi–Fe bonds are aligned along pseudo-cubic <111> direction. In the rhombohedral cells at x ≤ 4/6, the first shortest Bi–Fe length is ~0.31 nm, and the second shortest one is ~0.33 nm. In the tetragonal cells, the first shortest Bi–Fe length is ~0.33 nm, which is comparable to that in the rhombohedral one. We note that the second shortest length is as long as ~0.37 nm even at x = 0 (BiFeO

_{3}). This long Bi–Fe length does not allow the Fe_3d orbital to participate in the Bi_6p–O_2p hybridization, and thereby, the Bloch function is indeed formed by the Bi_6p–O_2p interaction in the tetragonal system, as shown in Figure 6.

_{s}. For the ferroelectrics in the rhombohedral (Figure 4a) and tetragonal systems (Figure 6i), the lone pair electrons of Bi are directed coherently along the c axis, which is the detrimental factor of ferroelectricity. The alignment of the lone pair electrons contributes to an enhancement of off-center displacements of cations, as in ferroelectric PbTiO

_{3}[57]. For the orthorhombic cell (Figure 7h), the lone pair electrons of Bi1 (y = 0.75) are directed opposite to those of Bi1 (y = 0.25) along the c axis (which is the symmetry constraint). This staggered accommodation of the lone pair electrons of Bi exclusively provides paraelectricity.

_{6}octahedra expressed by the Glazer notation [58]. The ferroelectric rhombohedral cells have an out-of-phase octahedral tilt expressed by ${a}^{-}{a}^{-}{a}^{-}$, which are accompanied by the short Bi–Fe bonds along with relatively large unit-cell densities of 8.595 g cm

^{−3}at x = 0 and 7.779 g cm

^{−3}at x = 0.5. In contrast, the ferroelectric tetragonal cells do not have any tilt or rotation of FeO

_{6}octahedra at x = 0 and 0.5, leading to the long (second shortest) Bi–Fe lengths and low densities of 8.027 g cm

^{−3}at x = 0 and 7.669 g cm

^{−3}at x = 0.5. The paraelectric orthorhombic cells have a tilt system of ${a}^{-}{b}^{+}{a}^{-}$, resulting in the short Bi–Fe lengths and high densities of 8.836 g cm

^{−3}at x = 0 and 7.807 g cm

^{−3}at x = 0.5.

## 4. Conclusions

_{1−x}La

_{x}FeO

_{3}in rhombohedral R3c and tetragonal P4mm symmetries by DFT calculations. In the rhombohedral system, a Bloch function arising from an indirect Bi_6p–Fe_3d hybridization via O_2p is the primary origin of P

_{s}. In contrast, the P

_{s}of the tetragonal phase stems from a Bloch function arising from a Bi_6p–O_2p mixing with a weak contribution of Fe-3d. The detrimental factor of the presence/absence of P

_{s}is an accommodation of stereo-active lone pair electrons of Bi. The paraelectric orthorhombic Pnma phase has a staggered accommodation of lone pair electrons of Bi, while the ferroelectric R3c and P4mm systems exhibit a coherent alignment of lone pair electrons of Bi. The rhombohedral system shows a monotonic decrease in P

_{s}with increasing x, which is directly associated with a weakening of the Fe_3d–O_2p–Bi_6p hybridization. In contrast, the tetragonal system displays a discontinuous drop of P

_{s}at ca. x = 0.3, which is ascribed to a transition from a 3D extension to an in-plane feature of the Bi_6p–O_2p mixed orbital.

## Supplementary Materials

_{1−x}La

_{x}FeO

_{3}; Figure S2: Bi and La arrangement on the A site in the tetragonal system with Bi

_{1−x}La

_{x}FeO

_{3}; Figure S3: Crystal structures (a), electronic density of states (DOS) (b–f), and band structures (of the majority spin band) in the valence band (g,h) of the rhombohedral cell with x = 2/6. The wavefunction of the band shown in the blue circle in h is displayed in i; Figure S4: Crystal structures (a), electronic density of states (DOS) (b–f), and band structures (of the majority spin band) in the valence band (g,h) of the rhombohedral cell with x = 4/6. The wavefunction of the band shown in the blue circle in h is displayed in i; Figure S5: Crystal structures (a), electronic density of states (DOS) (b–f), and band structures (of the majority spin band) in the valence band (g,h) of the tetragonal BiFeO

_{3}cell with x = 0. The wavefunction of the band shown in the blue circle in h is displayed in j; Figure S6: Crystal structures (a), electronic density of states (DOS) (b–f), and band structures (of the majority spin band) in the valence band (g,h) of the tetragonal cell with x = 2/8. The wavefunction of the band shown in the blue circle in (h) is displayed in i; Figure S7: Bi and La arrangement along with Bi pillars along the polar c axis of the tetragonal cells with a x = 2/8 and b 3/8; Figure S8: Bi and La arrangement along with Bi pillars along the polar c axis of the rhombohedral cells at a x = 2/6 and b 4/6; Figure S9: Bi–Fe bond lengths of BiFeO

_{3}in a rhombohedral and (b) tetragonal symmetries; Figure S10: Crystal structures in the (a) hexagonal and (b) primitive settings, electronic density of states (DOS) (c–f) and band structure (of the majority spin band) in the valence band (g) of the rhombohedral (x = 1/2) with an ordered Bi–La arrangement along the polar c axis. Table S1: Empirical formula, space group, numbers of Bi and La, and bond lengths of Bi–Fe in the unit cells of the rhombohedral cells; Table S2: Empirical formula, space group, numbers of Bi and La, and bond lengths of Bi–Fe in the unit cells of the tetragonal cells; Table S3: Empirical formula, space group, numbers of Bi and La, and bond lengths of Bi–Fe in the unit cells of the orthorhombic cells; Table S4: Structural parameters of the rhombohedral BiFeO

_{3}cell (space group R3) with an antiferromagnetic spin configuration; Table S5: Structural parameters of the rhombohedral (Bi

_{5}La)Fe

_{6}O

_{18}cell (space group P3) with an antiferromagnetic spin configuration; Table S6: Structural parameters of the rhombohedral (Bi

_{2}La)Fe

_{3}O

_{9}cell (space group P3) with an antiferromagnetic spin configuration; Table S7: Structural parameters of the rhombohedral (BiLa)Fe

_{2}O

_{6}cell (space group R3) with an antiferromagnetic spin configurations; Table S8: Structural parameters of the rhombohedral (BiLa

_{2})Fe

_{3}O

_{9}cell (space group P3) with an antiferromagnetic spin configuration; Table S9: Structural parameters of the rhombohedral (BiLa

_{5})Fe

_{6}O

_{18}cell (space group P3) with an antiferromagnetic spin configuration; Table S10: Structural parameters of the rhombohedral LaFeO

_{3}cell (space group R3) with an antiferromagnetic spin configuration; Table S11: Structural parameters of the tetragonal BiFeO

_{3}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S12: Structural parameters of the tetragonal (Bi

_{7}La)Fe

_{8}O

_{24}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S13: Structural parameters of the tetragonal (Bi

_{3}La)Fe

_{4}O

_{12}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S14: Structural parameters of the tetragonal (Bi

_{5}La

_{3})Fe

_{8}O

_{24}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S15: Structural parameters of the tetragonal (BiLa)Fe

_{2}O

_{6}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S16: Structural parameters of the tetragonal (Bi

_{3}La

_{5})Fe

_{8}O

_{24}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S17: Structural parameters of the tetragonal (BiLa

_{7})Fe

_{8}O

_{24}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S18: Structural parameters of the tetragonal LaFeO

_{3}cell (space group P4mm) with an antiferromagnetic spin configuration; Table S19: Structural parameters of the orthorhombic BiFeO

_{3}cell (space group P2

_{1}/m) with an antiferromagnetic spin configuration; Table S20: Structural parameters of the orthorhombic (BiLa)Fe

_{2}O

_{6}cell (space group P2

_{1}/m) with an antiferromagnetic spin configuration; Table S21: Structural parameters of the orthorhombic LaFeO

_{3}cell (space group P2

_{1}/m) with an antiferromagnetic spin configuration.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Optimized crystal structures in (

**a**) rhombohedral, (

**b**) tetragonal, and (

**c**) orthorhombic symmetries of the Bi

_{1−x}La

_{x}FeO

_{3}cells with x = 1/2. The tetragonal cells with x = 0, ½, and 1 do not have octahedral tilt, while those with x = 1/8, 2/8, 3/8, 5/8, and 7/8 have their distinct tilt modes. We adopt a rock-salt structure in an arrangement of Bi and La on the A site as much as possible, the details of which are displayed in Figures S1 and S2. The symmetry in parenthesis denotes the space group taking account of the antiferromagnetic spin configuration.

**Figure 2.**Phase stability and P

_{s}: (

**a**) total energy (E

_{total}) of the ABO

_{3}formula unit (f. u.); (

**b**) P

_{s}in the rhombohedral system; and (

**c**) P

_{s}and c/a in the tetragonal system as a function of La content (x). For comparing E

_{total}in different symmetries, we set the energies obtained by fitting the DFT energies of the rhombohedral cells by a quadratic function to zero.

**Figure 3.**Crystal structures (

**a**,

**b**), electronic density of states (DOS) (

**c**–

**g**), and band structures (of the majority spin band) in the valence band (

**h**,

**i**) of the rhombohedral BiFeO

_{3}(x = 0). The wavefunction of the band shown in blue circle in (

**i**) is displayed in Figure 4. The up (red) and down (blue) arrows in (

**c**) denote the majority and minority spin bands, respectively. The horizontal arrows (gray) in (

**c**–

**g**) correspond to the energy level of the blue circle in (

**i**).

**Figure 4.**Wavefunction (partial charge density) of the ↑ band shown in the blue circle in (

**i**) of the rhombohedral BiFeO

_{3}(x = 0): (

**a**) 3D plot and (

**b**) 2D visualization on the lattice plane including Bi1 and Fe1. This wavefunction arising from an indirect Bi1_6p–Fe1_3d hybridization mediated through O_2p is termed ‘Bloch function’. The roles of O1 and O2 in this band are almost identical, and therefore, we do not distinguish them.

**Figure 5.**Orbital interactions delivering the Bloch function of the majoring spin (↑) band shown in Figure 4 of the rhombohedral BiFeO

_{3}(x = 0): (

**a**) Fe1_3d–O_2p mixing leading to a bonding state in the valence band and an antibonding state in the conduction band; (

**b**) Bi_6p–O_2p mixing resulting in a low-lying bonding state and a high-lying antibonding state; and (

**c**) hybridization between Fe1_3d–O_2p and Bi_6p–O_2p leading to the Bloch function at a deep level in the valence band.

**Figure 6.**Crystal structure (

**a**), electronic density of states (DOS) (

**b**–

**f**), band structures (of the majority spin band) in the valence band (

**g**,

**h**), and

**i**wavefunction of the band shown in the blue circle of the tetragonal cell (x = 3/8). The up (red) and down (blue) arrows in (

**b**) denote the majority and minority spin bands, respectively. The horizontal arrows (gray) in (

**c**–

**g**) correspond to the energy level of the blue circle in (

**h**).

**Figure 7.**Crystal structure (

**a**), electronic density of states (DOS) (

**b**–

**e**), band structures (of the majority spin band) in the valence band (

**f**,

**g**), and (

**h**) wavefunction of the band shown in the blue circle of the orthorhombic cell (x = 1/2). The up (red) and down (blue) arrows in (

**c**) denote the majority and minority spin bands, respectively. The horizontal arrows (gray) in (

**c**–

**e**) correspond to the energy level of the blue circle in (

**g**).

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Noguchi, Y.; Matsuo, H.
Origin of Ferroelectricity in BiFeO_{3}-Based Solid Solutions. *Nanomaterials* **2022**, *12*, 4163.
https://doi.org/10.3390/nano12234163

**AMA Style**

Noguchi Y, Matsuo H.
Origin of Ferroelectricity in BiFeO_{3}-Based Solid Solutions. *Nanomaterials*. 2022; 12(23):4163.
https://doi.org/10.3390/nano12234163

**Chicago/Turabian Style**

Noguchi, Yuji, and Hiroki Matsuo.
2022. "Origin of Ferroelectricity in BiFeO_{3}-Based Solid Solutions" *Nanomaterials* 12, no. 23: 4163.
https://doi.org/10.3390/nano12234163