First Principles Study of Bismuth Vacancy Formation in (111)-Strained BiFeO₃

Abstract

Epitaxial strain is known to significantly influence the structural and functional properties of oxide thin films. However, its impact on point defect concentration has been less explored. Due to the challenges in experimentally measuring thin-film stoichiometry, computational studies become crucial. In this work, we use first-principles calculations based on density functional theory to investigate the formation and stability of Bi vacancies and Bi-O vacancy pairs in BiFeO₃ (BFO) under (111) epitaxial strain. Our results demonstrate that compressive strain (−4%) decreases the formation enthalpy of Bi vacancies by 0.88 eV, whereas tensile strain (4%) increases it by 0.39 eV. Out-of-plane (OP) Bi-O vacancy pairs exhibit enhanced stability under both compressive and tensile strain, with formation enthalpy reductions of 1.49 eV and 1.05 eV, respectively. In contrast, in-plane (IP) vacancy pairs are stabilized under compressive strain but are insensitive to tensile strain. Finally, we discuss how these findings influence Bi stoichiometry during thin-film growth and the role of local strain fields in the formation of conducting domain walls.

1. Introduction

BiFeO₃ (BFO) is a widely studied multiferroic material, exhibiting both ferroelectric and antiferromagnetic orders above room temperature (T_C = 830 °C, T_N = 370 °C), along with magnetoelectric coupling. Its large ferroelectric polarization (~90 µC/cm²), primarily driven by the displacement of Bi³⁺ along the [111] pseudocubic axis [1], makes BFO a candidate for device applications such as sensors, actuators, and memory devices [2,3]. The majority of technological applications of BFO rely on thin films [4,5]. Perovskite structures with rhombohedral space group R3c are readily grown as high-quality epitaxial thin films on a variety of different substrates. The resulting epitaxial strain leads to improved ferroelectric, piezoelectric and magnetic properties [6,7,8,9,10]. Although the structural effects of epitaxial strain are well understood [11,12], the influence of strain on compositional degrees of freedom, such as cation and anion vacancies, has received comparatively less attention [13,14]. This oversight is partly due to the difficulty of measuring subtle stoichiometric variations at the nanoscale [15,16].

Given that bismuth (Bi) is a volatile element and plays a critical role in determining the physical properties of BiFeO₃, understanding the effects of epitaxial strain on Bi vacancies is crucial. Recent research has shown that Bi volatility during film growth can lead to non-stoichiometry, influencing both ferroelectric and magnetic behaviors [17,18]. Specifically, local Bi deficiency has been linked to the formation of conductive domain walls, which have p-type conductivity and are significantly influenced by the epitaxial strain present during film growth [19,20]. Motivated by the observed dependence of vacancy formation on strain in other oxide systems [21,22], we investigated the formation of Bi and Bi-O vacancy pairs in (111)-strained BFO using DFT calculations. This orientation is of particular interest for device applications due to the out-of-plane alignment of ferroelectric polarization, providing a direct link between polarization and strain-driven defect formation. However, growing (111)-oriented thin perovskite films is considerably more challenging [23] than growing films in the more traditional (001)-orientation, calling for a better understanding of how strain in the (111)-plane affects vacancy formation and local stoichiometry.

2. Computational Methods

Density Functional Theory (DFT) calculations were performed with the projector-augmented wave (PAW) method [24] as implemented in the Vienna Ab initio Simulation Package (VASP 6.x) [25,26] We employed the GGA+U approximation [27,28] with the PBEsol functional [29], using Bi_d, Fe_pv and standard O PBE PAW potentials, and set the plane wave cutoff energy at 550 eV. Collinear G-type antiferromagnetism was imposed on the Fe sublattice, with a Hubbard U of 4 eV applied to the Fe 3d states [11]. Defect calculations were performed on a 120-atom supercell (2 × 2 × 1 of the 30-atom trigonal unit cell) with a gamma-centered 2 × 2 × 2 k-point mesh for reciprocal space integration. The spontaneous polarization is out-of-plane along the [111] pseudocubic direction, as illustrated in Figure 1.

Epitaxial strain in the (111) plane was simulated by fixing the in-plane lattice parameters to strain values defined as (a_h − a_h,0)/a_h,0 * 100%, where a_h,0 is the unstrained equilibrium in-plane lattice parameter. The out-of-plane lattice parameter c and atomic positions were relaxed until the forces on the ions were less than 0.01 eV/Å, and the resulting out-of-plane lattice parameter and unit cell volume for stoichiometric material is presented in Figure 2a. This setup allows us to accurately simulate the effects of strain on vacancy formation while avoiding complications from structural phase transitions.

3. Results and Discussion

In this section, we present and discuss the results of our first-principles calculation for BiFeO₃ thin films under (111) epitaxial strain. Our investigation focuses on the formation of Bi vacancies (V_Bi) and Bi-O vacancy pairs (V_Bi-V_O) and how strain affects their stability, electronic structure and overall impact on the material’s ferroelectric and magnetic properties. We structure this discussion sequentially, beginning with vacancy formation and strain effects, followed by an examination of the electronic structure and, finally, implications for real-world applications.

3.1. Bismuth Vacancies with Electronic Charge Compensation

The formation of bismuth vacancies was investigated by removing a single Bi atom from a 120-atom supercell, corresponding to a vacancy concentration of 4.2% (Bi_0.958FeO₃). Although this defect concentration is relatively high, it is within the range of experimental conditions [30]. Our analysis here focuses on electronic charge compensation, which occurs under oxygen-rich conditions. In this scenario, the charge from a single V_Bi is compensated by the oxidation of three Fe³⁺ ions to Fe⁴⁺, generating three holes. The formation energy of a ${\mathrm{V}}_{\mathrm{Bi}}$, denoted as $E_{{\mathrm{V}}_{\mathrm{Bi}}}$, for a neutral cell is calculated using the following equation:

$$E_{{\mathrm{V}}_{\mathrm{Bi}}}=E_{tot,{\mathrm{V}}_{\mathrm{Bi}}}-E_{tot,stoich}+{\mu }_{\mathrm{Bi}}$$

(1)

where $E_{tot,{\mathrm{V}}_{\mathrm{Bi}}}$ and $E_{tot,stoich}$ are the total energies of the defect-containing cell and the stoichiometric (perfect) cell, respectively. The chemical potential of Bi, ${\mu }_{\mathrm{Bi}}$, is introduced to maintain mass balance and is explained in detail in the Supplementary Materials. The formation energy of V_Bi under oxygen-rich conditions in unstrained bulk BFO was calculated to be 2.46 eV, increasing to 5.54 eV under oxygen-poor conditions, in agreement with previously theoretical studies [31]. Strain has a significant impact on Bi vacancy stability. Compressive strain strongly stabilizes Bi vacnacies, with formation energy decreasing by 0.88 eV as strain increases from 0 to 4%, exhibiting monotonic, non-linear scaling (Figure 2b). Conversely, tensile strain destabilizes V_Bi, increasing the formation energy by 0.39 eV for a strain of +4%.

The strain sensitivity of Bi vacancy formation can be understood as the inverse of the lattice contraction that occurs upon vacancy formation in bulk material. The full geometry optimization of an unstrained Bi₂₃Fe₂₄O₇₂ supercell reveals a volume contraction of 0.56%, which can be considered negative chemical expansion [32]. The charge-compensating oxidation of Fe³⁺ (0.645 Å, C.N. = 6) to Fe⁴⁺ (0.585 Å, C.N. = 6) further contributes to lattice contraction. Although the removal of a Bi³⁺ cation reduces the local shielding of repulsive forces between O²⁻ anions, the overall effect of V_Bi formation is lattice contraction. As a result, bismuth vacancy formation alleviates compressive strain, while tensile strain suppresses Bi vacancy formation.

3.2. Bismuth Vacancies with Ionic Charge Compensation

Under oxygen-poor conditions, Bi vacancy is compensated not by electronic holes but by the formation of oxygen vacancies. In this case, we consider an intermediate scenario where one Bi vacancy is charge-compensated by one hole and one oxygen vacancy, leading to mild hole doping. The resulting supercell stoichiometry becomes Bi_0.958FeO_2.958. The mechanism can be described by the following reaction:

$${\mathrm{BiFeO}}_3 \to {\mathrm{Bi}}_{1-\mathrm{x}}{\mathrm{Fe}}_{1-\mathrm{x}}^{3+}{\mathrm{Fe}}_{\mathrm{x}}^{4+}{\mathrm{O}}_{3-\mathsf{\delta }}+\mathrm{xBi}\left(\mathrm{g}\right)+\mathrm{x}/2 {\mathrm{O}}_2\left(\mathrm{g}\right)$$

(2)

In Kröger–Vink notation, this reaction is equivalent to

$$1/2 {\mathrm{Fe}}_2{\mathrm{O}}_3+1/4 {\mathrm{O}}_2\left(\mathrm{g}\right)\stackrel{{\mathrm{BiFeO}}_3}{\to }{\mathrm{V}}_{\mathrm{Bi}}^{‴}+{\mathrm{Fe}}_{\mathrm{Fe}}^{·}+2{\mathrm{O}}_{\mathrm{O}}^{\mathrm{x}}+{\mathrm{V}}_{\mathrm{O}}^{··}$$

(3)

Here, we investigated four types of Bi-O vacancy pairs, depending on their relative orientations to the strain plane as well as the distance between the two missing atoms. The different vacancy pairs are illustrated in Figure 3a. The formation enthalpies calculated for four types of Bi-O vacancy pairs in unstrained bulk BFO are detailed in

Table 1. The formation enthalpies of different Bi-O vacancy pairs in unstrained bulk BiFeO₃.

Type	Orientation of VBi-VO Pair	VBi-VO Distance (Å)	Formation Enthalpy of Vacancy Pair (eV)
I	OP	2.32	3.60
II	OP	3.35	3.68
III	IP	2.45	4.19
IV	IP	3.23	3.97

.

V_Bi-V_O pairs that are orientated relatively parallel or perpendicular to the (111) strain plane are labeled as ‘in-plane’ (IP) and out-of-plane (OP), respectively. We found that both the orientation and the distance between missing atoms in the perfect structure play a critical role in determining the stability of Bi-O vacancy pairs. For shorter vacancy pairs, the OP orientation is favored over the IP configuration, similar to the preferred alignment of V_Pb-V_O vacancy pairs in tetragonal PbTiO₃, which form so-called symmetry-conforming defects [33]. In contrast, no such orientation preference is observed for longer vacancy pairs. In addition, IP vacancy pairs are generally less stable than OP pairs but are more sensitive to the distance between the missing atoms. Shorter IP vacancy pairs are favored over longer ones by approximately 0.2 eV.

Furthermore, IP and OP vacancy pairs respond differently to strain (Figure 3b); OP pairs remain the most stable across all strain values, with their formation energy significantly reduced by both compressive and tensile strain: 1.49 eV under 4% compressive strain and 1.05 eV under 4% tensile strain. In contrast, IP pairs are primarily stabilized by compressive strain, with a reduction in formation energy of 0.76 eV from 0% to 4%. Tensile strain, however, weakly destabilizes IP pairs, with a small increase in formation energy of 0.08 eV.

As discussed previously, the stabilization of V_Bi under compressive strain can be attributed to the mitigation of lattice contraction and the removal of large Bi³⁺ cations. However, the stabilization of V_Bi-V_O pairs under both compressive and tensile strain is less straightforward, due to the competing effects at play; both the removal of Bi³⁺ and the oxidation of Fe³⁺ to Fe⁴⁺ promote lattice contraction, while the formation of oxygen vacancies tends to favor lattice expansion.

3.3. Structural Changes and Polarization

The out-of-plane axis expands and contracts under compressive and tensile strain, respectively, with a net volume increase under tensile strain, as shown in Figure 2a. In addition, the introduction of V_Bi and V_Bi-V_O pairs leads to significant structural changes in the surrounding lattice. In unstrained stoichiometric BFO, Fe-O bonds have two distinct lengths, 1.96 Å and 2.09 Å, resulting in an octahedral volume of 10.84 ų. Upon Bi vacancy formation, the nearest-neighbor FeO₆ octahedra contract to 10.67 ų due to both the removal of Bi³⁺ and the partial oxidation of Fe³⁺ to Fe⁴⁺. Meanwhile, second-neighbor octahedra expand to 11.12 ų to accommodate the local strain caused by the contraction around Fe⁴⁺ (Figure 2d). This localized distortion indicates a relatively short structural screening length of approximately 8 Å, consistent with previous observations of strain-induced lattice distortion [32].

Epitaxial strain further enhances the polar displacements of Bi and Fe from their centrosymmetric positions. Compressive strain increases Bi displacements, while Fe displacements increase monotonically from compressive to tensile strain. Interestingly, the presence of V_Bi reduces Bi displacement, particularly under tensile strain, but leaves Fe displacements largely unaffected (Figure 2c and Figure 3c). Despite these local distortions, the overall ferroelectric polarization remains largely unchanged, indicating that BFO can maintain its ferroelectric properties even with significant Bi vacancy concentration.

3.4. Electronic Structure

The electronic density of states (DOS) for stoichiometric and vacancy-containing BFO thin films presented in Figure 4 provides further insights into how strain influences the material’s electronic properties. For stoichiometric BFO, the band gap is calculated to be 2.3 eV, with the top of the valence band dominated by O 2p states and the bottom of the conduction band by Fe 3d states [34]. Under tensile strain, the band gap narrows to 2 eV at 4% strain, while compressive strain causes only minor changes. In cells containing Bi vacancies, the charge-compensating oxidation of Fe causes a portion of the Fe 3d states to shift into the bad gap, creating gap states with Fe 3d-O 2d character. These defect states are located just below the Fermi Energy, suggesting that Bi vacancies induce p-type conductivity, consistent with experimental observations of bulk BFO [35] and conducting domain walls [19]. The gap states induced by Bi vacancies and Bi-O vacancy pairs dominate the electronic structure and band gap changes compared to the isolated effect of strain on the stoichiometric cells, as illustrated in Figure 5. The electronic band gap is relatively insensitive to compressive strain, while in contrast, tensile strain significantly reduces the band gap from 2.31 eV to 2.20, 2.07 and 1.96 eV at 2, 4 and 6% strain, respectively. The shift in absolute values of the valence band maximum as a function of strain is illustrated in Supplementary Figure S1.

3.5. Implications for Thin Film Synthesis and Domain Walls

Our results suggest that BiFeO₃ thin films will become increasingly susceptible to Bi deficiency as the strain magnitude increases, except under tensile strain in oxygen-rich conditions, where V_Bi formation is somewhat suppressed. Therefore, the likelihood of Bi loss during thin-film growth and processing increases with higher strain magnitudes, particularly under inert or reducing conditions. In addition to the more readily observable changes in crystal symmetry by diffraction methods, cation vacancy formation should also be anticipated in epitaxial films subjected to greater epitaxial strain [36].

These findings may also help explain reports of Bi-deficient ferroelectric domain walls in BFO [19]. Domain walls inherently induce local strain fields due to the discontinuities in polarization direction. Our results on coherently strained BFO strongly suggest that bismuth vacancies are also more likely to form in the strained local structure in the vicinity of domain walls. Furthermore, strain lowering of V_Bi formation energy also implies that domain walls can act as sinks for diffusion Bi vacancies during high-temperature annealing below the Curie temperature (T_C), in agreement with the experimental findings of Rojac et al. [19].

4. Conclusions

In this study, we used DFT calculations to investigate the stability of Bi vacancies and Bi-O vacancy pairs as a function of epitaxial strain in (111)-oriented BiFeO₃ (BFO) thin films. Our results show that Bi vacancies in charge-neutral cells are more energetically favorable under compressive strain than tensile strain when charge compensation occurs through the oxidation of Fe (no oxygen vacancy). Bi-O vacancy pairs aligned out-of-plane with respect to the (111) strain plane exhibit enhanced stability under both compressive and tensile strain, whereas in-plane Bi-O vacancy pairs are stabilized by compressive strain but remain relatively insensitive to tensile strain. Out-of-plane vacancy pairs, which are more aligned with the polarization direction, are favored over in-plane pairs for all strain values investigated. The promotion of Bi vacancies, with or without accompanying oxygen vacancies, is particularly relevant for thin film growth, as strain tends to enhance Bi deficiency. Finally, we expect the local chemical composition around ferroelectric domain walls to be Bi-deficient due to the localized strain fields surrounding these regions.