Dimension-Dependent Bandgap Narrowing and Metallization in Lead-Free Halide Perovskite Cs₃Bi₂X₉ (X = I, Br, and Cl) under High Pressure

Abstract

Low-toxicity, air-stable cesium bismuth iodide Cs₃Bi₂X₉ (X = I, Br, and Cl) perovskites are gaining substantial attention owing to their excellent potential in photoelectric and photovoltaic applications. In this work, the lattice constants, band structures, density of states, and optical properties of the Cs₃Bi₂X₉ under high pressure perovskites are theoretically studied using the density functional theory. The calculated results show that the changes in the bandgap of the zero-dimensional Cs₃Bi₂I₉, one-dimensional Cs₃Bi₂Cl₉, and two-dimensional Cs₃Bi₂Br₉ perovskites are 3.05, 1.95, and 2.39 eV under a pressure change from 0 to 40 GPa, respectively. Furthermore, it was found that the optimal bandgaps of the Shockley–Queisser theory for the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites can be reached at 2–3, 21–26, and 25–29 GPa, respectively. The Cs₃Bi₂I₉ perovskite was found to transform from a semiconductor into a metal at a pressure of 17.3 GPa. The lattice constants, unit-cell volume, and bandgaps of the Cs₃Bi₂X₉ perovskites exhibit a strong dependence on dimension. Additionally, the Cs₃Bi₂X₉ perovskites have large absorption coefficients in the visible region, and their absorption coefficients undergo a redshift with increasing pressure. The theoretical calculation results obtained in this work strengthen the fundamental understanding of the structures and bandgaps of Cs₃Bi₂X₉ perovskites at high pressures, providing a theoretical support for the design of materials under high pressure.

1. Introduction

Organic–inorganic hybrid perovskites have attracted much attention from the researcher community because of their remarkable photoelectric properties, including high absorption coefficients in the visible light region, tunable bandgaps, high quantum yields, high carrier mobilities, and low effective carrier quality [1,2,3,4,5,6]. In addition, their processing is economical and utilizes simple solution treatments [7,8]. The power conversion efficiency (PCE) of perovskite solar cells (PSCs) has increased dramatically from 3.8% in 2009 to 25.5% [9,10]. Lead hybrid perovskites have the outstanding photovoltaic properties and a high PCE. However, the high toxicity of lead is a major challenge for the large-scale fabrication and commercialization of lead-based PSCs. Recently, nontoxic, all-inorganic, lead-free Bi-based perovskites have attracted significant attention owing to their stability, high photoluminescence, high quantum yield, and tunable bandgaps [11,12,13,14,15,16,17]. They have been widely used in applications, such as memory devices, photodetectors, solar cells, and X-ray detectors [18,19,20,21,22,23,24,25,26]. However, many studies have shown that there are some challenges to overcome for the use of Cs₃Bi₂X₉ (X = I, Br, and Cl) perovskites in photovoltaic devices. The large bandgap (>2 eV) is the most important factor causing a low PCE, which limits their absorption efficiency and carrier transport performance. According to the Shockley–Queisser theory, a semiconductor with a bandgap in the range of 1.3–1.5 eV is an ideal material for solar cells. Although the PCE of Bi-based PSCs has improved slightly with the improvement of the thin-film technology, it still lags far behind that of lead-based perovskites. The traditional chemical modification cannot overcome this inherent limit. Therefore, tuning the bandgap with the aim of improving the photovoltaic performance of the Cs₃Bi₂X₉ perovskites has become a key challenge.

High pressure (HP), which is a non-polluting tuning method, is widely used to modulate the physical and chemical properties of materials without changing their chemical composition [27,28]. In recent years, the properties of halide perovskites have been widely studied under HP; the properties studied include piezochromism, bandgap engineering, structural phase transitions and optical properties, [29,30,31,32]. In addition, various novel physical phenomena have been observed under HP conditions. For example, the organic–inorganic hybrid perovskite nanocrystals present the comminution and recrystallization under HP and exhibit higher photoluminescence quantum yield and a shorter carrier lifetime [29]. A reversible amorphization has been observed for the CH₃NH₃PbBr₃ perovskite under a pressure of approximately 2 GPa; during this transition, the resistance increased by five orders of magnitude, and the material still retained its response to the visible light and semiconductor characteristics up to a pressure of 25 GPa [33]. Notably, pressure-induced structural changes and optical properties are reversible upon decompression, and a semiconductor–metal transition can be observed at 28 GPa [34]. It is notable that there are few studies on Cs₃Bi₂X₉ perovskite systems of different dimensions at HP [34,35], and no reports exist on the one-dimensional halide perovskite Cs₃Bi₂Cl₉.

In this work, we calculated the lattice constants, band structures, density of states (DOS), and optical properties of the one-dimensional perovskite Cs₃Bi₂Cl₉ via the density functional theory (DFT) under HP for the first time and compared with the zero-dimensional perovskite Cs₃Bi₂I₉ and the two-dimensional perovskite Cs₃Bi₂Br₉. We discussed the relationship between the bandgap of the Cs₃Bi₂X₉ perovskites and the HP and focused on the optimal bandgap of the Shockley–Queisser theory of the Cs₃Bi₂X₉ perovskites. The Cs₃Bi₂I₉ perovskite completed the transition from semiconductor to metal at 17.3 GPa, this finding indicated that HP is an effective means to induce the semiconductor–metal transition. Moreover, it was found that the lattice constants and bandgaps of the Cs₃Bi₂X₉ perovskites are dependent upon dimension, that is, the changes in the lattice constants and bandgap gradually decrease as the dimension increases from zero to two under the same pressure. Our calculated results obtained in this work strengthen the basic understanding of different structures of the Cs₃Bi₂X₉ (X = I, Br and Cl), providing theoretical guidance for the structure and bandgap regulation of Bi-based perovskites under HP.

2. Computational Model and Method

The DFT was performed in the Vienna Ab-initio Simulation Package (VASP) using the projected augmented wave (PAW) framework [36,37]. DFT is derived from the Schrodinger equation under the Born–Oppenheimer approximation, described by the Hohenberg–Kokn theorem and the Kohn–Sham equation. The pseudopotential is a hypothetical potential energy function used in place of the inner electron wave function to reduce the computation. The electron exchange–correction function was obtained via the generalized gradient approximation (GGA) parameterized using the Perdew–Burke–Ernzerhof (PBE) formalism [38]. The cut-off energy of the plane wave was set to 500 eV [34]. The convergence criteria for the energy and force were set to 10⁻⁵ eV and 0.01 eV/Å, respectively. The Brillouin zone integration were sampled with 4 × 4 × 4, 4 × 4 × 4, and 4 × 4 × 2 Gamma-pack k-point meshes during the structure optimization of Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉, respectively. The entire optimization of the structures was completely relaxed [39]. The valence electronic configurations of the Cs, Bi, I, Br, and Cl atoms are 5s²5p⁶6s¹, 5d¹⁰6s²6p³, 5s²5p⁵, 4s²4p⁵, and 3s²3p⁵, respectively. The Γ Brillouin zone center has a highly symmetric path, with coordinates Γ (0, 0, 0) to the M (0.5, 0, 0), K (0.333, 0.333, 0), Γ (0, 0, 0), A (0, 0, 0.5), L (0.5, 0, 0.5), and H (0.333, 0.333, 0.5). The spin–orbit coupling is not considered due to the high computational cost. Previous studies have shown that higher levels of calculation (including the spin–orbit coupling, the GW method, or hybrid functionals) obtain more accurate bandgaps; However, they induce little change in the band structure of the heavy-metal halide perovskites [40].

3. Results and Discussion

The structures of the three halogenated perovskite crystals Cs₃Bi₂X₉ (X = I, Br, and Cl) are shown in Figure 1. The one-dimensional Cs₃Bi₂Cl₉ perovskite has an orthogonal-crystal structure; the two-dimensional Cs₃Bi₂Br₉ perovskite and the zero-dimensional Cs₃Bi₂I₉ perovskite have hexagonal cells [41,42]. The numbers of atoms in the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites in the primitive cell are 28, 14, and 56, respectively. The Bi atom is located at the center of the octahedron in the Cs₃Bi₂X₉ perovskites and is surrounded by six halogen atoms. Unlike in lead-based perovskites, in the Cs₃Bi₂Br₉ and Cs₃Bi₂Cl₉ perovskites, two octahedrons share one X atom (X = Br or Cl), whereas two octahedrons share three X atoms in the Cs₃Bi₂I₉ perovskite. X atoms with different sizes and the Bi atoms form a novel double-perovskite structure. Figure 2 shows the changes in the lattice parameters and volume of the Cs₃Bi₂X₉ perovskites under HP. In the primary cell of the Cs₃Bi₂I₉ and Cs₃Bi₂Br₉ perovskites, the lattice constants a and b are equal. The lattice constant and volume of Cs₃Bi₂X₉ clearly decrease with an increase in pressure, and the slope also decreases gradually. As is well established, pressure induces a reduction in the lattice constant. From a microscopic point of view, the pressure shrinks the distance between two atoms. The strong Coulomb force makes it increasingly difficult to further compress the material as the pressure increases.

Based on the semiconductor theory, analyzing the band structure and the DOS very closely, the Fermi level is important for establishing the possible applications of a material in the photoelectric and photovoltaic fields. We therefore calculated the band structure near the Fermi level (from −5 eV to +5 eV) under different pressures. The calculated bandgaps at a pressure of 0 GPa of the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ are 2.38, 2.60, and 3.08 eV, respectively. The scissor value approach was used to establish the band structure of the Cs₃Bi₂X₉ perovskites in order to obtain the accurate value for the Cs₃Bi₂X₉ perovskites when they reached the optimal bandgap of the Shockley–Queisser theory and the transition from semiconductor to metal; this methodology overcame the limitations of the GGA–PBE calculation method and has already been applied in the study of the photoelectric properties of the CsSnCl₃ perovskite at HPs [40]. Many experimental studies have been conducted regarding the bandgaps of the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites. Among these, Daniel R. Gamelin et al. synthesized Cs₃Bi₂X₉ perovskites nanocrystals via the thermal injection method, and the bandgaps of the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites were measured as 2.07, 2.76, and 3.26 eV, respectively [43]. Therefore, the scissors values of −0.31eV, 0.16eV, and 0.18eV were used for the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites, respectively.

Figure 3 shows the change in the band structure of the Cs₃Bi₂I₉ perovskite under the pressures of 0 (a), 4 (b), 10 (c), and 40 GPa (d). Without external pressure, the conduction band minimum (CBM) and the valence band maximum (VBM) of Cs₃Bi₂I₉ perovskite are located at the Γ- and M-points, respectively. We found the Cs₃Bi₂I₉ perovskite to be an indirect bandgap material, which is consistent with the existing literature [44]. The semiconductor with the optimized band gap energy of 1.34 eV is critical to achieve the efficiency limit of 33.7% based on the Shockley–Queisser theory [34]. In Figure 3, it can be seen that the bandgap of the Cs₃Bi₂I₉ perovskite decreases sharply with the increase in pressure and reaches its optimal bandgap value given by the Shockley–Queisser theory at 2–3 GPa. In addition, with the high pressure further increasing, the CBM continues to decrease and the VBM moves from the M-point to near the K-point. The Cs₃Bi₂I₉ perovskite completed the transition from semiconductor to metal at 17.3 GPa, and this finding indicated that HP is an effective means to induce the semiconductor–metal transition. The band structure of the Cs₃Bi₂Br₉ perovskite under the pressures of 0 (a), 4 (b), 10 (c), and 40 GPa (d) is presented in Figure 4. As can be seen from Figure 4, when the pressure is 0 GPa (a), the calculated band structure of the Cs₃Bi₂Br₉ perovskite is consistent with the results reported by Brent C. Melot et al. [45] They found a low-lying 2.52 eV indirect transition as well as a slightly larger direct gap of 2.64 eV, which are essentially in agreement with our calculated results. This structural characteristic becomes increasingly evident with the increase in pressure. The Cs₃Bi₂Br₉ perovskite reaches the optimal bandgap given by the Shockley–Queisser theory at 21–26 GPa, which means that the electron transitions from the valence band to conduction band become easier. Figure 5a–d shows the change of the band structure of Cs₃Bi₂Cl₉ perovskite with HP. The Cs₃Bi₂Cl₉ perovskite reaches the optimal bandgap in the range of 25–29 GPa; it is an indirect bandgap material. The CBM is at the Γ- point, and the VBM moves from the Γ-point to the Y-point, which is consistent with the results reported previously [41]. However, it was found that the Cs₃Bi₂Br₉ and Cs₃Bi₂Cl₉ perovskites did not metallize under HP despite the pressure reaching 40 GPa in both cases, which may be related to their unique structure. The bandgap changes in the Cs₃Bi₂X₉ perovskites under a range of HP (0, 2, 4, 6, 8, 10, 20, 30, and 40 GPa) are shown in Supplementary Figures S1–S3.

Figure 6 shows the changes in the bandgaps of the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites under HP. It can be seen that the bandgap of the Cs₃Bi₂X₉ perovskites decreases with the increase in pressure (Figure 6a); the bandgap of the Cs₃Bi₂I₉ perovskite takes a negative value, which is a typical indicator of metallic behavior. Figure 6b shows the change in the bandgap of the Cs₃Bi₂X₉ perovskites after using the scissor values under HP. When the pressure is varied from 0 to 40 GPa, the bandgap differences of the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites are 3.05, 1.95, and 2.39 eV, respectively. It was found that with the increase in dimension, the bandgap differences of the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites decrease in turn, which indicates that the bandgap of the Cs₃Bi₂X₉ perovskites depends on dimension.

To explain the dimension-dependent bandgap of the Cs₃Bi₂X₉, we investigate the construction of these perovskites. Based upon the primitive cell of Cs₃Bi₂I₉, Cs₃Bi₂Br₉ and the Cs₃Bi₂Cl₉ perovskites, the 2 × 2 × 1, 2 × 2 × 2, and 3 × 1 × 1 supercells are shown in Figure 7a–c, respectively. In Figure 7d, ΔL represents the difference in the lattice constants of the Cs₃Bi₂X₉ perovskites between 0 and 40 GPa along the a, b, and c coordinate axes. In general, for centrosymmetric perovskites, the ΔL values along the a- (ΔL_a), b- (ΔL_b), and c- (ΔL_c) axes are equal under HP. However, we found that ΔL_a, ΔL_b, and ΔL_c for the double perovskites were not equal; in other words, ΔL is anisotropic along the a-, b-, and c-axes. The ΔL_a, ΔL_b, and ΔL_c values for the Cs₃Bi₂I₉ perovskite (zero-dimensional) are 1.57, 1.57, and 5.2. It can be seen that ΔL_c is much larger than both ΔL_a and ΔL_b. The reason for this difference is that the Bi₂I₉³⁻ frame of the double-perovskite is continuous along the a- and b-axes but not along the c-axis. Along the c-axis, only the Cs⁺ atoms are above or below the Bi₂I₉³⁻ frame, which indicates that the change in the lattice constant along the c-axis is bigger than that observed along the a- and b-axes when the zero-dimensional Cs₃Bi₂I₉ perovskite is placed under HP.

The previous analysis shows that if the double-perovskite frame expands regularly in one direction, the ΔL in this direction will be smaller than in the other directions under HP. This indicates that the lattice constants of the Cs₃Bi₂X₉ perovskites are dependent on dimension. In general, the lattice constant and the bandgap decrease as the pressure increases [34]. It has also been found that the changes in the bandgap of the zero-dimensional Cs₃Bi₂I₉, one-dimensional Cs₃Bi₂Cl₉, and two-dimensional Cs₃Bi₂Br₉ perovskites between 40 and 0 GPa are 3.05, 2.39, and 1.95 eV, respectively. These results illustrate that the bandgaps of the Cs₃Bi₂X₉ perovskites are also dependent upon dimension; that is, the changes in the bandgap decrease gradually as the dimension increased from zero to two under the same pressure.

The partial density of states (PDOS) of the zero-dimensional Cs₃Bi₂I₉ (a), one-dimensional Cs₃Bi₂Cl₉ (b), and two-dimensional Cs₃Bi₂Br₉ (c) perovskites were shown in Figure 8. It can be seen that the VBM of the Cs₃Bi₂X₉ perovskites is dominated by p-X states, whereas the CBM is dominated by the p-Bi and p-X states (see Figure 8a–c). The changes in the DOS under HP are shown in Figure 8d–f. It is clear that many valence bands in the Cs₃Bi₂X₉ perovskites move to a deep level, and the conduction bands approach to the FE with an increase in pressure for Cs₃Bi₂X₉ perovskites; the shift of the conduction bands under the HP will induce the changes in the bandgap. In contrast to those of the Cs₃Bi₂Br₉ and Cs₃Bi₂Cl₉ perovskites, the forbidden band width of the Cs₃Bi₂I₉ gradually decreases and subsequently disappears, which indicates that the Cs₃Bi₂I₉ perovskite is no longer a semiconductor; it becomes a metal at 17.3 GPa. This conclusion is agreement with the calculated results for the band structure.

A large absorption coefficient is of great significance in photoelectric and photovoltaic applications. Such a property improves the PCE of solar cells and the luminous efficiency. The absorption coefficient is usually described by the dielectric function according to the following expression [46]:

$$\alpha =2\omega {\left[\frac{{\left({\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)\right)}^{1/2}-{\epsilon }_1\left(\omega \right)}{2}\right]}^{1/2}$$

where the $\omega$ is the frequency of light, and ${\epsilon }_1^{}$ and ${\epsilon }_2^{}$ are the real and imaginary parts of the dielectric function, respectively. The calculated ${\epsilon }_1^{}$ and ${\epsilon }_2^{}$ values of the Cs₃Bi₂X₉ perovskites are presented in Supplementary Figures S4–S6 along the a-, b-, and c-axes. The static dielectric function of the Cs₃Bi₂X₉ perovskites increases gradually with increasing pressure from 0 to 40 GPa. The ${\epsilon }_2^{}$ value is closely related to the optical absorption and usually used to describe the absorption behavior of materials.

The calculated absorption coefficients of the Cs₃Bi₂X₉ perovskites are shown in Figure 9 and Figure 10 along the a-, b-, and c-axes. It was found that the Cs₃Bi₂I₉ perovskite has the same absorption coefficient along the a- and b-axes with the increase in pressure, as is the case for the Cs₃Bi₂Br₉ perovskite. The absorption coefficients of the Cs₃Bi₂Cl₉ perovskite are unequable along the a-, b-, and c-axes under HP. The ΔL value of the Cs₃Bi₂X₉ perovskites in Figure 7d is consistent with these behaviors, which suggests that the changes in the structure of the Cs₃Bi₂X₉ perovskites under HP will affect the optical properties considerably. The Cs₃Bi₂X₉ perovskites exhibit a redshift with the increase in pressure, which indicates that the Cs₃Bi₂X₉ can absorb the low-energy photons. Moreover, the absorption coefficients of the Cs₃Bi₂X₉ perovskites increase gradually in the ultraviolet region as the pressure increases from 0 to 40 GPa. It was also found that the Cs₃Bi₂X₉ perovskites have a large absorption coefficient in the visible region (on the order of 10⁵ cm⁻¹). Therefore, the Cs₃Bi₂X₉ perovskites are an attractive candidate in applications of photoelectric and photovoltaic devices.

4. Conclusions

In summary, we investigated the lattice constants, band structure, DOS, and optical absorption of the cesium bismuth iodide Cs₃Bi₂X₉ (X = I, Br and Cl) perovskites under HP by using the DFT. It was found that the optimal bandgap of the Shockley–Queisser theory for the Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites can be obtained at 2–3 GPa, 21–26 GPa, and 25–29 GPa, respectively. The changes in the bandgap of Cs₃Bi₂I₉, Cs₃Bi₂Br₉, and Cs₃Bi₂Cl₉ perovskites are 3.05, 1.95, and 2.39 eV under a pressure of 40 GPa, respectively. The Cs₃Bi₂I₉ perovskite was found to transform from a semiconductor into a metal at 17.3 GPa. Furthermore, the dimension-dependent lattice constants, unit-cell volumes, and bandgaps of the Cs₃Bi₂X₉ perovskites were studied. Our calculations show that HP is an effective way to tune the photovoltaic and optoelectronic properties of the Cs₃Bi₂X₉ perovskites by modifying the crystal structure, which provides a promising method for material design and applications.