Structural and Optoelectronic Properties of Two-Dimensional Ruddlesden–Popper Hybrid Perovskite CsSnBr₃

Abstract

Ultrathin inorganic halogenated perovskites have attracted attention owing to their excellent photoelectric properties. In this work, we designed two types of Ruddlesden–Popper hybrid perovskites, Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2, and studied their band structures and band gaps as a function of the number of layers (n = 1–5). The calculation results show that Cs_n+1Sn_nBr_3n+1 has a direct bandgap while the bandgap of Cs_nSn_n+1Br_3n+2 can be altered from indirect to direct, induced by the 5p-Sn state. As the layers increased from 1 to 5, the bandgap energies of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 decreased from 1.209 to 0.797 eV and 1.310 to 1.013 eV, respectively. In addition, the optical absorption of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 was blue-shifted as the structure changed from bulk to nanolayer. Compared with that of Cs_n+1Sn_nBr_3n+1, the optical absorption of Cs_nSn_n+1Br_3n+2 was sensitive to the layers along the z direction, which exhibited anisotropy induced by the SnBr₂-terminated surface.

1. Introduction

Perovskites have become competitive candidate materials for photovoltaic and optoelectronic applications, such as solar cells, optically pumped lasers, detectors and light emitting diodes [1,2,3,4,5]. At present, the power-conversion efficiency of perovskite solar cells has reached 25.4% [6]. Over the past several years, the high-performance perovskite light-emitting diodes have developed rapidly, reaching high external quantum efficiencies of over 20% [1]. At room temperature, the optically pumped laser made from the lead halide perovskite nanowires can be tuned in the whole visible spectrum region (420−710 nm) with high-quality factors and low lasing thresholds [3]. The advantages of these materials include their long carrier lifetimes and diffusion lengths, large absorption coefficients in the visible spectrum, small effective masses, tunable bandgaps and high quantum efficiency [7,8,9,10,11,12]. In addition, compared with other materials, perovskites have the advantage of low-cost and facile processing [13,14]. Although many achievements in the application of lead-based perovskites have been obtained, there are still many challenges, especially due to the presence of toxic lead. The toxicity of lead can cause serious damage to life and the environment. Therefore, the use of less toxic materials, such as tin and other alkaline earth metals, to replace lead has been widely investigated [15]. An inorganic substitute can reduce hysteresis loss that is caused by the presence of methylammonium. Previous work showed that CsPbBr₃ solar cells are as efficient as and have higher levels of environmental stability than CH₃NH₃PbBr₃ solar cells, even after aging for 2 weeks [16].

Recently, two-dimensional (2D) Ruddlesden–Popper (RP) hybrid perovskite materials have attracted intensive attention because of their wide applications [17,18,19,20,21,22,23]. Compared with bulk perovskites, 2D perovskites show many unique properties. For example, 2D RP perovskites have a narrow full width at half maximum and blue-shift photoluminescence signal with the decrease in the number of layers owing to the quantum confinement of carriers [24]. Two-dimensional RP perovskites have a higher chemical stability because of the linking of organic molecules, such as CH₃(CH₂)₃NH₃ and CH₃NH₃, on the terminated surface [25]. Ultrathin 2D RP perovskite solar cells can also reduce production costs from the point of view of mass production [26,27]. Until recently, the layer numbers and the size of synthesized ultrathin 2D RP films have been accurately controlled experimentally. Importantly, the bandgap of 2D RP perovskites can be easily adjusted by changing the number of layers [28]. Therefore, the layer-dependent properties of perovskites further enrich their applications in optoelectronic and photovoltaic fields [28,29,30]. In addition, it is generally known that different terminated surfaces of 2D RP perovskites are important for optoelectronic properties and applications, such as chemical activity of the surfaces, sensing effects, and ultrathin film production. Previous work has reported that the iodine defect improves the efficiency of photoabsorbance in a SnBr₂-terminated surface perovskite solar cell by lowering its energy gap [31]. Ab initio calculations of the structural and electronic properties of the CsBr- and CaBr₂/GeBr₂/SnBr₂-terminated (001) surfaces of CsMBr₃ (M = Ca, Ge, Sn) perovskites indicated noticeable changes of the surface properties in comparison with those for bulk materials [32]. Thus far, there have been many literatures about 2D perovskites with different surface terminations, including the structural and electronic properties, surface relaxations, energetics, bonding properties, ferroelectrics and dipole moment [33,34,35,36,37,38,39,40]. The SrTiO₃ (100) surface relaxation and rumpling have been calculated with two different terminations (SrO and TiO₂) [39]. Experimentally, the researchers have already studied the ferroelectric relaxation, surface relaxation, polar oxide and surface rumpling of perovskites with different surface terminations [41,42,43,44,45,46]. Particularly, A. Ikeda et al. determined surface rumpling and relaxation of TiO₂-terminated SrTiO₃ (001) using the medium energy ion scattering, and found that the occupation fraction of the TiO₂ face ranged from 85 to 95% [45].

Herein, we have designed two models based on CsSnBr₃ with different surface terminations: Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2. The terminated surfaces are CsBr and SnBr₂ for the two models, respectively. We investigated the structural and optoelectronic properties, including band structures, surface relaxation effects, density of states and optical absorption spectra of the two models using the first-principles method. Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 with different numbers of layers showed different properties in the bandgap and absorption coefficient. The theoretical calculation results obtained in the present work can serve as a guideline in the design of different structures and for the improvement of the efficiency of optical absorbance for 2D inorganic perovskites.

2. Computational Model and Method

The first-principles calculation in this work is based on density functional theory (DFT) [47]. The calculation method was the all-electron-like projector augmented wave method and the exchange correlation potential realized by Perdew–Burke–Enserch (PBE) in the Vienna Ab Initio Simulation Package (VASP) [48,49,50]. The electron exchange correction function was described by the generalized gradient approximation parameterized by PBE. The cut-off energy of the plane wave was set to 500 eV. All atoms were allowed to relax until the Hellmann–Feynman forces reached the convergence criterion of less than 0.01 eV/Å. The convergence threshold of energy was set at 10⁻⁵ eV. The Monkhorst-Pack scheme was used to sample k-points in the Brillouin zone [51]. The k-point meshes were set to 6 × 6 × 1 and 8 × 8 × 1 for the electronic structure and density of states, respectively. The HSE06 hybrid functional used to calculate the bandgap and the fraction of exact exchange in the Hartree–Fock/DFT hybrid functional-type calculation was 25%. The spin–orbital coupling (SOC) interaction of the Sn atom is weaker than that of heavy atom lead, so the SOC interaction is ignored in our calculation.

As mentioned earlier, a differently terminated surface will change the band structure, optical absorption and bandgap energy of perovskites. In order to further explore the layer dependence of different surface terminations, we designed two models, namely, Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 (n = 1–5), as Type 1 and Type 2 based on the cubic phase structure (Pm3m space group) of three-dimensional CsSnBr₃ [52], shown in Figure 1. A two-layer Type 2 molecule (Cs₂Sn₃Br₈) was structured by a one-layer Type 1 molecule (Cs₂Sn₁Br₄), which added one plane composed by one Sn atom and four Br atoms on both the top and bottom surfaces, respectively. The same rule can be applied to multiple-layer structures, namely Cs_nSn_n+1Br_3n+2 (n = j + 1), which contain Cs_n+1Sn_nBr_3n+1 (n = j). A vacuum region of 10 Å in the z direction was set on the bottom and top of the models to avoid interaction between the atoms. The electronic configurations of the chemical elements of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 included 4s²4p⁵ (Br), 5s²5p⁶6s¹ (Cs) and 5s²5p² (Sn) [32].

3. Results and Discussion

In our simulation, the structures of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 was optimized and all atoms were allowed to relax. The degree of surface rumpling was quantitatively described to reveal the difference between the two structures. A variable d_i,i+1 can be defined as the interplanar distance between the neighboring atomic planes. The index i labels the atomic layers of Type 1 and Type 2 in Figure 1. The relative displacements are described using the following nondimensional quantity equation [32]:

$${\delta }_{i,i+1}=\frac{d_{i,i+1}-a_0/2}{a_0},$$

(1)

where a₀ is the theoretical lattice constant calculated for bulk CsSnBr₃.

δ_i,i+1 is related with the vacuum layer. For example, in Cs₃Sn₂Br₇ (n = 2) of Type 1, the relative displacements δ_1,2 and δ_2,3 were −1.05 and 1.69%, respectively. All relative displacements, δ_i,i+1, are shown in

Table 1. Calculated relative displacements, δ_i,i+1 (%, in unit of the calculated bulk CsSnBr₃ lattice constant), and layers rumpling, η_i (%), for Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 (n = 1–5).

Layer (n)			1	2	3	4	5
δi,i+1 (%)	Type 1	Br1Sn2	0.6	−1.05	−1.56	−1.64	−1.71
		Sn2Br3		1.69	2.28	2.62	2.6
		Br3Sn4			0.004	−0.49	−0.69
		Sn4Br5				0.62	0.79
		Br5Sn6					0.02
	Type 2	Sn1Br2	−1.98	−2.68	−2.87	−2.9	−3
		Br2Sn3		0.24	1.14	1.52	1.69
		Sn3Br4			−0.74	−1.01	−1.13
		Br4Sn5				0.055	0.33
		Sn5Br6					−0.25
ηi (%)	Type 1	Cs1Br1Cs1	−6.44	−8.28	−8.89	−8.78	−8.89
		Br2Sn2Br2	0	−1.78	−1.94	−1.89	−1.94
		Cs3Br3Cs3		0	−2.44	−3.11	−3.44
		Br4Sn4Br4			0	−0.17	−0.33
		Cs5Br5Cs5				0	−0.72
		Br6Sn6Br6					0
	Type 2	Br1Sn1Br1	−3.22	−3.28	−3.17	−2.94	−2.83
		Cs2Br2Cs2	0	−2.11	−2.83	−3.28	−3.67
		Br3Sn3Br3		0	−0.078	−1	−1
		Cs4Br4Cs4			0	−0.89	−1.28
		Br5Sn5Br5				0	−0.22
		Cs6Br6Cs6					0

for layers 1 to 5. The structures of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 were centrosymmetric and δ_i,i+1 gradually decreased from the terminated surface to the symmetric center. This trend implies that the stability of the octahedral structure near the center of symmetry was better than that of the terminated surface. We also found that the relative displacements δ_1,2 for Type 1 were bigger than that for Type 2, which means Cs_n+1Sn_nBr_3n+1 with CsBr-termination had better stability than Cs_nSn_n+1Br_3n+2. Our conclusion is consistent with the previous calculation [32].

The ratio of angle change before and after optimization can be defined to correspond with the degree of surface rumpling with bond angles in the same atomic plane. The equation is as follows:

$${\eta }_{i,i+1}=\frac{{\theta }_{i,i+1}-\pi }{\pi },$$

(2)

where θ is the degree of Cs_iBr_iCs_i or Br_iSn_iBr_i (i = 1–5); namely, two Cs atoms are nonadjacent in the CsBr plane or two Br atoms are along the y direction in the SnBr₂ plane. Compared with the angles of CsBrCs, the angles of BrSnBr forming octahedral frames change slightly. When n = 4 and 5, the layers rumpling, η_i, of CsBrCs and BrSnBr decreased to zero from the terminated surface to the symmetric center. In Type 1, the structure of Cs_n+1Sn_nBr_3n+1 contained one or more complete perovskite structures (ABX₃), in which the shape of the band structure had no change and only the value of the band energy varied with the layers from 1 to 5. It is noticed that Cs_nSn_n+1Br_3n+2 contains SnBr₂ termination on the top and bottom surfaces in addition to one or more ABX₃ in the Type 1 model. Therefore, compared to the structures of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2, we inferred that the change of band structure shape of Cs_nSn_n+1Br_3n+2 was induced by the SnBr₂-terminated surface.

Figure 2 shows the electron density difference of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 in the slice plane (0.5, 0, 0) with different layer numbers. Obviously, the lost charge of the Sn atom transferred to the six adjacent Br atoms, and it is asymmetric along the z direction. The length of the Sn–Br bond is inversely proportional to the electron density; that is, the larger the electron density, the shorter the bond. This asymmetry gradually decreases along the z direction from the terminated surface to the symmetry center. This is consistent with the changes in δ_i,i+1, which gradually decreased from the terminated surface to the center of symmetry.

The calculated bandgap energies of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 (n = 1–5) are listed in

Table 2. Calculated bandgap energies (in eV) of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 (n = 1–5). The d and ind in parentheses represent direct and indirect bandgap, respectively.

Layer (n)	1	2	3	4	5	Bulk
Type 1	1.209 (d)	1.036 (d)	0.928 (d)	0.851 (d)	0.797 (d)	0.641
Type 2	1.310 (ind)	1.266 (ind)	1.198 (ind)	1.100 (d)	1.013 (d)

. As n increases from 1 to 5, the bandgap energies of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 decrease from 1.209 to 0.797 eV and 1.310 to 1.013 eV, respectively. Our calculations are almost the same as those calculated by Anu et al. Their results showed that the bandgap energies of Cs_n+1Sn_nBr_3n+1 are 1.2 eV, 1.04 eV, 0.92 eV, 0.85 eV and 0.79 eV with the layers from 1 to 5 [53]. In experiments, it was also found that the bandgap energies of (PEA)₂(MA)_n–1Pb_nBr_3n+1 (2D) and (CH₃(CH₂)₃NH₃)₂(CH₃NH₃)_n−1Pb_nI_3n+1 (2D) gradually shrink with the increase in the layer number [41,42]. There are many publications which support our models [31,32,54,55,56]. As shown in Figure 3a–e, the conduction band minimum (CBM) and the valence band maximum (VBM) of Type 1 appear at the R point (0.5, 0.5, 0.5). The bandgap of Cs_n+1Sn_nBr_3n+1 presents a direct bandgap and gradually shrinks with the increase in the layer number. According to this trend, the bandgap of Cs_n+1Sn_nBr_3n+1 will reach 0.641 eV with the increase of n [54]. The bandgap of Cs_nSn_n+1Br_3n+2 also decreases with the increase of n. However, unlike that of Cs_n+1Sn_nBr_3n+1, the band structure of Cs_nSn_n+1Br_3n+2 in Figure 4 shows an indirect bandgap because the CBM of Cs_nSn_n+1Br_3n+2 does not appear at the M point (0.5, 0.5, 0) when n = 1, 2 and 3 (Figure 4a–c). It can be seen that the band structure at the bottom of the conduction band along the M→X and M→Γ directions is W-shaped (n = 1, 2 and 3) instead of parabolic, as shown in Figure 4. When n = 1, 2 and 3, the differences between the M point (0.5, 0.5, 0) and the lowest point are 26.6, 8.7 and 1.8 meV, respectively. With the increase of n, the differences gradually decrease to zero when n = 4 and 5 (Figure 4d,e), which means the band structure of Cs_nSn_n+1Br_3n+2 turns into a direct bandgap. The band structures of Cs_nSn_n+1Br_3n+2 (n = 1) calculated by PBE and HSE06 DFT are shown in Supplementary Figure S1. It is easy to see that the band structure of Cs_nSn_n+1Br_3n+2 (n = 1) calculated by HSE06 has a shift up compared with that by the PBE calculation, while the shape of the band does not change. In view of the high computational cost, PBE is used in this work.

To carefully examine which atom induces the emergence of the indirect bandgap of Cs_nSn_n+1Br_3n+2, the density of states (DOS) of both types with n = 1 and 5 were calculated, as shown in Figure 5. Figure 5 indicates that the Sn state plays a dominant role, while the Cs state is negligible for the valence band top and conduction band bottom in both models. It can be seen that a small peak appears at the bottom of the conduction band of Cs_n+1Sn_nBr_3n+1 (Type 1, n = 1; indicated by an arrow), which is induced by the degeneracy of energy levels (Figure 5a). Figure 5b shows that the valence band top and conduction band bottom of Cs_nSn_n+1Br_3n+2 (Type 2) are dominated by Br and Sn states. A sharp and strong peak (indicated by an arrow) appears for the DOS of Cs_nSn_n+1Br_3n+2 (n = 1), which indicates that the generation of an indirect bandgap is induced by Sn atoms. The intensity of the peak (indicated by an arrow) of Cs_nSn_n+1Br_3n+2 decreases gradually with layer numbers from 1 to 5, which means a transition from indirect to direct bandgap for Cs_nSn_n+1Br_3n+2. Therefore, according to the DOS in Figure 5, Cs_n+1Sn_nBr_3n+1 is a direct bandgap and Cs_nSn_n+1Br_3n+2 is an indirect bandgap led by Sn atoms.

Further efforts were made to separately calculate the partial density of states (PDOS) of Sn and Br atoms at the terminated surface in Figure 6 (the two atoms are indicated by arrows in Supplementary Figure S2). The results show that the conduction band bottom of Cs_nSn_n+1Br_3n+2 is mainly dominated by the 5p-Sn state. The peak of total DOS of Sn atoms in Figure 5b is 2.87, and the peak of PDOS of the Sn atom in Figure 6a is 1.41; there is about a twofold relationship between the size of the peak. It is also found that the contribution of the 4p-Br state to the conduction band bottom of Cs_nSn_n+1Br_3n+2 is negligible according to Figure 6b. Therefore, the 5p-Sn state at the terminated surface induced the generation of an indirect bandgap for Cs_nSn_n+1Br_3n+2. In addition, we calculated the orbital-projected band structures of Cs_nSn_n+1Br_3n+2 (n = 1) in Supplementary Figure S2a, which coincides completely with the calculated band structures in Figure 4a. This alignment indicates that the conduction band bottom of Cs_nSn_n+1Br_3n+2 is dominated by the 5p-Sn state, which agrees well with the PDOS calculations for Sn and Br atoms.

A suitable bandgap energy and large absorption coefficient are important for photoelectric and photovoltaic devices. Therefore, the optical absorption of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2, with different layer numbers, have also been studied in this work. The optical absorption is generally calculated using the complex dielectric function, which is expressed as ɛ(ω) = ɛ₁(ω) + iɛ₂(ω), where ω is the frequency of light, ε₁ and ε₂ are the real and imaginary parts of the dielectric function, respectively. ε₂ is usually used to describe the light absorption behavior and its specific description is given by the following equation [57]:

$${\epsilon }_2\left(\omega \right)=\frac{V{e}^2}{2\pi \hslash m^2{\omega }^2}{\displaystyle \int d^3}k{{\displaystyle \sum _{n{n}^{\prime }}\left|\langle kn|p{|kn\rangle }^{\prime }\right|}}^{2}f\left(kn\right)\ast \left(1-f\left(k{n}^{\prime }\right)\right)\delta \left(E_{kn}-E_{k{n}^{\prime }}-\hslash \omega \right),$$

(3)

where V is unit volume, e represents electron charge, m is the electron rest mass, p is the momentum transition matrix, ħ is the reduced Planck Constant; kn and kn′ are the wave functions of the conduction band and valence band, respectively. In order to rapidly distinguish these physical variables, we show them in the Supplementary Table S1. Moreover, by using the Kramers–Kronig relationship, the real part of the dielectric function is obtained as follows [58]:

$${\epsilon }_1\left(\omega \right)=1+\frac{2}{\pi }P{\displaystyle {\int }_0^{\infty }\frac{{\epsilon }_2\left({\omega }^{\prime }\right){\omega }^{\prime }d{\omega }^{\prime }}{{\omega }^{\prime }{}^2-{\omega }^2}},$$

(4)

where P is the principal value of the integral. The absorption coefficient is given as follows [59]:

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

(5)

The absorption coefficient is a key parameter and is of great significance in photoelectric and photovoltaic applications. Figure 7 shows the absorption spectra of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 with different layer numbers along the x, y (Figure 7a,c) and z (Figure 7b,d) directions. Both Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 have large light absorption coefficients in the visible and infrared regions. With the decrease in layer number, the absorption coefficients of both Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 have redshift. Different from that of bulk CsSnBr₃, the light absorption of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 show anisotropy. In the visible region, the absorption coefficients for Cs_nSn_n+1Br_3n+2 along the x and y directions are very close to and smaller than those for bulk CsSnBr₃ (Figure 7c); the layer dependence is not strong. This behavior is related to the absorption of Sn and Br atoms in the terminated surface. Cs_nSn_n+1Br_3n+2 has a larger absorption coefficient than Cs_n+1Sn_nBr_3n+1 along the x and y directions (Figure 7a,c). The SnBr-terminated surface model provides an ideal model for the design of 2D RP perovskites for photovoltaic and optoelectronic devices.

4. Conclusions

In conclusion, based on the cubic CsSnBr₃, we designed two models in this work, including Cs_n+1Sn_nBr_3n+1 with CsBr-termination and Cs_nSn_n+1Br_3n+2 with SnBr₂-termination. Their bandgap energies, structural and optoelectronic properties of the two models were calculated using DFT. The calculated results indicated that the band structure of Cs_n+1Sn_nBr_3n+1 is a direct bandgap. Additionally, the band structure of Cs_nSn_n+1Br_3n+2 can be altered from an indirect to direct bandgap with the increase in the layer numbers. With the variation of the layer number from 1 to 5, the bandgaps of Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 decreased from 1.209 to 0.797 eV and 1.310 to 1.013 eV, respectively. Furthermore, we calculated the DOS of Sn and Br atoms and the orbital-projected band structures of Cs_nSn_n+1Br_3n+2 (n = 1) in the terminated surface. It was found that the 5p-Sn state was responsible for the appearance of the indirect bandgap of Cs_nSn_n+1Br_3n+2. In addition, both Cs_n+1Sn_nBr_3n+1 and Cs_nSn_n+1Br_3n+2 have as large of an absorption coefficient as bulk CsSnBr₃ and show anisotropy. Nevertheless, Cs_nSn_n+1Br_3n+2 exhibits an insensitivity to the layer number along the x and y directions. The calculated results obtained in this work may provide new ideas for the design of photovoltaic devices.