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. 2021 Nov 5;6(45):30752–30761. doi: 10.1021/acsomega.1c04806

Density Functional Theory Analysis of Structural, Electronic, and Optical Properties of Mixed-Halide Orthorhombic Inorganic Perovskites

Hamid M Ghaithan †,*, Zeyad A Alahmed †,*, Saif M H Qaid †, Abdullah S Aldwayyan †,‡,§,*
PMCID: PMC8600628  PMID: 34805703

Abstract

graphic file with name ao1c04806_0012.jpg

Inorganic metal-halide perovskites hold a lot of promise for solar cells, light-emitting diodes, and lasers. A thorough investigation of their optoelectronic properties is ongoing. In this study, the accurate modified Becke Johnson generalized gradient approximation (mBJ-GGA) method without/with spin orbital coupling (SOC) implemented in the WIEN2k code was used to investigate the effect of mixed I/Br and Br/Cl on the electronic and optical properties of orthorhombic CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 perovskites, while the Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) method was used to investigate their structural properties. The calculated band gap (Eg) using the mBJ-GGA method was in good agreement with the experimental values reported, and it increased clearly from 1.983 eV for CsPbI3 to 2.420 and 3.325 eV for CsPbBr3 and CsPbCl3, respectively. The corrected mBJ + SOC Eg value is 1.850 eV for CsPbI3, which increased to 2.480 and 3.130 eV for CsPbBr3 and CsPbCl3, respectively. The calculated photoabsorption coefficients show a blue shift in absorption, indicating that these perovskites are suitable for optical and optoelectronic devices.

1. Introduction

Because of their superior thermal stability compared to their organic–inorganic hybrid counterparts, inorganic perovskites have emerged as one of the most appealing research hotspots in the field of perovskite photovoltaics over the last 5 years.1−3 Perovskite compounds have the chemical formula ABX3, where A is a monovalent cation such as CH3NH3 (MA), HC(NH2)2 (FA), or Cs, B is a divalent cation such as Pb or Sn, and X is an anion such as I, Br, or Cl.4 Inorganic mixed halides have recently been used to create various nanophotonic components due to their electroluminescence in the green5,6 to blue7 optical ranges. The broad tunability of halide perovskites has emerged as promising demonstrations for appealing solar cells, light-emitting diodes (LEDs), and laser applications, with the possibility of manipulating energy-efficient fluorescent lighting by replacement of the cations (MA, FA, and Cs) or halide components (I, Br, and Cl).8 Numerous density functional theory (DFT) calculations have been performed in recent years to investigate the structural, electronic, and optical properties of organic–inorganic perovskites.4,9−32 The local density approximation (LDA) method was used to conduct theoretical studies of organic–inorganic perovskites33 or Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA)34,35 as a result of their low computational cost.36 Because the obtained Eg values were much lower than the experiment values, the LDA and PBE-GGA potentials were unable to calculate the accurate Eg.34,35,37−39 Furthermore, the theoretical lattice parameters calculated with PBE-GGA outperformed the experimental lattice constants.36 The LDA potential typically underestimated the lattice constants, resulting in an underestimation of Eg.36 To overcome the significant shortcomings of the LDA and PBE-GGA potentials, our previous work demonstrated that the modified Becke Johnson (mBJ-GGA) potential is the most accurate method for studying the optoelectronic properties of CsPbBr3 perovskites.40 Because of its additional dependence on the kinetic energy density, the mBJ-GGA potential can be used to calculate Eg with good agreement with experimental values.41−44 Structural and electronic properties of mixed inorganic cubic symmetry at higher temperature were studied using DFT-based full-potential linear augmented plane wave (FP-LAPW) approach.23,45,46 Chen et al. studied the electronic band gaps for a 1 × 1 × 2 supercell of CsPb(I1–xClx)3 and CsPb(I1–xBrx)3 cubic phases at higher temperature.23,45,46 Castelli et al. investigated the trends over band gaps for 240 perovskites composed of organic–inorganic cations, Sn and Pb as B-ion, and halides as anions.23 The optoelectronic properties of mixed inorganic perovskites at the orthorhombic phase (Pnma) have not yet been investigated in detail, particularly the mixed halide from I to Br and Cl. It is advantageous to investigate the optoelectronic properties of mixed orthorhombic (Pnma) inorganic perovskites, which are available at room temperature and have applications in solar cells, LEDs, and lasers, using the most accurate DFT calculation methods.

In this study, the mBJ-GGA method without/with spin orbital coupling (SOC)47−50 was used to look into the impact of halide composition on the electronic and optical properties of mixed orthorhombic perovskites 1 × 1 × 2 CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 (x = 0.00, 0.25, 0.50, 0.75, and 1.00), while the PBE-GGA method was used to investigate their structural properties. The mBJ-GGA method demonstrated the evolution of band structure, optical absorption, and energy band gap (Eg) with increasing x in CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3. The lattice constants and Eg were computed and found to be consistent with previous research.10,11,14,20,21,27,28,31,43,51−60 Furthermore, the effective masses of charge carriers, absorption, optical dielectric, and reflectivity were precisely calculated.

2. Results and Discussion

2.1. Optimized Structures

At room temperature, CsPbX3 (X = I, Br, Cl) perovskites have orthorhombic structures with space group Pnma.17,61−71 Using VESTA software,72 a supercell 1 × 1 × 2 with 40 atoms was used to simulate CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 perovskites. Starting with an orthorhombic inorganic CsPbI3 structure, a supercell of CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 was created. The iodide was gradually replaced by an appropriate x concentration of Br, and Br was gradually replaced by an appropriate x concentration of Cl as shown in Figure 1. The structural properties of mixed-halide perovskites are investigated using first-principles DFT with the PBE-GGA method, which is implemented in the WIEN2k code.33,73 The structural information is presented in Tables S1–S9, Supporting Information. Figure 2a–i shows the fitting of total energy as a function of volume using the Murnaghan equation of state74 to compute the lattice constants of the perovskites. Table 1 displays the calculated lattice parameters (a, b, and c), unit-cell volume (V), pressure derivatives (B′), bulk modulus (B), and total energy (E) along with measured and previously predicted values for comparison. Our calculated data are roughly consistent with those measured and previously predicted values, indicating the reliability of the current computational scheme.15,16,61,62,67,75−79Figure S1 shows that as the concentration of Br and Cl increases, the volume of the unit-cell decreases.

Figure 1.

Figure 1

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Crystal structure of mixed-halide inorganic perovskites.

Figure 2.

Figure 2

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Variation in total energy with volume for (a) CsPbI3, (b) CsPbI2.25Br0.75, (c) CsPbI1.50Br1.50. (d) CsPbI0.75Br2.25, (e) CsPbBr3, (f) CsPbBr2.25Cl0.75, (g) CsPbBr1.50Cl1.50, (h) CsPbBr0.75Cl2.25, and (i) CsPbCl3 obtained using the PBE-GGA potential.

Table 1. Theoretical Lattice Parameters (a, b, and c), Unit-Cell Volume V (Å)3, Pressure Derivatives (B′), Bulk Modulus B (GPa), and Total Energy (E) for Mixed-Halide Perovskites. Note: The lattice Parameter, c has been doubled.

2.1.

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3.2. Electronic Properties

The electronic properties of mixed-halide perovskites are investigated using the mBJ-GGA method without/with SOC, which is implemented in the WIEN2k code.33,73 The electronic band structures without/with SOC follow the high-symmetry k-point path R → Γ → X → M → Γ as shown in Figure 3a–i. The SOC had a significant effect on the conduction band (CB) region, with a sharp reduction in the bottom of the CB,47,82 whereas there was no significant change in the valence band (VB)40 (see Supporting Information, Figure S2, SOC). The conduction band minimum (CBM) and valence band maximum (VBM) were found to be localized at the Γ point. Table 2 shows the Eg calculated with mBJ-GGA ranging from 1.983 eV for CsPbI3 to 3.325 eV for CsPbCl3, while Eg calculated with mBJ + SOC ranges from 1.066 eV for CsPbI3 to 2.182 eV for CsPbCl3, which is consistent with previous theoretical predictions.15,16,75,79,80,83−87 Because of its proper band gap of 1.983 eV, CsPbI3 is suitable for light-absorber applications, whereas CsPbBr3 and CsPbCl3 with band gaps of 2.420 and 3.325 eV, respectively, show promising application prospects in solar cells, LEDs, lasers, and photodetectors.88 Because the Eg values with SOC are small in comparison to the experimental results, the alloy formula was used to correct the Eg determined by the mBJ-GGA + SOC method14,89−92

3.2. 1

where ΔEg(A1–xBx) is the change in band gap for the mixed CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 perovskites, ΔEg(A) is the change in band gap for pure CsPbI3/CsPbBr3, and ΔEg(B) is the change in band gap for pure CsPbBr3/CsPbCl3. Table 2 shows that the corrected mBJ-GGA + SOC Eg ranges from 1.850 eV for CsPbI3 to 3.130 eV for CsPbCl3. Figure 3j shows the dramatic increase in Eg caused by replacing I with Br, followed by Cl. Electron (me*) and hole (mh) effective masses are important indicators of photovoltaic material transport properties.10 Lead, halides, and the symmetries of the perovskite structure all play important roles in determining the effective mass of electrons and holes. The CBM band edges were significantly flatter than the VBM, indicating that electrons had far more mass than holes.60 The effective masses, me* and mh, were calculated using the equation Inline graphic where m* is the effective mass of the electron or hole, i and j denote the reciprocal components, εn(k⇀) is the energy dispersion function of the nth band, and k⇀ represents the wave vector. The effective masses, me* and mh, without SOC ranged from 0.13579 to 0.23119m0 and from 0.07099 to 0.09354m0, respectively, and those with SOC ranged from 0.06019 to 0.08938m0 and from 0.06828 to 0.08354m0, respectively. As indicated by Figure 4, the calculated reduced effective mass82Inline graphic without/with SOC increased significantly with increasing Br and Cl concentration in CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 respectively. Table S10 in the Supporting Information contains additional information on the effective mass of the electron and hole and the reduced mass (μ) for mixed-halide perovskites.

Figure 3.

Figure 3

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Band structure of (a) CsPbI3, (b) CsPbI2.25Br0.75, (c) CsPbI1.50Br1.50, (d) CsPbI0.75Br2.25, (e) CsPbBr3, (f) CsPbBr2.25Cl0.75, (g) CsPbBr1.50Cl1.50, (h) CsPbBr0.75Cl2.25, and (i) CsPbCl3 obtained using mBJ-GGA potential without/with SOC. The band gap values versus the concentration of Br and then Cl calculated using mBJ, mBJ + SOC, and the corrected mBJ + SOC (j). The black dashed line represents the band structure calculated with mBJ + SOC. The VBM is set as zero.

Table 2. Band gap Eg (eV) Values for Mixed-Halide Inorganic Perovskites Compared With Previous Experimental and Theoretical Studies.

  this work
  
mixed-halide perovskites mBJ mBJ + SOC corrected mBJ + SOC other DFT(exp.)
CsPbI3 1.983 1.066 1.850 1.831,77,86 1.48,87 (1.75),88 (1.85)81
CsPbI2.25Br0.75 2.029 1.123 2.064 (2.010)81
CsPbI1.50Br1.50 2.112 1.151 2.079 1.93,86 (1.97),76 (2.17)81
CsPbI0.75Br2.25 2.281 1.343 2.276 (2.17),88 (2.23)81
CsPbBr3 2.420 1.482 2.480 2.32,90 2.40,86 (2.38),94 (2.479)81
CsPbBr2.25Cl0.75 2.623 1.602 2.593 (2.670)81
CsPbBr1.50Cl1.50 2.841 1.815 2.791 (2.720),76 (2.800)81
CsPbBr0.75Cl2.25 3.033 1.973 2.933 (2.940)81
CsPbCl3 3.325 2.182 3.130 3.05,10 (2.91),16 (2.78),95 (3.132)81
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Figure 4.

Figure 4

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Reduced effective masses μ for mixed inorganic perovskites.

3.3. Density of States

Figure 5 shows that the total density of states (TDOS) remained unchanged as the concentration (x) increased from 0.00 to 1.00 of Br and Cl in CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3 perovskites, but the edges were shifted up, where the VBM was shifted to 0 eV. The partial DOS (PDOS) for the mixed perovskite demonstrates that Cs+ has no effect on the VBM or CBM but only maintains overall load neutrality and structural stability11,14,28,40,76,93−97 as shown in Figure S3, Supporting Information. The VBM is primarily derived from the p orbitals of I, Br, and Cl, with contributions from the s orbitals of Pb. The CBM was formed mostly by the p states of Pb and minor contribution by the s and p states of I, Cl, and Br. An in-depth look at the band structure of CsPbI3, CsPbI1.50Br1.50, CsPbBr3, CsPbBr1.50Cl1.50, and CsPbCl3 with respect to PDOS is shown in Figure S4, Supporting Information.

Figure 5.

Figure 5

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TDOS of mixed-inorganic perovskites. Inset: TDOS in the range −0.8 to 3.75 eV.

3.4. Electron Density

To visualize the charge distribution and bonding nature of CsPbI3, CsPbI2.25Br0.75, CsPbI1.50Br1.50, CsPbI0.75Br2.25, CsPbBr3, CsPbBr2.25Cl0.75, CsPbBr1.50Cl1.50, CsPbBr0.75Cl2.25, and CsPbCl3 perovskites, the electron density distribution is investigated and presented in Figure 6. The atoms Cs, Pb, I, Br, and Cl have electronegativity values of 0.79, 2.33, 2.66, 2.96, and 3.16 on the Pauling scale, respectively. The difference in electronegativity (X1 – X2) is critical for describing the bonding character.98 The following equation is used to calculate the percentage of ionic character (IC) of the bonding obtained31,98,99

3.4. 2

Figure 6.

Figure 6

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Calculated electron density distribution of (a) CsPbI3, (b) CsPbI2.25Br0.75, (c) CsPbI1.50Br1.50, (d) CsPbI0.75Br2.25, (e) CsPbBr3, (f) CsPbBr2.25Cl0.75, (g) CsPbBr1.50Cl1.50, (h) CsPbBr0.75Cl2.25, and (i) CsPbCl3.

The electronegativities of the 1 and 2 atoms are represented by X1 and X2, respectively. The % IC of Cs–Br, Cs–I, and Cs–Cl was 69.18, 58.28, and 75.44, whereas for Pb–Br, Pb–I, and Pb–Cl, was 9.45, 2.69, and 15.82, respectively. The electron clouds surrounding Cs atoms are spherical and free of distortion, indicating that they are mostly ionic and partially covalent with the surrounding atoms.21,31,100 In contrast, the bond between Pb and I, Br, or Cl is mostly covalent and partially ionic, with electron clouds around these atoms distorted and overlapping significantly.

3.5. Optical Properties

We investigate the optical properties of mixed-halide perovskites, including dielectric function, refractive index, extinction coefficient, reflectivity, and optical absorption for energy ranging from 0 to 10 eV. The calculated dielectric functions ε1(ω) and ε2(ω) are shown in Figure 7a,b. The dielectric function describes how a material responds to incident photons as a function of energy. The real part of the dielectric function ε1(ω) value at zero frequency is known as the static frequency ε1(0), and it varies between 4.72 and 3.12 as shown in Figure 8. Figure 7b shows the behavior of the imaginary part of the dielectric function ε2(ω), where it represents the radiation absorbed by the compound,31,101 with main peaks between 3.42 and 6.68 eV. It is worth noting that ε2(ω) has a zero value until absorption begins after the photon energy reaches the band gap energy, which establishes the threshold for a direct optical transition between the VBM and the CBM.

Figure 7.

Figure 7

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(a) Real dielectric function ε1(ω) and (b) imaginary dielectric function ε2(ω) of mixed-halide perovskites.

Figure 8.

Figure 8

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Static dielectric constant ε1(0), refractive index n(0), and reflectivity R(0)% of mixed-halide perovskites with different dopant concentrations x.

The refractive index n(ω) and extinction coefficient k(ω) were calculated, as shown in Figure S5a,b, Supporting Information. n(ω) is a critical feature of semiconductors that indicates how much light is bent or refracted.101 The value of n(ω) increases as the energy increases up to 2.87 and 2.14 for CsPbI3 and CsPbCl3, and then it begins to decrease to 1.27 and 1.44 showing a nonlinear behavior as shown in Figure S5a. For CsPbI3, CsPbBr3, and CsPbCl3, the calculated n(0) values were 2.17, 1.95, and 1.77 which agree well with the previous theoretical and experimental values31,55,102 as shown in Figure 8. Figure S5b shows that k(ω) is proportional to Br/Cl concentration, similar to ε2(ω), with the local maximum occurring between 3.89 and 6.72 eV when moving from CsPbI3 to CsPbCl3.

The calculated reflectivity R(ω) in relation to incident energy is shown in Figure S6, Supporting Information. The mixed perovskites behaved as semiconductors; the value of R(ω) was not unity at zero energy.103 At zero frequency, CsPbI3 has a static reflectivity R(0) value of 13.7%, which then decreases to 10.4 and 7.7% for CsPbBr3 and CsPbCl3, respectively. When moving from CsPbI3 to CsPbBr3 and CsPbCl3, the maximum R(ω) occurs between 3.22 and 4.71 eV, and it begins to fluctuate and decrease at higher energies. As shown in Figure 8, the calculated R(0) at zero energy was approximately 13.65, 12.78, 12.42, 11.01, 10.43, 9.66, 8.42, 8.33, and 7.67% for mixed perovskite when transitioning from CsPbI3 to CsPbBr3 and CsPbCl3.

Figure 9a exhibits the absorption coefficient, α(ω), as a function of the energy. α(ω) peaks shifted to higher energies with increasing Br and Cl concentrations in CsPb(I1–xBrx)3 and CsPb(Br1–xClx)3. The wide absorption coefficient range from visible to ultraviolet indicates that they are useful for a variety of optical and optoelectronic applications.31Figure 9b shows the optical conductivity σ(ω) characteristics, which are analogous to α(ω) and provide information on how external parameters affect the electronic structure. The optical properties of the studied perovskite were consistent with those previously measured and reported.11,31

Figure 9.

Figure 9

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(a) Calculated absorption spectra α(ω) and (b) conductivity σ(ω) of mixed-halide perovskites with different dopant concentrations x.

3. Conclusions

In this study, the influence of halide composition on the optoelectronic properties of mixed-halide perovskites was investigated. The structural properties were calculated using the PBE-GGA method, and the lattice parameters are well comparable to previous experimental and theoretical work. When iodide (I) was replaced with bromide (Br) and then chloride (Cl), the unit-cell volume decreased linearly. The calculated band gap (Eg) using the mBJ-GGA method is in good agreement with the experimental values reported, and it increased clearly from 1.983 eV for CsPbI3 to 2.420 and 3.325 eV for CsPbBr3 and CsPbCl3, respectively, due to the increase in electronegativity of Br and Cl. Because the Eg values with mBJ + SOC are small compared to the experimental results, the alloy formula was used to correct the Eg values. The corrected mBJ + SOC Eg value is 1.850 eV for CsPbI3 and 2.480 and 3.130 eV for CsPbBr3 and CsPbCl3, respectively. The reduced masses (μ) are correlated with the energies of Eg, VBM, and CBM. When moving from CsPbI3 to CsPbBr3 and CsPbCl3, μ ranges from 0.046618mo to 0.066595mo without SOC and from 0.031569mo to 0.043181mo with SOC. As the Br and Cl content increases, the calculated photoabsorption coefficients show a blue shift in the absorption coefficient. According to the calculations, these perovskites can be used in solar cells, LEDs, and laser applications.

3.1. Computational Method

The FP-LAPW method within the framework of DFT as implemented in the WIEN2k code was used to optimize the structure of the mixed inorganic perovskites.33,73 The PBE-GGA method was used to calculate the structural properties of the mixed perovskites. The mixed inorganic perovskite structures were created by building a 1 × 1 × 2 supercell with a binary perovskite’s Pnma space group. Because of the presence of a heavy element (Pb) in the structure, the SOC interaction47−50,104,105 was used with mBJ-GGA, as described in our previous work.40 To match the experimental values, the calculated band gap with SOC was corrected by using the alloy formula.14,89,90 During the calculation, RKmax = 9 and k-points = 100 were used, and the total energy was converged to 10–4 Ry.

Acknowledgments

The authors thank the Deanship of Scientific Research at King Saud University for funding this work through Research Group No. RG-1440-038.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c04806.

  • Structural parameters; unit-cell volume versus Br and Cl contents; SOC; energy level splitting diagram for the orthorhombic phase of inorganic perovskites; PDOS of mixed inorganic perovskites; band structures; calculated effective mass of the electron and hole and reduced mass for mixed-halide perovskites; and calculated refraction indices and extinction coefficients and reflectivity spectra of mixed-halide perovskites with different dopant concentrations (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao1c04806_si_001.pdf (1.1MB, pdf)

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