2 Calculations Combined with a Statistical

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Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 4245−4251

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Relativistic DFT-1/2 Calculations Combined with a Statistical Approach for Electronic and Optical Properties of Mixed Metal Hybrid Perovskites Diego Guedes-Sobrinho,*,†,‡ Ivan Guilhon,*,‡ Marcelo Marques,*,‡ and Lara K. Teles*,‡ †

São Carlos Institute of Physics, University of São Paulo, P.O. Box 369, 13560-970 São Carlos, SP, Brazil Grupo de Materiais Semicondutores e Nanotecnologia, Instituto Tecnológico de Aeronáutica, DCTA, 12228-900 São José dos Campos, São Paulo, Brazil

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‡

S Supporting Information *

ABSTRACT: We overcome the great theoretical computational challenge of mixed perovskites, providing a rigorous and efficient model by including quasiparticle, spin−orbit coupling, and disorder effects. As a benchmark, we consider the mixed MAPb1−xSnxI3 perovskites. The calculations are based on the generalized quasichemical approach and the DFT-1/2 approximated quasiparticle correction. Both cubic and tetragonal structures are investigated. By mapping the entire range of compositions, we correctly describe the bowing-like behavior for the energy gaps with 1.24 eV as the minimum value at x = 0.70, in very good agreement with the experimental data. Furthermore, while the tetragonal alloy reaches the maximum absorbance with a limit for the red shift at x = 1.0, the cubic alloy sets a maximum absorbance/red shift for the optimal composition at x = 0.70.

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optical properties.14−16 As such, the theoretical prediction of electronic properties of MHP alloys requires that we overcome an additional challenge, that is the bowing-like curve with respect to the composition for the energy gap of MHP alloys. The challenge of rigorous first-principles calculation of a MHP mixture is threefold: (i) the inclusion quasiparticle (QP) effects, (ii) strong spin−orbit coupling (SOC), and (iii) disorder effects due to alloying. Therefore, standard density functional theory (DFT) calculations are incapable of providing reliable electronic and optical properties across the range of compositions.17−20 An alternative would be the relativistic GW approximation, which is accurate for electronic structure description,21,22 but its application in a statistical approach is infeasible due to the extremely high computational cost. In this sense, an efficient and accurate theoretical approach for obtaining the electronic and optical properties is highly desirable and could open an avenue for a deeper understanding of mixed MHP, such as MAPb1−xSnxI3. In this Letter, we present a protocol for calculating accurate band gap energies and absorption coefficients for perovskite alloys throughout the entire composition range, which is also applied by regarding different symmetries for the structure, e.g., cubic and tetragonal. Our framework combines a computationally efficient first-principles method based on the DFT-1/2 method for approximated quasiparticle corrections,

etal halide perovskites (MHP) as photovoltaic materials, e.g., the methylammonium (MA) lead iodide, CH3NH3PbI3 (MAPbI3), with efficiencies in excess of 22%, have attracted a significant amount of interest in the photovoltaic solar cell community.1−3 Replacing or complementing Si-based devices with MHP offers intriguing possibilities with respect to higher energy conversion efficiencies and lower processing costs.4,5 However, significant additional effort has been spent on finding alternatives for toxic Pb by preserving the material stability against moisture and air, among others.6−8 An alternative is to replace Pb with other nontoxic or less toxic cations. Among the possible candidates, Sn is one of the most viable, because it provides fewer lattice distortions. Moreover, the mixed MAPb1−xSnxI3 perovskites have been extensively investigated to extend the lightharvesting region. Experiments have revealed the stabilization of MHP mainly as tetragonal (I4/mcm) and pseudocubic (P4mm) structures, so that the increase in symmetry from I4/mcm to P4mm (second-order phase transition)9,10 is correlated with the increase in temperature. For instance, MAPbI3 is stable as tetragonal (lattice parameters given by a = 8.84 Å, b = 12.58 Å, and c = 8.55 Å) between 163 and 328 K,11,12 moving toward pseudocubic (a = 6.31 Å) for temperatures above 328 K.13 Nevertheless, for MHP alloys, e.g., the Pb/Sn mixture, the structural motif is also dependent on the composition. For MAPb1−xSnxI3 over a few compositions, experiments have revealed the I4/mcm to P4mm phase transition for x > 0.5, promoting a deviation from linearity of its electronic and © XXXX American Chemical Society

Received: May 24, 2019 Accepted: July 13, 2019 Published: July 13, 2019 4245

DOI: 10.1021/acs.jpclett.9b01499 J. Phys. Chem. Lett. 2019, 10, 4245−4251

Letter

The Journal of Physical Chemistry Letters including SOC, with a statistical treatment named generalized quasichemical approximation (GQCA), which describes the statistical ensemble of a solid solution of MHP alloys with a cluster expansion approach.23 This approach yielded electronic and optical parameters in excellent agreement with those from experiments with MAPb 1−x Snx I3 alloys for the whole composition range and an arbitrary growth temperature. All of the total energy calculations and structure relaxation were DFT-based24,25 within the Perdew−Burke−Ernzerhof (PBE)26 approximation for the electronic exchange-correlation functional (Exc). We also considered SOC for the valence electrons, which is an important relativistic treatment for nonspherical atomic orbitals in Pb-based perovskites. Thus, SOC affects the directionality of the bonds,27 and yields the Rashba−Dresselhaus splitting in band edges.17,28,29 We used the VASP30,31 code to solve the Kohn−Sham (KS) equations, from which a plane-wave cutoff energy of 500 eV was used, and to integrate over the Brillouin zone was considered a Monkhorst−Pack k-mesh of 4 × 4 × 4 (3 × 6 × 4) for a cubic structure. Shape, lattice, and atomic positions were fully optimized via minimization of Hellmann−Feynman forces on every atom, for values of 0.5. In view of the electronic properties of the MAPb1−xSnxI3 alloys determined via the DFT-1/2+SOC approach, we turn to the average optical absorption within our statistical treatment at some compositions aiming to observe its correspondence with the lowest Eg(x). Figure 5a shows the absorbance spectra

Figure 4. Energy gaps (solid lines) and their standard deviation (shaded regions) calculated within the GQCA statistical approach for the MAPb1−xSnxI3 alloy as cubic (tetragonal), employing the standard (a) DFT and (b) SOC-DFT-1/2 approaches. The empty diamonds are the experimental values (diamond from ref 14 and both square and circle from refs 15 and 16) varying as bowing-like for x = 0.0, 0.25, 0.50, 0.75, and 1.0 compositions.

DFT also fails in describing the empirically observed bowing, from which the minimum of the energy gap lies in the window Eg(min) = 1.17−1.23 eV for x = 0.75.14−16 Conversely, the relativistic interaction through the DFT+SOC enlarges the difference between the average Eg values for the cubic and tetragonal structures (Figure S4a), reproducing the bowing shape for the Eg curves. However, DFT+SOC strongly reduces the Eg values over a range of compositions, such that the magnitude of the energy gap is very far from the experimental results. Via application of the quasiparticle correction method, here as DFT-1/2+SOC in Figure 4b, average Eg values for the MAPb1−xSnxI3 alloys, connecting the MAPbI3 and MASnI3 pure perovskites (from which the accuracy has already been discussed in Figure 1), are in good agreement with the experimental values. DFT-1/2+SOC describes the bowing observed by the experiments, wherein we find a minimum Eg(min) of 1.24 ± 0.21 eV, i.e., average and standard deviation (shaded regions in Figure 4), through the GQCA approach for the cubic structure. This result lies at x = 0.70, differing between 0.81 and 5.98% in excellent agreement with the experimental result (1.17−1.23 eV) at x = 0.75.14−16 On the basis of that, we estimated a bowing parameter (b) for the energy gap dependency with composition as Eg(x) = xEg,MASnI3 + (1 − x)Eg,MAPbI3 − bx(1 − x). We found for the cubic (tetragonal) form a bowing value of b = 0.69 eV (0.54 eV), from which the error does not exceed 0.007 eV (0.006 eV). Therefore, the relativistic and quasiparticle approach to our statistical method shows an excellent agreement with empirical

Figure 5. Average optical absorbance for MAPb1−xSnxI3 alloys in (a) cubic and (b) tetragonal forms for the x = 0.0−1.0 composition range within the GQCA statistical approach at 300 K. The small charts expand the photon energy region between 1.2 and 1.7 eV, which permit the observation of the maximum red shift for the cubic form at x = 0.7 (due to the minimum energy gap from the bowing-like curve) and the tetragonal form at x = 1.0 (due to the gradual increase in the optical absorption from x = 0.0 to x = 1.0).

for the cubic structure, where notably one observes a maximum red shift for 0.5 < x < 0.8 compositions in close correlation with their correspondent lowest Eg(x) values. Our optical absorption results computed via DFT-1/2+SOC are in agreement with empirical investigations14−16 through transformed Kubelka−Munk treatment from near-infrared absorption spectra. Because the lowest Eg(x) values from the bowinglike curve are closer to the x = 1.0 value for the tetragonal structure (panel b), the maximum red shift lies on the solid solution compositions with a large excess of Sn, nearly as the MASnI3 pure perovskite. Finally, to provide an overview of all of the DFT-1/2+SOC results for the statistical ensemble of the MAPb1−xSnxI3 alloys, we correlate the energy gap, lattice parameter, and total absorbance average in Figure 6 as a complete and accurate representation of the systems. The cubic structure, panel a, depicts the coincident maximum red shift with the minimum energy gap from the bowing-like curve (Eg = 1.24 eV), from which the lattice constant around 6.31 Å correlates with the x 4248

DOI: 10.1021/acs.jpclett.9b01499 J. Phys. Chem. Lett. 2019, 10, 4245−4251

Letter

The Journal of Physical Chemistry Letters

skite alloy at x = 0.70, i.e., composition with the smallest amount of Pb presenting the best photovoltaic and optical performance, suggests the importance of seeking eco-friendly materials from the reduction of toxic chemical species. Therefore, our protocol for mixed perovskites can assist theorists and experimentalists dedicated to the design of mixed perovskites immersed in solar cells aiming to achieve high photovoltaic performance. The approach presented here can be directly applied to similar complex systems, such as 3D and 2D mixed metal halide perovskites.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01499.

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Additional results and methodological details about GQCA (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: *E-mail: *E-mail: *E-mail:

Figure 6. Correlation between the energy gap and the total average for the lattice constant calculated within GQCA at 300 K for the MAPb1−xSnxI3 alloy in the (a) cubic and (b) tetragonal structures. The color map corresponds to the total absorbance average obtained from GQCA for the photon energy interval of 0.0−3.5 eV.

[email protected]. [email protected]. [email protected]. [email protected].

ORCID

Diego Guedes-Sobrinho: 0000-0002-3313-2822 Ivan Guilhon: 0000-0001-8882-4490 Lara K. Teles: 0000-0002-8713-2094

= 0.70 composition (as shown in Figure 3). For the tetragonal form, even though its energy gap has a minimum at 8.60 Å for the lattice constant (correlated with the x = 0.90 composition), one observes that its spectral absorption increases gradually from pure MAPbI3 to MASnI3 (point of the maximum red shift). In summary, we performed a systematic and rigorous investigation via first-principles calculations of the MAPb1−xSnxI3 perovskite alloys within the DFT-1/2 approximated quasiparticle correction, considering the spin−orbit coupling and addressing the disorder effects with a cluster expansion approach. The spin−orbit coupling is mandatory for the correct description of the bowing parameter of the energy gap with respect to the composition and for distinguishing the electronic properties of cubic and tetragonal structures. In MHP, the spin−orbit coupling is very strong, resulting in a significantly smaller energy band gap. Therefore, the inclusion of QP correction is indispensable for a proper description of the electronic band structure. The DFT-1/2+SOC approach provides precise results for energy band gaps of the pure compound and mixed compositions, as well as its bowing-like behavior, wherein the minimum of the cubic configurations at x = 0.70 under 300 K is in excellent agreement with experimental findings. Our computed minimum energy gap of 1.24 eV differs by only 0.81% from the experimental value. This allows the calculation of reliable composition-dependent absorption spectra. We identify the correlation among the lattice parameter, energy gaps, and the spectral absorption over the entire composition range. Our results indicate for the cubic MAPb1−xSnxI mixed perovskite at x = 0.70 (optimal composition for a minimum energy gap) a maximum optical absorption/red shift, which is a crucial parameter for photovoltaic performance. As such, the MAPb1−xSnxI3 perov-

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Brazilian funding agency Coordination for Improvement of Higher Level Education (CAPES) (PVE Grants 88881.068355/2014-01 and 88887.145962/2017-00), the National Council for Scientific and Technological Development (CNPq) (Grants 308742/2016-8 and 306322/ 2017-0), and the São Paulo Research Foundation (FAPESP) (Grants 2006/05858-0 and 2012/50738-3) for financial support. The authors also thank the Scientific Computation National Laboratory for providing Santos Dumont Supercomputer resources for the performance of some of the calculations.

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DOI: 10.1021/acs.jpclett.9b01499 J. Phys. Chem. Lett. 2019, 10, 4245−4251