1. Introduction:

In recent years the cost-effective perovskite solar cells have gained tremendous attention from the academic community due to its outstanding progress in performance since its inception (NREL, 2020). Structurally, perovskite is defined with the stoichiometry of ABX₃, where A and B are cations and X is an anion. A majority of perovskite solar cells reported so far typically consist of the following combination, methylammonium (MA), formamidinium (FA) or cesium (Cs) or a combination of the three at the A site, lead (Pb) at the B site and iodine (I) or bromine (Br) or a combination of the two at the X site [(Cs/MA/FA)Pb(I_xBr_y)]. A major drawback preventing the global commercialization of perovskite solar cells is the poor long-term stability of the device in severe environmental conditions. Hence, continuous efforts are being made to develop devices with better stability by improvisations to the composition of the cell and its structural design. However, the study of potential alternatives for sites A, B, and X in the device structure is very limited.

2. Objectives:

The goal of this research is to design cost-effective perovskites by exploring potential analogs for sites A, B and, X in the ABX₃ chemical structure while maintaining their stability. The specific objectives to achieve this research goal are: (1) Collect existing information through literature review (2) List potential options for A, B, and X sites (3) Formulate an optimal synthesis problem that minimizes cost and maintains stability (4) Solve using mixed-integer linear programming, and (5) Find the ranked list of best combinations.

3. Methodology:

We propose a methodology that can select an optimal combination of different ions for the perovskite solar cell. Stability features that include, the tolerance factor (t) and octahedral factor (µ ) proposed by Goldschmidt provide guidelines for selecting ions for perovskite solar cells (Green, Ho-Baillie, and Snaith 2014).

For an idealized solid sphere model,

t = R_A+R_X/(2)^1/2(R_B+R_X) (1) and

µ = R_B/R_X (2)

where R_A: Ionic radii of cation A

R_B: Ionic radii of cation B

R_X: Ionic radii of anion X.

A base case is developed by selecting three potential candidates that are deemed most interesting to study each for site A, B, and X. We have an optimization problem with an objective function to select the possible combination of ions for the perovskite cell based on the stability factors and simultaneously minimize the material cost, which is considered a significant criterion for large scale commercialization of perovskite solar cells. The objective function was formulated and solved through General Algebraic Modeling Systems (GAMS) and MATLAB. The solution provides the optimum perovskite crystal structure with minimum cost. The motivational case was further extended to a large set of combinations (over 2000). A systematic representation of our evaluation framework is illustrated in Figure 1.

Summary:

The results indicate that the optimal perovskite crystal structure is ammonium-magnesium-formate with a cost of $6.33, while the percentage variation in cost from the first-best combination to the second-best is 24.7%. Thus, an optimization problem that attempts to find an optimal perovskite based on Goldschmidt stability factors and simultaneously minimizes cost is solved successfully. The mathematical model presented in this study can provide an excellent starting point for exploring the undiscovered combination of materials for perovskite solar cell synthesis.

References

Green, Martin A., Anita Ho-Baillie, and Henry J. Snaith. 2014. “The Emergence of Perovskite Solar Cells.” Nature Photonics 8 (7): 506–14. https://doi.org/10.1038/nphoton.2014.134.

Hoye, Robert L. Z., Philip Schulz, Laura T. Schelhas, Aaron M. Holder, Kevin H. Stone, John D. Perkins, Derek Vigil-Fowler, et al. 2017. “Perovskite-Inspired Photovoltaic Materials: Toward Best Practices in Materials Characterization and Calculations.” Chemistry of Materials 29 (5): 1964–88. https://doi.org/10.1021/acs.chemmater.6b03852.

Kieslich, Gregor, Shijing Sun, and Anthony K. Cheetham. 2014. “Solid-State Principles Applied to Organic–Inorganic Perovskites: New Tricks for an Old Dog.” Chemical Science 5 (12): 4712–15. https://doi.org/10.1039/C4SC02211D.

Kieslich, G., Sun, S., K. Cheetham, A., 2015. “An Extended Tolerance Factor Approach for Organic–Inorganic Perovskites.” Chemical Science 6 (6): 3430–33. https://doi.org/10.1039/C5SC00961H.

National Renewable Energy Laboratory (NREL) (2020) https://www.nrel.gov/pv/cell-efficiency.html (accessed 02.12.20).