We propose a two-stage multi-objective optimization framework for full scheme solar cell structure design and characterization, cost minimization and quantum efficiency maximization. We evaluated structures of 15 different cell designs simulated by varying material types and photodiode doping strategies. At first, non-dominated sorting genetic algorithm~II (NSGA-II) produced Pareto-optimal-solutions sets for respective cell designs. Then, on investigating quantum efficiencies of all cell designs produced by NSGA-II, we applied a new multi-objective optimization algorithm~II (OptIA-II) to discover the Pareto fronts of select (three) best cell designs. Our designed OptIA-II algorithm improved the quantum efficiencies of all select cell designs and reduced their fabrication costs. We observed that the cell design comprising an optimally doped zinc-oxide-based transparent conductive oxide (TCO) layer and rough silver back reflector (BR) offered a quantum efficiency ($Q_e$) of $0.6031.$ Overall, this paper provides a full characterization of cell structure designs. It derives a relationship between quantum efficiency, $Q_e$ of a cell with its TCO layers doping methods and TCO and BR layers material types. Our solar cells design characterization enables us to perform a cost-benefit analysis of solar cells usage in real-world applications
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