## 1 Introduction  
Solar panels with more than 40% power conversion efficiency in the real world will change our society, including running most electric vehicles on solar energy [1]. The potential for conversion efficiency of solar cells is one of the most popular research topics in photovoltaic science and has been studied intensively by many people, suggesting a bright future for the potential of photovoltaic energy conversion [2; 3; 4]. This is based on solid scientific background with ideal but trustworthy preconditions. However, the materials and processes in the real world are not ideal, and the record efficiency valuesfor photovoltaic cells are less than the values predicted by the scientists, for example, 28.5 % for Si solar cells [2; 5; 6]. For example, [PERSON] et al. predicted more than 45% efficiency in field concentrator solar cells intensively studied for the application of CPV (concentrator photovoltaic) [2], but the highest efficiency ever achieved was 44.2% in 2013 by the Sharp Corporation [5; 6]. More recently, research based on the practical limit of material improvement to various materials like Si, III-V, II-VI thin films, organic, and Perovskite, as well as various configurations such as quantum dots, hetero-junction, and multi-junction, has been published [7; 8; 9; 10; 11]. These kinds of efficiency-limit studies tend to present a decreasing record number following the improvement of the model, namely by increasing constraints and taking into account inherent limitations (which are small but non-negligible). However, taking the example of energy conversion efficiency, namely efficiency of conversion from sunlight (ASTM G173 AM1.5G standard solar spectrum) to electricity power, the highest-efficiency solar cells are a group of multi-junction cells [1; 5; 6; 7]. The principles of multi-junction cells were suggested by [PERSON] in 1955 [12], and [PERSON] et al. investigated them in 1960 [13]. However, the efficiency of multi-junction cells did not make significant progress until 1975 because of inadequate thin-film fabrication technologies. The liquid-phase and vapor-phase epitaxy brought AlGaAs/GaAs multi-junction cells into the 1980s, with tunnel junctions by [PERSON] et al. [14] and metal interconnections by [PERSON] et al. [15], [PERSON] [16] and [PERSON] et al. [17]. [PERSON] et al. predicted efficiency of close to 30% at that time [18], but this was not achieved because of difficulties with high-performance, stable tunnel junctions [19] as well as oxygen-related defects in the AlGaAs at that time [20]. [PERSON] et al. developed high-performance, stable tunnel junctions with a double-hetero (DH) structure [21]. [PERSON] et al. introduced InGaP for the top cell [22], and [PERSON] et al. achieved a 29.5% efficiency by a 0.25 cm\({}^{2}\) GaInP/GaAs multi-junction cell [23]. Recently, 37.9% efficiency and 38.8% efficiencies have been achieved with InGaP/GaAs/InGaAs 3-junction cell by Sharp [24] and with a 5-junction cell by Spectrolab [25]. Historically, high-efficiency multi-junction cells have been used for concentrator photovoltaic (CPV) cells. The energy conversion efficiency substantially increases under concentration operation [26]. Significant cost reduction was predicted in the 1960s [27]. The Wisconsin Solar Energy Center investigated the performance of solar cells under concentrated sunlight [27]. Research and development programs under the DOE (US Department of Energy), EC (European Commission), and NEDO (New Energy and Industrial Technology Development Organization, Japan) realized high conversion efficiencies using the CPV module and system. A 44.4% efficiency was demonstrated with the InGaP/GaAs/InGaAs 3-junction concentrator solar cell by Sharp [24]. The CPV system increased its installation in dry areas of the world after 2008. By 2017, total installation in the world reached 400 MW [28].