Practical limits of multijunction solar cells

Multijunction solar cells offer a path to very high conversion efficiency, exceeding 60% in theory. Under ideal conditions, efficiency increases monotonically with the number of junctions. In this study, we explore technical and economic mechanisms acting on tandem solar cells. We find that these mechanisms produce limitations that are the more pronounced the greater the number of junction is and, hence, limit the ideal number of junctions, as well as the corresponding efficiencies. Spectral variations induce current losses in series‐connected tandem solar cells. For Denver, we find that these losses reduce achievable harvesting efficiencies to 51% for non‐concentrated light and that they restrict the ideal number of junctions to less than nine. Independently operated solar cells suffer from optical losses with similar consequences. Even high optical efficiencies of 99% restrict the ideal number of junctions to below 10 and reduce achievable efficiencies by more than 10%. Only architectures with a sequential cell illumination are more resilient to these losses. Restricting available materials reveals that a sufficiently low band gap for the bottom cell of 0.9 eV or below is expedient to realize high efficiencies. Economic considerations show that five junctions or less are economically ideal for most conceivable applications.


| INTRODUCTION
Multijunction solar cells, in the following also referred to as tandems, combine absorbers with different band gaps to reduce two principle loss mechanisms occurring in single junction solar cells: thermalization and sub-band gap losses. 1 Increasing the number of junctions towards infinity monotonically increases the detailed balance efficiency limit to more than 65% without concentration and more than 85% with concentration 2 -about twice the limit of a single junction solar cell. 3 Yet, these values are obtained from calculations that use very few constraints. Neither optics nor interconnections are considered, the solar cells are illuminated with one unwavering spectrum at one given intensity and they operate at one set temperature. Such conditions can be approached in the lab, with some effort, but they cannot be realized in the field. Introducing constraints results in the formation of losses, and these losses, for fundamental reasons, affect tandems with different numbers of junctions differently. In this study, we explore a few of these constraints, and we show that they result in the formation of a finite optimum number of junctions exhibiting the greatest limiting efficiency. For the definition of the included constraints, it is useful to distinguish between different architectures of tandem solar cells and different interconnection schemes. In the next section, we introduce four different tandem architectures. We then explore limiting efficiencies without and with constraints due to available materials, variation in spectrum and imperfect optics. In a final investigation, we explore economic constraints using a simple bottom-up cost model for a perovskite-based multijunction solar cell.

| TANDEM ARCHITECTURES
In this study, we distinguish four types of tandem architectures based on the optical mechanism used for spectral splitting. For each architecture, there are several possibilities of electrical integration, which we will discuss later. The four architectures are displayed in Figure 1.
The first architecture ( Figure 1A) is termed stacking. In this architecture, cells are stacked on top of each other in order of decreasing band gap. Spectral splitting is achieved via selective absorption in the semiconductor absorbers; no additional optical components are required. This architecture is the most commonly realized tandem with achieved efficiencies of above 47% 4,5 under concentration and above 39% 4,6 without concentration. The second architecture is termed optical splitting ( Figure 1B). In this architecture, spectral selection is achieved via an optical element like a grating, 7,8 a prism 9 or a holographic element. 10,11 The optical element splits white light spatially into different wavelengths. Cells can be placed underneath such that each cell receives the part of the spectrum most suitable for it. The use of an optical element allows decoupling the aperture area from the cell area, which can be used for concentration or de-concentration. The highest efficiency reported for this configuration is 42.7% 9 for separately operated solar cells. A variation of this concept is the stacked fluorescent concentrator, 12 in which light is split by selective absorption and emission and is spatially separated via total internal reflection.
The third architecture utilizes spatial randomization and spectral selection via filters, and it is termed randomized spectral selection ( Figure 1C). Light falls through a small aperture into a body covered F I G U R E 1 Different tandem architectures. (A) Cells with decreasing band gaps are stacked on top of each other. Each cell acts as a filter for subsequent cells and absorbs the part of the spectrum in which it is optically active. (B) Cells are arranged sequentially, typically ordered by band gap. An optical element splits the spectrum such that each cell is illuminated with the part of the spectrum in which it is active. (C) Cells are equipped with selective filters that are transmitting the spectral range in which the cell is active and reflect all other light. Cells are arranged such that light reaches each cell after a series of scattering and reflection events. (D) Cells are equipped with selective filters but are arranged geometrically that light is guided from one cell to the next.

| EFFICIENCY LIMITS WITH BAND GAP CONSTRAINTS
A first limitation explored here concerns the availability of materials with suitable band gaps. In practice, the range of band gaps that can be used for tandem solar cells will be restricted, either for fundamental or for practical reasons. One example for such restrictions are III-V materials. In a lattice-matched configuration, using a germanium substrate and GaAs, AlAs combinations, a band gap range between 0.6 eV and 2.12 eV is available. This range can be expanded by adopting a technique referred to as lattice-mismatched- 19,20 or metamorphic 19 growth, or by direct wafer bonding. 21

| SERIES CONNECTION AND SPECTRAL VARIATIONS
As mentioned, all tandem architectures shown in Figure 1 can be realized with a series connection of the used cells, either through circuit interconnection or through monolithic integration. Series connection entails a current limitation and design of a series connected tandem will aim for current matching to maximize efficiency. In outdoor operation, the spectrum changes over the course of a day and with seasons 26 due to variations of air mass, atmospheric composition and albedo scattering. 27 The list of papers discussing spectral effects for tandem (and single junction) solar cells is too long for this bibliography. A few noteworthy papers are provided. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] Deviations from the design spectrum, typically AM1.5g, result in some junctions producing greater currents and other junctions producing smaller currents. The greater the number of junctions, the higher the probability that one junction will produce a significantly lower current than under reference conditions and will, hence, For all cases, Denver and Singapore, band gaps optimized or not, mismatch losses increase with the number of junctions. A consequence of these losses is that efficiency no longer increases monotonically with the number of junctions, but converges and even decreases, leading to the formation of a maximum. In Denver, the maximum emerges for eight junctions with harvesting efficiencies of 53% and 70% for concentrated light. In Singapore, the maximum is very flat. After nine junctions, no further significant increase appears, and after 15 junctions, a decrease is notable.
Maximum harvesting efficiencies are 58% and 74% for concentrated light.
In the radiative limit, temperature does not affect the ideal combi-

| INDEPENDENT OPERATION AND OPTICAL EFFICIENCY
Current matching is no concern if cells are operated and contacted individually. Yet, independent operation requires the introduction of optically active elements. For the architectures shown in Figure 1B-D, these elements are the optical splitter and filters. Yet even in the stacking architecture, contacts and a transparent insulating layer are required to separate cells electrically in a 2N-terminal configuration. These optical elements will never work with perfect optical efficiency. Optical losses will affect tandem efficiency and will be greater for tandems with a greater number of junctions. In the following, we discuss how optical losses will affect the different tandem architectures:

| Splitting
Optical splitting requires an element to distribute the spectrum onto the various solar cells ( Figure 1B). The efficiency of this optical element is defined by how accurately the spectrum is distributed spatially, which can be determined by the divergence of the transmitted beam. This divergence will create an error that will affect each junction. Consequently, losses will increase with the number of junctions:   Figure 2A. For Singapore, two calculations were carried out, one for the band gap combination in Figure 2A (solid lines) and one for a band gap combination that was optimized for the spectral conditions in Singapore (dotted lines). More information in Appendix S1.

| Randomized spectral selection
In this configuration, there are three contributions to optical efficiency: reflectance of the side walls, unwanted reflectance of the selective filter in the range where it should be translucent and unwanted transmission through the filter were it reflects. For simplicity, we use the same optical efficiency value for all these mechanisms, which could also be viewed as a lumped efficiency. Absorption on the sidewalls reduces light intensity overall; reflection of the filters prevents light from entering the solar cell. The optical efficiency in this configuration is calculated iteratively. In each step holds In these equations, R W,i is the weighted reflectance of the walls during the ith path (R W,0 ¼ 1Þ, R C,i is the weighted reflectance of the light on the area covered with the targeted cell R C,0 ¼ 1 ð Þand η C,i is the absorptance of light by the targeted on the ith path, A C is the total area covered with solar cells and A is the total inner surface of the sphere. In the calculated example ( Figure 5, centre), half the body is covered with cells ( AC A ¼ 0:5).

| Geometric selection/stacking
In these configurations ( Figure 1A,D), light encounters multiple optical elements or layers on its path to the different solar cells. In the stacking architecture, optical efficiency is determined by the transmission through each junction, in the geometric selection architecture by transmittance and reflectance through the filter. Optical losses increase exponentially as the light beam interacts with more optical elements on its path: Losses are calculated separately for each junction, and efficiencies are summed up in the end. For the geometric selection architecture, it may be possible to reduce losses by realigning band gaps. We have not succeeded in finding a combination that increases the efficiency, though.
Note that in all cases, optical losses affect primarily current, though a reduced current also results in a slight reduction of the produced voltage. This change in voltage was neglected in all calculations and results in a small overestimation of the shown efficiencies. Figure 5A plots the radiative efficiency limit as a function of the number of junctions N for a single-path optical efficiency η o ¼ 0:99 for concentrated and non-concentrated light. Figure  Exemplarily, this is shown for a double junction perovskite solar cell in F I G U R E 5 Limiting efficiency for the different configurations with separate cell connections in Figure 1. Figure 5A shows the limiting efficiency for an optical efficiency of 99%, Figure 5B shows the ideal number of junctions as a function of optical efficiency. All shown results utilize the band-gap combination shown in Figure 2A.  35 The abbreviation TCO stands for transparent conductive oxide, HTL stands for hole transport layer, ETL for electron transport layer and JB stands for junction box.  Figure 7B on top.
Note that for the subsequent calculations, we only use the numbers for monolithic integration. From MSP and efficiency, the $/W P cost of a module can be calculated. These are shown in Figure 7B on the bottom for illumination without concentration. Fabrication costs outweigh efficiency increases, resulting in tandems with more junctions having a greater $/W P cost, which is consistent with modelling for double junction solar cells. 42 The true value of a tandem is only revealed in a system. Efficiency Calculations for 1 year in Singapore and Denver suggest that spectrum variations limit the achievable efficiency to below 75% (90% of the infinite tandem limit) and that no further harvesting efficiency improvements are possible after about nine junctions.
While independent cell operation is not affected by spectral variations in this way, the introduction of optical elements necessary for spectral selection and cell isolation induces a similar effect. Imperfections of the optical elements introduce losses that are greater for configurations with more junctions. How a non-ideal optical efficiency affects efficiency depends on the tandem architecture, yet in most cases, we find that an optical efficiency of 99% reduces the ideal number of junctions to below 10 and reduces achievable efficiencies by more than 10% compared to the infinite tandem limit.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available in Spectral Solar Radiation Data Base at https://www.nrel.gov/grid/solarresource/spectral-solar.html. These data were derived from the following resources available in the public domain: -Spectral Solar Radiation Data Base, https://www.nrel.gov/grid/solar-resource/spectralsolar.html.