1. Introduction

Detailed-balance theories, started by Shockley and Queisser [1], predict the conversion-efficiency limits (η_sc) and optimum bandgaps (E_g) of multi-junction solar cells [2], which are η_sc = 30% with E_g = 1.3 eV (1 junction), η_sc = 42% with E_g’s of 1.9 eV and 1.0 eV (2 junctions), and η_sc = 49% with E_g set of 2.3 eV, 1.4 eV, and 0.8 eV (3 junctions), for unconcentrated 6000K blackbody sun. Over many years, calculations have been extended to various radiation conditions (6000K blackbody, AM0, AM1.5 global and AM1.5 direct solar spectrum, etc. with 1- to 46000-fold concentrations), reflector configurations (ideal reflectors between the interfaces of subcells, with back-mirror, or without back-mirror), and electrical interconnections (series or independent connections) [2–9]. These calculations [1–9] have so far been referenced as design targets and templates. However, all the above calculations were achieved under the assumption of the radiative limit, that is, by neglecting non-radiative recombination loss. In fact, carrier recombination events in almost all semiconductor materials at room temperature are dominated by non-radiative recombination, hence it is necessary to consider the impact of non-radiative recombination losses to predict conversion efficiency limit and optimal bandgap for semiconductor solar cells.

Shockley and Queisser incorporated the effect of non-radiative recombination processes on single junction cell performance by using the external radiative efficiency (denoted hereinafter as η_ext), which is defined as the ratio of the radiative recombination current due to external emission from the cell versus the total recombination current [1]. Green [10,11] surveyed state-of-the-art single junction cells under AM1.5 G spectrum and explained their conversion efficiencies assuming various η_ext values. Chan [12] presented the calculations for optimal bandgaps for 1- to 3-junction solar cells while incorporating the effects of η_ext by using the two-diode model for considering non-radiative Shockley-Read-Hall (SRH) recombination. Although the quantity η_ext reflects cell material quality, it is affected also by cell structures, such as surface reflectivity, cell thickness, substrate material, and so on.

In contrast, internal radiative efficiency (η_int), defined as the fraction of the internal radiative recombination rate over the total recombination rate, is a suitable property that directly represents material quality independent of cell structure factors. Yablonovitch et al. [13–17] investigated conversion efficiency limits of single-junction solar cells, as a function of internal radiative efficiency to include a finite non-radiative recombination. They pointed out that a very slight deviation of η_int from 1 caused conversion efficiency η_sc to degrade significantly and sharply from the radiative-limit value.

Multi-junction solar cells consist of a series of subcells with different materials. It is unrealistic to assume that all the materials in the stack have ideally high quality. Indeed, materials with η_int close to 1 are rare, except for GaAs [18]. Moreover, under high concentration, the operating temperature of multi-junction cells would rise significantly, which further increase non-radiative recombination losses in each subcell, thereby causing the realistic subcell η_int to deviate from 1. Hence, it is important to evaluate the effect of all the subcells’ quality on η_sc and the multi-junction cell design. For this aim, we previously formulated and analyzed basic trends in conversion-efficiency limit η_sc and optimized E_g sets for the 2-junction tandem solar cells with finite subcell internal radiative efficiency η_int1 (top-cell) and η_int2 (bottom-cell) for unconcentrated 6000K blackbody sun [19]. We found that η_sc drops sharply when η_int1 or η_int2 slightly reduces from 1 to 0.9, that the optimized E_g set increases as η_sc decreases, and that the geometric mean η_int* = (η_int1η_int2)^1/2 is a very useful parameter to represent the whole subcell quality. However, major attention in research and development is presently directed toward η_sc > 40% or 3-, 4-, and 5-junction structures for satellite or terrestrial concentrator uses. Thus, our formalism needs to be applied urgently to multi-junction cells and concentrator solar cells.

Considering that conversion efficiency (η_sc) is also affected by the structure and geometry of solar cells inevitably, in this work, we investigate η_sc and optimized subcell bandgap energies in three well-known versions of current-constrained 1- to 5-junction solar cells with no, air-gap and perfect intermediate reflectors for various η_int’s for all subcells and their geometric mean η_int* under the 1-sun AM1.5G and 1000-sun AM1.5D conditions. The results show that η_sc of the three versions of multi-junction solar cells drop significantly as η_int* decreases from 1 to 0.1 and then logarithmically as η_int* are reduced below 0.1. Differently from the well-investigated ideal multi-junction solar cells with η_int* = 1, the intermediate reflectors no longer boost η_sc in thick multi-junction solar cells, if η_int* is less than 0.1. Moreover, optimized bandgap energies in multi-junction solar cells made of low η_int values turned out to be much larger than those optimized for the familiar ideal cases with η_int = 1. The results provide realistic targets of efficiency limits and improved design principles of practical multi-junction solar cells.

2. Model

Figure 1 shows three versions of two-terminal multi-junction solar-cell structures studied in this work: (a) without intermediate reflectors, (b) with prefect selective intermediate reflectors that only reflect photons with energy greater than the subcell bandgap energy, and (c) with air-gap intermediate reflectors sandwiched by antireflection (AR) coatings. The structures (b) and (c) are designed for reducing external emission losses from subcell back surface to improve subcell voltage. We also assume that the three versions of multi-junction solar cells with parallel planar surfaces are all covered by AR coatings on topmost subcells and have prefect rear mirrors at bottommost subcells, thus, the reflectivity of AR coating is 0 and that of the rear mirrors is 1. Especially for single junction solar cells, the three cases are exactly the same. The solar cells consist of a stack of the materials with band gap E_gi and an internal radiative efficiency η_{int i}. Subscript $i$ represents a certain subcell. Subscripts 1 and n represent the uppermost- and bottommost-subcells, respectively. In typical multi-junction solar cells made of III-V compound semiconductors, refractive indices of subcells in the stack are close. Thus, each subcell refractive index (n_i) can be reasonably and simply assumed to be equal to 3.5.

 

Fig. 1 Structures of the multi-junction solar cells considered in this work (a) no intermediate reflectors, (b) with perfect intermediate reflectors (PIR) and (c) with air-gap intermediate reflectors. All of three versions of solar cells have planar surfaces, perfect front-surface AR coating on topmost cell and perfect rear mirror in bottommost cell.

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We define $\overline{{\alpha }_{suni}}$ as the energy-averaged absorption coefficient for sun-light absorption between this subcell E_gi and the above subcell E_gi-1 or $\infty$, and separately define $\overline{{\alpha }_{lumii}}$ as the energy-averaged absorption coefficient at luminescence energy of this subcell, respectively given as

As for subcell thickness (l_i), it is firstly important for high efficiency that the absorption of solar photons is strong. Thus, we choose $\overline{{\alpha }_{sun}}l$ = 5 or exp($-\overline{{\alpha }_{sun}}l$) = 0.007, in most calculations here, and $\overline{{\alpha }_{sun}}l$ = 2 or exp($-\overline{{\alpha }_{sun}}l$) = 0.14, for comparison as a thinner case. In accordance with the above evaluated ratio $\overline{{\alpha }_{lumii}}$/$\overline{{\alpha }_{suni}}$, we choose $\overline{a_{lumi}}l$ = 1 and 2 in most calculations, and analyzed effects of $\overline{{\alpha }_{lumi}}l$ later in some cases. We also assume infinite carrier mobility in all subcells, which means that minority carrier diffusion lengths are much larger than subcell thicknesses.

All subcells are connected in series, and the outflow currents (J) of all subcells are equal. According to carrier balance in each subcell, the current flux flowing out from each individual subcell would be equal to the difference between the carrier generation flux and carrier recombination flux. Thus, all of the subcell I-V characteristics in $n$-junction solar cells ($n$ = 1, 2, 3, 4, and 5) are given by the following equations, for the case without intermediate reflectors:

For the cases with the air-gap and perfect intermediate reflectors, Eq. (3)b) should be modified to

In this study, we use solar illuminations of incident power per unit area P_in = 100 mW/cm² or P_in = 90 W/cm² intensity conditions with the spectrum of AM1.5G (1-sun) and AM1.5D (1000-sun). For P_in on the front surface of the cell, we calculated the highest value of η_sc (the conversion-efficiency limit η_sc) by optimizing E_gi set in the stack, for respective values of η_int1, η_int2, …, η_intn, as,

We define the geometric mean ${\eta }_{int}{}^{*}=\sqrt[n]{{\eta }_{int1}{\eta }_{int2}\cdots {\eta }_{intn}}$ for an arbitrary set of (η_int1, η_int2, …, η_intn). We first studied various combinations of (η_int1, η_int2, …, η_intn) and confirmed that n-junction solar cells with the same geometric mean η_int* have almost the same conversion efficiency limit η_sc and optimized E_gi set, provided that all subcells have η_int <0.3. This feature was previously found in the simplest 2-junction case [19]. Taking 3-junction solar cells for instance, curves in Fig. 2(a)-(b) show η_sc and optimized E_g set against η_int* under various combinations of (η_int1, η_int2, η_int3). The three colorized curves in Fig. 2(a), representing η_sc at the case of η_int2 = η_int3 = 0.3 (red), η_int1 = η_int3 = 0.3 (blue), and η_int1 = η_int2 = 0.3 (green), almost overlap with each other, which in fact are the upper bounds of evaluated η_sc at arbitrary combination of (η_int1, η_int2, η_int3) when all subcell η_{int i} <0.3. The black curves are η_sc under the case of η_int1 = η_int2 = η_int3, which are close to the three color curves. We also have chosen more than 15 random sets of (η_int1, η_int2, η_int3) that independently vary from 0.3 to 10⁻¹⁰ (the lower bound of η_{int i} is around 10⁻¹⁷ [23]) but their geometric means η_int* are fixed as 0.1 (circles), 0.01 (triangles), 0.001 (inverted triangles), and 0.0001 (squares), respectively. It is clear that all of the symbols always sit in a narrow area between green (upper bound) and black curves (lower bound) with a deviation less than 0.2-0.3%. Thus, if η_int < 0.3 is satisfied in all subcells, η_int* can be deemed to a representative measure of all subcell material qualities for more-junction solar cells. To summarize the effects induced by various subcell qualities combinations, we only have to calculate the case of all the subcell η_int in the stack having the same values η_int1 = η_int2 = … = η_intn = η_int* to estimate the η_sc limit of n-junction solar cells for an arbitrary set of (η_int1, η_int2,…, η_intn) almost giving the same η_int*. The calculated results shown below were obtained in this way.

 

Fig. 2 (a) The efficiency η_sc and (b) optimized E_g set of 3-junction solar cells without intermediate reflectors as function of η_int* under various combinations of (η_int1, η_int2, η_int3). Black, red, blue, and green curves in (a) respectively represent the values of ${\eta }_{sc}$ at the case of η_int1 = η_int2 = η_int3, η_int2 = η_int3 = 0.3, η_int1 = η_int3 = 0.3, and η_int1 = η_int2 = 0.3. Circles, triangles, inverted triangles and squares represent more than 15 random combinations of (η_int1, η_int2, η_int3) that independently vary between 0.3 and 10⁻¹⁰ but their geometric means η_int* are fixed as 0.1, 0.01, 0.001, and 0.0001, which were amplified in (c)-(f).

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3. Results and discussion

Figure 3 summarizes the optimum η_sc of three versions of 1- to 5-junction solar cells, for various η_int* between 1 and 0.00001 at the four conditions: (a) $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1 under 1-sun irradiation, (b) $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1 under 1000-sun irradiation, (c) $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 2 under 1-sun irradiation, and (d) $\overline{{\alpha }_{sun}}l$ = 2 and $\overline{{\alpha }_{lumi}}l$ = 1 under 1-sun irradiation for analyzing the effects of imperfect sunlight absorption. The dashed, solid, dotted curves in Fig. 3 show η_sc of multi-junction solar cells of without intermediate reflector, with perfect and with air-gap intermediate reflectors. The three kinds of black curves show η_sc at η_int* = 1, or in the radiative limit assuming ideal materials without non-radiative loss. These agree well with the previously reported results [1–9] except for slight dissimilarities due to differences in detailed calculation parameters. If we select $\overline{{\alpha }_{sun}}l\ge$5 and $\overline{{\alpha }_{lumi}}l\ge$5 for calculation, we will get the same η_sc limits as these pervious detailed balance limits. As mentioned in the introduction, practical materials with η_int close to 1 are very rare, and the black curves for η_int* = 1 are unrealistic to estimate η_sc for practical designs. The lower colored curves in Fig. 3 show η_sc of the three versions of multi-junction solar cells for η_int* = 0.1, 0.001, and 0.00001. Note that η_sc drops very drastically as η_int* decreases from 1 to 0.1, revealing that η_sc is very sensitive to η_int* in this region. If η_int* is below 0.1, the reduction in η_sc is almost constant, as η_int* drops by one order of magnitude. In other words, the efficiency penalty $\Delta$η_sc is almost linear to log₁₀η_int* in this low-η_int* region. Furthermore, the efficiency penalty $\Delta$η_sc is increased as the junction number n increases. Taking the multi-junction cells in Fig. 3(a)-(b) with $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1 for instance, decrease in η_sc per decade decrease in η_int* of three versions of multi-junction solar cells for n = 2, 3, 4, and 5 in the region of η_int* close to 0.001 was around 2.49%, 2.82%, 2.85% and 2.95% under 1-sun, while that was around 2.89%, 2.99%, 3.37% and 3.44% under 1000-sun, respectively. Figure 3(a)-(b) clearly illustrate that subcell material quality affects η_sc very similarly in no and high concentration operations.

 

Fig. 3 The optimized efficiency η_sc [%] of multi-junction solar cells with no (dashed curves), air-gap (dotted curves) and perfect intermedium reflectors (solid curves) as function of junction number at various values of η_int* = 10⁻⁵, 10⁻³, 0.1, and 1 for (a) $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1 (1-sun), (b) $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1 (1000-sun), (c) $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 2 (1-sun) and (d) $\overline{{\alpha }_{sun}}l$ = 2 and $\overline{{\alpha }_{lumi}}l$ = 1 (1-sun).

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We compared η_sc for the three versions of solar cells in Fig. 3(a)-(d) and found clear increase in η_sc for 2- to 5-junctinon solar cells with η_int* = 1, by adding air-gap or perfect intermediate reflectors. These results are consistent with the previous researches showing the boosts of η_sc caused by the air-gap or perfect reflectors [9]. However, it turned out in Fig. 3(a)-(c) that η_sc with different reflectors are very close to each other when η_int* are less than 0.1. For a thin cell with insufficient sunlight absorption with $\overline{{\alpha }_{sun}}l$ = 2, Fig. 3(d) shows that η_sc with perfect reflection is obviously higher than that with no reflection even for η_int*<0.1, but is still similar to that in Fig. 3(a). Such improvement in η_sc stem from simple improvement in the sunlight absorptivity from $\overline{a_{suni}}$ = 85% for single pass without reflection to $\overline{a_{suni}}$ = 98% for double pass with perfect reflection. Therefore, we may summarize that intermediate reflectors no longer cause large η_sc boost for η_int*<0.1, in stark contrast to the ideal case with η_int* = 1, except for trivial compensation effect for insufficient sunlight absorption in thin solar cells. When solar cells are thick enough for single pass sunlight absorption, as is the case with typical practical cells, the advantage of intermediate reflectors is washed away by low η_int*<0.1. The reason is clear: for the solar cells with extremely high materials quality, the intermediate reflectors assist in photon recycling in the subcells that leads to higher internal voltage; however, for the cells with moderate or low quality, carrier recombination losses are dominated by non-radiative recombination (the 3rd term in Eq. (3)), and the external emission from the subcells (the 4th term in Eq. (3)) becomes very small. Thus, the intermediate reflectors in practical multi-junction solar cells become ineffective.

These results in Fig. 3 indicate that the impact of η_int* on efficiency η_sc is significant and must be included in the assessment or design of high-efficiency multi-junction solar cells. Our recent experimental study [20] on commercially available satellite-use InGaP/GaAs/Ge 3-junction solar cells indicated that η_int* was as low as 0.01 at the maximum-power operation condition under a 1-sun condition. Note that η_int*’s in this paper mean those at the maximum-power operation condition. Note also that η_int* is strongly bias-voltage (bias-current) dependent, and that η_int* of the present sample was about 0.1 at the open-circuit condition under a 1-sun condition. We may regard subcell materials with η_int* = 0.1 and 0.001 at the maximum-power operation condition, for example, as “good” and “moderate” materials respectively. η_sc limit of “good” and “moderate” multi-junction solar cells are clearly lower than that in the radiative limit (η_int* = 1) by several to 10% absolute, which reasonably explain why η_sc’s of realistic multi-junction solar cells [24] empirically deviate from the detailed-balance-theory predictions. Therefore, the colored curves for η_int* below 0.1 in Fig. 3(a-b) provide more realistic targets of η_sc than the prevailing data calculated for η_int* = 1.

Note that Fig. 3(c) for $\overline{{\alpha }_{lumi}}l$ = 2 with stronger self-absorption of luminescence showed slightly lower η_sc than that in Fig. 3(a) for $\overline{{\alpha }_{lumi}}l$ = 1. According to the results in Fig. 3, it is necessary to discuss the effects of material absorptivity for sunlight and subcell self-luminescence on η_sc limit.

Figure 4 plots η_sc as function of $\overline{{\alpha }_{lumi}}l$ for the three versions of 3-junction solar cells with η_int* = 1 (“ideal”), 0.1 (“good”), and 0.001 (“moderate”) having (a) strong sunlight absorption $\overline{{\alpha }_{sun}}l$ = 5 and (b) incomplete sunlight absorption $\overline{{\alpha }_{sun}}l$ = 2. Note that η_sc decreases gently as luminescence absorptivity ($\overline{{\alpha }_{lumi}}l$) increases, for fixed $\overline{{\alpha }_{sun}}l$. The reason is clear in Eqs. (3)-(4): as $\overline{{\alpha }_{lumi}}l$ increases, $\overline{a_{lumii}}$, $4\overline{{\alpha }_{lumii}}{l}_i{n}_i^2/\overline{a_{lumii}}$, and dark current density J_0i = qR_ext(E_gi,$\infty$, 0) increase, and both radiative and non-radiative recombination losses increase to lower η_sc. On the other hand, it is trivial that high values of $\overline{a_{suni}}$ can generate more photocurrents that leads to increase in η_sc. High-efficiency solar cells prefer large $\overline{a_{suni}}$ and small $\overline{a_{lumii}}$, and have optimum thickness.

 

Fig. 4 The efficiency ${\eta }_{sc}$ of 3-junction solar cells with no (dashed curves), air-gap (dotted curves) and perfect intermedium reflectors (solid curves) at ${\eta }_{int}{}^{*}$ fixed as 1, 0.1, and 0.001 as function of $\overline{a_{lumi}}l$ for (a) $\overline{a_{sun}}l$ = 5 and (b) $\overline{a_{sun}}l$ = 2.

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The bar graph in Fig. 5(a)-(b) shows the optimized bandgaps combination (E_g1, E_g2, …, E_gn) at η_int* = 1~0.00001, for n-junction (n = 1, 2, 3, 4, and 5) solar cell without reflectors under (a) 1-sun and (b) 1000-sun conditions. Since intermediate reflectors make negligible difference under η_int*<0.1 with $\overline{{\alpha }_{sun}}l$ = 5, we only show the case of no intermediate reflectors. Note that the optimized bandgaps blue-shifted as η_int* decreased. In the case of (a) 1-sun AM1.5G, blue-shifts in the optimized bandgaps are larger between η_int* = 1 and 0.1, than those occurred between an order of magnitude decrease in η_int* below 0.1, which are similar to the feature of η_sc limit against η_int* in Fig. 3(a). Note also that the bottommost bandgap E_gn of n-junction solar cell tend to be pinned at around 0.70, 0.93, 1.12 and 1.38eV, corresponding to rising edge on the AM1.5 spectrum gaps due to atmospheric gas absorption. As a mixed result of the two trends of blue-shifts and the bandgap pinning, some anomalous jumps in the optimized bandgaps were found. In the case with (b) 1000-sun AM1.5D, the optimal bandgap sets are obviously smaller than those under (a) 1-sun. Since short circuit currents increase linearly with concentration ratio (C), while open circuit voltages roughly increase logarithmically with C, concentrator multi-junction cells with smaller bandgap combination materials tend to achieve higher efficiency. The two trends of blue-shifts and the bandgap pinning in the optimized E_gi set with η_{int i} clarified in (a) were also found under concentrating operation (b), where the effect of bandgap pinning was very obvious and the larger blue shifts between η_int* = 1 and 0.1 than those below 0.1 was not significant.

 

Fig. 5 The optimized set of bandgaps $E_{gi}$ [eV] as function of junction number at various values of ${\eta }_{int}{}^{*}$ = 0.00001~1, for $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1 under (a) AM1.5G (1-sun) and (b) AM1.5D (1000-sun).

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The blue-shifts of the optimized E_gi set with η_{int i} decrease are the same with those found in our previous basic study on 2-junction solar cells under 6000K blackbody irradiation, that is a continuous spectrum without any gaps, and can be ascribed to the following reason. Subcell open circuit voltage (V_oci) is determined by subcell bandgap and the term of voltage penalty due to imperfect material quality (η_{int i} <1) independent of bandgap, expressed as V_oci$\approx$E_gi + kT_cln(η_{int i})-C, where C is a constant determined by subcell structure and material properties, such as refractive index [21]. If η_{int i} becomes low, subcell V_oci should decrease. Therefore, larger E_gi should compensate for this penalty in V_oci. Due to the series connection of subcells, the extracted current is limited by the least photo-generated current among all the subcells in the stack, and the optimal bandgaps must be chosen so as to satisfy the current-matching condition. It is interesting but reasonable that change in η_{int i} of only one subcell i causing a change in η_int*, which affects the optimization of all bandgaps in the stack.

The present work provides instructive conclusions and proposals for the practical design of high-efficiency multi-junction solar cells. The efficiency limits for 2- and 3-juntion cells without intermediate reflectors under 1-sun are high, which are respectively 38.2% and 43.5% for η_int* = 0.1, and 33.0% and 37.6% even for η_int* = 0.001 at $\overline{{\alpha }_{sun}}l$ = 5 and $\overline{{\alpha }_{lumi}}l$ = 1, provided that a properly blue-shifted optimal-bandgap set is chosen. This result reveals that 2- and 3-juntion cells still can be implemented as a potential alternative for higher-efficiency solar cells, since the efficiency of them can be further improved if we use proper bandgap sets optimized in accordance with the material quality. We additionally note that the improvements in η_sc becomes small: 2.5% and 2.1% when junctions are increased from 3- to 4-junctions and from 4- to 5-junctions, respectively, under 1-sun with η_int* = 0.001. That is to say, additional efficiency add-on is very small for a high number of junctions. Though this trend has already been proven [2, 9] for the case of η_int* = 1, here we confirmed it generally for all the cases of η_int* = 1, 0.1, and η_int*<<1.

Under concentration conditions, basic trends are similar to the above case without concentration: even for moderate materials with η_int* = 0.001, the η_sc limits for 2- and 3-juntions under 1000-sun at are high, 40.1% and 45.6%, respectively, but with an obvious blue-shifted optimal-bandgap set. Improvements in η_sc are not large, 3.0% and 2.1%, for junctions increases from 3 to 4 and from 4 to 5, respectively, for 1000-sun and η_int* = 0.001. For practical 3-junction cells, it is empirically considered that popular InGaP/(In)GaAs/Ge or InGaP/GaAs/InGaAs cells have already reached their practical efficiency upper limits under concentrated light, limited by subcell materials and lattice mismatch [25–29]. Therefore, several research groups explore new materials [25], new design [27] and new process [29,30] for 3- or more-junction structures. Our results and calculation model presented here provide useful design targets on subcell bandgap combination and efficiency reference for high-efficiency multi-junction solar cells.

Additional improvement in η_sc may occur at increased concentrations or pumping power density, with the help of improvements in η_int of each subcell. We believe that such improvement in η_int partly contributes to improvement in η_sc of practical solar cells at high concentrations, based on the results shown in Fig. 3(b). On the other hand, we note that the temperature of the multi-junction solar cells would increase under high concentrating operations, which usually causes significant reduction in η_sc obversed in experimental studies [31–33].

We here remind the readers that all the above results were obtained for $\overline{{\alpha }_{sun}}l$ and $\overline{{\alpha }_{lumi}}l$ being some assumed values, and a refractive index of 3.5. With different input values, the values of η_sc and optimized bandgaps may change. However, within realistic input values, the resulting values of optimal bandgap set of reasonably thick solar cells usually change slightly. Moreover, the qualitative features obtained in this study remain valid. In this study, we used another assumption of infinite carrier mobility to obtain the conversion-efficiency limits. Effects of finite mobility can be incorporated into the present theory, if we introduce phenomenological parameters t_i, which represent the probability that a photon absorbed in the subcell i produces an electron-hole pair available as short-circuit current, as was included in the seminal paper by Shockley and Queisser [1] for single-junction cells. In the paper, we have assumed all of the surfaces and interfaces of multi-junction solar cells are flat. Randomly textured surfaces have been proposed to increase subcell external radiative efficiency [9]. For the randomly textured multi-junction solar cells, we should change absorptivity expression mentioned in [13].

4. Conclusion

In conclusion, we studied the theoretical efficiency limit η_sc and subcell bandgap energies including their dependence on the geometric mean η_int* of all the subcells’ internal radiative efficiencies η_int in the multi-junction solar cells with no, air-gap and perfect intermediate reflectors. We confirmed that η_int* is a very useful parameter that summarizes η_int of all the subcells with η_int <0.3 and that η_sc calculated for η_int1 = η_int2 = … = η_intn represents η_sc for arbitrary combinations of (η_int1, η_int2, …, η_intn) giving the same η_int*. As η_int* decreases from 1 to 0.1, the efficiency limit η_sc of the three versions of cells decrease drastically. Semi-logarithmic relationships were found between η_sc and η_int* for η_int* below 0.1. Differently from the case with the well-investigated ideal multi-junction solar cells, the intermediate reflectors no longer boost η_sc of the practical thick multi-junction solar cells with η_int* less than 0.1. The optimized bandgap energies were obtained for finite η_int* below 1, which were significantly blue-shifted from those within the radiative limit (η_int* = 1) and affected by the edges of AM1.5 solar-spectrum gaps. We found that high η_sc are possible for 2- and 3- junction cells with realistic materials with low η_int*, if we chose the optimized designs obtained in this work. Our present results provide realistic efficiency targets and design principles for multi-junction solar cells made of realistic subcell materials.