Introduction

Multijunction solar cells (MJSC) are the most successful photovoltaic technology in using the solar resource efficiently. The current highest efficiency ever achieved by November 2019 is 47.1% demonstrated with a monolithic series-connected six junction solar cell under concentration1. However, MJSC are devices expensive to produce2 and for this reason they were successful in niche applications in which they are cost-effective, such as for space3 and in terrestrial use under concentrated sunlight4. On the other hand, modern techniques can drastically reduce production costs by the use of inexpensive materials5,6,7,8, high quality heteroepitaxy between groups II-VI, III-V, and IV materials9,10,11,12, the reuse of the growth substrate13 and the saving in epitaxy costs7,14, which can push MJSC to a cost level similar to flat-plate Si technology in the future2,15. Nonetheless, it is important to highlight that this optimistic estimative is strictly based in the high efficiencies still to be achieved for this kind of device. In this context, basic studies establishing the efficiency limits and defining the materials to be used in highly efficient devices are mandatory.

In 1961, Shockley and Queisser developed an elegant theory based on the detailed balance model to calculate the maximum efficiency of photovoltaic conversion by single pn junctions16. They found the limit to be 30% for a bandgap energy of 1.34 eV under a blackbody spectrum simulating the sun and 44% for a bandgap energy of 1.12 eV under maximum solar concentration. Their paper has been the trigger for several other theoretical studies which analyzed the conversion efficiency limit under different solar device architectures. In 1980, two independent studies conducted by Henry17 and De Vos18 found limiting conversion efficiencies for MJSC under concentrated sunlight to be 72% (1,000 suns and 36 junctions) for an optically and electrically series connected device and 86.8% (45,900 suns and infinite junctions) for an optically series-connected device with no electrical restriction, respectively. Several other investigations have followed taking into consideration more fundamental arguments such as thermodynamical backgrounds on entropy-energy generation19,20,21,22,23,24, including other physical effects, such as angle and energy restriction21, boundary conditions25, analyzing the influence of the series resistance26 and of materials dominated by other type of recombination processes, such as silicon27, and presenting new concepts, such as hot carrier solar cells28, intermediate band solar cells29, among others, enriching the literature and our comprehension of the photovoltaic conversion processes.

More recently, new studies on the theoretical efficiency of MJSC have taken into account the role of the optical interconnection among the junctions under the photon recycling and the luminescent coupling effects30,31,32,33,34,35,36,37. It is well understood that the recycling of the emitted photons by the same emitting junction increases the device open circuit voltage (VOC) and that the luminescent coupling among the junctions can increase the device short circuit current (ISC) allowing the device to approach the current matching condition. To these effects to positively contribute to the total generated power, the materials have to be limited by radiative recombination and the connection between the junctions needs to be optically transparent. Considering a more realistic scenario, Zhu et al.38 took into consideration the influence of non-radiative recombination using the internal radiative efficiency as a parameter in the efficiency calculations. They showed that conversion efficiencies are reduced and that the optimum bandgap energy sets change under the increase of the amount of non-radiative recombination.

The production of devices trying to fulfil the optical and mechanical requirements for a high-quality device, as predicted by theoretical studies, has followed up. The first demonstrated triple junction solar cell (3JSC) was a monolithic lattice matched GaInP/GaAs/Ge (1.90/1.42/0.67 eV) by Spectrolab in 2000 with an efficiency of 32.3% at concentrated sunlight39 (hereon the efficiencies are presented under concentrated sunlight except when explicitly mentioned). At that point, several different configurations of dual junction solar cells (2JSC) already existed fabricated with the most mature-to-produce monolithic lattice matched materials, such as GaInP/GaAs, InP/GaInAs, GaAs/GaInAsP, showing efficiencies up to 32.6%39. The following next decades have been marked with important innovations in the MJSC development yielding a rapid increase in their efficiencies. In 2001, Bett et al. demonstrated a 33.5% efficient mechanically stacked GaInP/GaInAs/GaSb (1.74/1.17/0.73 eV) 3JSC40. In 2005, Wanlass et al.41 presented a revolutionary new technique to produce non-lattice matched monolithic structures based on (inverted) metamorphic growth. Therefore, they showed a 37.9% effici ent monolithic metamorphic GaInP/GaAs/GaInAs (1.87/1.42/1.02 eV) 3JSC. After this new technique, the progress have followed quickly among initiatives using approaches with lattice matched materials, such as GaInP/GaInAs/Ge (1.87/1.4/0.67 eV)42, GaInP/GaAs/GaInNAs (1.86/1.42/1.0 eV)43, inverted metamorphic materials, such as GaInP/GaInAs/GaInAs (1.83/1.34/0.89 eV)44, and metamorphic materials, such as GaInP/GaInAs/Ge (1.74/1.17/0.67 eV)45. The current 3JSC highest efficiency since 2013 is 44.4% for an inverted metamorphic GaInP/GaAs/GaInAs structure developed by Sharp Co.1. In 2014, researchers from Fraunhofer ISE demonstrated a GaInP/GaAs//GaInAsP/GaInAs (1.88/1.42/1.12/0.74 eV) four junction solar cell (4JSC) with an efficiency of 44.7%1,46 based on a new technique: wafer bonding. This technique allows bandgap engineering without the need of the lattice matching constraint due to the direct bonding between crystals (the//symbol indicates where the bond is located). An improved version of this device has been further demonstrated with an efficiency of 46%1,47. Still in 2014, NREL also demonstrated a very high efficiency of 45.7% with a GaInP/GaAs/GaInAs/GaInAs 4JSC based on the inverted metamorphic approach1,48. The development of solar cells with more than four junctions is still in its very beginning. Nevertheless, the highest efficiency ever achieved up to November 2019 has been obtained with a AlGaInP/AlGaAs/GaAs/GaInAs/GaInAs/GaInAs (2.15/1.72/1.41/1.17/0.96/0.70 eV) six junction solar cell (6JSC) developed by NREL1,49. In addition, two results worth to mention are a 38.8% efficient wafer bonded 5JSC at 1-sun by Spectrolab1,50 and a 39.2% efficient 6JSC at 1-sun by NREL1,51.

The development history of MJSC have established an important guideline for the optimization of MJSC structures: theoretical calculations shall guide the choice of materials. However, it is possible to see an ambiguity in the definition of the optimum bandgap energy for the bottom junction (which in last instance determines the others). While some authors defend a bandgap energy of ~0.7 eV20, others say that ~1.0 eV would lead to the highest efficiency41,52 or even point both values when evaluating for different spectra4,53. In this context, one could blame the theoretical approach to generate different values for the optimal bandgap combination leading to the highest efficiencies, but this is not the case. Actually, both values for the bandgap energy of the bottom junction lead either to a global or a local maximum in the efficiency of MJSC. This is due to the particular illumination condition impinging in the solar cell regarding the solar spectrum and the concentration factor.

As already observed by many authors20,38,52,53,54, the molecular absorption of portions of the solar spectrum by the atmospheric gases is responsible for the reduction of the photon flux in some wavelength ranges. McMahon et al.54 correlated the disruption of the current matching in MJSC to this effect. They indicated the wavelength bands in which the efficiency is altered leading to the global and local maxima. The high energy edge of each band is of great importance, as they are the energy of the efficiency peaks. As such energies are of interest in this work, we labeled them as EA = 0.52 eV, EB = 0.70 eV, EC = 0.93 eV, ED = 1.12 eV, and EE = 1.34 eV and use them hereon.

The bottom junction bandgap energies leading to the historically highest experimental efficiencies for MJSC have varied between the corresponding values of EB and EC, as discussed. Despite McMahon et al.54 and other authors have correctly observed and explained the influence of the solar spectrum shape in the efficiency of MJSC, they did not consider that the variation in the concentration factor also has an impact in the MJSC efficiency and its optimum configuration. As it is going to be demonstrated, even for the same application (regarding spectrum and number of junctions), the optimum bandgap energy set changes with concentration. This is the reason for the divergence in the results of the different theoretical studies.

In this work, we revisit the theoretical study on the conversion efficiency of MJSC. The numerical methods used in this work (described in Methods) are based on the detailed balance model applied to MJSC with 2 to 6 junctions under several illumination conditions. Rather than focusing only in the maximum achievable efficiency, we thoroughly study the efficiency distribution in the N-dimensional bandgap energy space, in which N is the number of junctions, to investigate the more promising choice of materials in the design of highly efficient devices. As results, we reinforce previous studies conclusions showing that the highest efficiencies for terrestrial application take place for recurrent values of the bottom junction bandgap energy for any configuration regarding number of junctions and illumination condition. Moreover, we demonstrate that the optimum values for the bandgap energy sets change with the concentration factor even for the same spectrum and number of junctions, which directly impacts the device design for the same application. Finally, the observed recurrence for the best bottom junction bandgap energy allowed us to suggest more than a hundred promising materials to compose this junction in highly efficient MJSC for any terrestrial application.

Results and Discussion

MJSC efficiency under AM0 and AM1.5 g spectra

Figure 1 presents color mappings and level curves for the efficiency of 2JSC and 3JSC as a function of the bandgap energies of their composing junctions for two different illumination conditions: (a) 2JSC under AM0, (b) 2JSC under AM1.5 g, (c) 3JSC under AM0, and (d) 3JSC under AM1.5 g. The junctions are numbered increasingly (i = 1..N) from the bottom to the top, i.e., EGJ1 is the bottom junction bandgap energy. In Fig. 1c,d, the transparent gray spheres indicate the swept xyz relevant points (as described next) whereas the colorful projections in the axis planes facilitate the observation of the corresponding bandgap energies leading to such results.

Figure 1
figure 1

Multijunction solar cell efficiency color maps. (a,b) 2JSC efficiency (color mappings and level curves) as a function of EGJ1 and EGJ2 under AM0 and AM1.5 g, respectively. (c,d) 3JSC efficiency (color mappings) as a function of EGJ1, EGJ2 and EGJ3 under AM0 and AM1.5 g, respectively. The efficiency peaks are marked with symbols and are numbered increasingly according to the energies EA – EE.

Full size image

From Fig. 1, it is possible to see that the efficiency for the AM0 spectrum (Fig. 1a,c) has a single peak whereas for the AM1.5 g spectrum (Fig. 1b,d), it presents a multi-peak behavior, as expected due to atmospheric absorption53,54. In the figures, the efficiency peaks are marked with symbols within the xyz point distribution. Furthermore, Peaks 1 to 5 in AM1.5 g show up at EGJ1 close to the same energies EA to EE (the peak numbering is in accordance to the energies EA – EE), as specified by the vertical dashed lines in Fig. 1b and the horizontal dashed lines in Fig. 1d. It is important to notice that the correlation takes place exclusively for EGJ1.

Another result noteworthy is the broad range of bandgap energy combinations leading to high efficiencies (relevant points with efficiency higher than 90% of the efficiency peak). In Fig. 1, such regions are delineated by the contour lines separating the green and the blue areas (second last contour line in the figures), specifically at 37.5% and 40.4% for 2JSC under AM0 and AM1.5 g, respectively, and at 43.0% and 45.7% for 3JSC under AM0 and AM1.5 g, respectively. This is an important result concerning the design of highly efficient 2JSC and 3JSC, which allows an extensive choice of materials to compose the device junctions not necessarily attaining only to the bandgap energy values leading to the global efficiency peak.

The analysis for MJSC with more than three junctions followed. Although, as the N-dimensional domain with N > 3 is not possible to show in a single graph as done for 2JSC and 3JSC, we present an analysis of the efficiency over EGJ1 to further evaluate the correlation between the peaks and the absorption energy bands in the solar spectrum. Figure 2 shows the dependence of the best MJSC efficiencies on EGJ1 for (a) AM0 and (b) AM1.5 g and the number of junctions N varying from 1 to 6.

Figure 2
figure 2

Best MJSC efficiencies for AM0 and AM1.5 g. (a,b) Dependence of the best efficiencies on EGJ1 in MJSC with 1 to 6 junctions for AM0 and AM1.5 g spectra, respectively. In (a), the single peaks are marked with featured symbols and the corresponding bandgap energies are shown in the legend. In (b), the highest efficiencies are marked with stars and the energies EA – EE are marked with vertical black dashed lines.

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Over again, the same correlation between the EGJ1 values leading to the efficiency peaks for the terrestrial spectrum and the energies EA – EE is observed. Moreover, from Fig. 2, it is possible to see that the highest efficiency peak is redshifted and increases as the number of junctions increases for both illumination conditions. This is expected as a result of a more balanced division of the solar photons among the junctions and to a reduction of the losses related to transmission and thermalization. Furthermore, the highest efficiency peak for the AM1.5 g spectrum is not deviated continuously as for the AM0 spectrum. Instead, the peaks are interchanged only among the EA – EE values.

The gain in efficiency by adding a new junction is much higher when the number of junctions is low. From 2JSC to 3JSC under AM1.5 g, the maximum efficiency raises by 5.8% in absolute (13% relative) whereas from 5JSC to 6JSC only 1.7% (3% relative) is added. By a first estimative (whose methods to find are described in Methods), the maximum efficiency of a 10JSC is around 61%, a gain of only 2.8% in absolute (4.8% relative) compared to a 6JSC, for example. Thus, the stacking of too many junctions in a device to increase the efficiency even further shall be evaluated from the cost-effectiveness point-of-view.

Efficiency of MJSC under concentrated AM1.5d spectrum

The occurrence of multi-peaks in the MJSC efficiency distribution and the recurrence of the peak positions for values of EGJ1 close to the energies EA – EEalso take place for the AM1.5d spectrum under concentration, as can be seen by the colorful solid lines in Fig. 3 for (a) 2JSC, (b) 3JSC, (c) 4JSC, (d) 5JSC, and (e) 6JSC. Moreover, it is possible to see that the highest efficiency peak for AM1.5d, marked with colorful solid stars in the figures, changes depending on the concentration factor C, but only among the most recurrent values for the best EGJ1. To further investigate this behavior, Fig. 3 also shows how the efficiency peak values depend upon the concentration factor for (f) 2JSC, (g) 3JSC, (h) 4JSC, (i) 5JSC, and (j) 6JSC.

Figure 3
figure 3

Efficiency of MJSC under concentrated AM1.5d spectrum. (a–e) Dependence of the MJSC best efficiency on EGJ1 for different illumination conditions for 2JSC, 3JSC, 4JSC, 5JSC, and 6JSC, respectively. (f–j) Dependence of the MJSC peak efficiency on the concentration factor for AM1.5d spectrum for 2JSC, 3JSC, 4JSC, 5JSC, and 6JSC, respectively. The points in the figures on the right-hand side are related to the peaks present in the figures on the left-hand side.

Full size image

From Fig. 3, it is possible to notice that the highest efficiency is also redshifted and increases logarithmically under concentration. To understand such relations, we shall explore the dependence of the conversion efficiency η on the concentration factor C, as described by Eq. (1)55:

$$\eta (C)=\eta (1)+\frac{(kT/q)}{{V}_{OC}^{MJSC}(1)}ln(C)$$
(1)

in which \({V}_{OC}^{MJSC}(1)\) is the MJSC open-circuit voltage for 1-sun, k is the Boltzmann constant, q is the electronic charge and T is the cell temperature. As Eq. (1) indicates, the higher slopes are expected to take place for the bandgap energy combinations with lower \({V}_{OC}^{MJSC}(1)\). In fact, from Fig. 3, it is possible to see that the steeper increases are for the combinations in which the bandgap energies are lower which, in fact, lead to lower \({V}_{OC}^{MJSC}\). This tendency shifts the efficiency peak to low EGJ1 (and all other EGJi) values for high concentration.

Peak 1 (depicted as black squares in Fig. 3f–j) is the highest one for 4JSC under concentrations above 60 suns, for 5JSC under concentrations above 20 suns, and for 6JSC under concentrations above 160 suns. For the same devices, Peak 2 (red circles) is the highest one at the complementary concentration ranges, for 2JSC under concentrations above 1600 suns and under any concentration factor for 3JSC. Peak 3 (green triangles) is relevant under two conditions: it is the highest peak for 2JSC under concentrations below 1600 suns and it is the second highest peak for 3JSC under concentrations below 100 suns. Peak 4 (blue diamonds) and Peak 5 (not in the figures) do not appear as the highest peak in any MJSC configuration. However, Peak 4 still shows up in the graphs of Fig. 3 because it fills the requirement of high efficiency (>90% of the highest efficiency peak). Nonetheless, it plays an important role for low concentration in the 2JSC configuration.

Summary of MJSC configurations leading to the highest efficiencies

Table 1 summarizes the highest efficiencies of the MJSC studied in this work under several illumination conditions regarding spectrum and concentration factor. For each condition, the bandgap energy combinations of the peaks leading to efficiency higher than 90% are shown.

Table 1 Summary of the MJSC highest efficiencies and its configurations.
Full size table

Choice of materials for the bottom Junction of MJSC

The invariance of the EGJ1 values leading to the multiple efficiency peaks for terrestrial solar spectra in several MJSC configurations motivates the indication of materials to compose the bottom junction. In Table 2, we propose a list of promising materials for this purpose. The material parameters are in accordance with Vurgaftman et al.56 for III-V semiconductors, with Kudrawiec57 for nitrides semiconductor, and with Wei and Zunger58 for II-VI semiconductors. The III-V materials are presented in reference to a substrate just for orientation of the lattice parameter. Along with that, we remind the innovative technologies developed in the last two decades to allow mechanical integration of different materials either with or without the necessity of lattice matching, such as the (inverted) metamorphic growth and the direct bonding. For such reason, the comments column indicates whether the proposed material is lattice matched (LM) or lattice mismatched (LMM) to the reference substrate with a mismatching up to 3.0%. For the II-VI semiconductors, two cases are considered. In the first, no reference substrate is designated as the materials can be grown in any type of substrate (glass or plastic) in its polycrystalline form. However, in the second case, an III-V or IV semiconductor substrate is considered. This is based on some important demonstrations of integrating monocrystalline monolithic II-VI to such materials for solar cells that deserve attention11,12,59,60. Other possibilities of LMM materials within these systems are also possible and are not shown here but can be obtained by interpolation of the data shown elsewhere56,57,58 as well as other promising materials, such as perovskites, as Hörantner et al.8 point out.

Table 2 Materials for the bottom junction of MJSC for use in terrestrial applications.
Full size table

Conclusions

In conclusion, we have shown that the bandgap energy combinations leading to high efficiencies in optically and electrically series-connected MJSC are abundant. For the AM0 spectrum, the efficiency shows a single peak on the bandgap energy domain whereas for terrestrial spectra a multi-peak distribution is observed. In general, the peaks are redshifted in respect to the bandgap energies and increase with the number of junctions. This is expected due to a well-balanced division of solar photons among the junctions, reducing losses by transmission and thermalization. For a given number of junctions, the efficiency increases logarithmically with concentration, as expected, and the efficiency distribution is also redshifted. We have shown that this is due to the fact that the device configurations presenting lower bandgap energies have steeper increases in the efficiency with the increase of the concentration factor, which bring benefits for its use at high concentrations.

Additionally, we have shown that the optimum values for the bandgap energy of the bottom junction are recurrent for terrestrial spectra at energies close to EA = 0.52 eV, EB = 0.70 eV, EC = 0.93 eV, ED = 1.12 eV, and EE = 1.34 eV, matching the high energy edges of the atmospheric absorption bands. Moreover, when the concentration factor changes, the MJSC peak efficiency shifts but the optimum value for the bottom junction bandgap energy is always within the set of recurrent values described before.

Based on the overall results, we have suggested more than a hundred promising III-V, II-VI, and IV semiconductor materials to compose this junction and to serve as basis for the stacking of junctions in a MJSC for any terrestrial application. Silicon is a feature from this list. The increasing interest in integrating III-V, II-VI and perovskites to silicon seems to be a very promising way for further photovoltaic conversion efficiency improvements and cost reduction. We highlight the use of modern techniques, such as direct bonding and (inverted) metamorphic growth, to allow the mechanical integration of what would be formerly called incompatible materials. On the other hand, innovations in the material growth, such as the reuse of substrates, high quality heteroepitaxy, the use of higher growth rates and the use of more efficient processes in the incorporation of precursors, support further reduction in the cost production. The use of such techniques associated with a better choice of materials, based on the results of this work, can push multijunction solar cells to the next level, surpassing the 50% efficiency barrier in a very near future, thus, decreasing costs and making III-V photovoltaics more affordable.

Methods

Detailed balance model applied to MJSC

The conversion efficiency of solar cells was herein calculated using the detailed balance model (DBM)16 through numerical methods. In this model, the electrical current I flowing through a photovoltaic device results from the balance between generation G and recombination R, as in Eq. (2).

$$I=q(G-R)$$
(2)

in which q is the electronic charge. To calculate the ultimate efficiency of solar cells by the DBM, some assumptions are commonly made and were used in this work: (i) each absorbed photon generates only one electron-hole pair; (ii) each electron-hole pair that recombines only generates one photon; (iii) there is only radiative recombination taking place; (iv) electrical carrier mobility is infinite; (v) material absorptivity A(E) is unity for photons with energy E higher than EG and zero otherwise; (vi) there is no reflection at front surface; (vii) the back-surface is perfectly reflective; and (viii) emitted photons can only leave through front surface in which no angle restrictor is applied. Assumptions (iv) and (v) lead to a quantum efficiency QE(E) of one for photons with energy E higher than EG and zero otherwise. Assumptions (vii) and (viii) mean that the device luminescence is spread out solely to the full upper hemisphere, and lateral emission is neglected.

The application of the DBM to an optically and electrically series-connected MJSC with N junctions is straightforward. First, we have to consider that each junction Ji (i = 1..N increasing from the bottom junction upwards) of bandgap energy \({E}_{G}^{{J}_{i}}\) is subjected to a different incident photon flux \({\rho }_{\gamma }^{i}(\lambda )\) which is filtered from the solar spectrum by the set of junctions above. With the bandgap energies increasing from J1 upwards and considering only normal incidence, i.e. θ = 0, \({\rho }_{\gamma }^{i}(\lambda )\) can be described as in Eq. (3a):

$${\rho }_{\gamma }^{i}(\lambda )=\{\begin{array}{ll}0, & \lambda < {\lambda }_{Ga{p}_{i+1}}\\ {\rho }_{\gamma }^{i+1}(\lambda ), & \lambda \ge {\lambda }_{Ga{p}_{i+1}}\end{array},\,(i=1..N)$$
(3a)
$${\rho }_{\gamma }^{N+1}(\lambda )=\frac{{W}_{sun}(\lambda )}{hc/\lambda }$$
(3b)

In Eq. (3a), \({\lambda }_{Ga{p}_{i}}\) is the material cutoff wavelength (corresponding to \({E}_{G}^{{J}_{i}}\)) of Ji, and \({\lambda }_{Ga{p}_{i+1}}\) is the material cutoff wavelength of Ji+1, the junction immediately above Ji. In the case of the uppermost junction JN, \({\lambda }_{Ga{p}_{N+1}}=0\) shall be considered. Equation (3b) defines \({\rho }_{\gamma }^{N+1}(\lambda )\) as the solar photon flux calculated from the solar spectral irradiance \({W}_{sun}(\lambda )\), using h for the Planck constant, and c and λ as the speed and wavelength of light, respectively.

In this context, each junction Ji has its own generation rate Gi and recombination rate Ri. Gi is the temporal rate of absorbed photons per unit of area in Ji, which depends upon the incident photon flux and the quantum efficiency (QEi), as in Eq. (4):

$${G}_{i}={\int }_{0}^{\infty }Q{E}_{i}(\lambda ){\rho }_{\gamma }^{i}(\lambda )d\lambda =\,{\int }_{{\lambda }_{Ga{p}_{i+1}}}^{{\lambda }_{Ga{p}_{i}}}[\frac{{W}_{sun}(\lambda )}{hc/\lambda }]d\lambda $$
(4)

Ri can be calculated by the temporal rate of photons spontaneously emitted by Ji per unit of area through radiative recombination. Let \({\rho }_{BB}^{i}(\lambda ,V,T)\) denotes the blackbody spectral photon flux at temperature T under applied voltage V, and \({A}_{i}(\lambda )\) the material absorptivity of Ji. Then, Ri can be expressed as in Eq. (5)17:

$${R}_{i}(V)={\int }_{0}^{{\theta }_{E}}\,{\int }_{0}^{\infty }{A}_{i}(\lambda ){\rho }_{BB}^{i}(\lambda ,V,T)d\lambda d\varOmega =2\pi {\int }_{0}^{{\lambda }_{Ga{p}_{i}}}\frac{2c}{{\lambda }^{4}}\frac{d\lambda }{[{e}^{(\frac{hc}{\lambda }-q\cdot V)/kT}-1]}$$
(5)

in which \({\theta }_{E}=2\pi \) sterradian is the solid angle of emission (full upper hemisphere). The first equalities in Eqs. (4) and (5) are valid in a general case whereas the second ones are obtained using assumptions (i) to (viii) and Eq. (3).

The application of the DBM to a MJSC was carried out by considering the electrical series-connection constraint. Thus, the same electrical current IMJSC, which is the least of the N junctions, was imposed to flow through the junctions (Eq. 6) whereas the total voltage of the N series-connected generators at each point was summed (Eq. 7).

$${I}_{MJSC}(V)={I}_{min}(V)={{\rm{\min }}}_{i=1..N}[{I}_{i}(V)]$$
(6)
$${V}_{MJSC}({I}_{MJSC})={\sum }_{i=1}^{N}{V}_{i}({I}_{MJSC})$$
(7)

The relevant part of the IV characteristic was accessed by sweeping the parameter V in the range:

$$-{\sum }_{i=1}^{N}{V}_{O{C}_{i}}\le V\le {V}_{O{C}_{low}}$$
(8)

In Eq. (8), \(\,{V}_{O{C}_{i}}\) and \({V}_{O{C}_{low}}\) are the open circuit voltage of Ji and of the junction with the lowest current density, respectively.

Finally, from the described procedure, the figures of merit, such as the short-circuit current density \({I}_{SC}^{MJSC}\), the open circuit voltage \({V}_{OC}^{MJSC}\), and the fill factor \(F{F}_{MJSC}\), are extracted from the calculated MJSC IV curve. Using the incident power Pin, the efficiency \({\eta }_{MJSC}\) can be calculated with Eq. (9).

$${\eta }_{MJSC}=\frac{{P}_{MPP}^{MJSC}}{{P}_{in}}=\frac{F{F}_{MJSC}\cdot {I}_{SC}^{MJSC}\cdot {V}_{OC}^{MJSC}}{{P}_{in}}$$
(9)

In our study, standard ASTM solar spectra61 are used and temperature is set to T = 25 °C. No optical coupling was assumed, which means that the luminescence coming from the radiative recombination of a junction is not used by the other ones. If so, Eqs. (4) and (5) would be coupled between the N junctions, as Gi would depend on the several Ri’ (i′ > i). Under this condition, the method of calculating the total MJSC IV curve by the application of Eqs. (6) and (7) for N IV curves obtained independently could not be followed. Nonetheless, as Martí and Araujo20 discuss, the limiting efficiency is higher for a situation with no luminescence coupling in which perfect filters are put in between the junctions in order to reflect the luminescence only allowing it to exit from the front side. This reflects the case under consideration here.

The model has been validated through the comparisons of the results with other theoretical studies. Details can be found in the Supplementary Material.

Search methodology for the highest efficiency of MJSC

The procedure described above correlates a particular set of bandgap energies of a MJSC to its figures of merit. The search for the highest efficiencies for each device condition is performed by sweeping up the possible bandgap energy combinations in the N-dimensional space. However, the best configuration is expected to be reached for a singular condition: when the maximum power point current of the junctions \({I}_{MP{P}_{i}}\) is the same55. As a first estimative for the best candidate to the highest efficiency, we defined a simple test to be performed prior to the entire calculation. Instead of using the equality of the \({I}_{MP{P}_{i}}\), which can only be performed after the calculation of the N IV curves, we used the generation current in each junction \({I}_{{G}_{i}}\) as the parameter to be equalized, which can be obtained directly from a simple analysis of the solar spectrum. This is justified by the great similarity between such values, as recombination at the maximum power point is normally very low. Thus, for a given bandgap energy of the bottom junction EGJ1 (which defines its cutoff wavelength \({\lambda }_{Ga{p}_{1}}\)), an optimized maximum power point current density \({I}_{MPP}^{MJSC}\) would be obtained by a perfectly balanced division of solar photons through the N junctions, as Eq. (10) describes:

$${I}_{MPP}^{MJSC}\approx \frac{{\sum }_{i=i}^{N}{I}_{{G}_{i}}}{N}=\frac{q}{N}{\int }_{0}^{{\lambda }_{Ga{p}_{1}}}[\frac{{W}_{sun}(\lambda )}{hc/\lambda }]\,d\lambda $$
(10)

After this, the best candidate is found by calculating the bandgap energies leading to the same currents \({I}_{{G}_{i}}={I}_{MPP}^{MJSC}\) through Eq. (11):

$$q{\int }_{0}^{{\lambda }_{Ga{p}_{N}}}[\frac{{W}_{sun}(\lambda )}{hc/\lambda }]\,d\lambda =q{\int }_{{\lambda }_{Ga{p}_{N}}}^{{\lambda }_{Ga{p}_{N-1}}}[\frac{{W}_{sun}(\lambda )}{hc/\lambda }]\,d\lambda =\ldots =q{\int }_{{\lambda }_{Ga{p}_{2}}}^{{\lambda }_{Ga{p}_{1}}}[\frac{{W}_{sun}(\lambda )}{hc/\lambda }]\,d\lambda ={I}_{MPP}^{MJSC}$$
(11)

The first estimative is then the starting point for the search of the best bandgap energy combination leading to the highest efficiencies. Under all conditions tested in this work concerning number of junctions (from 2 to 6) and illumination (AM0, AM1.5 g, AM1.5d under concentrations up to 3000 suns), the maximum difference between the correct bandgap energies leading to the highest efficiencies and the ones from the estimative is less than 0.05 eV and the difference between the optimum and the estimated value for \({I}_{MPP}^{MJSC}\) is less than 5%. The sweeping resolution in the bandgap energy was 0.01 eV in all optimization processes performed in this work.