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# Shockley–Queisser triangle predicts the thermodynamic efficiency limits of arbitrarily complex multijunction bifacial solar cells

Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved October 15, 2019 (received for review June 24, 2019)

## Significance

A solar cell converts sunlight directly to electricity and is an important source of renewable energy. The thermodynamic efficiency of a solar cell defines the ultimate limit of photoconversion and suggests strategies to achieve it. Recent progress in manufacturing, efficiency, and reliability has led to an astounding proliferation of the fixed-tilt single-junction PV technologies. The progress can be accelerated by a variety of innovative strategies (e.g., bifacial tandems, agro-PV, etc.). In a simple, single-line scaling formula, this paper unifies isolated and scattered results derived over the last 50 years and generalizes them to technologies whose thermodynamic limits are unknown. This work identifies promising concepts and their performance gain over traditional technologies.

## Abstract

As monofacial, single-junction solar cells approach their fundamental limits, there has been significant interest in tandem solar cells in the presence of concentrated sunlight or tandem bifacial solar cells with back-reflected albedo. The bandgap sequence and thermodynamic efficiency limits of these complex cell configurations require sophisticated numerical calculation. Therefore, the analyses of specialized cases are scattered throughout the literature. In this paper, we show that a powerful graphical approach called the normalized “Shockley–Queisser (S-Q) triangle” (i.e.,

- solar cells
- thermodynamic efficiency
- Shockley–Queisser
- scaling theory
- tandem, concentrator, bifacial cells

The efficiency of single-junction monofacial solar cells has been rising steadily over the years (1⇓–3) and in some cases they are beginning to approach the fundamental limits for single-junction solar cells predicted by Shockley and Queisser (S-Q) (4, 5). In addition, the knowledge gained from volume manufacturing, reduction in the installation costs, and the steadily increasing lifetime, etc., have reduced the levelized cost of electricity (LCOE) and accelerated the worldwide adoption of photovoltaic (PV) technology as the lowest-cost source of electricity in many locations of the world (6, 7). The LCOE can be further reduced by the increased use of tracking, bifacial solar cells, and multijunction technologies (8, 9). In particular, the intrinsic bifaciality of silicon heterojunction cells, the relatively simple conversion of silicon passivated emitter rear cell (PERC) to bifacial PERC+ technology, and the availability of large-bandgap perovskite and organic solar cells have encouraged experimentation involving new cell structures and module and farm topologies (e.g., bifacial tandem, luminescent solar concentrator tandem, agro-PV systems, etc.). A conceptual framework that can calculate the thermodynamic efficiencies of these emerging technologies, with the same level of rigor and generality of the original Shockley–Queisser model, will be of broad interest because it can be used to assess the performance limits and relative (intrinsic) merits of the new and complex PV systems.

As is well known, the original S-Q paper (4) offered a powerful incentive for efficiency improvement of single-junction solar cells by highlighting the opportunity of efficiency gain toward its thermodynamic limit. Similar work by Henry (10) and others has helped define the thermodynamic limits for multijunction tandem cells. Recent works on thermodynamic limits of the 2-junction (2-J) tandem cell (silicon, perovskite) (8); *N*-junction bifacial solar cells; and 3-J, 4-J, and 5-J concentrator PVs including the effect of series resistance have been discussed (11, 12). Other topics involving optimization for food, energy, and water (FEW) and the hydrolysis of water by multijunction tandem PVs have also been analyzed (13, 14). A literature review shows that the calculation of the relative performance gains of new PV concepts are nontrivial and require complicated numerical analysis. Such an analysis cannot transparently establish the functional relationship between the design parameters and ultimate photoconversion efficiency.

In this paper, we develop an intuitive but powerful graphical approach called the Shockley–Queisser triangle (S-Q triangle). This approach unifies the thermodynamic efficiency results of various types of solar cells scattered in the literature through simple scaling relationships. It also predicts the efficiency limits of emerging solar cell concepts (e.g., bifical tandem solar cells) for which the thermodynamic results are unknown. More importantly, it explains the intrinsic trends of nonlinear efficiency gain with cell number, how a 2-junction bifacial tandem cell outperforms a 4-junction monofacial tandem cell, the effect of series resistance on the choice of cell configuration, and so on. Although not the main focus of this paper, we demonstrate the power and the versatility of the S-Q triangle by calculating (and validating by experiments and numerical simulation whenever possible) the performance limits of relatively complex PV systems such as agro-PV, visibly transparent PV, etc. These results are summarized in *SI Appendix*.

## The Shockley–Queisser Triangle

The scaling analysis presented in this paper relies on two key observations related to the voltage and the current needed to produce the maximum output power of a solar cell, i.e., maximum power-point voltage (

It is well known that the maximum power-point current

Inserting Eq. **1** into Eq. **2**, and defining

In Fig. 1 *B* and *C*, Eq. **3** defines the S-Q triangle. Each point on the diagonal represents a material with bandgap **4b** expresses the fact that the tandem subcell currents (**1** identifies the material of interest with specific bandgap,

To calculate the overall efficiency, **4** and **5** is accurate within a few percent to the most sophisticated numerical analysis published to date.

## Model Validation by Results Scattered in the Literature

### Efficiency of Single-Junction PV with c = 1 .

The essential correctness of Eqs. **4** and **5** can be established by calculating the optimum efficiency of a single junction (SJ) cell under AM1.5G illumination. With

### Efficiency of Concentrated Solar Cells with c = 300 .

Since a SJ solar cell operates far below the Carnot limit (∼95%) and converts only one-third (∼34%) of the incident energy into useful power output, many solar cell innovations since the 1960s have focused on improving the efficiency of a photovoltaic converter. One of these approaches uses a parabolic mirror to concentrate sunlight onto a solar cell.

The calculation of efficiency limits of concentrator solar cells is difficult; the numerical results are available only for specific concentrations. Fortunately, the efficiency and bandgaps predicted by Eqs. **4** and **5** hold for any arbitrary concentration, and therefore the model can be validated by comparing with specific numerical results from the literature (11). For example, for

### Thermodynamic Efficiency of N-Junction Tandem Cell.

A second approach to improve the conversion efficiency of solar cells involves choosing a series of absorbers with different bandgaps so that they all produce an equal amount of current. The absorbers are then connected optically and electrically in series to improve photoconversion efficiency. The traditional optimization involves an iterative search to find the bandgap combination to maximize efficiency.

In contrast, Eq. **5** predicts that**6** with the most sophisticated numerical results available in the literature (10, 16). *SI Appendix*, Table S1 confirms that the maximum errors between Eq. **6** and numerical predictions are within a few percent (primarily attributed to the approximation **6** identifies the efficiency-gain scaling factor for tandem cells (i.e., *C* is fully tiled with boxes for *B*.

Finally, although the scaling factor is specifically derived for AM1.5G, the result is general and this model can capture the essential trends in other spectra (including black-body radiation) as well. The key point to remember is that the S-Q triangle method can be applied as long as there is linearity between

Let us analyze a few numerical examples. For **4** are 0.48 V, 0.96 V, and 1.44 V. The corresponding bandgaps are given by Eq. **1**: 0.83 eV, 1.33 eV, and 1.84 eV. The results are within 0.1 to 0.2 eV of the results reported in the literature (11). The results become even more accurate for larger N. The deviation reflects the nonlinearity of **6** anticipates the nonlinear dependence of

### Thermodynamic Efficiencies of 4- and 5-Junction Concentrator Tandem Cells with c = 300 .

The conversion efficiency is further improved when a tandem cell is placed under concentrated sunlight. Once again, the numerical optimization is so difficult that this has been reported for only a few specialized cases. The optimum bandgaps to maximize efficiency are obtained by a time-consuming iterative search over the bandgap space. Since Eqs. **4** and **5** apply to any N-junction tandem cell under arbitrary illumination, we can confirm its validity by comparing to a few specific results for 4-junction and 5-junction (11) cells. Table 1 and Fig. 3 show that both the bandgap sequence and the thermodynamic efficiencies compare well with the values reported in the literature.

### The Effect of Series Resistance on a Concentrated Solar Cell.

The early design of concentrator solar cells highlighted the need to account for the voltage drop in the series resistance in response to extremely large current in these systems. Once again, the problem is solved iteratively, and maximum efficiency associated with an arbitrary series resistance is not known.

The discussion in previous sections suggests that *c*, the solar concentration (Eq. **5**). In practice, the series resistance, **1** can be rewritten as **2**, so that Eq. **3** can now be rewritten with the following parameters:

With the renormalized axes, Eq. **5** anticipates the efficiency of a concentrator solar cell as a function of *A*. These values are also shown in Fig. 4*B* as a function of c—we can see a turnaround in efficiency at high c. The concentration-dependent efficiencies have been reported for *B* shows that the analytical results from Eq. **5** (open symbols) match very well the numerical results (solid symbols) reported in the literature.

Interestingly, Fig. 4 *A* and *B* anticipates a reduction in η beyond **5**, we find^{−2},*c*_{crit}=110.2. In other words, *C*. The corresponding efficiencies by Eq. **5** are 58.32% and 56.1%, respectively.

## Bifacial Tandem Solar cells: An Emerging Solar Cell Technology

Although the bifacial solar cell concept originated in the 1980s, recent technological innovations have made it competitive compared to monofacial cells. Bifacial panels are expected to capture 30% of the market share by 2030 (6). Despite its significant implications, the general thermodynamic limit of bifacial solar cells is not known (9). In this section, we show that the S-Q triangle not only captures the scaling trends, but also explains intuitively an unexpected discontinuous jump in efficiency when the cell number exceeds a critical value,

Fig. 5 shows the generalization needed to calculate the efficiency of a bifacial tandem cell. The generalized S-Q triangle accommodates the cells illuminated both from the top and from the bottom. With a “rear collected albedo” of R (this is the effective fraction of sunlight received by the rear surface, averaged over the spectrum, and including the asymmetric charge collection, or the “bifacial factor”, of the rear surface), the maximum normalized current for a rear subcell is **3** can therefore be rewritten as:

An interesting aspect of bifacial cells is that depending on the albedo, the cell with the smallest bandgap

In the bifacial tandem, there are U cells above and D cells below the **5**, the sum of the boxes gives the normalized power output, **11** in Eq. **8**, we find**9**–**13** define the maximum power from a stack of N cells illuminated at the back by the rear collected albedo, R.

In addition, Eq. **8** reduces to limiting expressions: **6**). The gain gradually diminishes at higher N as larger bandgap boxes tile the original triangle, consistent with Fig. 2. The triangles anticipate that the bottom cell can have the smallest bandgap (i.e.,

Using Eqs. **8**–**11** and **13** in Eq. **12**, we find that for **14** and **15** assuming **14** and **15** compare remarkably well with the numerical results published previously (9); see discussions in *SI Appendix*, section S3. The expression reduces to the limiting case of the traditional tandem cell for

Fig. 6 shows the relative output gain of monofacial vs. bifacial tandem solar cells. Bifaciality provides significant gain for a single-junction solar cell; however, the relative gain is reduced in a multijunction tandem cell. Recall that the upper cells convert a significant fraction of the sunlight that was previously lost to thermalization. However, the need for current matching with albedo-illuminated bottom cells requires bandgap reduction in the upper cells of the bifacial tandem compared to monofacial cells (equation 11 in ref. 9). The absolute energy output is still larger, but the fraction of thermalization loss is somewhat larger as well. The improvement of absolute normalized efficiency is still significant because albedo increases the energy output and simplifies current matching. For example, considering

The aforementioned conclusions regarding bifacial tandem cells apply strictly at the thermodynamic (radiative) limit with optimal bandgaps. In practice, fabrication constraints limit the bandgaps available for bifacial and monofacial tandem cells. The performance gain could be reduced for systems with nonoptimum bandgap sequence, finite absorption, nonradiative recombination, etc. We discuss these limitations in the next section.

## Discussion: Thermodynamic Limits of Nonideal Solar Cells

In sections *Model Validation by Results Scattered in the Literature* and *Bifacial Tandem Solar Cells: An Emerging Solar Cell Technology*, we used the S-Q triangle to calculate the thermodynamic (radiative) limit of ideal single-junction, tandem, bifacial, and concentrator solar cells. The results establish the S-Q triangle as a powerful tool to calculate the fundamental efficiency limits of a wide variety of cell technologies.

In practice, it is sometimes helpful to modify the S-Q analysis to calculate the corresponding “practical” thermodynamic limit that accounts for material-specific losses (e.g., incomplete absorption, nonradiative losses, self-heating, etc.) In this section, we show that the S-Q triangle can predict the performance of these cells as well. Indeed, the results in the literature are special cases of our general results.

### Imperfect External Quantum Efficiency and External Radiative Efficiency in a Tandem Solar Cell.

A solar cell cannot convert all of the incident above-bandgap photons to useful current. For a single-junction device, we can write

Imperfect ERE due to nonradiative recombination reduces the operating voltage by

For example, consider a conventional tandem solar cell where δ is the ERE loss of each of the subcells. By setting *SI Appendix*, section S4.C)**5**, as expected.

### Nonoptimum E g in Tandem PV.

In the previous discussions, we have assumed that the subcell bandgaps have been optimally chosen so that each subcell produces equal amount of current; i.e., the subcell currents are perfectly matched. If the optimum set of bandgaps is not available, the current in an N-junction tandem cell is limited by the subcell with the lowest current contribution (i.e., the lowest absorption). The corresponding efficiency is (see *SI Appendix*, section S4.B for details)*SI Appendix*, section S4.B, we show that Eq. **22** anticipates the experimental results shown in ref. 19. In addition, in *SI Appendix*, sections S4.C and S4.D, we explain how the S-Q triangle interprets other nonideal effects, such as self-heating.

## Conclusions

The S-Q triangle offers an efficient and powerful technique to derive the thermodynamic efficiency limits of a variety of classical (e.g., single-junction, tandem, and concentrators cells) and emerging (e.g., bifacial tandem cells) technologies. The sequence of optimum bandgaps and the thermodynamic limits of currents and voltages are easily derived and can serve as an intuitive check of the experimental data. The approach provides, as a function of subcell number, a scaling justification for the improvement in the tandem cell efficiency and abrupt increase in bifacial tandem cell efficiency. Moreover, the approach is easily modified to approximately account for the nonideal effects related to finite absorption and radiative and nonradiative recombinations.

The analysis presented in this paper would allow one to calculate the intrinsic performance limits and relative merits of a variety of technologies, including agro-PV and visibly transparent building-integrated PV. Just as in the original S-Q analysis, however, the thermodynamic analysis presented in this paper must be supplemented by a detailed opto-electronic (8) and techno-economic analysis of various technology options (6, 7, 20). After all, the remarkable success of the Si photovoltaics through LCOE reduction is attributed both to the efficiency gain toward its thermodynamic limit and to the improvements in volume manufacturing, installation, and reliability.

### Data Availability.

All data generated are included in this paper and in *SI Appendix*.

## Acknowledgments

We acknowledge Reza Asadpour, Tahir Patel, Mark Lundstrom, Peter Bermel, Jeff Gray, and in particular the anonymous reviewers for their detailed and thoughtful suggestions regarding the manuscript. This work was partly supported by the National Science Foundation under Award 1724728.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: alam{at}purdue.edu.

Author contributions: M.A.A. and M.R.K. performed research, analyzed data, and wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1910745116/-/DCSupplemental.

Published under the PNAS license.

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