## 1.

## Introduction

Thermophotovoltaics (TPV) convert heat to electricity via thermal radiation. Photons with energies below the bandgap of the photovoltaic (PV) diode, resulting from the broad spectrum of the Planck blackbody distribution, are generally the dominant source of loss in TPV systems. Fortunately, these energy losses can be eliminated by selective emitters that have near-blackbody emission above the PV bandgap and low emission below the PV bandgap.^{1} Several materials have been proposed for selective emission, including plasmonic metamaterials,^{2}3.^{–}^{4} refractory plasmonic structures,^{5} rare earth materials,^{6}7.^{–}^{8} and photonic crystals (PhCs).^{9}10.11.12.13.14.15.16.17.18.19.20.21.22.23.^{–}^{24} However, realistic selective emitters still have residual low energy emission near the bandgap that can considerably limit the conversion efficiency. Significant improvement can be achieved by the use of cold-side PhC filters, including plasma filters, quarter-wave stacks,^{25} and rugate filters.^{26} These filters essentially reflect the low-energy photons back to the selective emitter, in a process known as photon recycling.^{27}28.29.30.31.^{–}^{32} In order to achieve sufficient photon recycling, proximity between the emitter and filter is required.^{33} In the typical cold-side filter configurations, where the filter is attached to the PV diode as an entire receiver, this requirement can be quantified by the view factor from the emitter to the receiver, which is the probability that emitted photons reach the receiver. Certain strategies, such as micro-gap or nanoscale-gap TPV, in fact require extremely high view factors to achieve evanescent coupling.^{34}35.^{–}^{36} However, it is extremely difficult to achieve high view factors in experiments, since a constant gap $d$ must be maintained between two surfaces when the distance is orders of magnitude smaller than the lateral width $w$ of each surface. The angular tolerance ${\theta}_{\mathrm{tol}}={\mathrm{tan}}^{-1}(d/w)$ is impractically low. Although ultranarrow (scanning electron microscope-like) tips can achieve such small gaps, the associated power produced is extremely small.

Despite the great potential for the improvement of TPV associated with these various approaches, certain common challenges remain in translating these ideas into experiment. One major difficulty is in achieving and maintaining sufficiently selective thermal emission. While simpler structures can be fabricated with high fidelity, their selectivity is often lacking, while more complex monolithic structures with higher potential selectivity are more vulnerable to fabrication error and degradation, which becomes even more severe with finer features.^{37} Composite structures consisting of simple emitters and cold-side PhC reflectors also have potential for highly selective effective emissivity, but these give rise to a second common problem: achieving precise alignment between a hot emitter and cold receiver. Unfortunately, configurations requiring the greatest precision are the same as those with the highest power generation potential. In order to overcome this challenge, it is important to fundamentally reexamine the geometric configuration of the three key objects: emitter, filter, and PV diode. While it is exceedingly difficult to bring the PV diode in contact with the emitter, it is certainly conceivable to bring the emitter and filter together. However, it is quite likely that a filter made exclusively of insulating materials would be a poor conductor of heat and thus would experience a significant thermal gradient. Therefore, a different treatment is needed, which correctly accounts for both the thermal gradient and potential for incoherent radiative exchange between the emitter and filter in proximity.

Thus, in this work, we propose an alternative approach for high-performance TPV systems capable of simultaneously addressing problems with long-lasting selectivity and alignment, known as an integrated photonic crystal selective emitter (IPSE), and develop a corresponding framework to analyze its performance. The IPSE combines the PhC selective emitter and the filter as an integrated whole, such that the view factor from the emitter to the filter approaches unity and is independent of the view factor from the emitter to the receiver (PV diode). This approach also sidesteps the difficulties of the nanoscale-gap regime, since no gap is required. Photon recycling and reabsorbed power can therefore no longer be considered as a perturbation;^{33} instead, new physical effects should arise in this nonperturbative regime. Finite difference time-domain (FDTD)^{38} and current transport simulations have shown that an IPSE enables system conversion efficiencies to reach up to 41.8%, using the realistic model for GaSb PV diodes, by nearly eliminating sub-bandgap emission. This corresponds to showing that the 10.9% gap from the ideal efficiency value observed in previous work^{39} can now be reduced to 4.3% using IPSEs. Arguably, the IPSE creates a nonzero chemical potential for photon emission using a purely thermal input source, which is a fundamentally distinctive physical behavior akin to light-emitting diodes.^{40} Further, the IPSE is relatively amenable to fabrication in the near term, with key features already achieved in previous experimental work.^{20}21.^{–}^{22}^{,}^{25}^{,}^{26}

## 2.

## Design of the Integrated Photonic Crystal Selective Emitter

As shown in Fig. 1(a), the IPSE consists of a metal two-dimensional (2-D) PhC and a chirped quarter-wave stack (QWS), which are brought to an extremely small separation. The metal 2-D PhC consists of an array of holes filled entirely with low refractive index material, so that the bilayer structure can be fabricated and firmly attached to the emitter without greatly increasing sub-bandgap emission.^{41}^{,}^{42} The chirped QWS structure placed directly atop the PhC consists of two alternating dielectric materials with different refractive indices. In such an integrated design, where the view factor approaches unity, all thermal photons emitted from the front surface of the 2-D PhC would be filtered. For the highest possible performance, it is essential to have either a backreflector or another PV diode on the opposite side.

In this work, the IPSE is examined directly as a whole using a TPV simulation tool developed by the present authors.^{43}^{,}^{44} The emittance spectrum associated with the design is first calculated using an FDTD simulation method known as MIT electromagnetic equation propagation.^{45} Then the current–voltage $(J-V)$ relation for TPV cells is calculated from the emittance spectrum assuming zero series resistance, as in previous work.^{33} The effect of finite series resistance is considered later in this work. The power output is then calculated by solving for the maximum power point via $d(JV)/dV=0$. Based on the experimental performance of metalorganic vapor phase epitaxial grown low-bandgap III-V PV diodes from Lincoln Laboratories, an average external quantum efficiency of 82% is assumed.^{33} The thermal power input is calculated as the total thermal emission less the power reabsorbed by the emitter. The system conversion efficiency $\eta $ is then finally given as

## (1)

$$\eta =\frac{{I}_{\mathrm{sc}}{V}_{\mathrm{oc}}\mathrm{FF}}{{P}_{\mathrm{em}}-{P}_{\mathrm{re}}}.$$The geometry of the 2-D PhC emitter follows from our recent work on the performance of tungsten (W) PhC selective emitters.^{39} Previously, we found that for a W PhC emitter paired with a cold-side rugate filter,^{46} the highest estimated $\eta =35.2\%$ occurs at ${E}_{g}=0.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ and $T=1573\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{K}$, where ${E}_{g}$ is the bandgap energy of the PV diode and $T$ is the operating temperature of the emitter. A suitable geometry for selective emission is determined using the technique of quality factor-matching, in which the radiative coupling rate is matched with the material absorption rate to achieve full absorption. In this proposed structure, the array of holes is filled with ${\mathrm{SiO}}_{2}$; therefore, the optimal 2-D hole array geometry needs to be rescaled by the refractive index ${n}_{{\mathrm{SiO}}_{2}}=1.46$, giving $r=0.2604\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, $d=2.1172\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, and ${a}_{x}={a}_{y}=0.6356\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$. Here, $r$ is the hole radius, $d$ is the hole depth, and ${a}_{x}$ and ${a}_{y}$ are the periods of the array along the $x$ and $y$ directions. Maxwell’s equations make these photonic structures completely scalable in principle.^{10} The key constraint is to find a pair of materials with sufficient refractive index values and very low losses; this criterion generally excludes frequencies in ultraviolet or above.

It will now be shown that IPSEs can lead to performance improvements over bare W PhCs, which help approach the limits of an ideal selective emitter. For specificity, we consider the case of a bandgap ${E}_{g}=0.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$. The ideal selective emitter for such a bandgap has a top hat emittance spectrum depicted by the red curves in Fig. 2, corresponding to 100% emittance between 1300 and 1771 nm, and zero emittance elsewhere. This property fully suppresses sub-bandgap losses and greatly reduces carrier thermalization,^{47} yielding an ideal emission conversion efficiency ${\eta}_{\text{ideal}}=46.1\%$.

## 2.1.

### Integrated Photonic Crystal Selective Emitter with Chirped Quarter-Wave Stack

Now we consider the efficiency of an appropriate IPSE structure for comparison. The integrated ${\mathrm{SiO}}_{2}/{\mathrm{TiO}}_{2}$ (${n}_{{\mathrm{TiO}}_{2}}=2.50$) QWS has a cutoff wavelength at 1771 nm, matching the PV bandgap energy of 0.7 eV. The bilayer structure has an initial bilayer thickness of 497 nm, which is chirped linearly upward at a rate of 5.1% per bilayer to make a broadband filter. It is important to limit the chirping rate and material choices here to suppress the emergence of anomalies in the photonic density of states (analogous to a Wannier-Stark ladder in electronic structures). At the selected chirping rate, the emittance spectrum of a 15-period QWS IPSE is simulated and displayed as a black curve in Fig. 2(a). At $T=1573\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{K}$, the spectrally averaged effective emittance from 1771 to 9000 nm is ${\u03f5}_{\mathrm{eff}}=0.02$; for useful emission, it is ${\u03f5}_{\mathrm{eff}}=0.71$. These calculations assume that the material parameters are unchanged from literature values at the targeted temperature.^{48} Compared to the bare W PhC (without an integrated filter) results, shown as the blue curve in Fig. 2(a), it is immediately obvious that the IPSE strongly suppresses parasitic emittance from the cutoff wavelength of 1771 up to 4000 nm, but strong parasitic emission occurs between 4000 and 9000 nm, as shown in Fig. 2(a). The conversion efficiency $\eta $ calculated using the emittance spectrum up to 9000 nm is 38.2%. Thus, an increase in $\eta $ of 3% is achieved at 1573 K, since the major parasitic emission is beyond the blackbody peak.

The performance of the 15-period QWS IPSE here is still 7.89% below the ideal case. The reason for this difference can be analyzed by comparing with the ideal emitter and quantifying the loss mechanism. Three sources of efficiency losses are considered in Table 1: (1) emission at wavelengths shorter than 1300 nm, (2) imperfect emission between 1300 and 1771 nm, and (3) parasitic emission at wavelengths from 1771 to 9000 nm. Wavelengths beyond 9000 nm are emitted even at room temperature; the power is small enough in this region to be neglected. Even if we assume a constant emittance of 0.1 from 9000 up to 15,000 nm, the efficiency loss at 1573 K is $<0.1\%$. The primary nonideality comes from parasitic emission, particularly between 4000 and 9000 nm, which contributes a 4% efficiency loss. Therefore, further suppression of the parasitic emission is required. However, increasing the number of bilayers or the chirping rate will shift the high-order reflections to longer wavelengths, suppressing useful emission between 1300 and 1771 nm, and thus is not a viable option.

## Table 1

Classification of physical loss mechanisms affecting TPV efficiency η. By modifying the structure of the IPSE, parasitic emission is reduced while useful emission is enhanced, allowing efficiencies to more closely approach the ideal value of 46.1%.

IPSE structure | Conversion efficiency η (%) | Total losses in efficiency (%) | Short-wavelength emission (%) | Insufficient emission (%) | Parasitic emission (%) |
---|---|---|---|---|---|

15-period QWS IPSE | 38.2 | 7.89 | 0.80 | 1.71 | 5.38 |

15-period QWS + 30-period rugate filter IPSE | 41.8 | 4.29 | 0.89 | 1.66 | 1.74 |

## 2.2.

### Integrated Photonic Crystal Selective Emitter with Chirped Quarter-Wave Stack and Chirped Rugate Filter

In order to suppress parasitic emission while preserving useful emission, we instead introduce a chirped 30-period rugate filter on top of the 15-bilayer IPSE structure. Rugate filters use a different refractive index profile, more akin to a sinusoid, which eliminates higher-order reflections.^{26} In this case, our rugate filter is composed of six different materials, with varying indices of refraction from ${n}_{{\mathrm{SiO}}_{2}}=1.46$ to ${n}_{{\mathrm{TiO}}_{2}}=2.50$. Each layer thickness is determined by following the type C design from Carniglia,^{46} with the cutoff wavelength designed at 3500 nm and a linear chirping rate of 3.5%. The simulated emittance spectrum in Fig. 2(b) shows that the rugate filter provides effective suppression of parasitic emission from 4000 to 9000 nm. The effective emittance of parasitic emission from 1771 to 9000 nm is greatly reduced to ${\u03f5}_{\mathrm{eff}}=0.004$, while that of useful emission actually increases to ${\u03f5}_{\mathrm{eff}}=0.74$, using the same assumptions as before. Overall conversion efficiency $\eta $ is thus increased to 41.8%. Similar analysis of loss mechanism for the IPSE with 30-period rugate filter on top is summarized in Table 1. The efficiency loss due to the first two categories is close to the 15-bilayer IPSE structure, showing that the emission at wavelengths shorter than 1771 nm is preserved by the pass-band of the rugate filter. The parasitic loss is significantly reduced by the broader stop-band of the rugate filter. Overall, the efficiency gap from the ideal case is substantially reduced to 4.3%, which is less than two-fifths of the original gap previously seen in the literature.^{39}

Assuming the presence of a large-area passive heat sink with convective and radiative cooling available on its outer surface, the best IPSE design leaves the temperature of the PV diode at 1.5 K above ambient when illuminated. It was also assumed that the PV diode has zero series resistance. Here, we consider the impact of a finite series resistance on the system conversion efficiency. A typical GaSb PV diode produced by JX Crystals has a series resistance ${R}_{s}=0.01\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\Omega}$ (the value is fitted from the experimental $I-V$ curve under illumination of a 1200°C emitter provided by JX Crystals). The light $I-V$ curve of GaSb PV diode illuminated by the 15-period QWS + 30-period rugate filter IPSE at 1573 K is calculated with ${R}_{s}=0.01\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\Omega}$, as seen by the black curve in Fig. 3. Compared to the case where zero ${R}_{s}$ is assumed, the fill factor decreases from 80 to 77% when ${R}_{s}=0.01\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\Omega}$, reducing the conversion efficiency to $\eta =40.3\%$.

## 3.

## Perturbative and Nonperturbative Photon Recycling

Next, the new physical effects that arise in TPV systems that use IPSEs are examined from the aspect of radiation heat transfer. Generally, the differential view factor for the $n$’th reflection between the emitter and the receiver can be defined as follows:

## (2)

$$d{F}_{n}=\frac{{\mathrm{cos}}^{2}\text{\hspace{0.17em}}{\theta}_{n}}{\pi {d}_{n}^{2}}d{A}_{n+2},$$^{49}${\theta}_{n}$ is the angle between the surface normal of the differential area $d{A}_{1}$ and the line connecting two differential areas $d{A}_{1}$ and $d{A}_{n+2}$. The integrated view factor ${F}_{n}$ can be calculated as shown in Ref. 50.

The $n$’th contribution to the short-circuit current for a given wavelength $\lambda $ is then calculated by

## (3)

$${I}_{\mathrm{sc}}(\lambda )=\sum _{n=\mathrm{0,2}\dots}^{\infty}{A}_{1}\frac{2qc\mathrm{E}\mathrm{QE}(\lambda ){({R}_{1}{R}_{2})}^{n/2}\u03f5(\lambda )}{{\lambda}^{4}\left[\mathrm{exp}\right(\frac{hc}{\lambda kT})-1]}{F}_{n}=\sum _{n=\mathrm{0,2}\dots}^{\infty}{B}_{n}(\lambda ){F}_{n},$$When a cold-side filter is applied, photon recycling in the TPV system is expressed using a power series (in the perturbative regime). The recycled power at a given wavelength $\lambda $ can be calculated as

## (4)

$${P}_{\mathrm{re}}(\lambda )=\sum _{n=\mathrm{1,3}\dots}^{\infty}{A}_{1}{I}_{\mathrm{BB}}(\lambda ){R}_{2}^{(n+1)/2}{R}_{1}^{(n-1)/2}(1-{R}_{1})\u03f5(\lambda ){F}_{n}=\sum _{n=\mathrm{1,3}\dots}^{\infty}{C}_{n}(\lambda ){F}_{n},$$However, in the nonperturbative regime, the view factor from the emitter to the filter approaches unity, while the view factor from the IPSE to the PV diode is still ${F}_{0}$. The reabsorbed power ${P}_{\mathrm{re}}$ is now a constant instead of a function of ${F}_{1}$ (or ${F}_{0}$). Thus, ${F}_{\mathrm{eff}}={F}_{0}$, so Eq. (5) should be modified as

where ${B}^{\prime}=\int {B}_{0}^{\prime}(\lambda )\mathrm{d}\lambda $; ${B}_{0}^{\prime}(\lambda )$ is similarly defined by Eq. (3) but using different parameters. Since the photon recycling perturbation is eliminated, linear view factor dependence is thus expected in the nonperturbative regime.The system conversion efficiency of a TPV system using IPSE is now simulated at different view factors (with respect to the receiver), using a TPV simulation tool developed by the authors. For comparison, the same approach is also employed to simulate a TPV system with a cold-side rugate filter attached to the PV diode (in the perturbative regime). Equations (5) and (7) are then used to fit the results of the perturbative and the nonperturbative cases respectively, as shown in Fig. 4. The fits are in perfect agreement with the simulation results in each case. The linear view factor dependence of a TPV system where IPSE is applied shows that the photon recycling is no longer influenced by the spacing between the emitter and the receiver. Thus, an extremely high view factor for sufficient photon recycling is no longer necessary. The distinction between the behavior in the perturbative and nonperturbative regimes is very accurately described by Eqs. (5) and (7). Therefore, ${F}_{\mathrm{eff}}$ correctly captures the physics of photon recycling in TPV systems and can be used as a metric to describe most TPV systems (both in the perturbative and nonperturbative regimes).

## 4.

## Thermophotovoltaics System with Photonic Crystal Waveguide

Furthermore, ${F}_{\mathrm{eff}}$ can be increased much further by surrounding the gap between the emitter and the receiver with a waveguide as shown in Fig. 5(a). The requirements for high reflectivity at specific wavelengths and low thermal conductivity could be satisfied by the use of dielectric PhC waveguides. To include the impact of the waveguide on the TPV systems, the expressions for ${F}_{0}$ and ${F}_{1}$ can be generalized as

## (9)

$${F}_{1}^{\mathrm{wg}}\approx {F}_{1}+({F}_{0}^{\mathrm{wg}}-{F}_{1})\frac{{f}^{2}R\gamma}{1-Rf(1-\gamma )},$$## (10)

$${F}_{\mathrm{eff}}^{\mathrm{wg}}\equiv \frac{{F}_{0}^{\mathrm{wg}}(1-{\kappa}^{\mathrm{wg}})}{1-{\kappa}^{\mathrm{wg}}{F}_{1}^{\mathrm{wg}}},$$## 5.

## Conclusion

In conclusion, the IPSE and waveguide eliminate losses, due to the nonideal photon recycling between the emitter and the receiver, as well as the difficulty of close alignment between a hot emitter and cold receiver. This approach can increase the theoretical conversion efficiency of a TPV system from the prior value of 35.2% up to 41.8% at 1573 K. Such behavior is described by analytical expressions based on our extended definition of the effective view factor ${F}_{\mathrm{eff}}$. Our approach captures the physics of TPV systems where Kirchoff’s law cannot be applied directly, particularly in the presence of photon recycling and waveguides. It is also consistent with previous results in the field.^{20}^{,}^{41}^{,}^{42} This extended definition of effective view factor can be broadly applied to any TPV system, and could even lead to further improvements in system conversion efficiencies and experimental fabrication, through the use of dielectric PhC waveguides.

## Acknowledgments

We thank Qingshuang Chen for providing access to simulation capabilities for this study, as well as Enas Sakr and Chao Zhou for valuable discussions. Support was provided by the Department of Energy, under DOE Cooperative Agreement No. DE-EE0004946 (PVMI Bay Area PV Consortium), the Semiconductor Research Corporation, under Research Task No. 2110.006 (Network for Photovoltaic Technologies), and the National Science Foundation, under Award EEC1454315-CAREER: Thermophotonics for Efficient Harvesting of Waste Heat as Electricity. Computational resources on nanoHUB.org were provided by the Network for Computational Nanotechnology, which is funded by the U.S. National Science Foundation under Grant No. EEC-1227110.

## References

## Biography

**Zhiguang Zhou** is a graduate research assistant at Birck Nanotechnology Center, Purdue University. He received his BS degree in physics from Fudan University, China, in 2013. He is now a direct PhD student working with Dr. Peter Bermel in the School of Electrical and Computer Engineering at Purdue. His current research interests include high-performance (solar) thermophotovoltaic systems and novel selective thermal emitters.

**Omar Yehia** is an undergraduate research assistant at Maurice J. Zucrow Laboratories at Purdue University. He is currently completing his Bachelor of Science in mechanical engineering from Purdue University. His current research interests include thermophotovoltaic systems, microscale combustion, and nanoscale composite energetic materials.

**Peter Bermel** is an assistant professor of electrical and computer engineering at Purdue University. His work improves photovoltaic, thermophotovoltaic, and nonlinear systems using the principles of nanophotonics. He has widely published in scientific peer-reviewed journals, and has been cited 3500 times. Key topics include understanding light trapping in thin-film photovoltaics; fabricating three-dimensional inverse opal photonic crystals for photovoltaics; designing high-performance thermophotovoltaic systems; and enabling photon recycling for high-efficiency lighting.