Efficiency of 40% Alina LaPotin,1 Kevin L. Schulte,2 Myles A. Steiner,2 Kyle Buznitsky,1 Colin C. Kelsall,1 Daniel J. Friedman,2 Eric J. Tervo,2 Ryan M. France,2 Michelle R. Young,2 Andrew Rohskopf,1 Evelyn N. Wang,1 Asegun Henry1a 1Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA, 02139, USA 2National Renewable Laboratory, Golden CO, 80401, USA aAuthor to whom correspondence should be addressed: [email protected] Abstract We report the fabrication and measurement of thermophotovoltaic (TPV) cells with efficiencies of >40%. The TPV cells are 2-junction devices with high-quality 1.0-1.4 eV materials that target high emitter of 1900-2400°C. These cells can be integrated into a TPV system for thermal energy grid storage (TEGS) to enable dispatchable . With these new TPV cells, TEGS has a pathway to reach sufficiently high efficiency and sufficiently low cost to enable full decarbonization of the grid. Furthermore, the high demonstrated efficiency gives TPV the potential to compete with turbine-based heat engines for large-scale power production with respect to both cost and performance, thereby enabling possible usage in natural gas or hydrogen-fueled production.

Main Thermodynamics initially developed as a framework for understanding and improving the performance of heat engines as they fueled the industrial revolution. Since then, heat engines have gone on to play a pivotal role in modern society, as more than 90% of electricity today is generated by heat engines. In reflecting on this history, it is remarkable that one specific class of heat engines emerged as the primary workhorse of the power industry, namely those using the turbine, which is used for electricity production from coal, natural gas, nuclear and concentrated solar. Turbines proliferated because of their high efficiency (25-60%) and their low cost per unit power (CPP) generated ($0.5-1/W). However, they have also reached their practical limits in terms of cost and efficiency, barring a materials discovery that would allow them to operate at substantially higher turbine inlet temperatures than the current values of ~1500 °C for Brayton cycles and ~700 °C for Rankine cycles.1 However, since turbines intrinsically require moving parts, there are corresponding requirements on the high- mechanical properties of the materials of construction, since they are subject to centrifugal loads. Henry and Prasher1 highlighted that -state heat engines have an advantage in this sense. Thermal energy grid storage (TEGS), as described by Amy et al.,2 relies on (TPV), a solid-state heat engine, to convert heat to electricity above 2000°C which is a regime inaccessible to turbines. Although TEGS was initially conceived with a molten silicon storage medium in mind, a graphite storage medium is even lower cost ($0.5/kg), and the projected capital cost per unit energy (CPE) is less than $10/kWh.3 This cost is so low, it would allow TEGS to meet the proposed cost targets (< $20/kWh) for long duration energy storage that would enable renewable energy with storage to be cost-competitive with fossil fuels.4-6 As a result, the proliferation of TEGS could ultimately enable abatement of ~40% of global CO2 emissions, by decarbonizing the grid (~25% of emissions), and then enabling -free electricity to charge vehicles in the transportation sector (~15% of emissions).7 The work of Amy et al.2,8 showed initial feasibility for operating a thermal energy storage infrastructure above 2000°C, but the lynchpin to realizing the full system is the demonstration of high efficiency TPV cells. Like (PV), TPV uses the to convert to electricity. However, with TPV, the source of light is a terrestrial object that is heated to a high temperature that predominantly emits (IR) light to the TPV cell, as opposed to light from the sun. In 1980 Swanson showed that a silicon TPV cell with an integrated back surface reflector (BSR) and a tungsten emitter at 2000°C could reach 29% efficiency.9 In the ~40 years since Swanson’s initial demonstration (see Fig. 1), TPV fabrication and performance have greatly improved using III-V and the development of high-quality BSRs.10,11 This has opened the path to achieving much higher efficiency by replicating similar advancements with materials that have higher bandgaps, multiple junctions, and using higher emitter temperatures. However, despite the fact that a number of studies have presented model predictions that TPV efficiencies can exceed 50%,10,12,13 the demonstrated efficiencies are still only as high as 32%, but remarkably at much lower temperatures < 1300°C.12-14

Figure 1. Historic TPV efficiency11 with cell materials Ge15,16 (blue), Si9 (gray), GaSb17 (purple), InGaAs12,14,18-20 (green), InGaAsSb21 (light blue), and GaAs13 (yellow). The black line shows the average thermal efficiency of power generation in the U.S. using a steam turbine (coal and nuclear).22,23 Prior to 2000, turbine efficiencies shown also include natural gas. Here we report TPV efficiency measurements of > 40%. This is a major advancement, because it is higher than the average heat engine efficiency in the U.S. which is < 35% based on primary energy inputs and electricity output.24 An efficiency of 40% is also higher than most steam cycles, and is in the same range as simple cycle gas turbines.25 Thus, 40% represents a major step forward (see Fig. 1), as this is the first time another type of heat engine has the potential to compete with turbines by exhibiting comparable efficiency and potentially even lower CPP e.g., < $0.25/W.2,26 To properly contextualize why this has broad reaching implications, it should be appreciated that over the last century, a range of alternative heat engines such as thermoelectrics,27 thermionics,28 TPV,11 thermally regenerative electrochemical systems,29 thermoacoustic engines,30 and Stirling engines31,32 have been developed. All these technologies have some intrinsic advantage(s) over turbines, such as low maintenance, no moving parts, and/or easier integration with an external heat source, yet none of them can compete with the efficiency and CPP of turbines for large-scale heat to electricity conversion. The temperature regime examined here is accessible via combustion of natural gas or hydrogen, which could be made into an efficient power generation system by using recuperators made from refractory and oxides.17,33 With TPV now being a potentially cheaper alternative to a turbine with comparable performance, a broad range of potential applications, such as TEGS,2,34 small-scale natural gas or hydrogen-fueled power generation17,33,35- 40 and high-temperature industrial waste heat recovery are now feasible.

High Efficiency TPV Cells It is important to appreciate that the efficiency of a TPV cell is defined differently from that of a because, unlike a solar cell, a TPV system can preserve and later convert the energy associated with sub-bandgap . This is because in the contexts where TPV is envisioned to be used, the emitter has a high view factor to the TPV cell, meaning that sub-bandgap photons can be reflected back to the emitter by the TPV cell, in contrast to the situation of a solar cell and the sun. By doing so, the energy of the sub-bandgap light is preserved through reabsorption by the emitter. The reflected light helps to keep the emitter hot, thereby minimizing the amount of heat that is wasted by affording it another opportunity to be emitted and captured by the TPV cell. As a result, the efficiency of a TPV cell is given by,

푃 휂푇푃푉 = (1) 푃+푄푎푏푠

In Eq. 1, 푃 is the electric power generated by the TPV cell (i.e., 푃 = 푉표푐퐼푠푐퐹퐹 ), where 푉표푐 is the open circuit voltage, 퐼푠푐 is the short circuit current and 퐹퐹 is the fill factor. The total heat absorbed by the cell is denoted by 푄푎푏푠, which is comprised of the heat generated by parasitic absorption in the or reflector, thermalization losses due to excess incident energy, Joule heating losses due to current flow, and nonradiative recombination losses. Based on Eq. 1, to increase TPV efficiency, one must increase the power output 푃 relative to the amount of heat absorbed 푄푎푏푠. The high emitter temperatures targeted here for TEGS and other applications allow higher bandgap cells ≥1.0 eV to be used instead of the low bandgap, InGaAs- or GaSb-based cells traditionally used for TPV. This is key, because the spectrum of light redshifts toward longer wavelengths as the radiator temperature is lowered, which is why traditional TPV cells that are paired with <1300°C emitters are typically based on ~0.74 eV InGaAs or ~0.73 GaSb. Considerable work on low bandgap semiconductors has been undertaken with the envisioned application of converting heat from natural gas combustion,17,33,35-40 concentrated ,26 space power applications,41,42 and more recently energy storage.2,34,43 This pioneering body of work has led to the identification of three key features that now enable TPV to become a competitive option for converting heat to electricity commercially, namely: (1) the usage of higher bandgap materials in combination with higher emitter temperatures, (2) high-performance multi-junction architectures enabled by high-quality metamorphic ,44 and (3) the integration of a high reflectivity BSR for band-edge filtering.10,12 With respect to higher bandgaps, they increase efficiency because there is an almost constant penalty on voltage of ~0.3-0.4 V, due to the thermodynamic requirements on the radiative recombination rate.45 As a result, this unavoidable loss penalizes lower bandgap cells more than higher bandgap cells, because this loss comprises a smaller fraction of the voltage for higher bandgap materials. Using higher bandgap materials also needs to be accompanied by operation at higher temperatures to maintain sufficiently high power density, which scales with the emitter temperature to the fourth power. Operation at high power density is critical for TPV economics since the cell costs scale with their area, and if the power generation per unit area increases the corresponding CPP decreases.1

With respect to BSRs, a highly reflective BSR is critical to minimize 푄푎푏푠. Highly reflective BSRs provide the additional benefit of boosting open-circuit voltage, because they also improve recycling of luminescent photons.46-48 This effect has led to regular integration of BSRs with solar PV cells, which provides a template for their use in TPVs. With these important lessons from prior work in mind,14 the cells used here were 1.2/1.0-eV and 1.4/1.2-eV 2-junction designs intended for the TEGS application with emitter temperatures between 1900-2400°C.2 Multi-junction cells increase efficiency over single junctions by reducing hot carrier thermalization losses and reducing resistive losses by operating at a lower current density. The cells were based on the inverted metamorphic multi-junction (IMM) architecture pioneered at NREL.49-51 The first cell design uses lattice-mismatched 1.2 eV AlGaInAs and 1.0 eV GaInAs top and bottom junctions, where the lattice mismatch is with respect to the crystallographic lattice constant of the GaAs substrate. The second design uses a lattice-matched 1.4 eV GaAs top cell and a lattice- mismatched 1.2 eV GaInAs bottom cell, taking advantage of the inherently higher material quality of lattice-matched epitaxy in the GaAs cell. This lower bandgap 1.2/1.0-eV tandem offers the potential for higher power density than the 1.4/1.2-eV tandem because it converts a broader band of the incident spectrum, and consequently the requirements on the BSR are less stringent to obtain high efficiency.43 Higher power density can also be a practical engineering advantage. On the other hand, while the 1.4/1.2-eV tandem has a lower power output, the reduced current density of this bandgap combination potentially enables significantly higher efficiency than the 1.2eV/1.0-eV, due to lower resistive losses.

Figure 2 . a) Reflectance of the 1.4/1.2 - eV and 1.2/1.0 -eV tandems. The 2150 °C blackbody spectrum corresponding to the average emitter temperature in the range considered is shown. b) Internal efficiency (IQE) of the 1.4/1.2-eV and 1.2/1.0-eV tandems. c) Current density/voltage curves measured in the efficiency setup at varying emitter temperatures for 1.4/1.2- eV and d) 1.2/1.0-eV tandems.

TPV Efficiency Measurement Results The TPV cell fabrication, measurement, and modeling details are provided in the Methods section. The cell labeled MT546 is a 1.4/1.2-eV tandem and that labeled MT671 is a 1.2/1.0-eV tandem. Reflectance measurements are shown in Fig. 2a and internal (IQE) is given in Fig. 2b. The sub-bandgap spectral weighted reflectance for the 2150 °C blackbody spectrum is 93.0% for MT546 and 93.1% for MT671. Current density vs voltage measurements were performed under a tungsten halogen bulb emitter and results for a range of emitter temperatures relevant to the TEGS application (~1900 °C – 2400 °C) are shown in Fig. 2c and Fig. 2d. The non- monotonic change in 푉표푐 at the highest emitter temperatures is because of increasing cell temperature due to the presence of a heat flux sensor (HFS) used for the efficiency measurement, that undesirably also impedes heat flow. Figure 3a shows the key result, which is the efficiency measurement at the same range of emitter temperatures and was accomplished by directly measuring 푄푎푏푠. The results for MT546 show increasing efficiency with increasing emitter temperature, and the efficiency exceeds 40% at 2250 °C, which is within the target range of 1900-2400°C needed for the TEGS application. At 2400 °C, the efficiency is as high as 42.0% +/- 1%, while the average efficiency between 1900-2400 °C is 37.4%. The electrical power density is 2.39 W/cm2 at the maximum emitter temperature of 2401 °C. The reduction in slope in efficiency at high emitter temperatures is due to a reduction in FF due to increasing series resistance losses and a reduction in the increase in 퐽푠푐 due to the cell becoming current-limited by the bottom cell at ~2250 °C. The results for MT671 show greater efficiency than MT546 at lower emitter temperatures due to the lower bandgaps of MT671. The efficiency of MT671 reaches a maximum of 40.3% +/- 1% at 2127 °C, quite close to 2150 °C which is the temperature at which our device model predicted this bandgap combination is optimal.43 The reduction in efficiency beyond this temperature is due to the increasing series resistance and the reduction in the increase in 퐽푠푐 with temperature due to the cell becoming current-limited by the bottom cell at temperatures greater than 2150 °C. The electrical power density is 2.42 W/cm2 at the maximum emitter temperature measured of 2279 °C. Comparing the performance of the two cells across the range of emitter temperatures, they exhibit different characteristics which are advantageous for TEGS. The efficiency of MT671 is less sensitive to changes in emitter temperature and has a higher electrical power density at a given emitter temperature, while MT546 can reach higher efficiency at high emitter temperatures. Figure 3a also shows model predictions for efficiency and corresponding uncertainty of the model prediction. The agreement obtained between the modeled and measured performance suggests that the model can be extended to extrapolate how the performance would change with additional improvements, or other operating conditions. The most important TPV cell property that could be improved is its spectral-weighted sub-bandgap reflectance, 푅푠푢푏. Figure 3b shows how the efficiency would change if 푅푠푢푏 could be increased, assuming the same 퐽푠푐 as the Fig. 3a model and a cell temperature of 25 °C which is feasible at this 푉퐹푒푓푓 without the presence of the heat flux sensor. In this prediction, for a 2200 °C emitter temperature, the efficiency of MT546 exceeds 50% at 푅푠푢푏 = 97%. The reason this is noteworthy is because the present value of 푅푠푢푏 is considerably lower than what was achieved with the air bridge approach recently demonstrated by Fan et al.14 Their work demonstrating a reflectivity > 98% charts a pathway towards further efficiency improvements. If the air bridge approach developed by Fan et al. could be combined with the advancements demonstrated here, it could to efficiencies greater than 56% at 2250°C, or greater than 51% averaged over the 1900-2400°C temperature range.

Figure 3. a) TPV efficiency measured at different emitter temperatures ranging from 1880 °C to 2400 °C. The dashed lines show the model predictions, and the shaded regions show the uncertainty on the model predictions. b) Predicted efficiency of MT546 and MT671 as weighted sub-bandgap reflectance (푅푠푢푏) is extrapolated assuming the same 퐽푠푐 as the Fig. 3a model and a 25 °C cell temperature. The solid lines show the average efficiency within the TEGS operating temperature range of 1900 °C to 2400 °C. The shaded bands show the maximum and minimum efficiency within the temperature range. The vertical dashed lines show the average current value of 푅푠푢푏 based on the reflectance in Fig. 2a weighted by the TEGS spectrum.

Conclusions We report two-junction TPV cells with efficiencies of > 40% using a tungsten emitter with a temperature between ~1900-2400°C. The efficiency of MT546 (1.4/1.2-eV) reaches as high as 42.0% +/- 1% at 2400°C, with an average of 37.4% over the target temperature range. This new record high performance is enabled by the usage of multi-junction cells with bandgaps > 1.0 eV, which are higher bandgaps than have been traditionally used in TPV. The higher bandgaps enable the use of higher emitter temperatures, which correspond to the temperature range of interest for the low-cost TEGS energy storage technology.2 This temperature range is also applicable for natural gas or hydrogen combustion, and further demonstrations of integrated systems are warranted. Reaching 40% efficiency with TPV is remarkable from the standpoint that it now renders TPV as a heat engine technology that can compete with turbines. An efficiency of 40% is already greater than the average turbine-based heat engine efficiency in the USA, but what could make TPV even more attractive than a turbine, is its potential for lower cost (< $0.25/W),2,26 faster response times, lower maintenance, ease of integration with external heat sources, and fuel flexibility. This is noteworthy because turbine costs and performance have already reached full maturity, so there are limited prospects for future improvement since they are at the end of their development curve. TPV on the other hand is very early in its progress down a fundamentally different development curve. Consequently, TPV has numerous prospects for both improved efficiency (e.g., by improving reflectivity and lowering series resistance) and lowering cost (e.g., by reusing substrates and cheaper feedstocks). Thus, the demonstration of 40% efficiency represents an important step towards realizing the potential which can be achieved with increased attention and funding in the coming years as commercial applications emerge and become profitable.

Methods Efficiency Measurement To measure the TPV cell efficiency, we seek direct measurement of the two contributing quantities in Eq. 1, the power output 푃 = 푉표푐퐼푠푐퐹퐹 and the heat absorbed 푄푎푏푠. To test the cells under a well-controlled and relevant spectrum (emission from tungsten between 1900-2400°C for TEGS), a tungsten halogen lamp was used in combination with a concentrator. The concentrator consists of a silver-plated elliptical reflector behind the lamp and a compound parabolic reflector (CPC) obtained from Optiforms that further concentrates the light onto the cell. At the base of the CPC, a water-cooled aluminum aperture plate is suspended above the TPV cell (see Fig. 4). The area of the aperture is 0.312 cm2 and the active area of the cell is 0.7145 cm2. To keep the TPV cell cool it is mounted on a microchannel heat sink (M2, Mikros) that is water-cooled. To measure 푄푎푏푠, a heat flux sensor (HFS), model gSKIN XP obtained from greenTEG, is placed between the cell and the heat sink. Thermally conductive adhesive tape holds the HFS in place on the heat sink, and thermal paste provides thermal contact between the cell and the HFS. Electrical contact to the cell bus bars is accomplished using a pair of copper clips which are both electrically and thermally isolated from the heat sink using a piece of insulation. A pair of wires is connected to the bottom of each copper clip to perform a 4-wire measurement. The bottom side of the aluminum aperture plate is shielded with several layers of copper-coated Kapton and aluminum tape acting as a shield to reduce the radiative transfer between the aperture plate and the TPV cell. A DC power supply (Magna-Power) provides power to the tungsten halogen lamp and the voltage is controlled to achieve the desired emitter temperature. The lamp is rated for 5 kW at a 3200 K temperature, but the temperature and power are tuned down to the desired emitter temperature by controlling the voltage to the lamp using the power supply. The emitter temperature was determined by measuring the resistance of the tungsten heating element in the lamp and using published correlations on the temperature dependence of the electrical resistivity and resistance of tungsten filaments in incandescent lamps.52 First the cold resistance of the bulb was measured at the point of the bulb junction and at the point of contact with the power supply to determine the resistance of the electrical to the bulb. The hot bulb resistance was measured by subtracting the electrical lead resistance from the total resistance as determined from the voltage and current input to the DC power supply. The heat sink was mounted to the z-stage to allow for repeatable control of the TPV cell positioning with respect to the aperture, reflectors, and lamp. The TPV efficiency was measured by taking simultaneous measurements of 푃 and 푄푎푏푠. The electric power was measured using a source meter (Keithley 2430) by sourcing the voltage and measuring the current density at the maximum power point, and the heat absorbed by the cell was measured using the HFS beneath the cell. Due to the temperature-dependent sensitivity of the HFS an estimate of the average HFS temperature is needed, which is taken from the average of the hot and cold side temperatures. The cold side temperature is determined by measuring the average of the inlet and outlet cooling water to the heat sink and the hot side temperature is measured by a thermocouple placed underneath the cell. From the calibration certificate from the manufacturer, the sensitivity 푆 (μV/(W/푐푚2) is given by Eq. 2.

푆 = (푇푠 − 22.5)0.025 + 19.98 (2)

The impact of 푇푠 on 푆 is that 푆 changes by 0.125% per °C temperature change from the reference temperature of 22.5 °C. We estimate a temperature uncertainty of 푇푠 of +/- 10 °C which results in an equivalent heat flux measurement uncertainty of +/- 1.25%, which was included in the error bars for the efficiencies in Fig. 3a. It was found that when the aperture plate was moved such that the cell was completely shaded by the reflective aluminum plate and no light from the exit of the CPC could reach the cell, a small but non-negligible “background” heat absorption measurement (~0.1 W at the higher emitter temperatures) was measured when the lamp was on. A thermocouple was placed on the section of the aperture plate under the outlet of the CPC using reflective aluminum tape, and it was found that this section of the plate reached temperatures of ~85 °C. This causes parasitic background heat absorption in the cell, which is due to radiation from the aperture plate to the cell. To remove the impact of this heating, each efficiency measurement was first taken with the aperture opening positioned over the cell. Once the HFS heat absorption measurement reached steady state in this configuration, the aperture plate was moved to the side such that a background heating measurement could be taken with the lamp on. The steady-state background heating value was then subtracted from the total heat absorption measurement to obtain a slightly corrected 푄푎푏푠.

Figure 4. a) Schematic of the concentrator setup showing the relative placement of the ellipsoidal and compound parabolic reflectors, water-cooled aperture, TPV cell, HFS, and heat sink. b) Image of the concentrator setup. c) Schematic of the heat and electricity flows through the measurement device. Electric power is extracted by two copper clips which interface with the cell bus bars on the top surface of the cell and are thermally and electrically insulated from the heat sink.

Effective view factor calibration To compare the measured TPV cell performance to model predictions, the effective view factor, 53 푉퐹푒푓푓, was calculated according to the method described by Osterwald and shown in Eq. 3 and Eq. 4. We used an NREL-fabricated GaAs cell with measured EQE and a 퐽푠푐 that was measured at NREL on an XT-10 solar simulator (AM1.5D, 1000 W/m2) using a secondary calibration reference cell to set the intensity. Prior to an efficiency measurement, the GaAs cell was placed in the setup 푇푃푉 at the same location as the multi-junction cell using the z-stage. In Eq. 3, 퐽푠푐 is the short circuit 퐺173푑 current of the GaAs cell measured in the efficiency setup, 퐽푠푐 is the short-circuit current of the cell measured using the XT-10 simulator at NREL, 퐸푇푃푉(휆, 푇) is the Planck distribution blackbody spectral emissive power under the spectrum in the efficiency setup, and 퐸퐺173푑(휆) is the AM1.5D 2 spectrum. Both spectra are in units of W/m /nm. 푉퐹푒푓푓 is the ratio of the actual irradiance in the 푇푃푉 efficiency setup, 퐸푖푟푟푎푑푖푎푛푐푒, to the blackbody irradiance in the efficiency setup at the same test temperature, ∫ 퐸푇푃푉(휆, 푇)푑휆, (Eq. 4). 퐺173푑 ( ) ( ) 푇푃푉 퐽푠푐 ∫ 퐸푇푃푉 휆, 푇 퐸푄퐸 휆 휆 푑휆 퐽푠푐 = 푉퐹푒푓푓 × 2 × × ∫ 퐸퐺173푑(휆)푑휆 (3) 1000 W/m ∫ 퐸퐺173푑(휆) 퐸푄퐸(휆) 휆 푑휆 푇푃푉 퐸푖푟푟푎푑푖푎푛푐푒 푉퐹푒푓푓 = (4) ∫ 퐸푇푃푉(휆, 푇)푑휆 푇푃푉 Measurements of 퐽푠푐 were averaged across the range of emitter temperatures. For MT546, 푉퐹푒푓푓 = 12.20% and for MT671 푉퐹푒푓푓 = 13.06%. The differences are due to slight adjustments made to the setup between measurements of the two multi-junction cells. 푉퐹푒푓푓 was then used to form efficiency model predictions.

TPV Cell Modeling

Eq. 1 for TPV efficiency can also be written in terms of Eq. 5, where 푃푖푛푐 is the irradiance incident on the cell, 푃푟푒푓 is the flux reflected by the cell, 푃푖푛푐,푎 is the above-bandgap irradiance, 푃푖푛푐,푠푢푏 is the sub-bandgap irradiance, 푅푎 is the spectral-weighted above-bandgap reflectance, and 푅푠푢푏 is the spectral-weighted sub-bandgap reflectance.43 The denominator of the efficiency expression represents the net heat flux to the cell.

푃 푉표푐퐽푠푐퐹퐹 푉표푐퐽푠푐퐹퐹 휂푇푃푉 = = = (5) 푃 + 푄푎푏푠 푃푖푛푐 − 푃푟푒푓 푃푖푛푐 − 푃푖푛푐,푎푅푎 − 푃푖푛푐,푠푢푏푅푠푢푏 To model the numerator, or electric power portion of the efficiency expression, IV curves were taken using a tunable high-intensity pulsed solar simulator (T-HIPSS) at varying irradiance to determine predicted 푉표푐 and 퐹퐹 as a function of 퐽푠푐 for the simulated spectrum. The predicted 퐽푠푐 was calculated using the measured EQE, emitter temperature, and 푉퐹푒푓푓 according to Eq. 6.

∞ 푞 ∗ 푉퐹푒푓푓 퐽푠푐 = ∫ 퐸푄퐸(휆) ∗ 퐸푏(휆, 푇)휆푑휆 (6) 푐 ∗ ℎ 0

Modeling of 푉표푐 was performed assuming a 25 °C cell temperature, but to incorporate the impact 54 of cell temperature on 푉표푐, Eq. 7 was used. 퐸푔 is the bandgap energy as a function of temperature which was obtained from the literature.55,56 Cell temperature was measured using a thermocouple. Fig. 5 compares the measured 퐽푠푐, 푉표푐, and 퐹퐹 with the model predictions.

푑푉 1 3푘푇 푇 푑퐸푔 퐸푔 표푐 ≈ (푉 − + − ) (7) 푑푇 푇 표푐 푒 푒 푑푇 푒 The blackbody spectral emissive power, 퐸푏(휆, 푇) was used to determine 푃푖푛푐 as a function of the emitter temperature, 푇, and 푉퐹푒푓푓 (Eq. 8). ∞ 푃푖푛푐 = 푉퐹푒푓푓 ∫ 퐸푏(휆, 푇)푑휆 (8) 0 The reflectance, 휌(휆), was measured on two different instruments due to the range of the spectrum. The mid-IR sub-bandgap reflectance was measured using a Fourier-transform infrared (FTIR) spectrometer (Nicolet iS50) with an integrating sphere accessory (PIKE Mid-IR IntegratIR). A copper aperture with area ~0.35 cm2 was used over the sample port, and the spot encompassed both the cell and the front grids. The above-bandgap and NIR sub-bandgap reflectance was measured using a UV-Vis-NIR spectrophotometer (Cary 7000) with the diffuse reflectance 2 accessory and with a spot size ~0.4 cm encompassing the cell and the front grids. 푃푟푒푓 was then calculated according to Eq. 9. ∞ 푃푟푒푓 = 푉퐹푒푓푓 ∫ 퐸푏(휆, 푇)휌(휆) 푑휆 (9) 0 This approach to modeling the cells was used to predict the cell performance under the tungsten filament lighting conditions. The decomposition of reflectance into 푅푎 and 푅푠푢푏 portions (Eq. 5) enabled the subsequent predictions of efficiency at higher 푅푠푢푏 shown in Fig. 3b.

Figure 5. Modeled vs measured a) 퐽푠푐, b) 푉표푐, and c) 퐹퐹. Good agreement can be seen between the measurement and model predictions. For each device, the 퐹퐹 measurement and model exhibit the same trend and the small offset between the two can be explained by an additional series resistance of the electrical leads/contacts in the experimental setup.

Uncertainty Propagation Uncertainty in the efficiency measurement arises from the measurement of 푃 and the measurement of 푄푎푏푠 (Eq 1). There are two heat absorption measurements taken, a primary heat absorption measurement, 푄1, and a background heat absorption measurement, 푄2, so that 푄푎푏푠 = 푄1−푄2. From the manufacturer, the calibration accuracy of the HFS is +/-3% and the temperature sensitivity accuracy is +/-1.25%, leading to an RMS uncertainty of heat absorbed of 퐵푄1 = 0.0325푄1 and 퐵푄2 = 0.0325푄2. Therefore, the uncertainty in the heat absorbed is 퐵푄푎푏푠 = 2 2 √(퐵푄1) + (퐵푄2) . From the source meter, the voltage measurement uncertainty is 0.03% of the −4 voltage (퐵푉 = (3푥10 ) ∗ 푉) and the current measurement uncertainty is 0.06% of the current −4 (퐵퐼 = (6푥10 ) ∗ 퐼). This leads to an uncertainty on the electric power measurement of 퐵푃 = 2 2 √(퐼 ∗ 퐵푉) + (푉 ∗ 퐵퐼) , which is negligible due to the low uncertainty on voltage and current. The absolute uncertainty in measured efficiency, 퐵휂,푚푒푎푠푢푟푒, was calculated as,

휕휂 2 휕휂 2 √ ( ) 퐵휂,푚푒푎푠푢푟푒 = ( 퐵푃) + ( 퐵푄푎푏푠) 10 휕푃 휕푄푎푏푠

The uncertainty in the model prediction primarily arises from the uncertainty in predicted 퐽푠푐 (퐵퐽푠푐 ≈ 0.03 ∗ 퐽푠푐) from the uncertainty of the EQE measurement of the multi-junction cell, and from the uncertainty of the FTIR reflectance measurement leading to 퐵푅푠푢푏 ≈ 0.013. Propagating these errors through Eq. 5, the absolute uncertainty in modeled efficiency, 퐵휂,푚표푑푒푙, was calculated according to Eq. 11 and the model uncertainty is shown by the shaded regions in Fig. 3a.

휕휂 2 휕휂 2 √ ( ) 퐵휂,푚표푑푒푙 = ( 퐵퐽푠푐) + ( 퐵푅푠푢푏) 11 휕퐽푠푐 휕푅푠푢푏

The uncertainty on the emitter temperature measurement was calculated from the variation in resistance of the bulb measured at each emitter temperature and the uncertainty on temperature dependence of resistance from the literature expression that was used which is a 0.1% relative error on resistance as a function of temperature.52 The RMS of these two yields temperature measurement uncertainties of < 4 °C which had a negligible impact on model uncertainty.

TPV Cell Growth and Processing Details Fig. 6 shows the device structures of the tandem cells. All materials were grown by atmospheric pressure organometallic vapor phase epitaxy (OMVPE) using trimethylgallium, triethylgallium, trimethylindium, triethylaluminum, dimethylhydrazine, arsine, and phosphine. Diethylzinc and carbon tetrachloride were used as p-type dopants and hydrogen selenide and dislane were used as n-type dopants. Growth took place in a purified hydrogen gas flow of 6 lpm. Substrates were n- type (100) GaAs with a 2° offcut towards the (111)B plane, and all devices were grown in an inverted configuration. For both types of cells, the substrate was prepared by first etching in NH4OH : H2O2 :H2O (2:1:10 by volume). The substrate was then mounted on a graphite susceptor and heated inductively to 700°C under an arsine overpressure, followed by a ~10 minute deoxidization under arsine. Growth of the 1.4/1.2-eV tandem started with a 0.2 µm GaAs buffer, then a 0.5 µm GaInP etch stop layer. Then, 0.1 µm of GaInAsN:Se and 0.2 µm of GaAs:Se were deposited as the front contact layer. Then, the top cell was grown, starting with a 0.02 µm AlInP window layer, then a 0.1 GaAs:Se emitter, a 0.1 µm undoped GaAs layer, a 2.8 µm GaAs:Zn base layer, and a 0.12 GaInP back surface field (BSF) layer. Next, a AlGaAs:C/GaAs:Se/AlGaAs:Si quantum well tunnel junction was grown, followed by a GaInP compositionally graded buffer (CGB). The CGB consisted of 0.25 µm GaInP steps spanning the compositional range Ga0.51In0.49P to Ga0.34In0.66P at a rate of 1% strain/µm, with the final layer being a 1 µm Ga0.34In0.66P strain overshoot layer. For the bottom cell, a 1 µm Ga0.37In0.63P window, a 0.1 Ga0.85In0.15As:Se emitter, 0.1 Ga0.85In0.15As i- layer, and a 1.5 µm Ga0.85In0.15As:Zn base and 0.05 µm Ga0.37In0.63P:Zn BSF. Finally, a 0.05 µm Ga0.85In0.15As:Zn++ back contact layer was grown. For the 1.2/1.0-eV design,43 a 0.2 µm GaAs buffer layer was grown first, then a GaInP CGB consisting of 0.25 µm GaInP steps, spanning the range Ga0.51In0.49P to Ga0.19In0.81P, with the final layers being a 1 µm Ga0.19In0.81P strain overshoot layer and a 0.9 µm Ga0.22In0.78P step back layer lattice matched to the in-plane lattice constant of the Ga0.19In0.81P. A 0.3 µm Ga0.70In0.30As:Se front contact layer was grown next, followed by the top cell, starting with a 0.02 µm Ga0.22In0.78P:Se window, a 1 µm Al0.15Ga0.55In0.30As:Se emitter, an undoped 0.1 µm Al0.15Ga0.55In0.30As i-layer, a 2.1 µm Al0.15Ga0.55In0.30As:Zn base and a 0.07 µm Ga0.22In0.78P:Zn BSF. Then the tunnel junction, comprising a 0.2 µm Al0.15Ga0.55In0.30As:Zn, a 0.05 µm GaAs0.72Sb0.28:C++, and a 0.1 µm Ga0.22In0.78P:Se++ layer, was grown. Lastly the bottom cell was grown, comprising a 0.05 µm Ga0.22In0.78P:Se window, a 1.5 µm Ga0.70In0.30As:Se emitter, a 0.1 µm Ga0.70In0.30As:Zn i-layer and 0.02 µm Ga0.22In0.78P:Zn BSF. Finally, a 0.05 µm Al0.4Ga0.30In0.30As:Zn++ back contact layer was grown. After growth, a ~2 µm thick reflective gold back contact was electroplated to the exposed back contact layer (the last semiconductor layer grown). The samples were bonded with low viscosity epoxy to a silicon handle and the substrates were etched away in NH4OH : H2O2 (1:3 by volume). Gold front grids were electroplated to the front surfaces through a positive photoresist mask, using a thin layer of electroplated nickel as an adhesion layer. The grids were nominally 10 µm wide, 100 µm apart, and ≥ 5 µm thick. The samples were then isolated into individual devices using standard wet-chemical etchants and cleaved into single cell chips for characterization. The completed cells had mesa areas of 0.8075 cm2, with illuminated areas (discounting the single busbar but including the grid fingers) of 0.7145 cm2.

Figure 6. Device structures of the 1.4/1.2-eV and the 1.2/1.0-eV tandems.

Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request. Acknowledgements The authors thank W. Olavarria and A. Kibbler for MOVPE growth work and C. Aldridge for earlier processing work. We acknowledge financial support from the U.S. Department of Energy (DOE): Advanced Research Projects Agency – Energy (ARPA-E) cooperative agreement DE-AR0001005; Office of Energy Efficiency and Renewable Energy grants DE-EE0008381, and DE-EE0008375. This work was authored, in part, by Alliance for Sustainable Energy, LLC, the manager and operator of the National Renewable Energy Laboratory for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

Contributions A.L. conducted the efficiency measurement experiments and analyzed the data. A.L, K.B., and C.C.K. designed, built, and tested the experimental setup. K.L.S. developed and optimized the epitaxial growth of the cells. D.J.F., M.A.S., and K.L.S designed the cells, and M.A.S. and M.R.Y. fabricated them. D.J.F., M.A.S., and K.L.S along with contributions from R.M.F., E.J.T., A.L, and A.R. characterized and modeled the cells. A.H. supervised the work. All authors contributed intellectually to the work’s execution and to preparation of the manuscript. Competing interest declaration M.A.S. and E.J.T. also work on a similar project with a small company that could be perceived as competing.

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