Efficiency-optimized near-field thermophotovoltaics using InAs and InAsSbP

Waste heat above 600 K represents 15% of global total energy use,¹ and recycling this waste heat to electrical power with solid-state modules could be a general solution to improve energy-use efficiency. For 600–900 K heat sources, existing commercial waste-heat-to-electricity solid-state converters, thermoelectric generators (TEGs), have high power densities of 2 W/cm² but suffer from low efficiencies less than 12%.² Conversely, thermophotovoltaic (TPV) systems have high theoretical efficiencies up to 45% but low power densities of 0.2 W/cm².³ Near-field thermophotovoltaic (NFTPV) systems, which position a heat source (radiator) and a TPV cell in extreme proximity, present both high theoretical efficiency and power density, up to 40% and 10 W/cm² for practical devices.^3,4 Although NFTPV systems with cells operating at room temperature have experimentally achieved 40-fold increases in power density over TPV systems,⁵ they have not exceeded 1.5% efficiency for 600–900 K radiator temperatures. This low efficiency is due to the use of TPV cells that were not optimized for near-field operation^5,6 or InGaAs-based cells with larger-than-optimal bandgaps (0.73 eV at 300 K).^7,8 Conversely, Lucchesi et al. designed a low bandgap (0.23 eV at 77 K) InSb-based NFTPV cell that was cooled to 77 K and measured a 14% efficiency with a power density of 0.75 W/cm² with a 732 K radiator.⁹ Although they reached a high efficiency, these cells only function properly when cooled to cryogenic temperatures, making them impractical for most situations.

Although the spectral characteristics required for high performance in NFTPV systems are well known,^4,10–26 concurrent optimization of the interrelated optical and electronic properties of NFTPV systems remains largely unexplored. Prior studies have not solved the full two-dimensional (2D) drift-diffusion equations in the cell, although such simulation allows realistic representation of both lateral and vertical current flow and carrier collection and is industry-standard in many other contexts.^27–30 Many studies calculated NFTPV electrical performance using detailed balance analysis, assuming radiative recombination loss only^10,19–22 or also including nonradiative recombination mechanisms.^4,23–26 Other analyses have employed the diode equation with a saturation current calculated using experimentally measured lifetimes of the studied materials.^21–28 An improved electrical model, solving the drift-diffusion equations in the low injection limit, was also employed to analyze NFTPV performance.^31–35 Recently, studies have solved the full 1D drift-diffusion equations,^36–39 which leads to more realistic results,³⁶ though still without lateral transport effects.

In the present work, we investigate the performance of NFTPV cells based on InAs, which has an ideal bandgap (0.353 eV at 300 K) for 600–900 K radiator temperatures and proven room temperature operating performance.^40–46 We optimize the coupled electrical and optical properties of three NFTPV designs, all with back reflectors and two with InAs/InAsSbP double-heterostructures, by maximizing their efficiency under the illumination of a 750 K p-doped Si radiator separated by 0.1 μm. We compare their performances to a baseline design on a 500 μm substrate, shown in Fig. 1(a). We further investigate the performance for the highest-efficiency design for radiator temperatures from 600 to 900 K and with radiator-TPV gaps from 0.01 to 10 μm.

We propose three designs, presented in Figs. 1(b)–1(d). The pin and pin-Q cell designs have a p–i–n doping order from top to bottom. Conversely, the nip-Q design reverses this order to reduce lateral sheet resistance in the front-surface field (FSF) layer since electron mobilities are higher than hole mobilities (see Sec. 1 of the supplementary material). The pin-Q and nip-Q designs include the quaternary (Q) InAs_xSb_0.31(1-x)P_0.69(1-x) lattice matched to InAs with bandgap up to 0.495 eV at 300 K⁴⁷ in the FSF and base layers to reduce undesirable diffusion currents, but at the cost of higher growth complexity. All cells have a uniform temperature of 300 K (see Sec. 2 of the supplementary material) and are illuminated by high-temperature 5 × 10¹⁸ cm⁻³ p-type silicon radiators, separated by vacuum. The distance between gridlines is d_G-G while the distance between the bottom of the radiator and the top of the FSF layer is d_R-FSF. For comparison, we consider the baseline design [Fig. 1(a)], which is based on the fabricated TPV cell from Lu et al.⁴⁰ In contrast to the baseline design, we added cap and back-surface field (BSF) layers to each new design to minimize contact resistance^48–50 and reduce undesirable minority diffusion currents. In addition, our three proposed designs have their substrates removed, increasing fabrication complexity but minimizing parasitic sub-bandgap (SBG) photon absorption. Substrate-less devices with a back reflector (BR) layer reflect SBG photons for re-absorption within the radiator, increasing efficiency, and decreasing cooling requirements for the TPV cell.

Our coupled optoelectronic model simulates the radiative thermal transport and the electrical response of the NFTPV device by combining the results of two software packages. We model radiation transport using a custom fluctuational electrodynamics solver^51,52 that treats the devices as laterally infinite layered structures. We compute depth- and frequency-resolved radiation transfer, separated by the physical absorption mechanism, allowing us to calculate the depth-resolved electron-hole generation rate from interband absorption and total heat transfer from lattice, free-carrier, and interband processes (see Sec. 3 of the supplementary material for model details). We verify the optical model by reproducing the spectral absorption calculated in Ref. 8; the results are shown in Sec. 4 of the supplementary material.

We model the electrical transport for the NFTPV device using Synopsys TCAD Sentaurus. This software solves Poisson's equation coupled with electron and hole drift and diffusion equations to determine the TPV cell current–voltage curves as well as depth-resolved recombination rate profiles including radiative, Auger, Shockley–Read–Hall (SRH), and surface recombinations.²⁷ We apply the 1D electron–hole pair generation rate profiles, extracted from the optical model, uniformly across the illuminated portion of the 2D TPV cell, with no generation directly below the top contacts. We assume no contact and sheet resistance for both top and bottom electrodes, as they can be designed to be negligible.^48–50 We calculate radiative and Auger recombination coefficients following the method in Ref. 53, providing Auger coefficients for InAs (C_n = 0.48 × 10⁻²⁶ and C_p = 1.01 × 10⁻²⁶ cm⁶/s) within the experimental uncertainties of Ref. 54. We employ a constant SRH lifetime (3.7 × 10⁻⁷ s) for both n- and p-type InAs, which provided the best fit to the measured dark-I–V curve of the TPV cell from Ref. 40. Subtracting Auger and radiative contributions from the total lifetime of Ref. 55 using Matthiessen's rule, we estimate the SRH lifetime of InAs_xSb_0.31(1-x)P_0.69(1-x) for x ≠ 1 to be 3.2 × 10⁻⁸ s. We include a doping-dependent surface recombination velocity at the vacuum/FSF interface, discussed further in Sec. 5 of the supplementary material. Due to the low conduction band density of states of InAs and InAsSbP lattice matched to InAs, we use a non-parabolic band model to calculate the electron quasi-Fermi level.⁵⁶ We employ the model and parameters from Ref. 57 to calculate the bandgap and electron affinity for all stoichiometric compositions of InAsSbP. The energy band diagrams of baseline and optimized nip-Q cells at maximum power point voltage (V_mpp) are presented in Fig. 2, including conduction band (E_C) and valence band (E_V) edges and the Fermi levels of electrons (E_Fe) and holes (E_Fh). Figure 2(b) shows abrupt band offsets at the heterojunctions with the FSF and BSF layers, which effectively block parasitic hole and electron transport, respectively. We validate the electrical model, comparing measured and simulated dark current–voltage curves of p–i–n InAs⁴⁰ and p-InAsSbP/n-InAs/n⁺-InAs⁴¹ devices, with the results given in Sec. 6 of the supplementary material.

With a goal to efficiently recover 600–900 K waste heat while considering achievable radiator-TPV gaps (d_R-FSF) of present NFTPV devices,^5–9,58 we optimize the three NFTPV designs of Figs. 1(b)–1(d) for a radiator temperature of 750 K and d_R-FSF = 0.1 μm. We calculate the total optical power density absorbed by the TPV cell (P_in) and its maximum power point density (P_mpp), with power densities defined as power divided by the illuminated area. We then optimize the device structure to maximize device efficiency, η = P_mpp/P_in, using an iterative optimizer with parameter domain sampling defined by the face-centered central composite method.⁵⁹ We optimize for efficiency as opposed to power as it increases waste heat use while also lowering cell cooling requirements.

To speed up the optimization, we hold constant the parameters with fixed values listed in Fig. 1. This includes the cap and BSF layer doping concentrations, which are set at readily achievable values^48–50 to maximize current collection and quench contact resistance. This choice necessitates a thin BSF layer to minimize free-carrier absorption. We employ an intrinsic InAs absorber layer to maximize absorption³¹ while minimizing the Auger recombination that dominates in cells using p-InAs.⁴⁰ See Sec. 7 of the supplementary material for effects of varying Cap and BSF layer thickness and doping on performance.

We consider the radiator to be semi-infinite, but instead using a 500-μm thickness decreases total radiation transfer by less than 1%. A separate optimization showed that our chosen radiator doping of 5 × 10¹⁸ cm⁻³ maximizes above-bandgap (ABG) radiation transfer for realistic radiator thicknesses (on the order of a typical 500 μm Si substrate or less). Thinner radiators demand higher doping concentrations to maximize ABG radiation transfer, but such radiators also increase SBG radiation transfer, which hurts efficiency.

Layer thicknesses and doping not specified in Fig. 1 are included as parameters for optimization. The optimization domain and results for all designs illuminated by a 750 K radiator with d_R-FSF = 0.1 μm are given in Table I. The optimized As mole fraction (x) of InAs_xSb_0.31(1-x)P_0.69(1-x), for the FSF and base layers of the pin-Q and nip-Q designs, respectively, reached the lower bound of 0.4, which we constrained to high quality compositions^60,61 outside the miscibility gap.⁶² The p-type doping concentration for the FSF layer of the pin design reached the upper bound of 10¹⁹ cm⁻³, which we limited to readily achievable concentrations for these devices.^49–51 These bounded values minimize parasitic electron diffusion; the lower bound Q has the largest bandgap with favorable band-alignment as a p-type material (see Fig. 2) while the pin design requires high p-doping to perform the same task.

The FSF conductivity effects can only be captured with 2+ dimension drift-diffusion solvers, highlighting the importance of our electrical model. To minimize lateral series resistance, the p-type FSF layer must be thick and moderately doped (pin-Q) or thin and highly doped (pin). The n-type FSF layer of the nip-Q device further reduces top sheet resistance, since electron mobility is higher than hole mobility, allowing for a thin and moderately doped FSF layer with 60% larger d_G-G, increasing power output density.

Comparing parameters of the optimized designs to the baseline design, the optimized designs have smaller d_G-G values and are composed of much thinner layers. Thinner cells improve device performance due to higher carrier collection efficiency and lower parasitic free carrier absorption but reduce current generation. However, the thin designs have higher J_sc than the baseline due to shorter penetration depth of evanescent vs propagating waves and high reflection at the gold BR layer. The base layers optimally reached lower doping concentrations to reduce SBG power transfer (P_SBG) originating from free-carrier absorption.

Our optical model calculates a significant enhancement of useful ABG power transfer (P_ABG) over the blackbody radiative limit for all devices. Figure 3 shows the layer-resolved spectral absorbed energy flux for each device. A 750 K blackbody radiative spectrum represented by the solid black line is included for comparison, along with a vertical dashed black line denoting the bandgap of InAs (0.353 eV). The optimized devices have P_SBG approximately 10% of that in the baseline design. P_SBG is not converted to useful power and instead raises the TPV cooling requirements. P_SBG could be further reduced with a more reflective BR.^16,63 The optimized designs [Figs. 3(b)–3(d)] have 4–6 times higher total P_ABG and 0.68–0.85 times lower P_SBG compared to the blackbody limit. Although P_ABG of the efficiency-optimized designs are about 70% of the baseline design, we see much higher efficiency and power output (Table I) due to order-of-magnitude thinner absorption layers, which improves current collection.

Up to 80% of available current is collected at V_mpp for the optimized designs compared to just 39% for the baseline, highlighting the importance of optimizing electronic device properties for near-field operation. Figure 4 shows the current–voltage characteristics, normalized to the total photo-generated current (J_ph), of the four designs. We include extracted current (J), surface recombination at the vacuum/FSF and electrode/semiconductor interfaces (J_surf), as well as Auger (J_Aug,l), radiative (J_rad,l), and Shockley–Read–Hall (J_SRH,l) recombination currents in layer l. All currents that contribute less than 1.5% to J_ph at V_mpp are combined into J_other. Auger recombination is the main electrical loss mechanism for all designs, consuming 8%–54% of J_ph at V_mpp. However, the devices could be further improved using higher quality absorber layer material, i.e., lowering SRH recombination. Comparing Figs. 4(b) and 4(c), we find that the introduction of the quaternary InAsSbP in the FSF and base layers reduces surface recombination by two-third and eliminates recombination in all but the absorber layer, contributing to a 45% larger V_mpp. Designs with InAsSbP, nip-Q and pin-Q, have similar normalized current–voltage characteristics [see Figs. 4(c) and 4(d)]; therefore, nip-Q's performance enhancement (Table I) is due to better P_SBG management and larger d_G-G.

Using our best design, the optimized nip-Q device, we explore the impact of radiator temperature and d_R-FSF on the power output and efficiency in Fig. 5. The star represents the parameter values used during the nip-Q device optimization. We calculate a significant power output enhancement from near-field energy transfer for all radiator temperatures investigated, with the largest enhancement being 107-fold larger than the far-field (d_R-FSF = 10 μm), occurring with d_R-FSF = 0.01 μm and a 600 K radiator. We also calculate an improved efficiency under near-field illumination for all radiator temperatures within 600–900 K, reaching up to a 3.7-fold increase with a 600 K radiator and d_R-FSF = 0.12 μm compared to a far-field device.

The maximum efficiency for a given radiator temperature varies with d_R-FSF. The optimal d_R-FSF decreases as temperature increases, going from 0.12 to 0.09 μm for a 600–900 K radiator, respectively. This shift toward smaller d_R-FSF occurs because there is proportionally less near-field P_SBG transfer to the FSF layer for higher radiator temperatures. Finally, the dip in efficiency at d_R-FSF $≈$ 1 μm is caused by a lowered P_ABG relative to P_SBG due to propagative wave interference effects. Since the relative fraction of P_ABG vs P_SBG increases with radiator temperature, the highest efficiency and power output of 14.2% and 1.55 W/cm² occur at 900 K. P_ABG increases as d_R-FSF decreases, but P_SBG absorbed in the FSF layer also increases rapidly below 0.1 μm, which reduces efficiency without impacting power output. Therefore, the maximum efficiency and power output occur at different d_R-FSF of 0.09 and 0.01 μm (lower limit), respectively.

Simulation of our optimized nip-Q design significantly outperforms the simulated p-InAs/n-InAs design studied in Ref. 31. At 800 K and with d_R-FSF = 0.1 μm, we calculate approximately 2.7- and 3.3-times higher efficiency and power density relative to that device (compared to their efficiency that assumes no absorption in the substrate). The nip-Q device performance enhancement is attributed to our BR layer, use of Q material, and n–i–p doping configuration.

In summary, three NFTPV cell designs containing InAs and/or InAsSbP were optimized and compared under near-field illumination by a 750 K p-Si radiator at d_R-FSF = 0.1 μm, using a validated optoelectronic model solving full 2D drift-diffusion equations and fluctuational electrodynamics. The optimized devices have 4–6 times higher above-bandgap and 0.68–0.85 times less sub-bandgap radiation transfer than the blackbody limit. Over the 600–900 K radiator temperature range, we calculate up to 14.2% efficiency and 1.55 W/cm² power output for the best performing device design. According to these results, our best design could significantly outperform the best measured NFTPV device for the conversion of 600–900 K waste heat, providing important guidelines for the design of future NFTPV cells.

See the supplementary material for further details and validation of the optical and electrical model, the calculated temperature gradient of the device, and the impact on device performance of parameters that were fixed during optimization.

The authors thank University of Ottawa colleagues C. Zhang and M. Giroux for discussions on this topic. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (No. NSERC CGS-D) and by the New Frontiers in Research Fund (No. NFRFE-2019-00334). They are also grateful to CMC Microsystems for providing access to the Synopsys Sentaurus software (vS-2021.06).