Electroluminescent refrigeration by ultra-efficient GaAs light-emitting diodes

A light-emitting diode (LED) that approaches the limit of unity luminescence efficiency, emitting a photon for every electron injected, can also operate as a solid-state refrigerator.^1–4 When the LED is biased below its bandgap, the device emits photons whose energy exceeds the voltage used to supply it with charge carriers. This energy shortfall is made up by lattice vibrations, leading to a refrigeration effect.^5–8 Although this cooling process is inherent in electroluminescence, it must compete with the heat that is deposited back into the lattice as a result of internal losses, such as non-radiative recombination and parasitic optical absorption. Net cooling of the device is therefore contingent upon a high external luminescence quantum efficiency η_ext, defined as the fraction of injected charge carriers that are ultimately emitted out of the device as photons. LEDs that are efficient enough to achieve net cooling are yet to be experimentally realized, except at a very small forward bias (less than the thermal voltage kT/q), where the power density is too small to be practical as a solid-state refrigerator.⁵ 

More recently, it has been recognized that external luminescence plays an important role in photovoltaic devices just as in LEDs.⁹ The recent efficiency records in solar energy conversion^9–11 have been predicated on the efficient extraction of internal luminescence. Indeed, reaching the Shockley-Queisser limit of solar cell efficiency requires the device to operate as an ideal LED with η_ext = 100%.¹² 

Electroluminescent cooling draws upon advancements in both LED and photovoltaic technology to become viable as an option for solid-state cooling. The optimal configuration for refrigeration makes use of thermophotonics,^13,14 in which luminescent photons act as the working fluid that pumps heat from a cold LED to a hot photovoltaic absorber, as shown in Fig. 1. The photovoltaic device partially converts the photons it absorbs into electricity, which can be used to offset the power consumption of the LED. This feedback allows a given amount of cooling to be provided with a much smaller supply of external power, greatly enhancing the refrigeration coefficient of performance (COP), defined as the cooling heat flux per unit supplied work. With light as the working fluid, the hot and cold sides can be separated by a vacuum spacer, eliminating heat leakage by direct thermal conduction. In this paper, we assume a vacuum gap that spans many optical wavelengths. This is to be contrasted with near-field refrigeration, considered in a recent paper,¹⁵ which enhances the coupling of luminescent photons between devices, but also increases the coupling of parasitic thermal photons.

In this work, we build upon previous theoretical efforts in similar materials^14,16,17 by proposing a design for a GaAs LED and photovoltaic cell that maximizes the external luminescence efficiency, as demanded by the refrigeration application. With this design, we project the maximum possible refrigeration coefficient of performance that is within the limitations of presently available optoelectronic materials and technology. For applications near room temperature, we predict that electroluminescent cooling with existing material quality surpasses the COP of thermoelectric coolers by 1.7× when used at moderate power densities (1.0–10 mW/cm²). In the domain of cryogenic cooling, the refrigerator operates far closer to ideal, greatly exceeding the coefficient of performance attainable by both thermoelectric cooling and laser cooling. These results suggest the potential use of LED refrigeration as an efficient and lightweight solid-state option for cryogenic experiments, low-power portable cooling (e.g., of medicine), and general-purpose refrigeration.

Electroluminescent refrigeration is inherent to the operation of an LED, whose applied bias is normally less than the emitted photon energy. The “missing” energy is made up by the transfer of heat from the lattice phonons to the injected charge carriers,¹⁸ which are in turn in quasi-equilibrium with the emitted photons.¹⁹ When electrons and holes meet in the device active region, radiative recombination transfers the heat to the luminescent photon gas. The emitted photons from the LED possess entropy, and can therefore carry heat, by virtue of their spatial and temporal incoherence. For each photon that successfully escapes, the amount of heat extracted from the device is the difference between the photon energy and the work consumed to inject an electron-hole pair. Meanwhile, any loss of carriers or photons during this process leads to the dissipation of energy as heat inside the device, directly counteracting the cooling effect.

An LED alone therefore acts as a refrigerator, pumping heat from its lattice at temperature T_c into the emitted luminescence. We can construct a more energy-efficient refrigerator by using the thermophotonic configuration in Fig. 1, in which a hot photovoltaic absorber (T_h > T_c) absorbs the heat delivered by the luminescent photons, and also reclaims part of the optical energy as electricity. The devices are separated by a vacuum gap, across which direct heat conduction is forbidden. For this system, the net heat flow per area Q_c away from the cooled LED is

where $E¯$ is the average energy of a luminescent photon (close to the semiconductor bandgap E_g), Φ is the luminescent photon flux, J is the current density, and V is the voltage corresponding to the quasi-Fermi level separation of electrons and holes in the active region. The subscripts “c” and “h” denote the cold (LED) and hot (photovoltaic) sides of the system, respectively. Due to electrical series resistance in the LED, the applied voltage at the contacts exceeds the quasi-Fermi level separation voltage; this energy difference is lost to Ohmic dissipation, and contributes a Joule heating rate of Q_Ωc. The first three terms correspond to the difference between the luminescent power output and the electrical power input to the LED, while the fourth term is the luminescent power incident from the photovoltaic cell and absorbed by the LED. We also include a parasitic non-luminescent heat flux Q_leak from hot to cold. All power densities, photon fluxes, and current densities are expressed per unit area of the device surface that faces the vacuum gap, which we will call the “front” surface.

The net work consumed to drive the electroluminescent heat transfer is given by

where Q_Ωh is the Ohmic dissipation in the photovoltaic cell. The efficiency of the cooling system is characterized by its coefficient of performance (COP), defined as the ratio of the cooling heat flux to the work input. The COP is bounded by the Carnot limit

To determine the luminescent flux that is transferred from one device to the other, we apply the principle of detailed balance¹² 

where E is the photon energy, θ is the polar angle in vacuum, c is the speed of light, h is Planck's constant, q is the elementary charge, k is the Boltzmann constant, and a(E, θ) is the absorptivity (or emissivity) of the device. The luminescence possesses a chemical potential that is equal to the quasi-Fermi level separation qV in the light emitter.¹⁹ Within the closed system of Fig. 1, only those photons that are ultimately absorbed by the opposing device contribute to the emissivity. We assume that each device sees the other through only its front surface.

We chose the same material as both the cold emitter and the hot absorber. A higher temperature leads to a slightly smaller bandgap on the hot side, which is optimal for refrigeration. The difference in the bandgap strongly enhances the transfer of luminescent photons from cold to hot (Φ_c) relative to the flux from hot to cold (Φ_h). Meanwhile, because the bandgap deficit on the hot side is small, the photovoltaic cell can still produce a large voltage V_h to maximize the electrical power recovered.

The electrical current densities in the two devices are given by

where we have set the polarity of the photovoltaic current so that J_h > 0 if the cell operates in the power-producing quadrant of the J-V curve, and a positive voltage always denotes a forward bias. The first term in each equation is the forward current, which is converted to external luminescence with a quantum efficiency η_ext. The second term is the reverse current generated by the photovoltaic conversion of incident luminescence from the opposing device, with a quantum efficiency η_abs. We refer to η_ext as the external luminescence efficiency, which corresponds to the conventional definition of the external quantum efficiency of an LED²⁰ and is sometimes called the external radiative efficiency in a photovoltaic cell.²¹ Meanwhile, η_abs corresponds roughly to the external quantum efficiency of a solar cell, integrated over photon energy.

To establish the requirements on device efficiency required for refrigeration, we compare the cooling that results from electroluminescence to the heating caused by internal losses. Every external luminescence event, occurring with a probability equal to η_ext,c, liberates roughly (E_g – qV_c) of lattice heat from the device. On the other hand, every time an injected carrier fails to leave the LED as a photon, occurring with a probability (1 – η_ext,c), the full energy qV_c of the lost electron-hole pair is dissipated in the lattice. The cooling and heating rates associated with these processes are balanced when η_ext,c = qV_c/E_g. Accounting for resistive losses and heat leakage, an LED efficiency of η_ext,c > qV_c/E_g is needed to produce a net cooling effect in the device. We note that for a given value of η_ext,c, electroluminescent refrigeration is inherently more efficient at low bias, where each photon extracts a larger amount of heat and the resistive losses are small.

Meanwhile, in order to produce a sufficiently large photon flux and hence an appreciable amount of heat transfer (Q_c ≥ 1.0 mW/cm²), the quasi-Fermi level separation must be within about 300 meV of the bandgap. In GaAs, for example, this implies that we must operate with qV_c/E_g > (E_g – 300 meV)/E_g ≈ 80%. Under these circumstances, the amount of heat generated by a single failed luminescence event can offset the amount of cooling accumulated over many successful luminescence events. A high external luminescence efficiency of at least 80% is required to overcome this imbalance. If we wish to operate the refrigerator at a significant fraction of the Carnot coefficient of performance, and not merely to reach the cooling threshold, η_ext,c must be well above 80%. Therefore, electroluminescence as a mechanism of cooling can be made practical only with LEDs that approach the limit of ideal external luminescence efficiency, η_ext,c = 100%.

On the photovoltaic side, the demand on efficiency is less stringent. A high external luminescence efficiency η_ext,h in the photovoltaic cell maximizes the output voltage V_h. But since electricity is generated from the LED luminescence, the benefit of the photovoltaic cell is ultimately contingent upon a highly efficient LED, whose design should take precedence. Nonetheless, because the two devices are reciprocal, an efficient LED can be operated as an efficient photovoltaic cell, and vice versa.

We define the external luminescence efficiency as the ratio of the externally emitted photon flux to the electrical current injected into the LED terminals. It can be written as²⁰ 

where η_int is the internal luminescence efficiency, the probability that an injected electron generates a photon in the active region, and C_ext is the light extraction efficiency, the probability that an internally generated photon escapes through the front surface of the device.

In this work, we consider devices that use GaAs (E_g = 1.42 eV at 300 K) as the active light-emitting material. We note that high efficiencies of η_ext > 80% have been demonstrated in visible-light GaN-based LEDs,²² and their design for electroluminescent cooling has been considered in other works.^6,23–25 However, the large bandgap in these materials leads to a greater heating penalty for failed luminescence, and thus, a considerably higher quantum efficiency η_ext is needed to achieve the same cooling performance as GaAs. Low-bandgap materials, on the other hand, tend to suffer from low internal luminescence efficiency due to their intrinsically strong Auger recombination. Very high photoluminescence quantum efficiencies, meanwhile, have been observed in GaInP/GaAs/GaInP double heterostructures: 96% at room temperature²⁶ and 99.5% at 100 K.²⁷ GaAs is also the active material utilized in the record-efficiency single-junction solar cells.¹¹ These results, which are indicative of near-ideal internal luminescence efficiency, suggest that GaAs LEDs should be capable of achieving ultra-high external luminescence efficiency.

In this section, we project the practical limits of LED performance for this heterostructure. To obtain near-unity external luminescence efficiency, we propose the device structure shown in Fig. 2, whose design will be the subject of this section. In order to minimize photon losses at the electrical contacts, we use a lateral current injection scheme that spreads the carriers uniformly across the full device area at low to moderate current densities, allowing most of the carriers to recombine far away from the electrodes. The electrodes are made up of a two-dimensional array of point contacts, whose minimal surface coverage reduces their impact on light extraction. The device can be realized in practice by using the epitaxial lift-off technique, which is in widespread manufacturing use for solar cells and LEDs, to release the heterostructure thin film from its growth substrate²⁸ and allow further processing of both the front and rear sides. The viability of the device in Fig. 2 for electroluminescent cooling will be analyzed in Sec. IV.

Current is supplied to the device in Fig. 2 from metallic gridlines on the front side and a metallic backplane on the rear side, entering the semiconductor through an array of point contacts. The carriers spread out laterally in the GaInP cladding layers before entering the GaAs active region. Thereafter, electrons and holes are confined to the active region by potential barriers at the GaInP/GaAs interfaces and eventually recombine. The internal luminescence efficiency is the fraction of the total recombination rate that is radiative

where the total supplied forward current per area (of the front surface) is resolved into components that contribute to: radiative recombination J_rad, Shockley-Read-Hall (SRH) recombination J_srh, and Auger recombination J_Auger. The structure in Fig. 2 is repeated over a large lateral area, so that non-radiative recombination at the sidewalls of the active region is negligible. Recombination outside of the active region due to carrier leakage, calculated using the formalism in Ref. 29, is also found to be negligible at all biases below the bandgap due to large confinement potentials at the heterojunctions.

To calculate the recombination rates, we assume a uniform quasi-Fermi level separation qV over the full thickness of the LED heterostructure. The net rate of radiative recombination can be found from the van Roosbroeck-Shockley detailed-balance relation³⁰ 

where n_r = 3.5 is the refractive index of GaAs, α is the inter-band absorption coefficient, d is the thickness of the GaAs active region, n_i is the intrinsic carrier density, and n and p are the electron and hole densities in the active region, calculated using the effective mass approximation. The second equality defines the radiative recombination coefficient B.

The absorption coefficient is given by

where α₀(E) is the absorption coefficient of intrinsic bulk GaAs in equilibrium, obtained from the experimental results in Ref. 31 at room temperature. The second term accounts for the occupancies of the conduction and valence band electronic states in the radiative transition, which in general depend on qV and the doping density N_D. For qV ≲ E_g – 3kT, the absorption coefficient and hence the radiative recombination coefficient B are constant with the quasi-Fermi level separation. In this regime, we obtain B = 8 × 10⁻¹⁰cm³/s for the GaAs active layer at 300 K, which is consistent with previous reports.³² 

The Auger recombination current is given by

where C_n and C_p are the coefficients of two-electron and two-hole Auger processes, respectively. We use a total Auger coefficient of C = 7 × 10⁻³⁰ cm⁶/s for GaAs at room temperature,³³ with C_n and C_p each assumed to equal half of this value.

We model the SRH recombination current as

where we have decomposed J_srh into the recombination rates in the bulk active region (inversely proportional to the bulk SRH lifetime τ_srh), at the GaAs/GaInP interfaces (proportional to the surface recombination velocity S), and in the depletion region of the p-n junction, J_dep,srh. We assume that mid-gap defects dominate, and we use the same lifetime for both carriers; under low-level injection in a doped semiconductor, τ_srh should be interpreted as the SRH lifetime of the minority carriers (holes). Experimental results in Ref. 27 suggest that a bulk lifetime of at least τ_srh = 21 μs has been attained in GaAs, while Ref. 34 reports S = 1.5 cm/s for this heterostructure after lift-off, both at 300 K.

The rate of SRH recombination in the depletion region of the p-n junction generally scales with a diode ideality factor of 2, i.e., proportional to $exp (qV/2kT)$⁠.³⁵ The magnitude of this 2kT parasitic current can be parameterized by its saturation current density J₀₂, which can be extracted from experimental I-V data. The value of J₀₂ can also be estimated theoretically by determining the carrier densities n(z) and p(z) as a function of position z in the GaInP/GaAs heterostructure via an electrostatic analysis. For the p-n junction in Fig. 2, SRH recombination in the wide-bandgap p-GaInP layer is negligible, due to the very low carrier densities present there. Integrating the local SRH recombination rate over the n-side depletion region, we obtain

where z = 0 marks the p-GaInP/n-GaAs interface and z_n is the n-side depletion width of the junction. Using the measured lifetime of τ_srh = 21 μs in GaAs, this calculation projects a value of J_02,300 K = 3.5 fA/cm². By contrast, owing to non-ideal material quality, the record-efficiency GaAs solar cells appear to have performance corresponding to J_02,300 K ≈ 200 fA/cm².¹¹ In this work, we consider the performance of devices with J_02,300 K between these values, with the anticipation that improvements in material quality can yield a saturation current density closer to the projected J_02,300 K. Though depletion-region SRH recombination is the dominant parasitic current only at low bias (<1.0 V), it can nonetheless limit the internal luminescence efficiency η_int from approaching unity at moderate bias.

We chose an active region doping density of N_D = 2× 10¹⁷ cm⁻³, which approximately equalizes the rates of SRH and Auger recombination at moderate bias. Figure 3(a) shows the internal luminescence efficiency of the heterostructure in Fig. 2 for several values of J_02,300 K. The efficiency is shown at 263 K, anticipating its use on the cold side of a refrigerator operating near room temperature. The values of J₀₂ and various other parameters are adjusted for the lower temperature as described in more detail in Sec. IV B. We note a steep drop in η_int at low bias due to depletion-region SRH recombination, while at high bias the efficiency drops due to Auger recombination.

As explained in Sec. II, electroluminescent refrigeration is inherently more efficient at moderate voltage (∼1.2 V) than at high voltage, so the luminescence efficiency in the moderate bias regime should be maximized. In Fig. 3(a), we observe that when J₀₂ is at its projected value (J_02,300 K = 3.5 fA/cm²), a maximum value of η_int = 99.87% is accessible over a broad voltage range, from about 1.15 V to 1.35 V. With the presently more realistic value of J_02,300 K = 200 fA/cm², the peak efficiency drops slightly and moreover, the voltage range with high efficiency is greatly reduced. Optimization of J₀₂ is therefore critically important for the realization of high-performance electroluminescent refrigeration, which demands extremely high values of η_int.

Radiative recombination pumps light into the internal reservoir of luminescent photons. Extracting this luminescence out of the device is inherently challenging, due to the high refractive index of the semiconductor relative to the external vacuum. While some photons leave through the front surface immediately, most are bound inside the semiconductor by total internal reflection. These photons may be absorbed by the active region, re-generating an electron-hole pair, or they may be absorbed parasitically, generating heat. In GaAs, where the internal luminescence efficiency is high, the re-generated carriers can largely be recovered as photons by radiative recombination, and may undergo multiple re-absorption/emission events before they eventually enter the escape cone. The light extraction efficiency is given by the ratio of the front extraction rate to the total rate of photon removal from the device

where Φ_front is the flux of photons extracted from the front surface. Photon losses occur through: rear surface absorption Φ_rear, non-radiative recombination after being absorbed by the active region Φ_nr, free-carrier absorption Φ_fc, and absorption by the Ohmic contacts Φ_Ω. We discuss below the features of the LED in Fig. 2 that maximize photon extraction and minimize photon loss.

Texturing the front surface of the semiconductor is a well-known technique for enhancing light extraction in LEDs.³⁶ In a planar device, photons that are trapped by total internal reflection must be absorbed and re-emitted many times (and potentially lost) before eventually entering the escape cone. A textured surface, as shown in the device in Fig. 2, helps extract these photons more rapidly by randomizing their angle, giving them a renewed opportunity to escape with every pass through the film. Fast extraction also means that photons are less susceptible to being lost internally through parasitic absorption.

To model the effect of the texture, we make the ergodic assumption:^36,37 on being scattered many times by the texture, the photons experience complete angular randomization, so the internal luminescence that develops inside the LED is an isotropic photon gas. Though the real dynamics can be more complex,³⁸ the assumption holds well for wavelength-scale surface texturization³⁹ and allows the internal photon behavior to be studied using a statistical ray optics approach. We find the individual photon outflow rates in Eq. (13) by integrating the spectral brightness b_int of the photon gas (in photons per unit time, area, solid angle, and energy).

The portion of the isotropic internal luminescence that is extracted out of the front surface is given by

where θ_i is the internal polar angle inside the GaAs film, T(E, θ_i) is the front transmission coefficient averaged over polarization, and $θc=sin−1(1/nr)=16.6°$ is the critical angle of the front surface. To facilitate extraction, we include an optimized anti-reflection coating (145 nm MgF₂ with n_r = 1.375, 80 nm ZnS with n_r = 2.3) on the front surface, which has been used in solar cells.⁴⁰ The front transmission coefficient is 79.9% when averaged over energy, angle, and polarization and corrected for the fraction of the front surface that is not covered by metal gridlines, compared to 56.5% for an uncoated surface.

The rate of photon loss through the rear surface, meanwhile, is given by

where R is the rear reflectivity. To attain the exceptionally high reflectivity that is a prerequisite for electroluminescent cooling, we propose the rear reflector shown in Fig. 2, which is composed of an Al_0.2Ga_0.8As/Al_xO_y distributed Bragg reflector backed by a low-index MgF₂ (n_r = 1.375) layer and a metallic backplane. The Al_xO_y oxide layers (n_r ≈ 1.6) can be introduced by first growing an Al_xGa_1–xAs stack with alternating Al-rich and Ga-rich layers, and then converting the Al-rich layers to their native oxide through selective wet oxidation.⁴¹ To maintain an uninterrupted light-emitting surface over a large area, these layers can be made accessible for lateral oxidation through the rear contact openings of the device, prior to metal deposition.⁴² We calculate the reflectivity of the stack using the transfer matrix method, and Fig. 4 shows its angle dependence after averaging over the polarization and energy distribution b_int of luminescence. The MgF₂ layer provides total internal reflection above the rear critical angle with GaAs of 23°, while for most of the angular range below 23°, the high index contrast of the Bragg stack enables broadband total reflection.⁴³ The angle-, energy- and polarization-averaged reflectivity of the mirror is 99.990%. This reflectivity can be maintained in the presence of parasitic absorption within the stack of 0.1 cm⁻¹ or less, as is the case when the layers are left undoped. To achieve high reflectivity, we forego the conduction of current through the reflector.

When a photon is absorbed within the active region volume, generating an electron-hole pair, it may fail to rejoin the internal photon gas if the carriers recombine non-radiatively. The rate of photonic loss due to non-radiative recombination is given by

This loss channel can be suppressed with η_int close to unity and with surface texturing, which reduces the necessary thickness d of the semiconductor for a given emitted flux.

Free-carrier absorption in the semiconductor volume is an intrinsic loss mechanism, with a rate given by

where the free-carrier absorptivity is $afc=σn(NDcdnc+nd)+σp(NAcdpc+pd)$⁠. We use the free-carrier absorption cross sections of σ_n = 3 × 10⁻¹⁸ cm² and σ_p = 7 × 10⁻¹⁸ cm² by electrons and holes, respectively, for interband luminescent photons.⁴⁴ Though these values have been reported for GaAs, we also apply them to the other III-V layers. The n- and p-GaInP cladding layers have doping densities of N_Dc and N_Ac, respectively, both equal to 10¹⁸ cm⁻³. Their thicknesses are d_nc = d_pc = 100 nm.

Parasitic absorption by the LED contacts and front metallization is given by

We model the contact absorptivity as $aΩ=fΩ(1−RΩ)+fg(1−Rg)$⁠, where f_Ω and R_Ω are the surface coverage and reflectivity of the Ohmic contacts on both sides, and f_g and R_g are the corresponding quantities for the front metal gridlines. We assume that the Ohmic contacts are partially absorbing with a reflectivity R_Ω = 90% to the internal photons, which has been achieved in prior work.⁴⁵ Their otherwise detrimental effect on light extraction is mitigated with the use of a point contact array on both the n and p sides, similar to the point contact solar cell,⁴⁶ which confines the losses to a small fraction of the device area. For the arrangement in Fig. 2, the contacts have a coverage of f_Ω = 0.17%. The front gridlines have a larger coverage of f_g = 16.7%, but since the metal does not penetrate the anti-reflection coating except at the point contacts, total internal reflection by the front dielectric layers allows a high internal reflectivity. Using the transfer matrix method, we obtain R_g = 99.85% (averaged over energy, angle, and polarization).

To calculate each of the photon outflow rates, it remains to determine the spectral brightness b_int of the internal photon gas. This is found by equating the extracted photon flux given by Eq. (14) to the detailed-balance expression in Eq. (4) for the same quantity. Enforcing this consistency yields

where $T¯$ is the front transmission coefficient averaged over the escape cone. The luminescence emissivity a(E) is angle-independent, due to the complete angle randomization in the textured film. To find an expression for a(E), we first calculate the absorptivity a₀(E) of a device that emits and receives radiation from free space, rather than confined to an enclosure as in Fig. 1. This is given under the ergodic assumption by³⁷ 

where parasitic absorption is contained in the term $L(E)=afc+aΩ/4+(1−R¯)/4$⁠, where $R¯$ is the angle-averaged rear reflectivity. Equation (20) is strictly valid under weak internal absorption (αd ≪ 1), weak loss (⁠$L≪1$⁠), and a narrow cone of escape (⁠$T¯/4nr2≪1$⁠), all of which must hold to maintain ergodicity. However, the expression is also approximately valid for strongly absorbed photons.⁹ Relative to a planar device, surface texturing enhances the emissivity for photons near the band edge by up to a factor of $4nr2$⁠. We chose an optimal active layer thickness of d = 200 nm to achieve both a large emissivity and low non-radiative recombination.

For the closed system in Fig. 1, we must modify the emissivity to account for multiple incoherent reflections between the LED and the photovoltaic cell

where for each side $a0†$ is the total absorptivity, obtained by adding the term $T¯L$ to the numerator of Eq. (20). The photovoltaic emissivity a_h is found by interchanging the subscripts “c” and “h.”

Figure 3(b) shows the external luminescence efficiency η_ext of the LED structure in Fig. 2, again at 263 K. In general, η_ext inherits the voltage dependence of η_int, with additional penalties due to parasitic absorption. We obtain a broad efficiency peak at η_ext = 97.7% when using the projected value of J_02,300 K (blue), based on a SRH lifetime of 21 μs. For the presently more realistic value of J_02,300 K (red), the peak efficiency is only slightly smaller at η_ext = 97.6% but covers a much narrower voltage range near 1.3 V. Figure 5 shows the spectrum of the photon inflow and outflow rates for the device. While the majority of the internal luminescence is re-absorbed, the extraction rate is considerably larger than the parasitic loss rate, leading to a high extraction efficiency. The various parasitic absorption losses are similar in magnitude for the optimized device. Further reduction of free-carrier absorption and contact absorption is possible by using thinner cladding layers and contacts with smaller surface coverage, respectively. However, these strategies incur significant penalties in Ohmic dissipation, discussed below.

Meanwhile, the quantum efficiency of reverse current generation in the photovoltaic cell is given by the fraction of absorbed photons (from the LED) that produce electrons and holes, assuming that the extraction of generated carriers to the electrodes of the cell is ideal

where dΦ_c/dE is the spectrum of luminescence that is emitted by the LED and absorbed by the photovoltaic cell, given by the energy integrand in Eq. (4). If the LED luminescence lies fully above the photovoltaic bandgap, as is the case when the photovoltaic cell operates at an elevated temperature, we readily obtain η_abs,h > 99%.

In a real LED, electroluminescent refrigeration must compete not only with heating caused by non-radiative radiation and parasitic optical absorption, but also with Joule heating due to electrical resistances. The need for low contact absorption and near-perfect reflectivity, however, necessitates the lateral spreading of carriers across both sides of the device as shown in Fig. 2. This lateral injection scheme incurs a relatively large loss by Ohmic dissipation, since the carriers must travel significant lateral distances through the semiconductor current-spreading layers, which must be made thin to suppress free-carrier absorption. The trade-off between light extraction and Ohmic dissipation ultimately limits the system's refrigeration performance at high power densities.

For each device, the total Ohmic dissipation per unit surface area is given by

where the first term is the loss due to lateral current spreading in the semiconductor layers and the second term is due to series resistances at the device terminals. To calculate the former, we group the layers of the material stack in Fig. 2 into two composite surfaces: an n-type surface with sheet resistance R_◻n and total thickness d_n, and a p-type surface with sheet resistance R_◻p and thickness d_p. For a given bias, we find the lateral current densities $J→n(x,y)$ and $J→p(x,y)$ on the two surfaces by solving the equations of current continuity, treating the p-n junction as a spatially varying current path from the p- to n-surface. The integral is then taken over a unit cell of the device area (red dashed square in Fig. 2), which draws a total current I and whose dimension L_c = 30 μm is equal to the separation between neighboring contacts. The sheet resistances in the semiconductor layers are calculated using the empirical mobility model in Ref. 47, which accounts for both temperature and doping dependence. The series resistances R_sp and R_sn include the contributions of the Ohmic contacts (assuming a contact resistivity of ρ_c = 10⁻⁶ Ω cm²),⁴⁸ the front metallization, and the metal vias on the rear side. More details about the calculation of Q_Ω can be found in  Appendix A.

Solving for the lateral current distributions $J→n(x,y)$ and $J→p(x,y)$⁠, we obtain the expected result that the Ohmic dissipation increases with greater contact separation and with smaller thicknesses of the current spreading layers. This makes it difficult to reduce Joule heating without introducing too much additional losses due to contact and free-carrier absorption. A contact spacing of 30 μm and the layer thicknesses shown in Fig. 2 represent the optimum that balances these trade-offs.

In this section, we use the LED structure in Fig. 2 to investigate the practical limits of electroluminescence as a mechanism of refrigeration, both for room-temperature and cryogenic cooling applications. Since an efficient LED is also an efficient photovoltaic cell, we use an identical structure for both devices. The only modification we make for the photovoltaic cell is a thicker active region with d = 400 nm for more complete absorption of the incident LED illumination, and a larger p-GaInP layer thickness of d_pc = 400 nm to reduce the photovoltaic-side Ohmic dissipation.

Efficient electroluminescent refrigeration relies on the effective management of parasitic thermal leakage paths so that they become relevant only at the lowest power densities. To block the exchange of thermal photons between the two devices, we assume the presence of a metal mesh filter in the vacuum gap that is highly reflective in the far-infrared. We also operate the system with many LEDs and photovoltaic cells in series, building up a large operating voltage at the output of each side. This ensures that the system is subject to only small resistive losses in the electrical feedback connection between the two sides, permitting the use of a narrow wire that leaks minimal heat from hot to cold. For the results in this section, we assume that the implementation of these strategies suppresses the heat leakage to a realistic level of Q_leak = 0.1 mW/cm². Please see  Appendix B for more details.

We emulate a realistic steady-state cooling application near room temperature by choosing T_c = 263 K and T_h = 313 K (–10 °C and 40 °C), which results in a GaAs bandgap of 1.44 eV for the LED and 1.42 eV for the photovoltaic cell.⁴⁹ To characterize the system's most efficient operating point, we iterate through a range of LED voltages V_c, and for each value, we numerically search for the photovoltaic voltage V_h that maximizes the coefficient of performance given by Eq. (3). This voltage must be large enough to provide substantial recovery of electrical power, but small enough to suppress the undesirable photon backflow Φ_h from the photovoltaic cell to the LED. At the optimal value of V_h, Φ_h is generally about one order of magnitude smaller than Φ_c. Owing to the 20 meV bandgap difference, this is satisfied at a relatively large operating voltage V_h.

Figures 6(a) and 6(b) show the net heat flux Q_c and the Carnot-normalized COP, respectively, of the electroluminescent refrigerator as a function of the quasi-Fermi level separation voltage V _c in the LED. We use the projected value of J_02,300 K = 3.5 fA/cm² in both devices, and the characteristics are shown with and without the energy recovery action of the photovoltaic cell. In the latter case, the hot side is a passive surface at 313 K that fully absorbs the luminescent photons. The electroluminescent cooling flux follows the exponential increase in the photon flux with voltage, and rises above the level of parasitic heat leakage when V_c > 1.1 V. Using the projected value of J_02,300 K, the cooling performance at low voltage (V_c < 1.15 V) is limited more by heat leakage than by SRH recombination. In the moderate bias regime (V_c = 1.15 V to 1.25 V, or roughly J_c = 2 to 200 mA/cm²), the COP is determined chiefly by the external luminescence efficiency η_ext. Since the change in η_ext over this range is marginal, the COP decreases with voltage for the fundamental reason discussed in Sec. II: the heat liberated per external photon diminishes, while the heat generated for every lost photon increases. Cooling at larger voltages (V_c > 1.25 V) is limited by rapidly increasing Ohmic dissipation, which sets a maximum cooling flux for this system of Q_c,max = 62 mW/cm² (at V_c = 1.29 V, J_c = 1.0 A/cm²). At all biases, the small reduction in cooling flux in the presence of the photovoltaic cell indicates that the reverse photon flux Φ_h is not significant when the photovoltaic voltage has been optimized. However, the gain in COP due to the reclaimed power from the photovoltaic cell is substantial.

Figure 7 plots the two principal figures-of-merit of cooling against each other for the GaAs electroluminescent refrigerator with photovoltaic recovery. We show the characteristics for two values of J_02,300 K, which parameterizes the material quality of the diode. In both cases, a trade-off appears between the cooling heat flux and the COP: more efficient operation can be accessed at the expense of the heat flux, and vice versa, by adjusting the applied bias. The most practically relevant regime of operation lies roughly in the range from Q_c = 1.0 mW/cm² to Q_c = 10 mW/cm², which is well above the level of parasitic heat leakage but below the point where cooling efficiency plummets due to Joule heating.

The 2kT recombination current J_02,300 K emerges as a critically important quantity to be minimized in order to realize high-performance cooling in the moderate-flux regime. With a projected value of J_02,300 K = 3.5 fA/cm² (blue), based on the longest observed SRH lifetime in GaAs,²⁷ the system would operate with a Carnot-normalized COP of 24.7% at Q_c = 1.0 mW/cm², and 21.0% at Q_c = 10 mW/cm². With a value of J_02,300 K = 200 fA/cm² (red), which represents the material quality achieved in the record-efficiency GaAs solar cell, the Carnot-normalized COP is 16.2% at Q_c = 10 mW/cm².

We also include for comparison the refrigeration properties of an ideal GaAs system with η_ext = 100%, zero Ohmic dissipation, and no heat leakage (dashed). We immediately recognize two important deviations of the optimized GaAs system (blue) from the ideal case. First, much higher cooling fluxes can be reached in the ideal system due to the absence of Ohmic dissipation. The maximum cooling flux available with GaAs lies near Q_c,max ≈ 10 W/cm², and is ultimately limited by the rate at which photons inside the high-index semiconductor can be emitted into the optical modes of the surrounding vacuum. Heat transfer can be further enhanced by reducing the hot-cold separation to the sub-wavelength scale, so that the two high-index materials are directly coupled in the near field.⁴ This approach can increase Q_c,max by about an order of magnitude, but at a cost of greater radiative heat leakage.¹⁵ Second, we note that even though the optimized GaAs LED has a near-ideal luminescence efficiency of η_ext = 97.7%, the COP in the region of peak η_ext is considerably lower than that in the lossless case, η_ext = 100%. Large improvements to the COP are possible with even more ideal devices than the structure proposed in Fig. 2, and it is the small deviation from unity efficiency (1 – η_ext) that becomes the true indicator of performance. The electroluminescent cooling application, therefore, reaps large benefits from asymptotic approaches to material and device perfection.

For a technological comparison, Fig. 7 also shows the characteristics of thermoelectric cooling operated between the same temperatures. Thermoelectric materials are designed to fulfill the competing goals of a high Seebeck coefficient S, a low electrical resistivity ρ, and a low thermal conductivity κ; the overall performance for a given application is captured in a material figure-of-merit, defined by $ZT=(S2/ρκ)×12(Tc+Th)$⁠. For this comparison, we assume a value of ZT = 1, representative of the best commercially available devices at room temperature.⁵⁰ In terms of this quantity, it can be shown that the thermoelectric cooler achieves a maximum possible COP given by⁵¹ 

To represent the full range of cooling properties accessible with a ZT = 1 material, every point on the black curve in Fig. 7 corresponds to a different thermoelectric thickness and optimized bias current. The COP given by Eq. (24) can be accessed for cooling fluxes up to ∼1 W/cm², above which the performance is degraded by losses at the electrical and thermal contacts.⁵² To estimate the COP at these high fluxes, we assume for the thermoelectric material typical values of S = 260 μV/K, ρ = 2.4 × 10⁻⁵ Ω m, and κ = 0.81 W/m K, which together yield ZT = 1 at these temperatures. We also assume an electrical contact resistivity of 10⁻⁶ Ω cm² and a thermal contact conductance of 100 kW/m² K.

The thermoelectric cooler with ZT = 1 achieves a maximum COP equal to 9.3% of the Carnot limit, as given by Eq. (24). Inspecting Fig. 7, we conclude that for room temperature operation, electroluminescent cooling is the potentially more efficient choice for lower heat fluxes, Q_c < 50 mW/cm². In particular, for the moderate flux range between 1.0 mW/cm² and 10 mW/cm², the electroluminescent refrigerator operates with a considerably improved coefficient of performance. With the material quality available in existing devices (J_02,300 K = 200 fA/cm²), the COP of electroluminescent cooling can be up to 1.7× that of thermoelectrics in this range. With further improvements in material quality (reaching J_02,300 K = 3.5 fA/cm²), the refrigerator can operate with a COP between 2× and 3× greater than that of the best commercial thermoelectrics.

To further explore the comparison, we find the thermoelectric-equivalent ZT_eff that corresponds to the COP of electroluminescent cooling using Eq. (24). The result is plotted in Fig. 8 for several values of the temperature difference ΔT with a fixed cold-side temperature T_c = 263 K. We hold J_02,300 K to the projected value of 3.5 fA/cm². For the previously considered case of ΔT = 50 K, electroluminescent cooling attains a value of ZT_eff = 2.65 at a flux of 1 mW/cm², which greatly exceeds the values of ZT in the best commercial thermoelectric materials at room temperature.⁵⁰ For larger temperature differences, the equivalent ZT of electroluminescent refrigeration is even greater. As the difference ΔT decreases, however, the advantage of electroluminescent over thermoelectric refrigeration diminishes and eventually disappears for ΔT < 20 K. This is because as ΔT approaches zero, the thermal conduction leakage in thermoelectric coolers vanishes, eliminating one of the primary non-idealities in that system. Meanwhile, in luminescent heat transfer, the condition of ΔT → 0 does not necessarily reduce the non-radiative losses in the refrigerator, allowing thermoelectrics to gain the efficiency advantage.

Electroluminescent cooling becomes even more efficient at low temperatures, where laser cooling has also achieved great success.^53,54 Semiconductor diodes are more ideal light emitters at cryogenic temperatures, as non-radiative recombination is suppressed relative to radiative recombination, which is accelerated for a given carrier density. SRH recombination in this heterostructure is thermally activated, and we model the bulk recombination rate 1/τ_srh and surface recombination velocity S to vary with temperature as $exp (−EA/kT)$⁠, where E_A = 18 meV.³⁴ The 2kT recombination current density J₀₂ is suppressed by the same factor and additionally follows the strong temperature dependence of the intrinsic carrier density.³⁵ The Auger coefficient in GaAs is also thermally activated and decreases at low temperature, with E_A = 58 meV.^34,55

To model the absorption coefficient α₀(E) at low temperature, we fit the data in Ref. 31 using a piece-wise function above and below the bandgap as in Ref. 9 and account for the temperature dependence of the GaAs bandgap and the Urbach tail parameter.⁴⁹ We find that the radiative recombination coefficient B evaluated using Eq. (8) increases at low temperatures approximately as T^−1.78. Additionally, the carrier mobilities increase at low temperatures with a peak near 100 K,⁴⁷ and from this we also infer a corresponding decrease in the free-carrier absorption cross-sections σ_n and σ_p.⁵⁶ 

As a result of these favorable trends, the internal luminescence efficiency η_int asymptotically approaches unity when operated at cryogenic temperatures. The change in external luminescence efficiency with temperature is shown in Fig. 9(a), which indicates the contributions of various mechanisms to the luminescent loss, 1 – η_ext. Non-radiative recombination is strongly suppressed at low temperatures, and η_ext becomes limited mainly by parasitic optical absorption. In these plots, the active region doping density is set to N_D = 2 × 10¹⁷ cm⁻³, but at low temperatures, we limit N_D to be no greater than the effective conduction band density of states N_c (T) to avoid the band-filling effect. At each temperature, η_ext is evaluated at an LED bias that yields a fixed heat flux of Q_c = 1.0 mW/cm², which is not necessarily the peak-luminescence-efficiency operating point. At a cryogenic temperature of 113 K, the GaAs LED attains η_int = 99.99% and η_ext = 98.7% when biased for 1.0 mW/cm² heat flux.

Figure 9(b) shows the cryogenic refrigeration characteristics of the optimized GaAs system, with T_c = 113 K and T_h = 313 K. The device structure is identical to that used in the room-temperature analysis, other than a reduced LED active region doping density of N_D = 9 × 10¹⁶ cm⁻³. With the projected material limit of J_02,300 K = 3.5 fA/cm², we predict a Carnot-normalized refrigeration COP of 35.8% at Q_c = 1.0 mW/cm², and 24.0% at Q_c = 10 mW/cm². For the same temperatures, thermoelectrics would require ZT > 6.7 merely to reach the cooling threshold; this makes the technology impractical for cryogenic cooling, since known thermoelectric materials have much poorer performance at low temperatures.⁵⁷ 

At low temperatures, electroluminescent cooling is an alternative to solid-state laser cooling, whose COP also potentially benefits from photovoltaic energy recovery. In laser cooling, the semiconductor is optically pumped just below the band-edge and relaxes by the emission of photoluminescence, which peaks at a slightly higher energy. A primary challenge in laser cooling is the weak interband absorption of the pump photons, which faces increasing competition with parasitic absorption as the pump energy is reduced bellow the bandgap. This limits the amount of energy extracted from the lattice to approximately kT per photon.⁵³ The use of electrical rather than optical excitation of the light emitter offers a practical means to circumvent this issue. When operated at a sufficiently small LED bias, the electroluminescent refrigerator can therefore extract much more lattice heat per photon than kT, at the cost of being limited to smaller heat fluxes.

Figure 9(b) further shows that with presently more realistic diode quality corresponding to J_02,300 K = 200 fA/cm², the COP is 32.8% of the Carnot limit at a heat flux of 1.0 mW/cm²—only slightly inferior to the projected performance with J_02,300 K = 3.5 fA/cm². Since non-radiative recombination becomes a minor source of loss at low temperatures, the requirement on material quality is less stringent in cryogenic cooling compared to room-temperature cooling; this opens the refrigeration application to a larger class of optoelectronic materials with comparable or slightly lesser quality to GaAs. Lead halide perovskites, for example, have already been demonstrated as excellent light-emitters in laser cooling experiments.⁵⁸ 

As optoelectronic devices—the light-emitting diode and the photovoltaic cell—continue to improve, their design holds important implications not only for their traditional applications, but also for refrigeration. In this work, we have optimized the GaAs/GaInP heterostructure for electroluminescent cooling. The result (Fig. 2) represents our vision of the most efficient refrigeration device that is within the capability of presently available optoelectronic materials. However, it is not perfect. In order to minimize optical losses, we have introduced a lateral carrier injection scheme. The significant spreading resistance associated with this structure limits the cooling performance beyond moderate cooling fluxes of about 50 mW/cm². A device that relies on diffusion-based carrier injection, similar to the point contact solar cell⁴⁶ or more recently proposed LED architectures,^59,60 may mitigate the losses associated with current spreading, allowing greater cooling power densities while maintaining a high coefficient of performance.

Experiments have demonstrated the long non-radiative carrier lifetimes that are the basis of high internal luminescence efficiency in GaAs devices. A key parameter for electroluminescent cooling is the 2kT saturation current J₀₂, which represents Shockley-Read-Hall recombination in the depletion region. Very low values of J₀₂ have led to record-efficiency GaAs solar cells. Based on this material quality, the refrigeration performance of our device exceeds the COP of the best commercial thermoelectric coolers by 1.7×. With improved material quality, corresponding to the longest observed SRH lifetimes in GaAs, the COP advantage over thermoelectrics can increase to between 2× and 3×. These values are attained at room temperature, when operated at moderate heat fluxes of 1.0–10 mW/cm².

Electroluminescent refrigeration may be suitable for portable cooling applications, such as the active cooling of medicine, due to its superior efficiency at moderate power densities. This power density is also close to that used for household refrigeration and air conditioning. The technology can be more swiftly applied, however, to cryogenic cooling applications, which are perhaps more compelling. Most semiconductors become considerably more efficient light-emitters at low temperature. In this domain, electroluminescent refrigerators operate closer to ideal and their efficiency has no competition among solid-state refrigeration alternatives.

The authors thank Ryan Brandt (North Carolina State University) for his work on modeling Ohmic dissipation. This work is part of the “Light-Material Interactions in Energy Conversion” Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001293. T.P.X. was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1106400.

To calculate the Ohmic dissipation rates Q_Ωc and Q_Ωh, we model the n- and p-type layers as two parallel, planar sheets across which current can flow. Current travels from the positive contacts of the device on the p-surface to the negative contacts on the n-surface, and passes vertically between the two surfaces through the p-n junction diode. At a specific lateral position (x, y), the diode has a quasi-Fermi level separation equal to V_d(x, y) and draws a vertical current density equal to J_d(x, y). Here, we focus our attention on a unit cell of the device area (red dashed square in Fig. 2), with the approximation that current is not shared between adjacent unit cells. The unit cell has a side length equal to the contact separation of L_c = 30 μm, and we set the origin (0, 0) to the location of the positive contact.

We solve for the lateral current densities $J→n(x,y)$ and $J→p(x,y)$ by applying current continuity. On the p-type surface, lateral current decreases away from the positive contact (at the origin) as charge is drawn out vertically through the diodes. The reverse is true for the n-type surface, which receives the current injected through the diodes. Current continuity then yields for the two surfaces

The sheet resistance has the definition R_◻ = ρ/d for each material layer. The resistances R_◻n and R_◻p of the n- and p-type surfaces are found by adding the sheet resistances of the constituent layers in parallel. The resistivity of a layer j is found using $ρj=(qμjNj)−1$⁠, where μ_j and N_j are the mobility and the density of the majority carriers in the layer. Using Ref. 47 to determine the carrier mobilities in each layer, accounting for doping and temperature dependence, we obtain R_◻n = 240 Ω/◻ and R_◻p = 12 kΩ/◻ for the LED at 263 K.

For the purposes of this calculation, we relate the voltage V_d to the current J_d using the standard diode equation

where J₀ is the forward saturation current density and J_R is the reverse current density due to the absorption of luminescence from the opposing device, assuming that the absorption is laterally uniform. In our thermophotonic system, the first term dominates in the LED, while the second term dominates in the photovoltaic cell. We can extract the values of J₀ and J_R by matching the two terms with their corresponding components in Eq. (5). Equation (A4) is an approximation, since the actual current follows a more complicated dependence due to the voltage-dependent external luminescence efficiency, η_ext (V). Nonetheless, this method works well if either (1) the spatial variation in the diode voltage due to spreading resistance is small, or (2) η_ext varies slowly with voltage near the operating point. At the biases relevant for electroluminescent cooling, at least one of these conditions holds true.

In Sec. III, we calculate the recombination rates in the LED using a single value for the quasi-Fermi level separation qV_c. Therefore, we find it convenient to calculate the Ohmic dissipation as a function of this voltage, Q_Ωc (V_c). However, in the presence of spreading resistance, the quasi-Fermi level separation is non-uniform. To match the calculation of Q_Ω to a particular value of V_c, we ensure that the power consumption of the device with a non-uniform voltage distribution V_d(x, y) is equal to that of the same device with a uniform voltage distribution at V_c

where the current J_d(V_c) is evaluated using Eq. (A4), and the integral is taken over the surface area of the unit cell, dA = dx². At the end of this section, we evaluate the validity of using a uniform value of the quasi-Fermi level splitting to calculate the device recombination rates.

We solve the non-linear differential equation (A3) for V_d(x, y) using the successive over-relaxation method, with Eq. (A5) as a boundary condition. We evaluate the voltages on a grid whose resolution is 0.5 μm near the contacts, where the voltage changes most rapidly, and 2.0 μm everywhere else. From this solution, we can write a differential equation for the voltage of the n-surface by combining Eqs. (A1) and (A2)

With the right side known, we again use the successive over-relaxation method to solve for V_n(x, y). The p-surface voltage distribution is given by V_p = V_n + V_d. The lateral current densities can then be found readily using Eq. (A2).

The Ohmic dissipation due to the spreading resistance on each surface is found by integrating the contribution from each differential current element. The total dissipation also includes the losses from the series resistances R_sn and R_sp

where I is the total current injected into the unit cell

The differential lateral current is given by $dIp=|J→p|×dA⊥$⁠, where dA_⊥ = dx × d_p is the differential area normal to the lateral current flow. Inserting this into Eq. (A7), we arrive at Eq. (23). For the device in Fig. 2, Ohmic dissipation is dominated by the spreading resistance of the p-type surface, due to the low hole mobility in the p-GaInP layer.

The n- and p-side series resistances are estimated by

where the first term in each expression is the Ohmic contact resistance, assuming a contact resistivity of ρ_c = 1.0 × 10⁻⁶ Ω cm² for both the n-type and p-type contacts,⁴⁸ and d_c = 1.0 μm is the contact diameter. The second term in R_sn is the resistance associated with the front metal grid lines, where ρ_m = 1.6 × 10⁻⁸ Ω m is the resistivity of Ag, L_g = 5.0 μm is the grid line width, h_g = 5.0 μm is the grid line depth, and N_c is the total number of contacts connected to each grid line. Assuming a total dimension of 1.0 cm for the LED array, we have N_c = 333. The second term in R_sp is the resistance of the metal via connecting the p-GaInP layer to the rear Ag backplane, where h_via is the depth of the via trench, equal to the total thickness of the Bragg reflector. We assume negligible spreading resistance in the Ag backplane.

We find that the variation in V_d(x, y) over the LED unit cell due to spreading resistance is smaller than kT_c/q ≈ 22.7 mV up to an average diode voltage of V_c = 1.285 V (or approximately 0.87 A/cm² at T_c = 263 K). Above this voltage, the assumption of a laterally uniform recombination rate no longer holds, and the optoelectronic properties of the device cannot be modeled fully accurately using a single value of the quasi-Fermi level separation. However, the refrigeration COP at this voltage is already heavily degraded by Joule heating; see Fig. 6(b). Therefore, for the voltage range corresponding to moderate heat fluxes (1.0–10 mW/cm²), where electroluminescent cooling has the greatest advantage over present technology, the assumption of a laterally uniform quasi-Fermi level separation is valid.

Although separated by vacuum, passive radiative heat transfer from the hot to the cold side can occur alongside the active heat transfer by luminescent radiation. For our system, the amount of net heat transfer by thermal radiation is bounded by the blackbody limit: $max(Qleak,rad)=σ(Th4−Tc4)=27.3 mW/cm2$ for temperatures of T_c = 263 K and T_h = 313 K, where σ is the Stefan-Boltzmann constant. In reality, the heat leakage will be smaller than this value, since the devices have less than unity emissivity at the wavelengths of the thermal photons, which are in the far-infrared. The exact amount of heat transfer depends on the thicknesses and dielectric functions of every material layer in each device, including both the semiconductor and dielectric layers.

Since the most practical regime of operation of the electroluminescent refrigerator lies at moderate fluxes of 1.0–10 mW/cm², we must take additional steps to suppress the thermal radiative heat leakage. One strategy is to place a metal mesh in the vacuum gap between the two devices, which can act as an optical high-pass filter.⁶¹ The periodicity of the mesh is chosen to maximally reflect the low-energy thermal photons and transmit the high-energy luminescent photons. Assuming blackbody thermal radiators, the required far-infrared reflectivity is 99.6% in order to reduce the leakage to <0.1 mW/cm², so that the impact on moderate-flux operation becomes negligible. For temperatures of (T_c = 113 K, T_h = 313 K) or (T_c = 263 K, T_h = 338 K), the required reflectivity is 99.8%. These requirements can be relaxed after accounting for the below-unity far-infrared emissivity of the two sides.

In the thermophotonic configuration, the electrical connection that supplies the generated photovoltaic power to the LED is a source of heat leakage by direct thermal conduction. Suppressing this heat leakage using a narrower or longer feedback wire, however, would directly increase the Ohmic dissipation in the wire, which causes heating of the LED. To break this trade-off, we can connect an ensemble of individual devices on both the LED and photovoltaic sides of the system in series. A large voltage builds up across the ensemble, but the total current through it is reduced, permitting the use of a narrow wire to attain both low heat leakage and low resistive losses.

The heating of the LED due to the presence of the electrical feedback wire is the sum of the heat leakage along the wire and half of the total Joule heating in the wire (the other half heats the photovoltaic side)

where A is the total device area served by the feedback connection, κ_fb and σ_fb are the thermal and electrical conductivities of the wire, l_fb and A_fb are the wire length and cross-sectional area, and I_h is the total photovoltaic current that passes through the connection.

Suppose that a single feedback connection supplies the power generated in N photovoltaic cells of area $Lc2$⁠, all connected in series. The total current is equal to the current generated in a single cell, $Ih=Jh×Lc2$⁠, where J_h is the current density in the photovoltaic device, but the total area served by the connection is $A=NLc2$⁠. We notice that the two terms in Eq. (B1) depend in opposite directions on the wire geometric parameter (A_fb/l_fb). The value of Q_leak,fb is minimized when the two terms are equal in value. By choosing the value of the geometric parameter that minimizes the heating of the LED for a given J_h, we can simplify the above equation to

The ratio of thermal to electrical conductivity in a metal can be found from the Wiedemann-Franz law: $κfb/σfb=L×12(Th+Tc)$⁠, where L = 2.44 × 10⁻⁸W Ω K⁻² is a constant.

Using the GaAs device in Fig. 2, operating the system between T_c = 263 K and T_h = 313 K and a cooling flux of 10 mW/cm² (including Ohmic dissipation in the device) requires a current density of approximately J_h = 60 mA/cm² in the photovoltaic cell. To reduce the parasitic heating to Q_leak,fb = 0.1 mW/cm², where it is negligible compared to the electroluminescent cooling flux, we need to connect at least N = 16 cells in series. For the same electrical configuration (equal N and wire geometry, A_fb/l_fb) at a cooling flux of 1.0 mW/cm², the heat leakage will be Q_leak,fb ≈ 0.05 mW/cm². In practice, the LEDs and photovoltaic cells must be connected both in series and in parallel in order to properly match their total currents.