Entropic and Near-Field Improvements of Thermoradiative Cells

A p-n junction maintained at above ambient temperature can work as a heat engine, converting some of the supplied heat into electricity and rejecting entropy by interband emission. Such thermoradiative cells have potential to harvest low-grade heat into electricity. By analyzing the entropy content of different spectral components of thermal radiation, we identify an approach to increase the efficiency of thermoradiative cells via spectrally selecting long-wavelength photons for radiative exchange. Furthermore, we predict that the near-field photon extraction by coupling photons generated from interband electronic transition to phonon polariton modes on the surface of a heat sink can increase the conversion efficiency as well as the power generation density, providing more opportunities to efficiently utilize terrestrial emission for clean energy. An ideal InSb thermoradiative cell can achieve a maximum efficiency and power density up to 20.4% and 327 Wm−2, respectively, between a hot source at 500 K and a cold sink at 300 K. However, sub-bandgap and non-radiative losses will significantly degrade the cell performance.


Section I. The Thickness Dependence of the Thermoradiative Cells
The performance of the thermoradiative cells depend on the thickness of the InSb thin film. Both efficiencies and power densities are calculated for three configuration including (i) a thin-film InSb, (ii) a thin-film InSb and a selective surface of ∆ℏω = 0.01 eV, and (iii) a thin-film InSb and a bulk CaCO3 at a gap distance of 10nm. The optical constants of InSb and CaCO3 used in all the calculations are also shown in Fig. S1 [1][2][3]. For each configuration, the ideal case and the other two cases that take into account the sub-bandgap and non-radiative losses are calculated and shown in Figs. S2, S3, and S4. The ideal cases (red solid curve) only include the radiative process, which contributes to the emission where the photons have energy higher than the bandgap energy, to create the negative chemical potential and generate the power. For the real material, the emission include the photons with their energy below the bandgap energy. These photons cannot contribute to the formation of chemical potential and result in the sub-bandgap loss (blue solid curve).
However, there are also non-radiative processes such as Auger, Shockley-Read-Hall, and surface defect processes. These processes will result in the net charge generation rate when the chemical potential is negative, and the net generation will reduce the magnitude of chemical potential and cause the electronic loss or non-radiative loss (black solid curve).
There are several important points for each case. First, the efficiency is higher for thinner thermoradiative cells, especially for the configuration (iii) that relies on the near-field radiative transfer; however, power density increases with thickness for each configuration except for the near-field radiative transfer including both sub-bandgap and non-radiative losses (blue data shown in Fig. S4). Second, if the non-radiative rate can be significantly reduced, the efficiency is higher if a selective suface is used to control the emission. If the non-radiative rate is comparable with the radiative rate, the case using near-field radiative transfer has the highest efficiency due to its enhanced radiative rate. Third, there are some oscillations of efficiencies and power densities for various thicknesses, as shown in the Figs. S2 and S3.
They happen due to the interference inside the InSb thin film resulting in the 'thermal well' effect. [4][5]

Section II. The Formulas for the Near-Field Thermal Radiation Between Two Thin Films.
To calculate the near-field thermal radiation between a thin film and a semi-infinite substrate, we use the formulas that are derived from references 1 and 2. The Planck oscillator term,   θ ω,T , determines the thermal distribution of modes contributing to the radiative heat transfer and in its most general form includes the Bose-Einstein distribution and a zero-point energy term as follows, where is the reduced Planck constant, kB is the Boltzmann constant, is the frequency of the photon, and T is temperature of the emissive body. For transport calculations, the zero-point energy term is negated by emission from both the hot and cold media.
For TM and TE polarizations, respectively, the Fresnel coefficients for an interface are, where εi and εj is the complex dielectric permittivity for layers i and j, respectively. Likewise, kz,i and kz,j is the perpendicular wavevector for layers i and j. For a thin slab with index j, the reflection and transmission coefficients are, where dj is the thickness of the layer j.
To isolate the radiative heat flux from film 1 to film 3, a simple subtraction is needed using equations (S2) and (S3) as follows, '  2  2  2  2  TE  TE  TE  TE  z,0 z,2  z,4 z,2  20  20  24  24  22  k  z,2  z,4  1  rr  2  2  2ik g  TE TE  0  20 24 13 ω,prop 1 k k k k 1-r -t Re 1-r -t Re kk θ ω,T k dk ... π 4 1-r r e q ω,T =       is absorbed by a cold thin-film and a supporting substrate. However, to completely describe the system in Fig. S5, it is also necessary to consider thermal emission from the supporting substrate of the hot emitter.
In order to isolate this contribution, it is possible to make use of the formulation for radiative heat transfer between two semi-infinite media. Despite the presence of thin-films, previous studies [6][7][8] have proven that the formulation for two semi-infinite media can still be used to calculate the total radiative heat transfer for a hot emitter and cold absorber composed of an arbitrary number of layers assuming the hot and cold side temperatures are uniform across their respective layers.
The spectral radiative heat flux in this case will consist of the following, 2   2  2  2  2  TE  TE  TM  TM  k  20  24  20  24  1  rr  22  2  2ik g  2ik g  TE TE  TM TM  0  20 24  20 24   01 34  ω,prop  1 1-r 1-r 1-r 1-r θ ω,T k dk π 4 1-r r e 4 1-r r e q ω,T = TE  TE  TM  TM  20  24  20  24  -2k g  1  rr  22  2  -2k g  -2k g  TE TE  TM TM  k  20 24  20 24   01 34  ω,eva  1 Im r Im r Im r Im r θ ω,T k dk e π 1-r r e 1-r r e q ω,T = (S10) where it is assumed the temperature of substrate 0 is equal to the temperature of film 1. The reflection coefficients represent the total reflectance of the thin-film and substrate for the hot and cold sides. By combining Eqs. (S5) -(S10), the emission from thin film 1 to substrate 3 and 4 can be calculated. We now have a complete set of equations to describe radiative heat transport for a system composed of a single thin-film supported by a substrate on both the hot and cold sides.
This formulation will serve as the foundation in the manuscript to explore thin-film morphological effects on near-field radiative heat transfer using dielectrics and metals.

Section III. The Net Generation Rate through Surface Defect Process
In the neutral regions of a p-n junction, minority carriers are driven by the diffusion force so that the drift-diffusion equation in p-type neutral region can be simplified to = ∇ , where the Ie is the electron current density, De is the diffusivity of electron, and n is the electron concentration. [9][10] Under these assumptions, the minority carrier concentration profile in one dimension can be found by solving is the Auger lifetime, ℎ and ℎℎ are the Auger rate constants, ′ = − , is the electron non-equilibrium concentration, and are the electron and hole equilibrium concentration respectively, the and are the free carrier generation rates for radiative and Auger processes respectively, and the and are the free carrier recombination rates for radiative and Auger processes respectively.
To solve this equation, the generation and recombination rate of the radiative process are