Spatial and doping effects on radiative recombination in thin-film near-field photonic energy converters

Modeling radiative recombination is crucial to the analysis of photonic energy converters. In this work, a local radiative recombination coefficient is defined and derived based on fluctuational electrodynamics that is applicable to thin-film cells in both the near field and far field. The predicted radiative recombination coefficient of an InAs cell deviates from the van Roosbroeck-Shockley relation when the thickness is less than 10 um and the difference exceeds fourfold with a 10 nm film. The local radiative recombination coefficient is orders of magnitude higher when an InAs cell is configured in the near field. The local radiative recombination coefficient reduces as the doping level approaches that of a degenerate semiconductor. The maximum output power and efficiency of a thermoradiative cell would be apparently overpredicted if the luminescence coefficient (defined in this letter) were taken as unity for heavily doped semiconductors.

6][7] The phenomenon of electroluminescence that enables light-emitting diodes can also be used to transfer thermal energy from a colder region to a hotter region while consuming electrical power. 9,10ar-field radiative heat transfer can boost the photon flux by orders of magnitude as demonstrated both theoretically and experimentally. 11,12[19][20][21] The annihilation of electron-hole pairs through the radiative recombination process is responsible for the emission of photons or luminescence.3][24] Trupke et al. 25 studied the temperature dependence of radiative recombination coefficient using the van Roosbroeck-Shockley relation 26 considering the radiative properties of semi-transparent laminate samples.While external luminescence and photon recycling have been investigated for both far-field and near-field photonic energy converters, [27][28][29][30] the local radiative recombination coefficient is not clearly defined, especially in the near-field regime.Furthermore, the magnitude and impact of above-bandgap thermal radiation due to free carriers in heavily doped semiconductors on the device performance are yet to be explored.
In this work, fluctuational electrodynamics (FE) is used to connect the external luminescent emission to a local radiative recombination coefficient.A thin-film InAs cell is modeled in free space as well as in a thermophotovoltaic (TPV) setup with or without a back gapped reflector (BGR).
Comparison is made with the conventional van Roosbroeck-Shockley relation for a free-standing film with varying thicknesses.The effect of doping for cells with high dopant concentrations is considered using a luminescence coefficient, which also enables the distinction between thermal and nonthermal radiation above the bandgap energy.The effect of luminescence coefficient on the performance of a TR device is quantitatively examined.
Re i ( , ) Here, ω is the angular frequency, c is the speed of light in vacuum, i ) are the tensor components of the electric (E) or magnetic (H) dyadic Green's functions, and * denotes complex conjugate.The integration over the x y ′ ′ − plane is carried out by transforming the Green's function to the wavevector x y k k − space.For a multilayer structure, the spatial integration is with respect to z′ only.For thermal emission at equilibrium, ( ) ( ) ( ) , where ε ′′ is the imaginary part of the dielectric function at z′ and ( ) ( ) is the mean energy of a Planck oscillator at temperature T. Note that B k and  are the Boltzmann constant and the reduced Planck constant, respectively.Considering a direct bandgap semiconductor, the emission may be devided into a nonthermal portion due to interband transition that is affected by the chemical potential and a thermal portion due to other transitions such as free-carrier transitions and lattice vibrations. Therefore, where ( )  may be viewed as the mean energy of a photon mode based on the modified Bose-Einstein distribution considering photon chemical potential µ. ε ′′ = when g ω ω < .The significance of Eq. ( 2) is that it separates the luminescent emission originated from interband transition from the thermal emission at above bandgap energies.
The ratio of ib ε ′′ and ε ′′ is proportional to the luminescence emission over the total radiative emission: Djuric et al. 33 used the ratio of absorption coeffients in a similar way to analyze the quantum efficiency of InSb photodiodes.The quantity defined in Eq. ( 3) was called the cell internal quantum efficiency.13 In the present study, φ is termed luminescence coefficient since it is a factor that should be included in evaluating luminescent emission.For lightly doped semiconductors, φ is close to unity; however, the effect on φ must be considered for heavily doped semiconductors.The dielectric function model of InAs considering the effects of temperature and doping level is outline in the supplemantory material (SM) after Milovich et al. 16 The photon flux due to external luminescence from the cell is calculated by adding the processes + C and C − described in Fig. 1; therefore, where ( ) Here, 0 / k c ω = is the wavevector in vacuum, > regions, respetively. 31,34Note that 1,2 F depend on the dielectric function and thickness of each layer.
Applying Boltzmann approximation discussed in the SM, the local radiative recombination rate is expressed as where i n is the intrinsic carrier concentration that is a function of temperature.The radiative recombination rate of the film is expressed as FE 0 1 ( ) .
For a free-standing film in the far field, the radiative recombination coefficient of the cell is simplified as where R γ and T γ are the reflection and transmission coefficients of the thin film when light is incident from air.The superscript γ represents the polarization state of light (s or p).The summation in Eq. ( 7) represents the absorptance of the film and may be derived directly from FE or indirectly using Kirchhoff's law. 34,35   The effect of surrounding structure on ( ) B z′ is considered in two scenarios.One is a TPV and the other is a TPV with a back gapped reflector (TPV-BGR), as shown in the inset of Fig. 3.
The emitter is a bulk tungsten with a 30-nm-thick indium tin oxide (ITO) film to enhance plasmonic resonances with the 400-nm InAs cell in the near field. 6A BGR is placed at the lower region with a fixed 10 nm vacuum gap between the InAs cell and a 100-nm-thick gold film.The vacuum gap spacing (d) between the emitter and the InAs cell is taken as a variable.Similar structures have been investigated previous. 36,37The local radiative recombination coefficient expressed in Eq. ( 6) depends on the cell's temperature and dielection function as well as the dielectric functions and thicknesses of the surrounding materials.As discussed in the SM, for InAs with a doping level less than 10 17 cm −3 , the intrinsic dielectric function may be assumed.In the near field, the radiative recombination is highly localized and structure dependent.
As shown in Fig. 3, as d is reduced from 1 mm to 10 nm, ( ) B z′ increases by orders of magnitude, especially near the front surface due to surface plasmon polaritons (SPPs).SPPs are excited at large k  , resulting in very small penetration depth. 38The frustrated modes also contribute to the enhancement with a much large penetration, resulting in the enhanced ( ) B z′ for the TPV at d = 10 nm.The frustrated modes are associated with , where the refractive index m is between 3.5 and 3.9 for InAs in the spectral region of interest.Adding a BGR gives rise to a higher ( ) B z′ near the back as well as in the middle region of the cell due to multiple reflections between the plasmonic emitter and the BGR, which affect the TPV-BGR configuration for both d = 10 nm and 1 mm.In the far field (d = 1 mm) without BGR, the profile of ( ) predicted by the van Roosbroeck-Shockley relation.Some of the luminescently emitted photons are reabsorbed by the cell, resulting in photon recyclying.However, photon recycling is mainly due to frustrated modes and should not exceed int B .In evaluating the photon recycling portion using near-field formulism, 30 one must set up an upper limit or cutoff k  for the integration over k  to be bounded. 39,40Hence, the obtained ( ) B z′ based on fluctuational electrodynamics may be assumed as the local "internal" radiative recombination coefficient near the front surface by negnecting photon recycling.For the TPV-BGR configuiration at d = 10 nm, the radiative recombination coefficient averaged over the cell, FE B , is 16  3   6.9 10 m /s − × , which is nearly four times that of int B .The BGR can significantly enhance photon recycling and improve the performance as discussed previously. 37The associated large radiative recombination rate may also affect the dark current of the cell. 24, corresponds to the maximum power condition of this TR device.The ideal assumption with φ = 1 treats all the above-bandgap absorption as due to interband transition and overpredicts the luminescent emission as shown in Fig. 5(a).At a given frequency, thermal emission is scaled to . The difference between the ideal and acutal luminescent emission is scaled to , which is smaller than when 0 µ < as in this case.
The total net luminescent emission calculated with the ideal φ assumption is 7.4 kW/m 2 , which is 11% higher than that with the actual φ .This would cause more than 10% overprediction of both the maximum power density and the efficiency of the TR as shown in Fig. 5(b).Though the calculations are oversimplified and the structure used is not optimized for a TR device, the results provide evidence of the effect of thermal emission with high dopant concentrations.Hence, the spatial effect and doping effects must be considered for thin-film near-field photonic energy converter.In summary, this work defines and develops a formulism for calculating the local radiative recombination coefficient based on fluctuational electrodynamics that is applicable to both the near-and far-field photonic energy converters.SPPs can modify the radiative recombination coefficient near the surface by several orders of magnitudes.The spatial profile of radiative recombination coefficient can be tuned by modifying the surrounding structure and materials.The use of a luminescence coefficient allows the distinction of the luminescent emission from that of thermal emission due to the above-bandgap free-carrier contribution.It also enables more accurate modeling of photonic energy converters that employ heavily doped semiconductors.
It may be integrated over the frequency as shown in Eq. ( 6).The application condition of Boltzmann approximation depends on the injected carrier density and doping level of the semiconductor materials.The error of Boltzmann approximation has been discussed in Feng et al. 4 Rigorously speaking, B defined above is the "external" radiative recombination coefficient because it is related to external luminescene.However, the word "external" is omitted for brievity, unless clarifications are necessary.Portion of the emitted photons are reabsorbed by the cell and the reabsorption process is called photon recycling 5,6 .Hence, the "internal" radiative recombination coefficient should be greater than the radiative recombination coefficient given previously.Since B is related to electroluminescence and recombination lifetime, it is an important parameter in radiative energy converters. 7

S2. Expressions of the van Roosbroeck-Shockley relation
The van Roosbroeck-Shockley relation provides an expression of the internal radiative recombination coefficient with the absorption coefficient ( ) α ω as follows: 8 ( ) ( ) where ( ) A ω is the spectral absorptance of the cell layer considering multiple reflections at the interfaces.The absorptance is the same for incidence from either above or below the cell when it is a free-standing film.Here, the external luminescent emission considers both the upper and lower hemisphere.While multiple reflections are included in calculating ( ) A ω , the original formulation did not consider interference effects, which are important for thin films whose thickness is on the order or smaller than the wavelength of interest.
When the cell thickness h is small and, ( ) based on geometric or ray optics.Considering hemispherical emission/absorption, the factor 2 2n still holds for large enough n that makes the refraction angle in the cell to be sufficiently small.When 1 n → (the refractive index of the cell is about the same as the surroundings), surface reflection is negligibly small.Furthermore, integration of the path length / cos h θ (where θ is the polar angle) over the hemisphere gives ( ) It should be noted that Eq. (S8) is appropriate for most practical applications such as solar cells.Nevertheless, it will produce a large relatively error as the cell thickness is reduced to submicron regime as illustrated in Fig. 2. In the near-field regime, the radiative recombination processes also depend on the surrounding materials and geometric parameters such as the vacuum gap distance as discussed in the main text. 3,4,6,7

S3. Dielectric function model of doped InAs
The dielectric function of InAs is depicted by the summation of absorption due to interband transitions, lattice resonance (optical phonon), and free-carrier absorption.Milovich et al. 10 developed a comprehensive model for the dielectric function of InAs as a function of dopant concentration and temperature.Since the absorption edge of a narrow bandgap semiconductor can be affected under the heavily doped condition, the Moss-Burstein effect (or shift) needs to be considered. 11,12A brief summary of the dielectric function model is given in the following.
A Drude-Lorentz model is used to calculate the contribution by phonons (lattice vibration) and free carriers to the frequency-dependent dielectric function as 13 ( ) where ε ∞ is the high-frequency constant, LO ω and TO ω are the longitudinal and transverse optical phonon frequencies, γ and Γ are the damping rates for phonons and free carriers, respectively, and p ω is the plasma frequency divided by 1/2 ε ∞ .
The dielectric function due to interband transitions is calculated based on the interband absorption coefficient, where the Moss-Burstein shift is considered for heavily-doped narrow bandgap semiconductors.The Fermi energy level is solved as a function of dopant concentration using the relation given in Sijerčić et al. 14 with the InAs parameters.The interband absorption coefficient ib ( ) α ω as a function of frequency and dopant concentration is then evaluated using Eq. ( 16) or (20) in Ref. [11].A Kramers-Kronig transformation of ib

Thermoradiative cell
A thermoradiative (TR) device operates when the photovoltaic (PV) cell is at a temperature higher than the surroundings.The p-n junction is negatively biased so that it emits fewer photons than the corresponding case at thermodynamic equilibrium.There is an associated electron-hole pair generation to recover the equilibrium population, resulting in photocurrent and power output to a load.Detailed descriptions and derivations can be found from the literature. 16,17By neglecting

Fig. 1 .
Fig. 1.Schematic of a thin-film radiative energy converter with four regions: an upper semi-infinite region, vacuum gap region with a thickness of d, active (cell) region with a thickness of h, and a lower semi-infinite region.The processes A, B, and C describe irradiation into the cell, thermal emission from the cell, and external electroluminescent emission from the cell across its upper (+) and lower (−) boundaries, respectively.

and 2 F
represent the solution of multilayer Green's function for emission originated from z′ towards 0 z < and z h is about 30 times that of ext B for very small h, calculation based on fluctuational electrodynamics fully capture the wavy feature due to interference when the film thickness is less than 10 µm.

Fig. 3 .
Fig. 3. Local radiative recombination coefficient for a TPV and TPV-BGR configurations, as shown in the inset, with both the near-field (d = 10 nm) and far-field (d = 1 mm) vacuum gaps.The active region is modeled as intrinsic InAs at 300 K.
z′ for the TPV is nearly symmetric about the center of the cell with a valley in the middle region; this is caused by wave interference within the InAs film.Contour plots of ( , )B z ω ′ are shown in the SM to help understanding the spectral distribution.It should be noted that ( ) B z′ for a free-standing InAs film (not shown) is very similar to the case with the TPV when d = 1 mm, dispite that the tungsten-ITO emitter is highly reflecting in the far field.At the front surface of the cell for d = 10 nm, -BGR, which are about 75 and 126 times that of int B

Fig. 4 .
Fig. 4. Doping effect on B(z') for the near-field TPV-BGR configuration.The result for NA = 1×10 17 cm -3 is essentially the same as that for intrinsic InAs.

Fig. 5 .
Fig. 5. (a) The net spectral heat flux due to thermal and nonthermal emission for a TR cell configuration shown in the inset for 0.052 eV µ = − .The active region consists a p-n junction or diode, though, the dielectric function is treated as uniform with pdoped InAs with 19 3 A v @ 600 K 1.05 10 cm N N − = = × .The net spectral heat flux by assuming ideal luminescence coefficient is also shown for comparison.(b) Power density and efficiency as functions of the bias voltage for the actual and ideal luminescence coefficient.
photoluminescence spectrum to calculate the radiative recombination coefficient.Following their derivation, one obtains a corresponding "external" radiative recombination coefficient of a layer as follows.
Shockley relation that is applicable to thin films.The conventional van Roosbroeck-Shockley relation is discussed in SM.Take a free-standing intrinsic InAs film with g E = 0.354 eV at T = 300 K as an example.
Equation(7)provides a modified van Roosbroeck-FE B and ext B is 4.3 and 3.6 times when h = 10 nm and 20 nm, respectively.The calculated int B using Eq.(S7) is