Thermal Impacts on the Performance of Nanoscale-Gap ... - IEEE Xplore

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 2, JUNE 2011

Thermal Impacts on the Performance of Nanoscale-Gap Thermophotovoltaic Power Generators Mathieu Francoeur, Rodolphe Vaillon, and M. Pinar Mengüç

Abstract—The thermal impacts on the performance of nanoscale-gap thermophotovoltaic (nano-TPV) power generators are investigated using a coupled near-field thermal radiation, charge, and heat transport formulation. A nano-TPV device consisting of a tungsten radiator, maintained at 2000 K, and cells made of indium gallium antimonide (In0 .1 8 Ga0 .8 2 Sb) are considered; the thermal management system is modeled assuming a convective boundary with a fluid temperature fixed at 293 K. Results reveal that nano-TPV performance characteristics are closely related to the temperature of the cell. When the radiator and the junction are separated by a 20 nm vacuum gap, the power output and the conversion efficiency of the system are respectively 5.83 × 105 Wm−2 and 24.8% at 300 K, whereas these values drop to 8.09 × 104 Wm−2 and 3.2% at 500 K. In order to maintain the cell at room temperature, a heat transfer coefficient as high as 105 Wm−2 K−1 is required for nanometer-size vacuum gaps. The reason for this is that thermal radiation since thermal radiation enhancement beyond the blackbody from a bulk radiator of tungsten is broadband in nature, while only a certain part of the spectrum is useful for maximizing nano-TPV performance. In future studies, near-field radiation spectral conditions leading to optimal performance characteristics of the device will be investigated. Index Terms—Energy conversion, nanoscale-gap thermophotovoltaic, near-field thermal radiation, thermal effects.

h∞ ¯h i J J0 Jph Jsc k kb kcond Ldp m0 m∗(e,h) ni n(e,h) n(e,h)0 Na Nc Nd Nv

NOMENCLATURE cv D(e,h) e E Eg g

g

Speed of light in vacuum (=2.998 × 108 ms−1 ). Diffusion coefficient of electron/hole (m2 s−1 ). Electron charge (=1.6022 × 10−19 J eV−1 ). Wave energy (=¯ hω/e) (eV). Bandgap (eV). Generation rate of electron-hole pairs (m−3 s−1 ).

Manuscript received September 6, 2010; revised January 19, 2011; accepted February 11, 2011. Date of publication April 7, 2011; date of current version May 18, 2011. This work was partially sponsored by the Kentucky Science and Engineering Foundation under Grant KSEF-1718-RDE-011. Partial support for MPM was received from the FP-7-PEOPLE-IRG-2008 (No: 239382) and the TUBITAK 1001 (No: 109M170) projects. Paper no. TEC-00357-2010. M. Francoeur was with the University of Kentucky. He is now with the Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112 USA (e-mail: [email protected]). R. Vaillon is with the Centre Nationale de la Recherche Scientifique (CNRS), Université de Lyon, INSA-Lyon, UCBL, CETHIL, UMR5008, F-69621, Villeurbanne, France (e-mail: [email protected]). M. P. Mengüç is currently with Ozyegin University, Altunizade, Uskudar, 34662 Istanbul, Turkey, on leave from the Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEC.2011.2118212

Pm q Q QNRR QRR QT Re Sr S(e,h) T T∞ V0 Vf Vo c

Weyl component of the dyadic Green’s function (m−1 ). Convective heat transfer coefficient (Wm−2 K−1 ). Reduced Planck’s constant (=1.0546 × 10−34 J s). Complex constant (=(−1)1/2 ). Effective photocurrent (Am−2 ). Dark current (Am−2 ). Photocurrent generated (Am−2 ). Short-circuit photocurrent (Am−2 ). Wavevector (=k + ik ) (rad m−1 ). Boltzmann constant (=1.3807 × 10−23 J K−1 ). Thermal conductivity (Wm−1 K−1 ). Thickness of depletion region (m). Electron rest mass (=9.109 × 10−31 kg). Effective electron/hole mass (kg). Intrinsic carrier concentration (m−3 ). Electron/hole concentration (m−3 ). Electron/hole equilibrium concentration (m−3 ). Acceptor concentration (m−3 ). Effective density of states in the conduction band (m−3 ). Donor concentration (m−3 ). Effective density of states in the valence band (m−3 ). Maximum electrical power output (Wm−2 ). Radiative heat flux (Wm−2 ). Heat source (Wm−3 ). Heat source due to nonradiative recombination of electron-hole pairs (Wm−3 ). Heat source due to radiative recombination of electron-hole pairs (Wm−3 ). Heat source due to thermalization (Wm−3 ). Real part. Radiative heat source (Wm−3 ). Surface recombination velocity of electron/hole (m s−1 ). Temperature (K). Temperature of thermal management system (K). Equilibrium potential in depletion region (V). Forward bias (V). Open-circuit voltage (V).

Greek symbols εr εs εv

0885-8969/$26.00 © 2011 IEEE

Dielectric function (=εr + iεr ). Static relative permittivity (F m−1 ). Absolute permittivity (F m−1 ).

FRANCOEUR et al.: THERMAL IMPACTS ON THE PERFORMANCE OF NANOSCALE-GAP THERMOPHOTOVOLTAIC POWER GENERATORS

ΦPR ηc κ μ(e,h) Θ ρ, θ, z τ (e,h) ω

Photon recycling factor. Internal conversion efficiency. Absorption coefficient (m−1 ). Electron/hole mobility (m2 V−1 s−1 ). Mean energy of a Planck oscillator (J). Polar coordinates. Electron / hole lifetime (s). Angular frequency (rad s−1 ).

Subscripts, superscripts, and abbreviations ∗ abs avg Auger dp e emi EHP fc h ib j latt n NRR p RR SRH TPV v ω

Complex conjugate. Absorbed. Averaged. Auger recombination. Depletion. Electron. Emitted. Electron-hole pair. Free carriers. Hole. Interband. Control volume. Lattice. n-doped. Nonradiative recombination. p-doped. Radiative recombination. Shockley-Read-Hall recombination. Thermophotovoltaic. Vacuum. Monochromatic.

I. INTRODUCTION HE CURRENT world energy consumption is about 14 TW, among which less than 1% is coming from clean and renewable sources [1]. By 2050, it is expected that this global demand will reach about 25–30 TW. In order to minimize the environmental impacts of this energy consumption, experts estimated that about 20 TW should come from carbonfree renewable energy resources [1]. In an opinion paper recently published, Baxter et al. [1] pointed out the importance of nanoengineering to develop low-cost and high-efficiency renewable energy technologies, and emphasized, among other techniques, solar thermophotovoltaic (TPV) power generators that could greatly benefit from nanoscale design. In such TPV devices, solar irradiation is absorbed by a radiator, which, in turn, re-emits thermal radiation in a spectrally selective fashion toward a cell generating electricity. TPV power generators are not restricted to solar applications, as any kind of heat source can be used. In that sense, TPV devices are particularly interesting from an energy saving point of view, as wasted heat can be turned into electricity. Beyond their potential versatility, TPV systems are expected to be quiet, low-maintenance, modular, safe, and pollution-free [2], [3]. Typically, the electrical power output of a TPV system is about

T

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104 Wm−2 with an internal conversion efficiency of about 20% to 30% [3]. Cells with bandgaps of about 1.1 eV are usually employed in solar photovoltaic (PV) applications. TPV radiators are typically maintained between 1000 K and 2000 K, such that cells with bandgaps lower than 1.1 eV are required. For this reason, it is common to make the distinction between TPV and solar PV cells. So far, research on TPV cells has mainly focused on III-V binary compounds, such as gallium antimonide (GaSb) and gallium arsenide (GaAs), as well as their ternary and quaternary III-V alloys [3], [4]. In order to potentially improve the performance of TPV systems, Whale and Cravalho [5], [6] proposed to separate the radiator and the cells by a subwavelength vacuum gap. At subwavelength distances, radiation heat transfer is in the near-field regime, such that the energy exchanges can exceed the values predicted for blackbodies due to the contribution from evanescent waves [7]–[10]. For typical temperatures involved in thermal radiation, the near-field effects become dominant when the bodies are separated by few tens to few hundreds of nanometers. A TPV system using the near-field effects of thermal radiation is, therefore, referred as a nanoscale-gap TPV device, or more simply, a nano-TPV device. To date, theoretical and experimental research efforts on nano-TPV power generation remain very scarce. Whale and Cravalho [5], [6] considered two bulks separated by a vacuum gap, where the TPV cells were maintained at a temperature of 300 K. A fictitious low conductivity material was used for the radiator with a dielectric function described by a Drude model. Indium gallium arsenide (In1-x Gax As) with a bandgap varying from 0.36 to 1.4 eV was considered for the cells. The authors reported that nano-TPV devices provided a significant enhancement of the electrical power output combined with marginal gains in conversion efficiency. Whale [11] later showed that the conversion efficiency could be increased via interference of propagating waves in the gap and by using multijunctions, leading to maximum gains of about 10% and 5%, respectively. Narayanaswamy and Chen [12] proposed to use a radiator supporting surface phonon-polaritons in order to increase the power output and conversion efficiency of nano-TPV devices. They considered a bulk radiator made of cubic boron nitride maintained at a temperature of 1000 K, and the TPV cell was modeled as a 100-nm-thick layer maintained at 300 K. A fictitious dielectric function for the TPV cell was used to approximate the behavior of a direct bandgap semiconductor (Eg = 0.13 eV). The results showed that for a 20 nm gap, the power absorbed by the cell was about three orders of magnitude higher than absorption of solar irradiation. Laroche et al. [13] studied a nano-TPV system consisting of two bulks separated by a vacuum gap. The TPV cell, maintained at 300 K, was made of GaSb with a bandgap of 0.7 eV; it was assumed that all radiation with energy higher than this bandgap contributed to the photocurrent generation (i.e., quantum efficiency of 100%). Two types of radiator, maintained at 2000 K, were considered in the simulations: a tungsten emitter and a fictitious quasimonochromatic source described by a Drude model

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having a resonance matching the bandgap of the cell. For the tungsten radiator, calculations of the electrical power output showed an enhancement by a factor 50 for vacuum gaps varying from 10 μm down to 5 nm; for the fictitious Drude radiator, an enhancement factor of about 35, relative to a blackbody at 2000 K, was predicted for a 5-nm-thick vacuum gap. Finally, the authors reported conversion efficiencies varying nonmonotonically from 21% to 27% for gap values from 10 μm to 5 nm, respectively, when the tungsten radiator was considered; conversion efficiencies varying from 10% to 35% for the fictitious Drude radiator were calculated. The most sophisticated nano-TPV model was provided by Park et al. [14]. The analysis was performed by solving for the first time the coupled near-field thermal radiation and charge transport problem within the TPV cell. The radiator was modeled as a bulk tungsten maintained at 2000 K, while indium gallium antimonide (In0.18 Ga0.82 Sb) cells, maintained at 300 K with a bandgap of 0.56 eV, were considered as receivers. Numerical simulations showed that the conversion efficiency of the system increased as the gap decreased when the quantum efficiency was 100%. When quantum efficiencies lower than 100% were considered via the solution of the charge transport problem, the conversion efficiency was lowered by 5% to 10%. For vacuum gaps between 2 nm and 10 μm, the conversion efficiency varied nonmonotonically between 17% and 23%, respectively. Experiments on nano-TPV devices were conducted by DiMatteo et al. [15]. The system was comprised of a silicon radiator and indium arsenide (InAs) cells, and the gap between these two components was maintained using silicon dioxide (SiO2 ) spacers. The gap between the radiator and TPV cells was varied using a piezoactuator flexing the heater chip by fraction of microns. By decreasing the gap for initial radiator temperatures of 348 K, 378 K, and 408 K, it was observed that the short-circuit current increased by a factor 5. A dynamic test was also performed, where the piezoactuator was oscillating between frequencies of 200 Hz and 1000 Hz, thus causing the vacuum gap also to oscillate. Results showed that variations of the short-circuit current somehow followed in-phase the gap oscillation frequency, thus leading the authors to conclude that the increase of the current was due to tunneling of evanescent waves. The experimental device was later refined by DiMatteo et al. [16], where indium gallium arsenide (InGaAs) cells were used. Measurements of the electrical power output for a vacuum gap of about 200 nm showed an enhancement as the temperature of the radiator increased from 550 ◦ C to 850 ◦ C. Results also revealed that the enhancement in radiation absorption by the cells was proportional to the enhancement of the electrical power output. Note that DiMatteo is the founder, CEO and Chairman of MTPV (Micron-gap Thermal PhotoVoltaics) commercializing micron-gap TPV devices [17]. However, no data are available regarding these systems beyond the experimental investigations cited in this paragraph. More recently, an experimental investigation was conducted by Hanamura and Mori [18], where the radiator and cells were made of tungsten and GaSb, respectively. Measurement of the J-V characteristic for a radiator temperature of 1000 K showed


Fig. 1.

Schematic representation of the nano-TPV system under study.

an increase of the current as the vacuum gap decreased. For gaps less than 10 μm (down to 1 μm), the temperature of the radiator greatly decreased, which was explained by the increasing radiative heat flux from the radiator toward the TPV cells. While the literature showed that tunneling of evanescent waves can substantially improve the electrical power output of nano-TPV systems, with marginal gains in conversion efficiency, some important questions about the viability of this technology are still unanswered. In this paper, our objective is to analyze the feasibility of maintaining the temperature gradient between the radiator and the cells usually discussed in the literature (2000 K for the radiator and 300 K for the cells), and to quantify the impacts of the thermal phenomena on the performance of nano-TPV power generation. This analysis is accomplished by solving the coupled near-field thermal radiation, charge and heat transport problem within the cell. The paper is structured as follows. The nano-TPV system under study is first described. The modeling details for the coupled near-field thermal radiation, charge and heat transport problem are provided afterwards, along with the optical, electrical and thermophysical properties needed for the solution. The thermal impacts on nano-TPV system performance are subsequently analyzed, and concluding remarks are given. II. DESCRIPTION OF THE NANO-TPV DEVICE The nano-TPV power generator under consideration is shown in Fig. 1. Here, all layers are assumed to be parallel and perfectly smooth, homogeneous, isotropic, nonmagnetic, and described by a dielectric function εr (ω) local in space. The system is azimuthally symmetric and infinite along the ρ-direction, such that only variations along the z-axis need to be considered. A bulk radiator of tungsten (W), modeled as a half-space, and an indium gallium antimonide (In0.18 Ga0.82 Sb) cell of thickness tcell are separated by a subwavelength vacuum gap of length dc . The radiator is assumed to be maintained at a constant and uniform temperature T0 of 2000 K via an external heat input, while the operating cell temperature is given by Tcell (z). When the cell is illuminated, the absorption of a wave with


energy E (= h ¯ ω/e) equal or larger than the bandgap Eg of the semiconductor generates mobile charges by electron-hole pairs (EHPs). The bandgap Eg of In0.18 Ga0.82 Sb at 300 K is 0.56 eV, which corresponds to an angular frequency of 8.51 × 1014 rad/s and a vacuum wavelength of 2.21 μm. The TPV cell consists of a single p-n junction (p on n configuration), where the p-doped region is assumed to have a thickness tp of 0.4 μm and a doping level Na of 1019 cm−3 , while the n-doped material has a thickness tn of 10 μm with Nd = 1017 cm−3 . These parameters are the same as those used by Park et al. [14] for purpose of comparison. As the cell is likely to heat up from various sources, a thermal management system is assumed to be used to maintain the p-n junction around room temperature. Nano-TPV system performance characteristics are evaluated via the solution of the coupled near-field thermal radiation, charge, and heat transport problem; the modeling details are given in the next section. III. PHYSICAL AND MATHEMATICAL MODELING As a first step, the TPV cell is discretized into N control (p,n ) . The spatial discretizations of the p-doped and volumes Δzj n-doped regions are different, since tp is much smaller than tn . Details about the discretization are provided in reference [19].

Near-field radiant energy exchanges between the radiator and the cell are calculated using fluctuational electrodynamics, where a stochastic current density is added to the Maxwell equations to account for the random fluctuations of charges due to thermal agitation within a material [9], [20]. The monochromatic radiative heat flux at a particular z-location within a prescribed layer is determined by calculating the timeaveraged z-component of the Poynting vector and by applying the fluctuation-dissipation theorem, which provides the link between the stochastic current source and the local temperature of the emitting medium [20], [21]. The near-field radiative heat flux due to thermal emission by the radiator at temperature T0 (p,n ) absorbed by a control volume Δzj within the cell, delimited (p,n )

(p,n )

and zj +1 , is found by calculating the (p,n )

difference between the flux crossing the boundary zj

and the

(p,n ) zj +1 , as shown (1), at the bottom of

flux crossing the boundary this page, where the subscript l in (1) refers to the layer where (p,n ) is located (l = 2 for the p-doped layer and l = 3 for the Δzj n-doped layer) and the subscript α involves a summation over E H and g0l are, the three orthogonal components. The terms g0l respectively, the Weyl components of the electric and magnetic dyadic Green’s functions (DGFs) relating the fields in layer l

qωabs (p ,n ) ,Δ z j

(p,n )

(or zj +1 ) with frequency ω and wavevector at location zj kρ to a source located in medium 0. The explicit expressions of the Weyl components of the DGF, as well as the numerical procedure used to solve (1), are provided in reference [21]. Thermal radiation emitted by the cell toward the radiator has to be accounted for in the analysis. The near-field radiative heat (p,n ) due to the emitflux absorbed by a control volume Δzj ting radiator at temperature T0 is the same as the radiative heat flux absorbed by the radiator due to an emitting control vol(p,n ) at temperature T0 . Therefore, (1) normalized by ume Δzj the mean energy of a state Θ depends solely on the geometry of the system and the material properties. Given that, the near(p,n ) field radiative heat flux emitted by a control volume Δzj at temperature Tcell ,j and absorbed by the radiator, q em i ( p , n ) , ω ,Δ z j

is calculated by dividing (1) by Θ(ω,T0 ) and then by multiplying the resulting transfer function by the mean energy of an electromagnetic state at temperature Tcell ,j , Θ(ω, Tcell ,j ). Note that for the purpose of near-field thermal radiation modeling, the dielectric function of the thermal management system (medium 4 in Fig. 1) is assumed to be equal to unity. B. Minority Carrier Transport Modeling Minority carrier transport within the cell is modeled using the steady-state continuity equations written as [22], [23]:

A. Near-Field Thermal Radiation Modeling

by the boundaries zj

(p,n )

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⎧ ⎪ ⎪ ⎪ ⎪ ∞ 2 kρ dkρ kv Θ(ω, T0 ) ⎨ = Re iεr 0 ⎪ 2π 2 kz0 0 ⎪ ⎪ ⎪ ⎩

D(e,h)

d2 Δn(e,h),ω (z) Δn(e,h),ω (z) − + gω (z) = 0 dz 2 τ(e,h)

(2)

where the minority carriers in the p-doped and n-doped regions are electrons and holes, respectively. The term Δn(e,h), ω (z) is the local excess of minority carriers above the equilibrium concentration n(e,h)0 generated by absorption of radiation with frequency ω (i.e., Δn(e,h),ω (z) = n(e,h), ω – n(e,h)0 , where n(e,h), ω is the local carrier concentration). The equilibrium concentrations of minority carriers in the p-doped and n-doped regions are given, respectively, as ne0 = n2i /Na and nh0 = n2i /Nd , where ni is the intrinsic carrier concentration. In (2), τ (e,h) is the minority carrier lifetime that accounts for radiative recombination as well as nonradiative Auger and Shockley-Read-Hall (SRH) recombination processes. Emission by radiative recombination can either be re-absorbed within the cell, or be transmitted outside the semiconductor. Since radiation emitted by radiative recombination has E ≥Eg , the reabsorption of such waves generates another EHP within the cell; this phenomenon is referred in the literature as “photon recycling” [24]. Photon recycling is accounted for in this work using an approximate method, where a radiative lifetime multiplication factor ΦPR is introduced. Using this approach, an effective radiative lifetime ΦPR τ (e,h),R R

⎡

⎤⎫ ⎪ ⎪ ⎪ ⎥⎪ ⎢ (p,n ) (p,n ) ⎬ E H∗ ⎥ ⎢ , ω)g0lρα (kρ , zj , ω) ⎢ −g0lθ α (kρ , zj ⎥ (p,n ) (p,n ) ⎥ ⎢ E H∗ g0lρα (kρ , zj +1 , ω)g0lθ ⎪ ⎦⎪ ⎣ α (kρ , zj +1 , ω) ⎪ ⎪ − ⎭ (p,n ) (p,n ) E H∗ −g0lθ α (kρ , zj +1 , ω)g0lρα (kρ , zj +1 , ω) (p,n )

E g0lρα (kρ , zj

H∗ , ω)g0lθ α (kρ , zj

(p,n )

, ω)

(1)

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is defined, which accounts strictly for radiation emitted by radiative recombination that is transmitted outside the cell. The total minority carrier lifetime is calculated as τ (e,h) = (1/τ (e,h),A u g er + 1/τ (e,h),SRH + 1/ΦPR τ (e,h),RR )−1 . The minority carrier diffusion equations are coupled with near-field thermal radiation via the local generation rate of EHPs (p,n ) : calculated as follows at location Δzj ⎛ abs ⎞ q 1 ib ⎝ ω ,Δ z j( p , n ) ⎠ κ (3) gj,ω = hω j,ω Δz (p,n ) κj,ω ¯ j

where the subscript j refers to a given control volume where it is assumed that the local generation rate is constant. In (3), κj , ω is the local monochromatic absorption coefficient that accounts for absorption by the lattice and the free carriers, as well as the interband absorption process, while κib j,ω is the local monochromatic interband absorption coefficient, which is nonzero for E ≥ Eg . Strictly speaking, the generation rate of EHPs is given by the incident radiation power multiplied by the interband absorption coefficient [23]. The term in parentheses in (3) corresponds to the incident radiation power, which is then multiplied by κib j,ω . The solution of the minority carrier diffusion equations requires boundary conditions at z = Z2 and z = Z4 (see Fig. 1) where there is recombination of EHPs [23]: dΔn(e,h),ω (Z(2,4) ) = S(e,h) Δn(e,h),ω (Z(2,4) ) (4) dz where S(e,h) is the surface recombination velocity. At the edges p n and Zdp ), all minority carriers of the depletion region (i.e., at Zdp are assumed to be swept by the electric field at the p-n junction, (p,n ) such that Δn(e,h), ω (Zdp ) = 0 [23]. D(e,h)

C. Heat Transport Modeling for the TPV Cell Since the TPV cell is a few micrometers thick, it is possible to apply the Fourier law to the problem. The 1-D steady-state energy equation with heat generation applied to the cell is written as: d2 Tcell (z) + Q(z) = 0 (5) dz 2 where the thermal conductivity kcond is assumed uniform in the cell and where Q(z) is the local heat generation term given by Q(z) = −Sr (z) + QT (z) + QNRR (z) − QRR (z). The term Sr (z) is the local radiative heat source representing the balance between thermal radiation emission and absorption by the lattice and the free carriers. This term is computed within a given control volume j using the solution of the near-field thermal radiation problem: ⎤ ⎡ em i abs ∞ q (p ,n ) − q (p ,n ) ω ,Δ z ω ,Δ z j j c ⎦dω (6) ⎣ (κfj,ω + κlatt Sr,j = j,ω ) (p,n ) 0 Δzj κj,ω kcond

where the term within the square brackets is the net radiation power, which is multiplied by the sum of the absorption coeffifc cients due to the lattice κlatt j,ω and the free carriers κj,ω [23]. It is important to note that the net radiation power, as calculated in (6), does not account for the redistribution of energy inside the

cell due to radiative exchanges between the control volumes. This contribution is negligibly small compared to heat conduction within the layer. Moreover, for the temperatures involved in the simulations, near-field thermal radiation emitted by the radiator and absorbed by a control volume dominates the value of the heat generation term, such that radiative transfer between the control volumes does not affect in a perceptible manner the net radiation power. Radiation absorbed by the cell with E > Eg releases its excess of energy into heat. This contribution to the local heat generation term, QT (z), is called thermalization and is also calculated within a control volume j from the solution of the near-field thermal radiation problem as follows [23]: ⎡ abs ⎤ ∞ q (p ,n ) ω ,Δ z Eg j ib ⎣ ⎦ 1− κj,ω QT ,j = dω (7) (p,n ) (¯ hω)/e ωg Δz κj,ω j

where ω g = Eg e/¯h. The term within the square brackets is the incident radiation power, while the expression in parentheses corresponds to the fraction of energy in excess above the bandgap of the cell. Minority carriers that recombine before reaching the depletion region, through nonradiative Auger and SRH processes, contribute to raise the temperature of the TPV cell. The contribution of these nonradiative recombinations, QNRR (z), to the local heat generation term is calculated within a control volume j from the solution of the minority carrier diffusion equations as [23]: ∞ eEg Δn(e,h),j,ω dω (8) QNRR ,j = τ (e,h),NRR ωg where τ (e,h),NRR is the minority carrier lifetime due to nonradiative recombinations, calculated as τ (e,h),NRR = (1/τ (e,h),A u g er + 1/τ (e,h),SRH )−1 . Radiative recombination QRR (z) contributing to the local heat generation term is calculated in the same way as (8), except that τ (e,h),NRR is replaced by ΦPR τ (e,h),RR . Note that in using (8) to calculate QRR (z), it is implicitly assumed that all radiative energy generated by radiative recombination leaving the p-n junction is not reflected back toward the cell. At the boundaries of the cell, the internal heat conduction and surface recombination are balanced with an external heat flux. At z < Z2 , there is a vacuum, such that the external heat flux is equal to zero; this boundary condition can therefore be written as: ∞ dTcell (Z2 ) + Se eEg Δne,ω (Z2 )dω = 0. (9) −kcond dz ωg At the boundary z = Z4 , there is an external heat flux due to the thermal management system (modeled as a convective boundary condition). Moreover, surface recombination at z = Z4 can be neglected due to the relatively large thickness of the p-n junction [25]. The boundary condition at z = Z4 is consequently written as: −kcond

dTcell (Z4 ) = h∞ [Tcell (Z4 ) − T∞ ] dz

(10)


where h∞ and T∞ are respectively the heat transfer coefficient and temperature of the cooling fluid at z > Z4 .

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calculated as the difference between the total flux crossing the boundary z = Z2 and the boundary z = Z4 .

D. Modeling Nano-TPV Device Performance The photocurrent generated by the cell has to be calculated first in order to evaluate the performance of the nano-TPV device. In the depletion region, it is assumed that all created EHPs contribute to generate photocurrent: Zn dp Jdp,ω = e gj,ω dz. (11) Z dp p

For EHPs generated outside the depletion region, the minority carriers diffuse toward the region where they are majority carriers. If these EHPs reach the edge of the depletion region, photocurrents are generated. These photocurrents, proportional to the gradient of minority carrier concentration at the edges of the depletion region, are given by [23]: Je,ω = eDe

p dΔne,ω (Zdp )

(12a) dz n dΔnh,ω (Zdp ) Jh,ω = −eDh . (12b) dz The monochromatic photocurrent generated by the cell is given by: Jph , ω = Je , ω + Jh , ω + Jdp , ω . The total photocurrent Jph is then calculated by integrating Jph , ω over ω from ω g to infinity. When the cell is not illuminated, there is a current due to the applied voltage at the junction, called the dark current J0 . When the junction absorbs radiation, the generated photocurrent Jph flows opposite to J0 . The dark current is calculated by solving the minority carrier diffusion equations in dark conditions (i.e., gω (z) = 0) [23]. The boundary conditions at z = Z2 and z = Z4 are the same as (4) in illuminated conditions, except that there is no frequency dependence. The boundary conditions at the edges of the depletion region are on the other hand modified to account for the forward bias Vf (positive voltage from p relative to n) [22], [23]: eVf (p,n ) Δn(e,h),ω (Zdp ) = n(e,h)0 exp . (13) kb Tcell Once the minority carrier diffusion equations are solved in dark conditions, J0 can be calculated as a function of Vf . The dark current is the sum of currents due to minority carriers at the edges of the depletion region, and is consequently calculated using (12a) and (12b) (without the spectral dependence). The effective photocurrent generated by the nano-TPV device is then given by the difference between the total photocurrent and the dark current: J(Vf ) = Jph −J0 (Vf ). By calculating J0 for a series of Vf (starting with Vf = 0), the J-V characteristic of the TPV cell can be computed. For simplicity, J is taken here as positive even if, strictly speaking, J should be negative on the J-V characteristic. The conversion efficiency η c of the nano-TPV power generator is defined as the ratio of the maximum power output Pm over the total radiative heat flux absorbed by the cell qabs cell . The maximum power output Pm is easily determined using the J-V characteristic of the TPV cell, while the flux absorbed is

IV. MODELING OF OPTICAL, ELECTRICAL AND THERMOPHYSICAL PROPERTIES A. Dielectric Functions of the Radiator and TPV Cells The dielectric function of tungsten has been modeled by curve-fitting the data reported in reference [26]. For direct bandgap semiconductors, such as GaSb and InSb, radiation absorption above Eg is dominated by the interband process. For energy slightly lower than Eg , there is a spectral band of transparency where the absorption coefficient is very low. As the energy decreases below the transparency region, the absorption coefficient increases due to contributions from the lattice and the free carriers [5], [27]. For E ≥ Eg , the interband dielectric function of the TPV cell is modeled using the semi-empirical model proposed by Adachi [28]. It accounts for the various energy transitions within the semiconductor, and fitting parameters are required to calculate these contributions. The temperature-dependence of the fitting parameters, provided for both GaSb and InSb, is accounted for using Varshni’s equation: E(Tcell ) = E(0)−δTcell 2 /(Tcell + β), where E(0) corresponds to an energy level, such as the fundamental bandgap Eg , at 0 K. The variables δ (eV/K) and β (K) are fitting constants provided in reference [4]. The dielectric function of the ternary alloy of GaSb and InSb (In1−x Gax Sb) is calculated using the parameters for GaSb and InSb combined with Vegard’s law: Palloy (x) = xPGaSb + (1−x)PInSb −x(1−x)CB , where P refers to a given parameter, while CB is the so-called bowing constant that accounts for deviations from the linear interpolation due to lattice disorders [4]. Note that due to a lack of data in the literature, the doping-dependence of the interband dielectric function is not accounted for in the analysis. The lattice and free carrier contributions to the dielectric function of the TPV cell is accounted for via a Lorentz-Drude model. The parameters for GaSb have been found in [29], while the parameters for InSb have been determined using reference [27]. To calculate the dielectric function due to the lattice and the free carriers for In0.18 Ga0.82 Sb from the parameters for GaSb and InSb, Vegard’s law is used with x = 0.82 and CB = 0. The temperature-dependence of the dielectric function due to the lattice and free carriers is not accounted for. The dopingdependence is taken into account only for GaSb; the values do not directly depend on the doping level, but on whether if the semiconductor is p- or n-doped. The imaginary part of the dielectric function of In0.18 Ga0.82 Sb at 300 K is shown in Fig. 2, where it is assumed that GaSb is p-doped. The imaginary part of the dielectric function is very low directly below Eg , resulting in the so-called transparency spectral band. At Eg , the imaginary part of the interband dielectric function is much higher than for the lattice and the free carriers. Therefore, it is possible to assume that the contributions from the free carriers and the lattice for energies E ≥ Eg is negligible. The interband absorption coefficient of In0.18 Ga0.82 Sb, in m−1 , is shown in Fig. 3 at different temperatures.

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and −1.35; these values are valid for temperatures from 50 K up to 920 K [30]. For InSb, A and n are respectively given by 5729.7 Wm−1 K−1 and −1.028, values that are valid for temperatures equal or larger than 300 K [31]. The thermal conductivity of In0.18 Ga0.82 Sb is calculated via Vegard’s law with P = kcond , x = 0.82, and CB = 0. At 300 K, the thermal conductivity of In0.18 Ga0.82 Sb is 28.9 Wm−1 K−1 . C. Intrinsic Carrier Concentration

Fig.2.

Imaginary part of the dielectric function of In0 . 1 8 Ga0 . 8 2 Sb at 300 K.

Calculation of the intrinsic carrier concentration ni requires the knowledge of the effective density of states in the conduction and valence band, which, in turn, depends on the effective electron and hole masses. The effective electron and hole masses for In1−x Gax Sb are calculated respectively as m∗e = (0.015 + 0.01x + 0.025x2 )m0 and m∗h = (0.43 − 0.03x)m0 , where m0 is the electron rest mass (9.109 × 10−31 kg) [25]. Once the effective masses are determined, the effective density of states in the conduction and valence band are calculated respectively h)3/2 . as Nc = 2(m∗e kb Tcell /2π¯h)3/2 and Nv = 2(m∗h kb Tcell /2π¯ Finally, using Nc and Nv , the intrinsic carrier concentration is calculated as ni = (Nc Nv )1/2 exp(-Eg /2kb Tcell ) [22]. At 300 K, the intrinsic carrier concentration of In0.18 Ga0.82 Sb is 2.22 × 1013 cm−3 . D. Thickness of the Depletion Region The thickness of the depletion region is calculated as follows [22]: 1/2 2εs 1 1 V0 + , (14) Ldp = e Na Nd

Fig. 3. Interband absorption coefficient of In0 . 1 8 Ga0 . 8 2 Sb at various temperatures.

Using the model proposed by Adachi [28] to calculate the dielectric function of the TPV cell, interband absorption arises for E ≥ Eg . In reality, low interband absorption can be observed at energies slightly less than the bandgap threshold, especially in heavily doped materials [3]. The interband absorption at E < Eg is called the “exponential tail.” Since the exponential tail cannot be represented with the current model, we should expect spectral quantities to drop sharply at E = Eg . Experimental data are required to accurately represent the exponential tail. Despite the fact that Adachi’s model will induce some very low imprecision near the bandgap Eg , this will not affect at all the predictions of the performance of the nano-TPV device. B. Thermal Conductivity The temperature-dependent thermal conductivities of GaSb and InSb are calculated using kcond (Tcell ) = ATcell n [30]. For GaSb, A and n are given respectively by 7 × 104 Wm−1 K−1

where V0 is the equilibrium voltage at the p-n junction given by V0 = (kb Tcell /e)ln(Na Nd /ni 2 ). The term εs is the static relative permittivity calculated as εs = (16.8 – 1.1x)εv for In1−x Gax Sb, where εv is the absolute permittivity (8.85 × 10−12 Fm−1 ) [25]. At 300 K and for the doping levels given in section II, the thickness of the depletion region for In0.18 Ga0.82 Sb is estimated to be 99 nm. Under a forward bias Vf , the thickness of the depletion region should be calculated using (V0 − Vf ) instead of V0 as in (14) [22]. Instead, the procedure described by Vaillon et al. [23] is used in this work. The problem is solved by assuming that the thickness of the depletion region is given by (14). Once the calculations have converged, the dark current is calculated for a series of Vf , and the J−V characteristics of the nano-TPV system is thus generated. Alternatively, (14) with (V0 − Vf ) could have been used to calculate the thickness of the depletion region under a forward bias, such that the effective photocurrent would have been obtained directly. This procedure would however lead to excessive CPU requirements, since the convergence would need to be performed for each Vf value considered in order to generate the J–V characteristic. For the doping levels considered in this work, we determined that more than 99% of the depletion region is within in the n-region regardless of the temperature of the cell [32]. Consequently, it is assumed that the entire depletion region is located in the n-doped material.


E. Diffusion Coefficients and Surface Recombination Velocities The diffusion coefficients are calculated from the electron and hole mobility, μe and μh , via the Einstein relation D(e,h) = μ(e,h) (kb Tcell /e) [25]. The temperature- and doping-dependent electron and hole mobilities are computed using the relation reported in reference [33] and the parameters provided in [25] for GaSb and InSb. The carrier mobility of In0.18 Ga0.82 Sb is calculated using Vegard’s law with P = μ−1 , CB = 0, and x = 0.82. At 300 K, the electron and hole diffusion coefficients are given respectively by 35.2 cm2 s−1 and 18.3 cm2 s−1 for In0.18 Ga0.82 Sb. Due to the relatively large thickness of the cell, surface recombination velocity of holes in the n-doped region is neglected (Sh ≈ 0) [25]. For the p-doped region, it is difficult to determine a precise value for Se , as the surface recombination velocity depends not only on the material, but also on the surface treatment of the cell. Martin and Algora [33] suggested that a Se of about 2 × 104 m·s−1 can be used for GaSb, whereas Frank and Wherrett [34] suggested that Se for InSb should take values between 1 to 104 m·s−1 , depending on surface preparation. For the purpose of simulations, we used Se = 2 × 104 m·s−1 for In0.18 Ga0.82 Sb. F. Minority Carrier Lifetimes Minority carrier lifetime due to SRH nonradiative recombination is calculated as: τ (e,h),SRH = (1/σNt )[m∗(e,h) / (3kb Tcell )]1/2 , where Nt is the density of traps (1.17 × 1021 m−3 ), and σ is the capture cross-section of minority carriers (1.5 × 10−19 m2 ) [25]. Temperature-dependent minority carrier lifetime due to nonradiative Auger recombination is calculated via the model presented in reference [25]. At 300 K, the total nonradiative minority carrier lifetimes of electrons and holes are 9.8 ns and 31.1 ns, respectively, for In0.18 Ga0.82 Sb. The minority carrier radiative lifetime is calculated using the following simple model [3]: τ (e,h),R R = (BN(a,d) )−1 , where B is the bimolecular recombination coefficient (BGaSb = 8.5 × 10−11 cm3 s−1 [33] and BInSb = 5 × 10−11 cm3 s−1 [31]). The minority carrier radiative lifetime of In0.18 Ga0.82 Sb is calculated using Vegard’s law with P = B, CB = 0, and x = 0.82. A photon recycling factor ΦPR of 10 is used [33]. At 300 K, the effective minority carrier radiative lifetimes of electrons and holes are 13 ns and 1.27 μs, respectively, for In0.18 Ga0.82 Sb. Finally, the total minority carrier lifetimes of electrons and holes at 300 K are respectively 5.5 ns and 30.3 ns for In0.18 Ga0.82 Sb. V. EVALUATION OF NANO-TPV DEVICE PERFORMANCE A. Numerical Solution of the Coupled Problem The coupled near-field thermal radiation, charge and heat transport problem is solved using a standard finite-volume discretization method [35]; the numerical details are provided in reference [19]. The general algorithm used to predict the performance of nano-TPV power generation is: 1) Specify the initial temperature Tcell (z) of the TPV cell. 2) Prescribe a heat transfer coefficient and the temperature at z > Z4 (thermal management system).

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3) Calculate the properties of the cell at temperature Tcell (z). 4) Solve the near-field thermal radiation problem. 5) Calculate the local radiative heat source and thermalization term. 6) Calculate the local generation rate of EHPs. 7) Solve the minority carrier diffusion equations. 8) Calculate the photocurrent generated. 9) Calculate the nonradiative and radiative recombination source terms. 10) Solve the energy equation and obtain an updated temperature distribution of the cell. 11) Compare the temperature distribution obtained in step 10 with the temperature distribution from the previous iteration; if the maximum relative difference is greater than a specified convergence criterion, go back to step 3. Otherwise, go to step 12. 12) Solve the minority carrier diffusion equations in dark conditions for a series of forward biases. 13) Calculate the effective current density generated by the nano-TPV system, and plot the J-V characteristic. 14) Determine the point on the J-V characteristic where the power generated by the device is maximal. 15) Evaluate the performance of the nano-TPV power generation system. A part of the nano-TPV model presented in this paper was validated against the results of Park et al. [14], where the thermal effects within the TPV cell were not accounted for. These validation results can be found in reference [19]. B. Performance of Nano-TPV Power Generation as a Function of the Temperature of the Cell In this section, the coupled near-field thermal radiation and charge transport problem is solved for a fixed and uniform temperature of the cell Tcell . Fig. 4(a) and (b) show the radiation absorbed by the p-n junction qabs cell , the maximum electrical power output Pm , the conversion efficiency η c and the photocurrents Jph , Jdp , Jh , and Je of the nano-TPV system for gaps dc from 1 nm to 10 μm at Tcell = 300 K. It is worth noticing that gaps below 20 nm are very difficult to achieve and maintain in practice. The theoretical study of nano-TPV performance in the extreme near-field will help us to identify general trends and to interpret the underlying physical phenomena. As expected, the radiation absorbed by the cell, and, therefore, the electrical power output, increases as the gap decreases due to radiation tunneling. It is interesting to note that an increase of qabs cell and Pm is observable when dc increases from 800 nm to about 1 μm. This inverse trend is likely to be due to wave interference within the gap. Fig. 4(b) shows that the photocurrent due to diffusion of minority electrons in the p-region increases as dc decreases, while the photocurrents due to diffusion of holes in the n-region and within the depletion zone saturate, and decrease slightly below dc values of about 10 nm. Similarly, the conversion efficiency of the nano-TPV device decreases significantly below a gap of 10 nm. A maximum conversion efficiency of 24.8% is predicted for a 20-nm-thick gap, where the electrical power output is 5.83 × 105 Wm−2 . For a sufficiently

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Fig. 4. Performance of nano-TPV power generation as a function of dc for Tc e ll = 300 K: (a) Radiation absorbed by the cell, electrical power output, and conversion efficiency. (b) Photocurrent generated.

large gap (i.e., starting at about dc ≈ 5 μm), all the parameters shown in Fig. 4(a) and (b) become independent of dc as the far-field regime of thermal radiation is reached. At this limit, an electrical power output and a conversion efficiency of 2.40 × 104 Wm−2 and 19.8%, respectively, have been calculated. The behavior observed in Fig. 4(a) and (b) was discussed by Park et al. [14]. As the gap dc decreases, near-field radiative heat transfer is dominated by evanescent waves with decreasing penetration depths in vacuum, and therefore in the TPV cell, of the order of dc [36]–[38]. This implies that for small dc values, a large proportion of the radiative energy is absorbed near the boundary z = Z2 , and the EHPs thus generated are more likely to recombine before reaching the depletion region. As a consequence, the combination of increasing thermal radiation absorption by the cell and decreasing penetration depths of evanescent waves, as dc decreases, result in low conversion efficiencies. This also explains the results of Fig. 4(b), where the


Fig. 5. Performance of nano-TPV power generation as a function of Tc e ll for dc = 20 nm, 50 nm, and 100 nm: (a) Radiation absorbed by the cell and electrical power output. (b) Total photocurrent generated.

photocurrents generated in the n-doped and depletion regions, Jh and Jdp , decrease as dc becomes extremely small. Fig. 5(a) and (b) shows the radiation absorbed by the p-n junction, the electrical power output and the total photocurrent generated for Tcell varying from 300 K to 500 K and for gaps dc of 20 nm, 50 nm and 100 nm. It can be observed in Fig. 5(a) that thermal radiation absorption increases slightly as the temperature of the cell increases due to the slight increase in the interband absorption coefficient as Tcell increases, and most importantly due to the fact that Eg decreases with increasing the temperature (see Fig. 3). On the other hand, the electrical power output Pm of the TPV device decreases significantly when Tcell increases, regardless of the gap dc . For a dc value of 20 nm, the electrical power output is about 5.83 × 105 Wm−2 at Tcell = 300 K (η c = 24.8%), and drops significantly at 8.09 × 104 Wm−2 at Tcell = 500 K (η c = 3.2%). It is interesting to note that despite this significant drop in electrical power output, the photocurrent Jph is almost


Fig. 6. (a) J-V characteristic of nano-TPV power generation for dc = 20 nm and various temperatures Tc e ll . (b) Dark current J0 as a function of the forward bias Vf for various temperatures Tc e ll .

insensitive to the temperature of the cell, as shown in Fig. 5(b). Indeed, regardless of the gap dc , Jph slightly increases when Tcell increases from 300 K to about 425 K, and then slightly decreases for Tcell > 425 K. As mentioned above, an increase of Tcell leads to an increase of the absorbed thermal radiation by the cell. Therefore, when qabs cell increases, more EHPs are generated, thus leading to an increase of Jph . On the other hand, minority carrier lifetimes τ e and τ h decrease as Tcell increases, thus leading to a larger recombination rate of EHPs. Fig. 5(b) suggests that below a temperature of about 425 K, the increase in radiation absorption overcomes the increase in EHP recombination, and vice versa above 425 K. However, from a practical point of view, Jph can be considered as nearly constant as a function of the temperature. The trends observed in Fig. 5(a) and (b), independent of the thickness of the gap separating the radiator and the p-n junction, can be better understood by inspecting the J-V characteristic of the nano-TPV power generation device. In Fig. 6(a), J-V characteristics are shown for a nano-TPV device with dc = 20 nm and for various temperatures Tcell . In Fig. 6(b), the dark current J0 as a function of the forward bias Vf is shown for various Tcell values. Fig. 6(a) shows that the short-circuit current Jsc (current at Vf = 0) slightly varies with Tcell , while the open-circuit voltage

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Vo c (voltage at J = 0) significantly decreases with increasing Tcell . At Vf = 0, it is possible to assume that Jsc ≈ J ≈ Jph , since J0 is very small [22], such that the variations of Jsc as a function of Tcell observed in Fig. 6(a) follow the trends reported in Fig. 5(b). On the other hand, Vo c decreases significantly as Tcell increases, and therefore the area under the curve delimited by Vo c and Jsc also significantly diminishes. This behavior obviously leads to decreasing values of the maximum power output Pm and the conversion efficiency η c . To ensure a maximal value of the open-circuit voltage Vo c , the dark current J0 needs to be as small as possible [19], [22]. Fig. 6(b) shows clearly that for a given Vf , the dark current J0 increases when the temperature of the cell increases. Consequently, the decreasing power output and conversion efficiency of the nano-TPV device with increasing Tcell are fundamentally due to an increasing dark current. Indeed, as the temperature of the cell increases, the intrinsic carrier concentration ni increases. For example, ni = 2.22 × 1013 cm−3 and 3.66 × 1015 cm−3 for cell temperatures of 300 K and 500 K, respectively. Consequently, the equilibrium concentration of minority carriers, given by n(e,h)0 = ni 2 /N(a,d) , also increases when increasing Tcell . The boundary conditions of the minority carrier diffusion equations at the edges of the depletion region in dark conditions (13) show that the local excess of minority carriers above the equilibrium concentration is directly proportional to n(e,h)0 . The dark current is proportional to the local excess of minority carriers above the equilibrium concentration, which increases with increasing the temperature of the cell. The observations made in this section are crucial, as they show that the cell needs to be maintained around room temperature in order to have efficient nano-TPV power generation devices. C. Predictions of Nano-TPV Power Generation Performance Using the Coupled Near-Field Thermal Radiation, Charge, and Heat Transport Model As outlined in the beginning of section V, an iterative process is required to solve the coupled problem, and the convergence is evaluated via the temperature of the cell. A convergence criterion of 10−4 (i.e., relative difference of 0.01%) was used, since the temperature distribution and the performance of the nano-TPV device were not affected when decreasing this value. In all simulations, it is assumed that the temperature of medium 4 (T∞ ) is fixed at 293 K. Fig. 7 shows averaged cell temperature Tcell ,avg as a function of the heat transfer coefficient h∞ for various gaps dc . For all cases treated in this section, the temperature gradient within the cell was found to be very small, while the conduction heat flux in the cell was large. A maximum temperature difference of 0.5 K was calculated for dc = 20 nm and h∞ = 5 × 103 Wm−2 K−1 , such that it is justified to analyze the averaged cell temperature. The melting temperature of GaSb is about 985 K [39], while it is 800 K for InSb [31]. Using Vegard’s law with CB = 0 and P = Tm elt , the melting temperature of In0.18 Ga0.82 Sb is estimated to be 952 K. Due to the temperature-dependence of the interband dielectric function, the simulations do not converge when the TPV cell reaches a temperature higher than

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Fig. 7. Averaged temperature of the cell as a function of h∞ for various dc values.

the melting point; indeed, at such temperatures, the bandgap of In0.18 Ga0.82 Sb calculated using Varshni’s equation becomes negative, thus resulting in a nonconvergence of the interband dielectric function [28]. For dc = 5 μm, 100 nm, 50 nm, and 20 nm, the simulations did not converged for h∞ values equal or less than 2 × 102 Wm−2 K−1 , 103 Wm−2 K−1 , 2 × 103 Wm−2 K−1 , and 4 × 103 Wm−2 K−1 , respectively. The values of h∞ needed to maintain the TPV cell around 300 K are quite high. For a gap dc of 5 μm, a h∞ value of 104 Wm−2 K−1 is required to keep the p-n junction around room temperature, while a h∞ of 105 Wm−2 K−1 is needed for gaps dc of 100 nm, 50 nm, and 20 nm. Generally speaking, heat transfer coefficients h∞ up to 103 Wm−2 K−1 can be achieved via free convection, while h∞ up to about 2 × 104 Wm−2 K−1 can be reached by forced convection; heat transfer coefficients above this threshold is possible via convection involving phase change [40]. The results of Fig. 7 should not be surprising, as radiation with energy E below or above the bandgap Eg largely contributes to heat generation in the p-n junction. The use of a bulk radiator in the near-field provides a broadband enhancement of the flux, which contributes simultaneously to increase the electrical power output and to increase heat generation within the p-n junction. Even when the gap dc between the radiator and the cell is relatively large (5 μm), the temperature of the junction becomes high (331 K and 374 K) for the different h∞ values corresponding to free convection. This is why it is necessary for “macroscale-gap” TPV devices to use selectively emitting radiators or to employ filters between the radiator and the cell to reflect back unwanted radiation. The spatial distributions of the different contributions to the local heat generation term are shown in Fig. 8 for dc = 20 nm and a h∞ value of 104 Wm−2 K−1 . The heat source Q(z) is maximum at z = Z2 , which corresponds to the irradiated boundary, and is dominated by thermalization QT (z), a conclusion that is applicable to all cases analyzed in this paper. This is due to the fact that thermal radiation absorption is largely dominated by the interband process. It can also be observed that QT (z) decreases sharply as z increases, and does not dominate anymore


Fig. 8. Spatial distributions of the different contributions to the local heat generation term Q(z) for dc = 20 nm, T∞ = 293 K and h∞ = 104 Wm−2 K−1 .

Q(z) starting at a z value of about 3 μm. Indeed, for E Eg , the absorption coefficient is quite high (see Fig. 3), such that radiation mean free path is small. The electrical power output and the conversion efficiency of the nano-TPV devices investigated in Fig. 7 are presented respectively in Fig. 9(a) and (b) as a function of the heat transfer coefficient h∞ . The performance characteristics of the nanoTPV devices are significantly affected by the thermal boundary condition imposed at z = Z4 . For example, when dc = 20 nm, the conversion efficiency η c when h∞ = 106 Wm−2 K−1 (Tcell,avg = 294 K) is 25.4%, whereas this value drops to 6.9% when h∞ = 5 × 103 Wm−2 K−1 (Tcell,avg = 466 K). The power output and the conversion efficiency, shown in Fig. 9(a) and (b), do not include any insight into the power and associated means to obtain the desired heat transfer coefficient h∞ . Taking into account this parameter, which will be done in a future study, is likely to exhibit extremely poor performance of the nano-TPV device discussed in this work. VI. CONCLUSIONS The thermal impacts on the performance of nanoscale-gap thermophotovoltaic (nano-TPV) power generation were investigated for the first time. For this purpose, a coupled near-field thermal radiation, charge and heat transport model within the cell was solved. A system consisting of a tungsten radiator, maintained at 2000 K, and TPV cells made of indium gallium antimonide (In0.18 Ga0.82 Sb) was investigated. The analysis of the performance as a function of the temperature of the cell revealed that both the conversion efficiency and the electrical power output significantly decrease as the temperature of the cell increases. For a gap of 20 nm, a power output and a conversion efficiency of 8.09 × 104 Wm−2 and 3.2%, respectively, were calculated when the cell is maintained at 500 K, versus 5.83 × 105 Wm−2 and 24.8%, respectively, at cell temperature of 300 K. It was shown that the radiation absorbed by the cell is almost insensitive to its temperature, while the open-circuit voltage decreases significantly with increasing the


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between the radiator and the cells. These and other potential practical concerns are not discussed here, but will be considered in future research efforts. In this work, we rather focused on a relatively simple geometry in order to evaluate clearly the thermal impacts on the performance of nano-TPV system; the trends reported in this paper are of course applicable to more complex geometries. Finally, from a practical point of view, the nano-TPV device discussed in this paper involves a high temperature gradient between the radiator and the cell that is difficult to maintain. Low-temperature nano-TPV devices might be a viable alternative, where the radiator would be maintained between 350 K and 400 K. This technology would, however, require the use of cells with bandgaps much lower than the typical 0.5–0.7 eV employed for high-temperature TPV power generators. For low-temperature nano-TPV devices, materials supporting surface phonon-polaritons in the infrared, such as silicon carbide, could be employed for the radiator. By nanostructuring a material supporting surface polaritons in the infrared, as discussed in [36], [37], [41]–[43], it might be possible to obtain optimal nano-TPV device performance via a fine spectral tuning of near-field radiant energy exchange. REFERENCES

Fig. 9. Performance of nano-TPV power generation as a function of dc and h∞ (T∞ = 293 K): (a) Electrical power output Pm . (b) Conversion efficiency η c .

temperature of the p-n junction due to an increase of the dark current. The solution of the coupled near-field thermal radiation, charge, and heat transport problem showed that the nano-TPV devices proposed so far in the literature might be unpractical. The thermal management system was modeled using a convective boundary with a fluid temperature fixed at 293 K. Even in the far-field regime, a considerably high value of 104 Wm−2 K−1 for the heat transfer coefficient is needed to maintain the cell around room temperature. For gaps of 20 nm, 50 nm, and 100 nm, a heat transfer coefficient of 105 Wm−2 K−1 is required to maintain the cell at room temperature, a value that is extremely large. This is due to the fact that the near-field radiative heat transfer enhancement from a tungsten bulk is broadband, thus contributing not only in increasing the electrical power output, but also in increasing the heat source within the cell. For actual engineering applications, a number of practical details should also be considered. They include edge effects for 2-D and 3-D systems, layer thickness variations, material inhomogeneities, surface roughness and imperfect parallelism

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Mathieu Francoeur received the Bachelor’s and Master’s degrees with highest honors from the Department of Mechanical Engineering of Université Laval, Québec City, PQ, and the Ph.D. degree from the Department of Mechanical Engineering at the University of Kentucky, Lexington, in 2010. He joined the Department of Mechanical Engineering at the University of Utah, Salt Lake City, as an Assistant Professor in 2010. He received in 2009, the Young Scientist Award in the radiative transfer category from the Journal of Quantitative Spectroscopy and Radiative Transfer. His current research interests are directed toward nearand far-field radiative heat transfer, the control of thermal radiation emission and absorption in resonant nanostructures, thermophotovoltaic and photovoltaic energy conversion, surface polariton interactions in nanosize objects, and optical characterization of nanoparticles.

Rodolphe Vaillon received the Ph.D. degree from Poitiers University, Poitiers, France, in 1996. He is a CNRS researcher at the Thermal Science Centre of Lyon (CETHIL, CNRS-INSA LyonUCBL), Villeurbanne, France. He is an associate editor of the Journal of Quantitative Spectroscopy and Radiatıve Transfer (Elsevier). His current research interests include radiative heat transfer in semitransparent media, radiative properties of particles and gases, optical/radiative diagnostic of particles, heat transfer modeling of photovoltaic devices, and near-field thermal radiation.

¨ ¸ received the Ph.D. degree from M. Pinar Menguc Purdue University, West Lafayette, Indiana, in 1985. The same year he moved to the University of Kentucky, Lexington, as an Assistant Professor of mechanical engineering. He became full Professor in 1993, and was named Engineering Alumni Association Professor in 2008. Over the years, he held temporary positions at the Harvard-Massachusetts General Hospital, Boston, and at the Università degli Studi di Napoli Federico II, Napoli, Italy. In 2007, he was awarded an Honorary Professorship at ESPOL, Guayaquil, Ecuador. He has guided more than 35 Ph.D. and M.S. students. He has coauthored two books, more than 200 archival journal articles, book chapters, and conference papers, and has four awarded and two pending US patents. He has given more than 80 invited lectures around the globe. He is one of the three Editors-in-Chief of the Journal of Quantitative Spectroscopy and Radiatıve Transfer (Elsevier) and was an Associate Editor of the Journal of Heat Transfer (ASME). He has organized five International Symposia on Radiative Transfer, all in Turkey. He was Director of Nano-Scale Engineering Certificate Program and one of the founding members of the Honors Program on ¨ Nanotechnology at the University of Kentucky in the US before joining Ozye˘ gin University in Istanbul, Turkey, in 2009, where he is currently the Director of Center for Energy, Environment and Economy. Prof. Mengüç is a fellow of the American Society of Mechanical Engineers and the International Centre for Heat and Mass Transfer.