Moderate-temperature near-field thermophotovoltaic systems with thin-film InSb cells
MModerate-temperature near-field thermophotovoltaic systems withthin-film InSb cells
Rongqian Wang, ∗ Jincheng Lu,
1, 2 and Jian-Hua Jiang † School of physical science and technology & CollaborativeInnovation Center of Suzhou Nano Science and Technology,Soochow University, Suzhou 215006, China. Center for Phononics and Thermal Energy Science,China-EU Joint Center for Nanophononics,Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology,School of Physics Science and Engineering,Tongji University, Shanghai 200092 China (Dated: January 27, 2021)
Abstract
Near-field thermophotovoltaic systems functioning at 400 ∼
900 K based on graphene-hexagonal-boron-nitride heterostructures and thin-film InSb p - n junctions are investigated theoretically. Theperformances of two near-field systems with different emitters are examined carefully. One near-field system consists of a graphene-hexagonal-boron-nitride-graphene sandwich structure as theemitter, while the other system has an emitter made of the double graphene-hexagonal-boron-nitride heterostructure. It is shown that both systems exhibit higher output power density andenergy efficiency than the near-field system based on mono graphene-hexagonal-boron-nitride het-erostructure. The optimal output power density of the former device can reach to . × W · m − ,while the optimal energy efficiency can be as large as of the Carnot efficiency. We analyze theunderlying physical mechanisms that lead to the excellent performances of the proposed near-fieldthermophotovoltaic systems. Our results are valuable toward high-performance moderate temper-ature thermophotovoltaic systems as appealing thermal-to-electric energy conversion (waste heatharvesting) devices. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . a pp - ph ] J a n . INTRODUCTION Thermophotovoltaic (TPV) systems are solid-state renewable energy resource that areof immense potentials in a wide range of applications including solar energy harvestingand waste heat recovery [1–10]. In the TPV system, a photovoltaic (PV) cell is placedin the proximity of a thermal emitter and converts the thermal radiation from the emit-ter into electricity via infrared photoelectric conversion. However, the frequency mismatchbetween a moderate-temperature thermal emitter and the PV cell leads to significantly re-duced performance of the TPV systems at moderate temperatures (i.e., 400 ∼
900 K whichis the majority spectrum of the industry waste heat). To overcome this obstacle, materialswhich support surface polaritons have been used to introduce a resonant near-field energyexchange between the emitter and the absorber [6, 9, 11–14]. As a consequence, near-fieldTPV (NTPV) systems have been proposed to achieve appealing energy efficiency and outputpower [12, 15–22]. Near-field systems based on graphene, hexagonal-boron-nitride ( h -BN)and their heterostructures have been shown to demonstrate excellent near-field couplingsdue to surface plasmon polaritons (SPPs), surface phonon polaritons (SPhPs) and theirhybridizations [i.e., surface plasmon-phonon polaritons (SPPPs)] [11, 13, 23–27]. It wasdemonstrated that a heterostructure consisting of graphene- h -BN multilayers performs bet-ter than the monocell structure and the heat flux is found to be three times larger than thatof the monocell structure [25–27]. Here, we consider the graphene- h -BN-graphene sandwichstructure and graphene- h -BN-graphene- h -BN four-layer structure as the near-field thermalemitters to provide the enhanced radiative heat transfer.In order to convert the infrared radiation into electricity, the energy bandgap ( E gap ) ofthe NTPV cell (i.e., the p - n junction) must match the radiative spectrum generated bythe emitter. III-V group compound semiconductors like Gallium Arsenide (GaAs), Galliumantimonide(GaSb), Indium antimonide (InSb) and Indium Arsenide (InAs) have been useddue to the small bandgap energy, high electron mobility and low electron effective mass [9, 12,16, 23]. Recently, semiconductor thin-films have been explored in NTPV systems. In Refs. [9]and [22], a NTPV system based on a InAs thin-film cell has exhibits appealing performanceoperating at high temperatures. But the system suffers from low energy efficiency (below ) when operating at moderate temperature due to the parasitic heat transfer inducedby the phonon-polaritons of InAs. Here, we use InSb as the near-field absorber since thebandgap energy of InSb is lower compared to InAs and its photon-phonon interaction is muchweaker than InAs. For the temperature of the InSb cell, T cell = 320 K , which has been provedan optimal cell temperature in our previous work [28], the gap energy is E gap = 0 . eV andthe corresponding angular frequency is ω gap = 2 . × rad/s.In this work, we examine the performances of two NTPV devices: the graphene- h -BN-graphene-InSb cell (denoted as G-FBN-G-InSb cell, with the graphene- h -BN-graphene2 mitter 𝑇 emit TPV cell 𝑇 cell 𝑑 (a) h- BN Graphene h- BN Graphene h- BN (b)(d) InSb Thin-film ℎ InSb
Substrate (c)
FIG. 1. Schematic illustration of the NTPV systems. (a) A representative TPV device. A thermalemitter of temperature T emit is located at a distance d of a thermophotovoltaic cell of temperature T cell . Two compositions of the thermal emitter with (b) graphene- h -BN-graphene structure and (c)graphene- h -BN-graphene- h -BN structure. (d) The thermophotovoltaic cell with a InSb thin-film ofthickness h InSb supported by a semi-infinite substrate. sandwich structure being the emitter and the InSb thin-film being the cell) and thegraphene- h -BN-graphene- h -BN-InSb cell (denoted as G-FBN-G-FBN-InSb cell, with thedouble graphene- h -BN heterostructure being the emitter and the InSb thin-film being thecell). We study and compare the performances of these two systems and reveal their un-derlying physical mechanisms. We further optimize the performance of the near-field TPVsystems for various parameters.In this work, we address two issues: First, we try to optimize the design of the h -BN-graphene heterostructures as the emitter to improve the performance of the NTPV system.Second, we try to discuss the effect of finite thickness of the InSb cell on the performanceof the NTPV system.This work is structured as follows. In Sec. II, we describe our NTPV system and clarifythe radiative heat flux exchanged between the emitter and the cell. In Sec. III, we study andcompare the performances of the two NTPV systems and analyze the spectral distributionsof the photo-induced current density and indicent heat flux. We also study the photontunneling coefficient to further elucidate the physical mechanisms. In Sec. IV, we examinethe performances of the two NTPV systems for various InSb thin-film thicknesses and emittertemperatures to optimize the performances of the NTPV systems. Finally, we summarizeand conclude in Sec. V. 3 I. SYSTEM AND THEORY
In Fig. 1, we consider the graphene- h -BN-InSb NTPV systems. Fig. 1(a) is a schematicpresentation of a typical NTPV system, which consists of a thermal emitter and a ther-mophotovoltaic cell. The emitter and the thermophotovoltaic cell is separated by a vacuumgap with thickness d . The temperatures of the emitter and cell are kept at T emit and T cell ,respectively. The thermal radiation from the emitter is absorbed by the cell and then con-verted into electricity via photoelectric conversion. Figs. 1(b) and 1(c) present the twocompositions of the emitter. Fig. 1(b) is a graphene- h -BN-graphene sandwich structure andFig. 1(c) is made of two graphene- h -BN heterostructures. The thickness of h -BN thin film is h BN and the graphene monolayer is model as a layer of thickness h g . Fig. 1(d) is a thin-filmInSb p - n junction supported by a substrate. The thickness of the InSb thin-film is h InSb andthe substrate is set to be semi-infinite intrinsic InSb.When the InSb cell is located at a distance d which is on the order of or smaller thanthe thermal wavelength λ th = 2 π (cid:126) c/k B T emit from the emitter, the thermal radiation can besignificantly enhanced due to energy transfer via evanescent waves [12]. The radiative heatexchange between the emitter and the cell is given by [29, 30] Q rad = Q ω<ω gap + Q ω ≥ ω gap (1)where Q ω<ω gap and Q ω ≥ ω gap are the heat exchanges below and above the band gap of thecell, respectively.The below Q ω<ω gap and above-gap heat exchange Q ω ≥ ω gap are respectively given by Q ω<ω gap = (cid:90) ω gap dω π [Θ ( T emit , ω ) − Θ ( T cell , ω )] (cid:88) j (cid:90) kdkζ j ( ω, k ) , (2)and Q ω ≥ ω gap = (cid:90) ∞ ω gap dω π [Θ ( T emit , ω ) − Θ ( T cell , ω, ∆ µ )] (cid:88) j (cid:90) kdkζ j ( ω, k ) , (3)where Θ ( T emit , ω ) = (cid:126) ω/ [exp (cid:16) (cid:126) ωk B T emit (cid:17) − and Θ ( T cell , ω, ∆ µ ) = (cid:126) ω/ [exp (cid:16) (cid:126) ω − ∆ µk B T cell (cid:17) − are the Planck mean oscillator energies of blackbody at temperature T emit and T cell , respec-tively. ∆ µ is the electrochemical potential difference across the p - n junction, which describesthe effects of charge injection or depletion on the carrier recombination processes and hencemodify the number of photons through the detailed balance. k is the magnitude of thein-plane wavevector of thermal radiation waves. ζ j ( ω, k ) is the photon transmission coeffi-cient for the j -th polarization ( j = s, p ) , which consists of the contributions from both thepropagating and the evanescent waves [31] ζ j ( ω, k ) = ( −| r emit | )( −| r cell | ) | − r j emit r j cell exp(2 ik z d ) | , k < ωc ( r j emit ) Im ( r j cell ) exp(2 ik z d ) | − r j emit r j cell exp(2 ik z d ) | , k ≥ ωc (4)4here k z = (cid:112) ω /c − k is the perpendicular-to-plane component of the wavevector invacuum. r j emit ( r j cell ) with j = s, p is the complex reflection coefficient at the interfacebetween the emitter (cell) and the air. Here, the reflection coefficients of the emitter andcell are calculated by the scattering matrix approach [32, 33].Via the infrared photoelectric conversion, the above-gap radiative heat exchange is thenconverted into electricity. Based on the detailed balance analysis, the electric current densityof a NTPV cell is given by [1, 34] I = I ph − I [exp( V /V cell ) − , (5)where V = ∆ µ/e is the voltage bias across the NTPV cell, V cell = k B T cell /e is a voltage whichmeasures the temperature of the cell [1]. I ph and I are the photo-generation current densityand reverse saturation current density, respectively. In Eq. (5), an ideal rectifier conditionhas been used to simplify the nonradiative recombination [1]. The actual nonradiativemechanisms include Shockley-Read-Hall (RSH) and Auger nonradiative processes. Here, forthe sake of simplicity, we just follow the Shockley-Queisser analysis [1].The reverse saturation current density is determined by the diffusion of minority carriersin the InSb p - n junction, which is given by I = en (cid:32) N A (cid:114) D e τ e + 1 N D (cid:114) D h τ h (cid:33) , (6)where n i is the intrinsic carrier concentration, D e and D h are the diffusion coefficients ofthe electrons and holes, respectively. N A and N D are the p -region and n -region impurityconcentrations, respectively [35]. τ e and τ h are correspondingly the relaxation time of theelectron-hole pairs in the n -region and p -region. Numerical values of these parameters aretaken from Refs. [35] and [36].The photo-generation current density is contributed from the above-gap thermal heatexchange [16, 23] I ph = e π (cid:90) ∞ ω gap dω (cid:126) ω [Θ ( T emit , ω ) − Θ ( T cell , ω, ∆ µ )] × (cid:88) j (cid:90) kdkζ j ( ω, k ) (cid:0) − exp (cid:2) − k InSb z ) h InSb (cid:3)(cid:1) , (7)where k InSb z = (cid:112) ε InSb ω / c − k is the perpendicular-to-plane component of the wavevectorin the InSb p - n junction. ε InSb is the dielectric function of the InSb cell, which is given by ε InSb = (cid:16) n + i c α ( ω )2 ω (cid:17) . n InSb = 4 . is the refractive index and c is the speed of light invacuum. α ( ω ) is a step-like function describing the photonic absorption, which is givenby [23] α ( ω ) = 0 for ω < ω gap and α ( ω ) = α (cid:112) ω/ω gap − for ω > ω gap with α = 0.7 µ m − [23]. Since the NTPV cell is a thin film, the exponential decay characteristic of5he electromagnetic wave propagating in the InSb thin-film must be considered. The term (cid:0) − exp (cid:2) − k InSb z ) h InSb (cid:3)(cid:1) is the absorption probability of the incident radiation in theInSb film of thickness h InSb , which measures the actual availability of the above-gap photonsin the photon-carrier generation process.The dielectric function of h -BN is described by a Drude-Lorentz model, which is givenby [37] ε m = ε ∞ ,m (cid:32) ω , m − ω , m ω , m − iγ m ω − ω (cid:33) , (8)where m = (cid:107) , ⊥ denotes the out-of-plane and the in-plane directions, respectively. ε ∞ ,m is thehigh-frequency relative permittivity, ω TO and ω LO are the transverse and longitudinal opticalphonon frequencies, respectively. γ m is the damping constant of the optical phonon modes.The values of these parameters are chosen as those determined by experiments [38, 39].The effective dielectric function of the graphene monolayer is modeled as [40] ε g = 1 + i σ g ε ωh g , (9)where σ g is the optical conductivity [41].The output electric power density P e of the NTPV system is defined as the product ofthe net electric current density and the voltage bias, P e = − I e V, (10)and the energy efficiency η is given by the ratio between the output electric power density P e and incident radiative heat flux Q inc , η = P e Q inc , (11)where the incident radiative heat flux is given by the radiative heat exchange defined inEq. (1). The second law of thermodynamics imposes an upper bound on the energy efficiency,which is the Carnot efficiency, η c = 1 − T cell T emit . (12)When studying the energy efficiency of various NTPV systems with different temperatures,we use the Carnot efficiency to quantify and compare their energy efficiencies. III. PERFORMANCES OF GRAPHENE- H -BN-INSB NEAR-FIELD SYSTEMS We first examine the performances of the two types of NTPV cells. Fig. 2 shows theoutput power density and energy efficiency as a function of voltage bias for the two set-ups, respectively denoted as G-FBN-G-InSb cell (graphene- h -BN-graphene heterostructure6 b)(a) FIG. 2. Performances of two NTPV cells. (a) Electrical power density and (b) energy − conversionefficiency in units of Carnot efficiency ( η c ). The temperatures of the emitter and the cell are set at T emit = 450 K and T cell = 320 K, respectively. The vacuum gap distance is d = 20 nm, the h -BNfilm thickness is h BN = 20 nm and the thickness of the InSb thin film is h InSb = 1000 nm. Thechemical potential of graphene is µ g = 1 . eV. The Carnot efficiency is given by η c = 1 − T cell /T emit . as the emitter and thin-film InSb p - n junction as the receiver) and G-FBN-G-FBN-InSbcell (graphene- h -BN-graphene- h -BN heterostructure as the emitter and thin-film InSb p - n junction as the receiver). The output power density and energy efficiency are optimized forvarious physical parameters, including the temperature of the cell, the chemical potentialof graphene and h -BN film thickness [28]. The analysis in Ref. [28] shows that setting µ g = 1 . eV, h BN = 20 nm and T cell = 320 K provides roughly optimal performance, bothin terms of power and efficiency, for the NTPV systems considered in this work. Therefore,these parameters are kept as those constants in the main text. The thickness of the InSb thinfilm is set as h InSb = 1000 nm, which is close to the thickness of experimental samples [42].As shown in Fig. 2, the maximum output power density of the G-FBN-G-InSb cell isabout . × W · m − , nearly 1.1 times of the the maximum output power density of theG-FBN-G-FBN-InSb cell (about . × W · m − ). For the energy-conversion efficiency,the maximum values of the G-FBN-G-InSb cell is about η c , which slightly higher thanthe maximum efficiency of the FBN-G-FBN-InSb cell (about . η c ).In order to analyze the physical mechanisms responsible for such performances of thesetwo setups, we show the spectral distributions of the photo-induced current and incidentradiative heat flux at the maximum electric power density for the G-FBN-G-InSb cell andG-FBN-G-FBN-InSb cell in Fig. 3. The two shaded areas in Fig. 3(a) and (b) are the tworeststrahlen bands of h -BN [43]. As exhibited in Fig. 3(a) that higher photo-induced currentspectra of the two systems appear at and near the reststrahlen band due to the reststrahleneffect. The reststrahlen effect originates from the hyperbolic optical properties of the h -BNfilm due to the photon—optical-phonon interactions. In the reststrahlen band presented in7 b) (a) FIG. 3. (a) Photo-induced current spectra I ph ( ω ) and (b) incident radiative heat spectra Q inc ( ω ) at the maximum electric power density for the two configurations. The temperatures of the emitterand the cell are kept at T emit = 450 K and T cell = 320 K, respectively. The vacuum gap is d = 20 nm and the chemical potential of graphene is set as µ g = 1 . eV. The h -BN film thickness is h film = 20 nm. The chemical potential difference across the InSb p - n junction ∆ µ is optimizedindependently for maximum output power for each configuration. Fig. 3(a), the in-plane dielectric function of h -BN is negative, leading to strong reflectionand suppressed transmission of incident photons [43]. The near-field radiation effects areessentially caused by the evanescent propagation of the incident photons, which dramaticallyenhances the radiative heat transfer between two closed spaced objects [24–28, 37, 44]. Suchenhancement of the radiative heat transfer eventually leads to the significant increase of theinput heat flux, output electric power and the energy efficiency [28].It is noticed that the overall photo-induced current spectrum of the G-FBN-G-InSb cellis higher than the one of the G-FBN-G-FBN-InSb cell except in the frequency range from . × rad · s − to . × rad · s − . After integrating over ω , this higher photo-inducedcurrent spectrum gives rise to improved output power of the G-FBN-G-InSb cell.Contrary to the photo-induced current spectra, the overall incident radiative heat fluxof the G-FBN-G-FBN-InSb cell is higher than the G-FBN-G-InSb cell. Since the energy-conversion efficiency is defined as the ratio of the output electric power density and theincident radiation heat flux, the higher output power density and lower incident radiationheat flux give a higher energy efficiency of the G-FBN-G-InSb cell.To further elucidate the mechanism for the enhanced performance of the near-field sys-tems, we examine the photon tunneling coefficient ζ j ( ω, k ) (given by Eq. 4). As shown inFig. 4, the bright color indicates a high transmission coefficient. The green dashed linesare the light lines of vacuum and InSb, respectively. Note that the maximum transmissioncoefficient is 2 due to the contribution of both s and p polarizations. In the above-gap range,both G-FBN-G-InSb cell and G-FBN-G-FBN-InSb cell exhibit enhanced photon transmis-8 b) (a) 𝒌 = 𝝎/𝒄 𝒌 = 𝒏 𝐈𝐧𝐒𝐛 𝝎/𝒄 𝒌 = 𝝎/𝒄 𝒌 = 𝒏 𝐈𝐧𝐒𝐛 𝝎/𝒄 FIG. 4. Photon transmission coefficient ζ ( ω, k ) for (a) G-FBN-G-InSb cell and (b) G-FBN-G-FBN-InSb cell. The temperatures of the emitter and the cell are kept at T emit = 450 K and T cell = 320 K, respectively. The vacuum gap is d = 20 nm and the chemical potential of grapheneis set as µ g = 1 . eV. The h -BN film thickness is h film = 20 nm and the InSb thin-film thickness is h InSb = 1000 nm. sion, thanks to the SPPPs supported by the graphene- h -BN heterostructures. Fig. 4 showsthat the photon transmission spectra of the two near-field NTPV systems do not differ much.Therefore, the performances of the two NTPV systems are comparable. IV. OPTIMIZATION OF GRAPHENE- H -BN-INSB NEAR-FIELD SYSTEMS We now study the performances of the two near-field systems for various InSb thin-film thicknesses and emitter temperatures in Figs. 5 and 6. As presented in Fig. 5 thatboth of the output power density and the energy-conversion efficiency of the two setupsare improved with increasing InSb thin-film thickness. Especially for h InSb = 10000 nm,the maximum power densities of the G-FBN-G-InSb cell and G-FBN-G-FBN-InSb cell areabout . × W / m and . × W / m , respectively. The maximum values of thecorresponding efficiency are close to η C and η C , respectively. This enhancement canbe attributed to the increased absorption factor (cid:0) − exp (cid:2) − Imag( k InSb z ) h InSb (cid:3)(cid:1) because it isdependent on h InSb . However, when the InSb thin-film thickness is increased from 4000 nmto 10000 nm, the increase of the output power density and the energy-conversion efficiencyfor both systems are soon saturated. This is essentially due to the near-field nature of theheat transfer: the photons transferred from the emitter to the InSb cell is dominated by theevanescent electromagnetic waves which are highly localized at the surface of the emitter.Further increase of the thickness of the InSb cell does not change the amount of photonsreceived by the InSb cell.Fig. 6 shows that by increasing the emitter temperature, both the performances of the9 b)(a) (d)(c)
FIG. 5. Performances of two near-field TPV cells for various InSb thin-film thickness. (a) Elec-trical power density and (b) energy − conversion efficiency in units of Carnot efficiency ( η c ). Thetemperatures of the emitter and the cell are set at T emit = 450 K and T cell = 320 K, respectively.The vacuum gap distance is d = 20 nm, the h -BN film thickness is h BN = 20 nm and the chemicalpotential of graphene is µ g = 1 . eV. two setups can be improved by orders of magnitudes. Here, the InSb thin-film thickness hasbeen chosen as the optimal value with h InSb = 10000 nm. For the G-FBN-G-InSb cell, themaximum output power density and energy efficiency at T emit = 800 K are increased to . × W / m and η c , respectively. For the G-FBN-G-FBN-InSb cell, the maximum outputpower density and energy efficiency at T emit = 800 K are also improved to . × W / m and η c , respectively.We now examine the maximum output electric power density and the efficiency at maxi-mum power. We consider specifically the situation with h InSb = 10000 nm and T emit = 800 K.As shown in Fig. 7, both the maximum electric power density and the efficiency at the max-imum power of the two near-field NTPV systems dramatically increase as the vacuum gap d decreases. The best performances are achieved when the vacuum gap is d = 10 nm forboth systems. Specifically, the maximum electric power density reaches to . × W / m ,while the efficiency at maximum power goes up to of the Carnot efficiency. The two10 b)(a) (d)(c) FIG. 6. Performances of two near-field TPV cells for various emitter temperatures. (a) Electricalpower density and (b) energy − conversion efficiency in units of Carnot efficiency ( η c ). The temper-ature of the cell is set at T cell = 320 K. The vacuum gap distance is d = 20 nm, the h -BN filmthickness is h BN = 20 nm and the thickness of the InSb thin film is chosen as an optimal value of h InSb = 10000 nm. The chemical potential of graphene is µ g = 1 . eV. near-field NTPV systems perform nearly equally well. V. SUMMARY AND CONCLUSIONS
In conclusion, we investigated two NTPV devices, denoted as the G-FBN-G-InSb cell andG-FBN-G-FBN-InSb cell, which have different emitters. The purpose of the investigationis to find a high-performance emitter design with relatively simple structure. Indeed, wefind that both systems perform better than the near-field NTPV system based on monographene- h -BN heterostructure, i.e., graphene- h -BN-InSb cell [28]. The optimal outputpower density of the G-FBN-G-InSb cell can reach to . × W · m − , nearly twice of theoptimal power density of the graphene- h -BN-InSb cell. The optimal energy efficiency canbe as large as of the Carnot efficiency, which is higher than the optimal efficencyof the monocell-system. We analyze the underlying physical mechanisms that lead to the11 b)(a) FIG. 7. Optimal performances of two NTPV cells. (a) Maximum electrical power density and(b) energy − conversion efficiency at maximum electric power in units of Carnot efficiency ( η c ). Thetemperatures of the emitter and the cell are set at T emit = 800 K and T cell = 320 K, respectively. The h -BN film thickness is h BN = 20 nm and the thickness of the InSb thin film is h InSb = 10000 nm. Thechemical potential of graphene is µ g = 1 . eV. The Carnot efficiency is given by η c = 1 − T cell /T emit . excellent performances of the G-FBN-G-InSb cell. We further show that the performanceof the proposed NTPV system is affected negligibly by the finite thickness of the InSb cellwhich is due to the near-field nature of the heat transfer: the absorbed photons are highlylocalized at the surface of the InSb cell. Our study shows that NTPV systems are promisingfor high-performance moderate temperature thermal-to-electric energy conversion. VI. ACKNOWLEDGEMENTS
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