Designing a Concentrated High-Efficiency Thermionic Solar Cell Enabled by Graphene Collector
Xin Zhang, Xiaohang Chen, Jinchan Chen, Lay Kee Ang, Yee Sin Ang
DDesigning a Concentrated High-Efficiency Thermionic Solar Cell Enabled byGraphene Collector
Xin Zhang
School of Science, Jiangnan University, Wuxi 214122, China
Cong Hu, Xiaohang Chen, and Jincan Chen ∗ Department of Physics, Xiamen University, Xiamen 361005, China
Lay Kee Ang and Yee Sin Ang † Science, Math and Technology, Singapore University of Technology and Design, Singapore 487372 (Dated: February 8, 2021)We propose a concentrated thermionic emission solar cell design, which demonstrates a highsolar-to-electricity energy conversion efficiency larger than 10% under 600 sun, by harnessing theexceptional electrical, thermal and radiative properties of the graphene as a collector electrode. Byconstructing an analytical model that explicitly takes into account the non-Richardson behavior ofthe thermionic emission current from graphene, space charge effect in vacuum gap, and the vari-ous irreversible energy losses within the subcomponents, we perform a detailed characterization onthe conversion efficiency limit and electrical power output characteristics of the proposed system.We systematically model and compare the energy conversion efficiency of various configurationsof graphene-graphene and graphene-diamond and diamond-diamond thermionic emitter, and showthat utilizing diamond films as an emitter and graphene as a collector offers the highest maximumefficiency, thus revealing the important role of graphene in achieving high-performance thermionicemission solar cell. A maximum efficiency of 12.8% under 800 sun has been revealed, which issignificantly higher than several existing solid-state solar cell designs, such as the solar-driven ther-moelectric and thermophotovoltaic converters. Our work thus opens up new avenues to advancethe efficiency limit of thermionic solar energy conversion and the development of next-generationnovel-nanomaterial-based solar energy harvesting technology.
I. INTRODUCTION
Solar energy, an abundant and clean energy source, isexpected to play an essential role in easing the tense situ-ation of fossil energy supply, optimizing the energy struc-ture and reducing the environment pollution due to en-ergy production and consumption [1–3]. Nowadays, pho-tovoltaic (PV) cells [4], photochemical approach [5], andsolar thermal power conversion [6] are three main tech-niques for harvesting solar energy. Within a PV energyconversion process, photons with energies above the semi-conductor bandgap excite an electron-hole pairs, whichsubsequently diffuse into charge-selective electrodes toform an electrical current. The photochemical processconverts solar energy to storable chemical fuels, such ashydrogen. Solar thermal conversion, in which photonsare converted into thermal energy by solar-thermal ab-sorbers for subsequent electrical power generation, arewidely used in industrial production and residential facil-ities, such as the mechanical heat engines in large-scalepower plants and the solar water heaters in householdheat supply system [7, 8]. The major advantages of thelast approach are the exploitation of the entire solar spec-trum and therefore it enables higher conversion efficien-cies. More importantly, it is environmentally friendlywith almost the smallest carbon footprint among all the ∗ Corresponding author: [email protected] † yeesin˙[email protected] energy harvesting approaches [9].Based on thermionic emission, the thermionic energyconverter (TIEC) is an emerging heat-to-electric powergeneration system, with its performance ultimately lim-ited by the Carnot efficiency [10–12]. Operated at rela-tively high temperatures, typically over 1400 K, electronswith energies larger than the emitter surface work func-tion are thermally excited to escape from the emittersurface. When the system is connected to an externalload, these emitted electrons traverse across the vacuumgap and are subsequently absorbed by the collector main-tained at a lower temperature, thus generating an elec-trical power through the system. Compared with currenttechnologies for heat-to-electricity conversion, such as thethermoelectric generator (TEG) [13, 14] and thermopho-tovoltaics (TPV) [15–17], the TIEC operating at highertemperatures possesses substantially a higher theoreti-cal efficiency and is thus particularly well suited for con-centrated solar thermal-harvesting system and for high-grade waste recovery. For TEGs and TPVs, the high-temperature operation is plagued by multiple fundamen-tal device limitations such as maintaining large temper-ature gradients in TEGs while minimizing the detrimen-tal heat back-flow, and mitigating the large dark satura-tion currents in TPVs. In this aspect, the TIEC, whichinherently operates at high-temperature regime wherethermionic emission is profound, can significantly havemore advantageous than the TEG and TPV systems.Despite being an appealing energy conversion technol-ogy that has been widely investigated for both indus- a r X i v : . [ phy s i c s . a pp - ph ] F e b trial and aerospace applications, the demonstrated so-lar conversion efficiencies of TIECs are still typically be-low 10% [18]. Such a low efficiency originates from thehigh work functions of the electron emitter and collec-tor. For the routinely used tungsten and graphene, therather high work functions of roughly 4.5 eV forms amajor obstacle in obtaining high-power generation andconversion efficiency [19, 20]. Such materials require ex-tremely high temperatures to enable substantial electronemission current, which hampers the widespread appli-cations for solar concentrators owing to the predominantradiative losses that seriously reduces their performance.Another major challenge is the inevitable presence of thespace charge effect in practical vacuum thermionic de-vices [21, 22]. When the accumulation of space chargewithin the inter-electrode vacuum gap forms a potentialbarrier that suppress the electron emission, thus leadingto a much reduced electrical current and power output.A comprehensive understanding of how these effects in-fluence can quantitatively impacts the energy conversionefficiency of TIECs, and an optimal design strategy forachieving the efficiency larger than 10% remains largelyunanswered thus far.In this work, we propose a new design of high-efficiencysolid-state concentrated thermionic solar cell (CTSC),enabled by a spectrally selective receiver and a graphene-based TIEC (Fig. 1). We design a system prototypewith a parallel-plate geometry by incorporating advancedfunctional materials, such as thermally resilient hafniumcarbide (HfC) based solar receiver, diamond films actingas thermionic emitter, and graphene as an electron collec-tor. Importantly, we construct a comprehensive physics-based TIEC model by employing experimental param-eters to obtain a realistic description of the TIEC sys-tem performance. Our mode suggests that graphene is adesirable material choice for designing high-performancecollector, when compared to conventional bulk metal,due to the following facts: (i) Graphene quenches thereversed electron flux from the collector because of itslow electron emission current density as limited by thevanishing density of states near the Dirac point, thus in-creasing the net electrical current generation; (ii) Theexceptional physical properties, such as ultrahigh elec-trical mobility, low emissivity avoids heat radiation loss(2.3%), and exhibits superior thermal stability (up till4000 K), thus making graphene a robust and low-losscollector; and (iii) The work function of graphene can bereduced to 1 eV by a combination of electrostatic gatingand alkali metal deposition [23], thus resulting in a largeroutput voltages. Because of these valuable advantages,back-gated graphene serves a promising collector mate-rial for boosting the power generation and conversion ef-ficiency in TIECs, thus playing an important role in en-hancing the performance of CTSC proposed in this work.It should be noted that although an idealized CTSCbased on a graphene-based emitter has been predictedto yield a maximum efficiency of 21% under 800 suns inprevious work [24], important physical effects, such asthe space-charge effect, the non-Richardson thermionic emission behavior of graphene and other physical prop-erties and the temperature dependence of the electrodematerials, are not included in the prior model. The modelpresented in this work thus provide an important theoret-ical step-forward for the computational design of CTSCunder more realistic operating conditions.Particularly, our model takes into account two impor-tant, yet often missing in prior literature, effects: (i)Based on a full-band electronic band structure modelof graphene, a generalized analytical model of space-charge limited thermionic emission current across thegraphene/vacuum interface is developed and is incorpo-rated in the modeling; and (ii) We further include thevariation of the electrode temperatures by performing anenergy balance analysis between the two electrodes. Tounderstand the importance of graphene in the design ofCTSC, we systematically study four different emitter-collector configurations, namely: (i) diamond-graphene,(ii) diamond-diamond, (iii) graphene-graphene, and (iv)graphene-diamond configurations. Our model revealsthat the diamond-graphene emitter-collector configura-tion exhibits the largest solar-to-electricity conversion ef-ficiency limit of > II. SYSTEM DESIGN AND MODELLINGA. Design principle of concentrated thermionicsolar cells (CTSCs) using graphene as collector
The concept of the proposed CTSC is composed ofa radiation absorber and a TIEC unit, as depicted inFig. 1(a). A transparent window allows the concentratedsunlight to impinge on the absorber, which is surface-textured for enhancing solar absorption and pursues min-imum losses. The absorber harnesses the concentratedsolar spectrum and transforms them into thermal energy.The TIEC stage exploits the absorbed heat, operatingat the temperature range of 1000-1500 K, thus achiev-ing thermal-to-electric energy conversion via thermionicemission. Simultaneously, the low-grade heat ( <
800 K)rejected from the collector is released to the environmentvia thermal radiation and conduction. In order to followclosely the absorber temperature, the emitter composedof a suitable low-work-function material is deposited onthe absorber inner surface. A collector with a work func-tion lower than that of the emitter establishes an outputvoltage across an external load. The whole system shouldwork under a vacuum environment in order to ensure asufficiently long mean free path for the emitted electronsand to avoid chemical-physical degradation of active ma-terials when operated at high temperatures.The material selection of the various subcomponentsdescribed above is as follows. HfC is selected as theabsorber material in our design in order to realize thespectral selective cut-off absorber at high operating tem-peratures [25]. HfC possesses the high melting point(4153 K) and high electric conductivity (4 × − Ωcm atroom temperature), which are particularly advantageousin high-temperature operation and in reducing the seriesresistance and to avoid the bottlenecks for the refillingof emitted electrons. Excellent thermal conductivity (0.4Wcm − K − at 1773 K) enables the absorbed thermal en-ergy to be transported efficiently to the emitter surface,without producing a wide temperature gradient betweenthe light-receiving and electron-emitting surfaces. Moreimportantly, HfC also has an intrinsic spectral selectivity,with high reflectance in the medium and far IR ranges,and low solar reflectance [26].The material selected for the emitter is N-doped H-terminated diamond films which have excellent electricand thermal properties [27]. Combining the advan-tages of negative electron affinity and Fermi level upshiftthrough N-type doping, N-doped diamond films lowerthe effective work function to 1.7 eV and mitigate thespace charge effect, thus improving the thermally drivenelectron emission capability [28]. In addition, diamondfilms possess the highest thermal conductivity among thesolids (ranging from 10 to 20 Wcm − K − ), which is anessential property for closing to the receiver temperaturewith minimum thermal losses [29]. For maximizing theoutput power, a collector with work function substan-tially lower than N-doped diamond films is necessary toobtain a sizable output voltage. For this purpose, mono-layer graphene grown on copper foil by chemical vapordeposition and transferred onto Hafnium oxide (HfO )and silicon substrate for electrostatic gate-tuning capa-bility is selected as the collector material [18]. Here,graphene serves as a practical collector material becauseof its exceptional physical properties such as ultrahighelectrical mobility, gate-tunable Fermi level, and supe-rior thermal conductivity [30–32]. Recently, an ultralowwork function of 1.01 eV is obtained in an electrostati-cally back-gated graphene monolayer with Cs/O surfacecoating [23], which has been successfully incorporated asa collector in a TIEC prototype. Electrostatic gating ofgraphene via a 20 nm HfO dielectric layer allows thegraphene work function to be dynamically tuned by 0.63eV [18]. These superior physical properties of graphenejustify its role as a low work function collector in ourCTSC design.The inter-electrode vacuum gap of the TIEC is chosenin the micrometer regime [33] because an overly narrowinterelectrode gaps can lead to the collector overheat-ing caused by near-filed heat transfer, whereas an overlylarge vacuum gap is dominated by the space charge ef-fect which can negatively impact the performance of thethermionic emitter. Here we set the vacuum gap as 2 µ m,in order to weaken the space charge effect and optimize FIG. 1. (a) Schematic diagram of the thermionic solar cellenabled by back-gated graphene collector. The concentratingsolar radiation impinges on the surfaced-textured receiver,feeding the TIEC emitter with thermal energy. The elec-trons emitted by the emitter due to thermionic emission travelacross the vacuum gap, and then condensate at the graphene-based collector. (b) The corresponding energy-band diagram,where Φ E (Φ C ) is the work functions of the emitter material(collector), and E FE ( E FC ) is the Fermi level of the emit-ter (collector). The position-dependent potential distributionΦ( x ) along with the x direction yields the maximum valueΦ max . V is the voltage difference between two electrodes and q is the elementary charge. other parameters such as emitter temperature to achievehigh energy conversion efficiency. B. Modeling the space charge effect in theinterelectrode vacuum gap
The space-charge effect in a TIEC arises from theCoulombic repulsion caused by the electrons in-transit ininterelectrode region. Figure 1(b) illustrates the energydiagram of a space-charge-limited micron-gap vacuumTIEC. The formation of space charge raises the maxi-mum surface potential barrier Φ max . The space chargeeffect thus imposes a stricter requirement for electronsto escape from the emitter via the thermionic emissionpathway. As a result, the space charge effect significantlyreduces the output current density of the device. Wemodel the space charge effect by assuming that electronstransport across the interelectrode vacuum gap is colli-sionless. Under this assumption, the additional barrierheight generated by the space charge effect, i.e. Φ( x ),can be obtained from the Poisson’s equation d Φ dx = − q N ( x ) ε , (1)where N ( x ) = (cid:82) + ∞−∞ dv z (cid:82) + ∞−∞ dv y (cid:82) + ∞−∞ f ( x, v ) dv x de-notes the number density of electrons at position x , f ( x, v ) represents the distribution function of electronvelocity, and ε is the permittivity of free space. As-suming that the electron velocity distribution is a halfMaxwellian at the position of maximum motive, x m , Eq.(1) can be rewritten in terms of the dimensionless poten-tial barrier, γ , and the dimensionless distance, ξ , as2 d γdξ = exp( γ )[1 ± erf( (cid:112) γ )] , (2)where the upper signs apply for ξ < ξ ≥ γ = [Φ max − Φ( x )] /k B T E , ξ = ( x − x m ) /x , x = (cid:112) ε k B T E / q N ( x m ), anderf( x ) = 2 / √ π (cid:82) x e − t dt is the error function. k B isthe Boltzmann constant, and T E is the temperature ofthe emitter. Integrating Eq. (2) with the appropriateboundary conditions leads to ξ = ∓ (cid:90) γ dt (cid:113) e t ± e t erf( √ t ) ∓ (cid:112) t/π − . (3)We calculated the value of this integral numerically for awide range of γ . The value of Φ max depends on the op-erating voltage, which is listed, for the saturation, spacecharge, and the retarding modes of operation, asΦ max = Φ E , < V < V sat Φ E + γ E k B T E , V sat < V < V cr Φ C + qV, V > V cr (4)where Φ E (Φ C ) is the work function of the emitter(collector), and the output voltage is then given by qV = E FC − E FE . γ E is the value of γ at the emit-ter surface and is given by γ E = ln ( J sat /J E ), in which J sat = AT exp [ − Φ E / ( k B T E )] is the emitter saturationcurrent density and J E is the emitter current density atvoltage V . V sat and V cr are the saturation and critical-point voltage, respectively [21, 22]. Finally, for conven-tional metallic emitters, the thermionic electrical currentdensity is described by the Richardson-Dushman equa-tion: J E = AT exp (cid:18) − Φ max k B T E (cid:19) , (5)where A is a materials-specific Richardson-Dushman con-stant. C. Modelling thermionic emission from graphene:Non-Richardson-Dushman thermionic emission
In 2D material such as graphene, the thermionic emis-sion physics is radically different from the 3D materialcounterpart [20, 34–39]. Firstly, due to the carrier scat-tering effects, electron momentum conservation is vio-lated during the out-of-plane thermionic emission process[20, 34, 35]. Secondly, electrons undergoing thermionicemission are no longer described by the parabolic en-ergy dispersion as in the case of most 3D materials,and is instead described by a linear energy band struc-ture, commonly known as the Dirac cone approximation[36–39], for carrier energy lower than 1 eV. For electronthermionic emission from graphene that involves higherelectron energy as in the case of graphene/vacuum in-terfacial thermionic emission, the full-band tight-binding energy dispersion should be employed [40, 41]. Thethermionic current density flowing out of the graphenesurface can be modeled as J = 4 q (2 π ) (cid:88) n v ( n ) x ( k ( n ) x ) t Gr (cid:90) τ n ( k (cid:107) , k x ) f FD ( E k (cid:107) ) d k (cid:107) , (6)where the factor 4 represents the spin-valley degeneracy, t Gr = 0 .
335 nm is the thickness for graphene, f FD ( E k (cid:107) )is the Fermi-Dirac distribution function, k (cid:107) = ( k y , k z )is the electron wave-vector component lying in the 2D yz -plane, x denotes the direction orthogonal to the y − z plane of the 2D system, k ( n ) x is the quantized out-of-planewave-vector component of the n -th subband, v ( n ) x ( k ( n ) x ) = (cid:113) mE ( n ) k x is the cross-plane electron group velocity, m isthe free electron mass, E k x is the discrete bound-stateenergy level, and the summation (cid:80) n runs over all ofthe n -th quantized subbands. The nonconservation of k (cid:107) during the out-of-plane thermionic emission process leadsto the coupling between k (cid:107) and k x . Accordingly, the n -thsubband transmission probability becomes τ ( n ) ( k (cid:107) , k x );i.e., the cross-plane electron tunneling is dependent onboth k (cid:107) and k x .For thermionic emission, the minimum energy requiredto overcome the barrier equals Φ max so that the transmis-sion probability τ ( n ) ( k (cid:107) , k x ) = λ H ( E k (cid:107) + E ( n ) k x − Φ max );i.e., E k (cid:107) and E ( n ) k x are combined to overcome the inter-face barrier Φ max . Here, λ characterizes the strengthof 2D plane momentum-non-conserving scattering pro-cess. The term of H ( x ) denotes the Heaviside step func-tion. According to the electronic properties of graphene,the k (cid:107) integral is transformed rewritten as d k (cid:107) =(2 π ) D ( E k (cid:107) ) dE k (cid:107) , where D ( E k (cid:107) ) is the electronic den-sity of states (DOS). For a graphene-vacuum interfacebarrier, the Fermi-Dirac distribution function approachesthe semiclassical Maxwell-Boltzmann distribution func-tion since the emitted electrons are in the nondegener-ate regime. Because there is only one set of subbandsinvolve in the thermionic emission from graphene, the E k x of this subband is conventionally set to zero. Thisis very commonly performed in the density functionaltheory (DFT) simulation literature of graphene and itsheterostructure, in which the Dirac point is convention-ally set as the zero energy. Equations (6) can be sim-plified into single-subband thermionic emission electricalcurrent density for graphene as J = 4 qv x (cid:90) τ ( E k (cid:107) ) ζ ( E k (cid:107) ) D ( E k (cid:107) ) dE k (cid:107) , (7)where v x = (cid:112) mE k x is the cross-plane electron groupvelocity, ζ ( x ) = exp[ − ( x − E FC ) /k B T C ]. In the processof obtaining Eq. (7), the summation sign disappears be-cause in the case of graphene, there exists only one setof spin and valley degenerate subbands that are respon-sible for the thermionic emission of electrons. Becauseof the atomic thickness and the semimetallic electronicproperties of graphene, there is only one Dirac conic dis-persion in the low energy regime around the Fermi level,which evolve into a more complex (and highly nonlinear)band structure in the higher energy regime. This singu-lar set of energy bands thus allow the summation termto be omitted. The full-band (FB) tight-binding modelof graphene yields D ( µ ) = (cid:40) D √ Γ( µ ) K ( µ Γ( µ ) ) , < µ < D √ µ K ( Γ( µ )4 µ ) , < µ < µ = E k (cid:107) /E , D = (4 /A c π )( E k (cid:107) /E ), E = 2 . A c = 0 .
052 nm , Γ( x ) = (1 + x ) − ( x − / K ( µ ) = (cid:82) [(1 − x )(1 − µx )] − . dx is the completeElliptic integral of the first kind. By taking these intoaccount, the thermionic emission current density fromthe graphene collector is no longer described by the clas-sic Richardson-Dushman law in Eq. (5), and is, instead,described by the following relation [41] J C = ˜ λ qv x π t Gr E ϑ (Φ max ) = 4 qπ τ inj E ϑ (Φ max ) , (9)where τ jnj = t Gr / (˜ λv x ) is a charge injection characteris-tic time constant whose value is influenced by the qual-ity of the contact. Importantly, recent experiment ofgraphene/silicon Schottky contact has demonstrated theinsufficiency of the Richardson-Dushman model in de-scribing the emission behavior of graphene, and the non-Richardson-Dushman model is found to provide a moreappropriate explanation on the observed current-voltageand current-temperature characteristics [42]. This exper-imental work thus provide a strong assurance to the useof 2D thermionic emission model as developed above. Asshown below, τ inj plays an important role in determiningthe energy conversion efficiency of the device.Finally, ϑ (Φ max ) can be numerically solved from ϑ (Φ max ) = H ( E (cid:48) − Φ max ) × (cid:90) E Φ max E k (cid:107) dE k (cid:107) (cid:112) Γ( µ ) ζ ( E k (cid:107) ) K (cid:18) µ Γ( µ ) (cid:19) + (cid:90) E E E k (cid:107) dE k (cid:107) √ µ ζ ( E k (cid:107) ) K (cid:18) Γ( µ )4 µ (cid:19) , (10)The net current density, J , is the difference between thethermionic current densities from the emitter and thecollector, i.e., J = J E − J C . D. Modelling conversion efficiency and energybalance
The solar-to-electricity conversion efficiency of a CTSCcan be written as the product of the opto-thermal ef-ficiency of the absorber ( η ot ) and the TIEC efficiency( η TIEC ), which is expressed as η = η ot η TIEC , where η ot is the efficiency of converting the concentrated solar ra-diation ( Q sun ) into heat flux ( Q abs ) that flows into theemitter, i.e. η ot = Q abs Q sun = Q abs C (cid:82) ∞ I AM ( λ ) dλ . (11) FIG. 2. Solar irradiance and emissivity of an absorber varyingwith wavelength.
Here, I AM ( λ ) represents the solar irradiance varying withthe wavelength λ , whose corresponding radiation wave-length of AM 1.5 Direct solar spectra is 0.28 to 2.4 µ m[43], as indicated in Fig. 2. The energy balance equa-tion for the absorber yields Q sun = Q abs + Q ref + Q rad,A ,where Q ref = C (cid:90) ∞ r ( λ ) I AM ( λ ) dλ (12)represents the reflected radiation flow in the absorber,and Q rad,A = 2 πhc λ (cid:90) ∞ [Θ( T E ) − Θ( T amb )] I AM ( λ ) (cid:15) A ( λ ) dλ (13)denotes the spectral blackbody emissive power densityinto the environment. (cid:15) A ( λ ) is the spectral emissivity ofthe absorber surface, according to Kirchhoff law, whichequals the spectral absorptance, and r ( λ ) stands for thereflection coefficient. Moreover, C is the solar concentra-tion factor, Θ( T ) = ( e ( (cid:126) c/λk B T − − is the Bose-Einsteindistributions of photons at the equilibrium temperature,and T amb is the ambient temperature. For the thermionicpart, the TIEC efficiency is given by η TIEC = P out Q abs = JVQ abs . (14)where P out denotes the output electric power density.The energy losses existing in the emitter originate froma number of fundamental energy carriers: thermionicallyemitted electrons, radiative photons, and electron heatconduction in the leads. The net energy carried by emit-ted electrons from emitter flowing towards the collectoris given by Q ther,E → C = J Φ max + 2 k B T E J E − k B T C J C q (15)The factors 3 k B T and 2 k B T represent the excess energyof thermionically emitted electrons from the grapheneand conventional metal, respectively. It should be notedthat the average thermal energy per Dirac fermion is E = (cid:126) v F (cid:82) ∞ k (cid:107) e − (cid:126) v F k (cid:107) /k B T d k (cid:107) (cid:82) ∞ e − (cid:126) v F k (cid:107) /k B T d k (cid:107) = k B T (16)for each degree of freedom, which is different from k B T / k B T in Diracsystems with three degrees of freedom, while the thermalenergy is 2 k B T in typically parabolic systems [44].The radiative heat transfer between the emitterand the collector is described by the familiar Ste-fan–Boltzmann law at large inter-electrode gaps: Q rad,E → C = σ ( T − T )1 /(cid:15) E + 1 /(cid:15) C − , (17)where (cid:15) E and (cid:15) C are the emissivity of the emitter and thecollector, and σ is the Stefan–Boltzmann constant. Anal-ogously, the collector also includes three major energylosses, such as the electron energy flux due to thermionicemission, radiative energy flux, and heat conduction be-tween the collector and ambient. The net energy carriedby emitted electrons from collector flowing towards theemitter is given by Q ther,C → E = 3 k B T C J C − k B T E J E − J (Φ max − qV ) q (18)According to the Newton heat transfer law, the thermalloss caused by heat convection and conduction from theexternal surface of the collector towards the environmentis expressed as Q con = h ( T C − T amb ), where h is theglobal heat transfer coefficient. The temperatures of theemitter and collector, i.e., T E and T C , are determined bythe following energy balance conditions: Q sun − Q ref − Q rad,A = Q ther,E → C − Q rad,E → C , (19a) Q ther,C → E − Q con − Q rad,E → C = 0 . (19b)In Fig. 3, a flow chart summarizing the self-consistentiterative algorithm to implement the model developedabove is shown. The self-consistent algorithm is used tocalculate the temperature at different parts of the deviceas well as the different energy exchange channels. Thealgorithm takes material-and device- related parametersas input at the start of the iterative solution process. Tostart the iteration, an initial guess of the temperaturesis provided. With this initial guess, the algorithm thenchecks for convergence of the energy balance criterion atdifferent parts of the device and updates the tempera-tures accordingly until convergence is achieved. Here wenote that the various energy exchange channels, such asthermionic and radiative heat flux have strong nonlineardependences on the electrode temperatures. Therefore,adaptively updated coefficients have been used to updatethe electrode temperatures during the self-consistent cy-cles. III. RESULTS AND DISCUSSION
Based on the candidate materials of each subcompo-nents and the 1D energy transfer model described in pre-vious sections, we perform a numerical parameter opti-mization to obtain a sufficiently realistic estimation of
FIG. 3. Flowchart of the self-consistent numerical iterativemodel implemented in this work. the CSTC system performance and energy conversion ef-ficiency. The work functions of emitter and collector in aTIEC are critically important parameters because elec-tron emission depends on these parameters exponentially.To obtain high thermionic emission at practically achiev-able temperatures, the emitter work function should below. On the other hand, to maximize efficiency, a largevoltage difference between emitter and collector is re-quired, which, in turn, requires the collector work func-tion to be considerably lower than that of the emitter.However, if the collector work function is too low, signif-icant back emission from the collector will occur, whichlowers the efficiency. Given these complex requirements,a reasonable set of values for the emitter and collectorwork functions is chosen as Φ E = 2 eV and Φ C = 1 . A = 8 .
42 A cm − K − is considered in order toestimate the application potential [45]. (2) The emissiv-ity (cid:15) E of the emitter is assumed to be a function of tem-perature, which is given by (cid:15) E = 0 .
13 + 2 . × − T E [45]; (cid:15) C = 2 . h is assumed to be 0.1 Wcm − K − [47], which is the upper bound of heat transfercoefficient for cooling by free convection.The performance of a CTSC is strongly influencedby the output voltage. The functional dependence ofthe operating temperatures, the current density, the en-ergy fluxes and the energy conversion efficiency, thusoffer a useful perspective on the operation and perfor-mance limit of a CTSC. Based on the energy balanceat the emitter and collector, as defined in Eqs. 19(a)and 19(b), respectively, the emitter and collector tem-perature [Fig. 4(a)], net current density and maximummotive [Fig. 4(b)], different energy fluxes [Fig. 4(c)], andefficiency of different components [Fig. 4(d)] are numeri-cally obtained using the self-consistent approach [shownin Fig. 3] for different operating voltages. In the fol-lowing discussion, the incident spectrum is the AM1.5direct circumsolar spectrum multiplied by the flux con-centration C = 800. An inter-electrode distance of 2 µ mand τ inj = 20 ns are chosen. The values of these param-eters are used unless specifically mentioned. We derivethe current–voltage characteristics of the CTSC. In thesaturation region (0 < V < V sat ), the maximum motiveis equal to the emitter work function (Φ E ), which is theminimum energy required by the electron to be thermion-ically emitted from the emitter. All emitted electronscan reach the collector, as there is no potential barrierin the inter-electrode space, and thus the current doesnot change with voltage. In contrast, in the space-chargeand retarding regions ( V > V cr ), the maximum motiveis higher than the emitter work function and it gradu-ally rises with the output voltage. Consequently, only aportion of the emitted electrons with sufficient energy toovercome the additional energy barrier generated by thespace charge effect in the inter-electrode gap can reachthe collector. As a result, both the current density andthe energy flux from the emitter decrease as the voltageincreases, leading to an increase in emitter’s temperature[see Fig. 4(b)]. This increase in temperature of the emit-ter raises the radiative heat transfer from the emitter tothe collector [see Fig. 4(c)]. This loss mechanism be-gins to dominate the energy flux from the emitter as thedevice is driven deeper into the retarding region. In addi-tion, since the energy exchange of the collector is closelydependent on the emitted energy fluxes from the emitter,the collector’s temperature and the heat flux released tothe ambient from the collector ( Q con ) decrease monotoni-cally with the increase of the output voltage. Such a phe-nomenon is the direct consequence of the energy balancerequirement at the collector. The conversion efficiencyof the CTSC depends on the output power produced bythe TIEC, which is a product of the output current den-sity and the output voltage. Due to the interplay of suchtwo parameters, the CTSC reaches an efficiency of up to12.8% for a maximum barrier height of 2.9 eV and anoutput voltage of 1.76 V, which corresponds to the op-erating temperatures for the emitter and the collector of1707 K [Fig. 4(c)] and 352 K [Fig. 4(d)], respectively.To provide crystalized guidelines for the performanceimprovement of practical CTSC system, we study thevarious processes that underlie the conversion efficiencyand the energy loss pathways of the system. Figure 4(d)shows the efficiency of the TIEC and the radiation ab-sorber subsystems as a function of the output voltage.Individually, the opto-thermal efficiency ( η ot ) and theTIEC efficiency ( η TIEC ) reaches 39.8% and 32.2%, re-spectively, thus revealing the TIEC as the more domi-nant bottleneck that limits the conversion efficiency ofthe CTSC system. The product of these two individ-ual efficiencies yields the combined energy conversion effi- ciency of 12.8% for the CTSC is achievable under 800 sun(1 sun is 0.1 W cm − K − ). In Fig. 4(c), we show the fourmain energy fluxes existing in the TIEC. We found thatthe heat conduction and back-body radiative losses aretwo major loss mechanisms that significantly degrade sys-tem performance. To enhance the system performance,more efforts thus should be paid to improve the followingcomponents: (1) High-performance wavelength-selectivesolar receivers with an innovative design uses a high ther-mal concentration in an evacuated enclosure to preventair convection and conduction losses. (2) New generationTIEC possesses higher conversion efficiency and operat-ing temperature by fabricating low-work-function ther-mal stable, and low-emissivity nanostructured materials.Moreover, we study how different solar concentrationscan affect the optimal performance characteristics of theCTSC, as shown in Fig. 5. In general, larger solar con-centrations correspond to more input energy fluxes, andthus leading to higher maximum conversion efficiencies( η max ) [Fig. 5(a)] and electrode temperatures [Fig. 5(b)].Further increase in efficiency is possible with higher op-tical concentration for devices to operate at higher tem-peratures. The optimum values of the maximum motive(Φ max,opt ) and output voltage ( V opt ) that produces themaximum efficiency also monotonically increase with thesolar concentration [Fig. 5 (c) and Fig. 5 (d)]. How-ever, the optimal current density ( J opt ) decreases withincreasing solar concentration. The current density ismore strongly influenced by the maximum motive thanthe temperature (or solar concentration). Specially, whenthe temperature is relatively low, the effects of maximummotive on the electric current become dominant. Thus,there is an inevitable compromise between the higherconversion efficiency and the higher output power, whichmay affect the practical design of CTSCs. For exam-ple, for high-power energy generator where higher currentdensity and output voltage are desirable, the conversionefficiency could be inherently lower.To investigate the effects of inter-electrode gap widthon CTSC operation, the peak efficiency at the inter-electrode spacing region of 2 µ m to 10 µ m are shownin Figs. 6(a), where the output voltages are optimized.It can be seen that the inter-electrode distance has a sig-nificant impact on the performance of the CTSC and thepeak efficiency can be enhanced when the vacuum gapis decreased. As the interelectrode distance increases,the space charge effect starts to dominate, which signifi-cantly reduces electron flux from the emitter to the col-lector. As a result, the output power density decrease atlarge inte-electrode distances and radiative heat transfer(which is governed by the Stefan-Boltzmann law) grad-ually becomes dominant due to the rising temperaturedifference between the electrodes. It should be notedthat when the gap is very small, the near-field radiativeheat transfer between the electrodes, which is caused bythe coupling of evanescent waves between the two elec-trodes, becomes dominant. The energy flux carried bythe thermionic electrons is thus small at a very smallelectrode gap. In this case, the current density is low FIG. 4. (a) current density (solid line) and maximum motive (dash line), (b) electrode temperatures, (c) energy fluxes in theTIEC component, and (d) opto-thermal efficiency of the absorber, TIEC efficiency, and overall conversion efficiency of theCTSC as a function of the operating voltage. despite mitigation of the space-charge effect. We fur-ther study the τ inj -dependence of the peak efficiency inFigs. 6(b). The τ inj is related to the out-of-plane ve-locity of electrons in graphene with nonconserving scat-tering strength. Here we consider a representative rangefrom 1 ps to 1 ms, which is referenced to the values ofgraphene-semiconductor contact reported experimentally[36, 39, 42, 48]. A larger τ inj corresponds to the situa-tion in which the contact resistance across the graphene-vacuum interface is large [36]. In general, the key per-formance parameters of the CTSC is influenced by thevalues of τ inj . Particularly when τ inj is small, increasing τ inj leads to a significant improvement of the peak effi-ciency [see Fig. 5(a-c)]. Such improvement eventuallysaturates as τ inj is further increased. This analysis thussuggests that electrical contact engineering may offer aroute to improve the system performance of CTSCs.To understand the optimal electrode configurations ofthe thermionic emitter unit, we examine the optimalperformances for CTSCs operating in various emitter-collector configurations. Figure 7 shows the peak ef-ficiency for the four different configurations, which aredenoted as D-Gr (i.e. diamond films as emitter and graphene as collector), Gr-D (i.e. graphene as emitterand diamond films as collector), Gr-Gr (i.e. graphene asboth emitter and collector), and D-D (i.e. diamond filmsas both emitter and collector). For a solar concentrationranging between 500 to 1000 suns, we found that the bestperformance is delivered by the D-Gr setup [Fig. 7(a)].This demonstrates the important role of graphene as acollector material for achieving high-efficiency CTSCs.The physical mechanisms that underlies the better per-formance of metal-graphene configurations revealed inFig. 7(a) can be understood as followed. In contrastto diamond- or metal-based collector, graphene quenchesthe parasitic electron back-flux originating from the col-lector because of its low electron emission current den-sity – a direct consequence of the vanishing density ofstates near the Dirac point. Furthermore, low emissiv-ity of graphene avoids heat radiation loss (2.3%). Thecombination of these two key factors leads to a much-improved conversion efficiency when graphene is usedas a collector material. Fig. 7(b-c) also reveals that:(i) graphene-based emitter generally delivers lowest ef-ficiency because the thermionic emission current is in-herently low in graphene; (ii) while thermionic emission FIG. 5. (a) The peak efficiency η max (that is, the highest value of the efficiency as a function of voltage) and optimal currentdensity J opt , (b) optimal electrode temperature T E(C),opt , (c) optimal maximum motive Φ max,opt , and (d) optimal outputvoltage V opt under the 500-1000 solar concentrations.FIG. 6. (a) The peak efficiency η max varying with inter-electrode spacing d (inset shows a magnified view of the current-voltagegraph for d = 2 µ m and d = 5 µ m) and charge injection characteristic time τ inj under 800 suns. IV. CONCLUSION
In summary, we have established a new concept of theconcentrated thermionic solar cells enabled by graphenecollector for harvesting solar energy. We developedan analytical model that combines the unconventionalthermionic emission characteristics of 2D graphene andthe space charge effect in the vacuum gap. It has beenfound that CTSCs possess a peak efficiency of 12.8% un-der 800 sun, which is significantly higher than severalexisting solar cell architectures, such as thermoelectricand thermophotovoltaic based solar cells, and the opti-mal conversion efficiency and output electrical power canbe customarily made by changing the solar concentra-tions. By comparing the conversion efficiency of vari-ous combinations of graphene and metal in the emitter-collector configurations of the thermionic emission unit,we show that graphene as a collector material deliversthe highest conversion efficiency, thus revealing the roleof graphene as a key enabler to achieve high efficiencyin CTSCs. These findings provide new insights for thedesign of high-performance CTSCs and shall form theharbinger of 2D-material-based TIEC systems towardscleaner and more sustainable energy.
ACKNOWLEDGMENT
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