IEEE ELECTRON DEVICE LETTERS, VOL. 39, NO. 3, MARCH 2018
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Graphene-Based Thermionic Solar Cells Xin Zhang , Yanchao Zhang, Zhuolin Ye, Wangyang Li, Tianjun Liao, and Jincan Chen
Abstract — A model of the graphene-based thermionic solar cell (TSC) consisting of a concentrator, an absorber, and a thermionic emission device configured with graphene-based cathode is proposed, where the radiation and reflection losses from the absorber to the environment, the thermal radiation between the cathode and the anode electrodes, and the heat losses from the anode to the environment are considered. The performance characteristics of the TSC are analyzed by numerical calculations. It is found that the maximum efficiency can reach 21% when the area ratio is 0.24 and the voltage output is 2.01 V. In addition, the maximum efficiencies of the TSC under different concentrations and the optimal values of some key parameters are determined, and consequently, the corresponding optimally operating conditions are obtained. The results obtained here may provide guidance for the appropriate selection of electrode materials and the optimum design of practical TSC devices. Index Terms — Graphene, thermionic solar cell, irreversible loss, performance characteristics, optimum design criteria.
I. I NTRODUCTION
A
S ONE kind of advanced two-dimensional materials, graphene [1]–[4] has added a novel concept in achieving enhanced electric current and efficient energy conversion, which can serve as the electrode material of thermionic energy converters (TECs) for harvesting heat energy due to many unique features such as the linear band structure [5], excellent mobility [6], and ultrahigh electrical conductivity [7]. Different from metal materials, graphene increases the emission electrons of TECs operated at high temperatures and is more suitable for acting as the cathode material of TECs. Numerous theoretical and experimental studies on graphenebased thermionic emission devices have been reported. For example, Massicotte et al. [1] experimentally demonstrated that the photo-thermionic effect enables the detection of sub-bandgap photons, which possesses size-scalable, electrically tunable, broadband, and ultrafast features. Liang and Ang [8] investigated the thermionic emission from a single layer graphene and derived the analytical expression. Kahaly et al. [9] established a formalism to address Manuscript received December 14, 2017; accepted January 2, 2018. Date of publication January 5, 2018; date of current version February 22, 2018. This work was supported by the National Natural Science Foundation under Grant 11675132. The review of this letter was arranged by Editor J. Moon. (Corresponding author: Jincan Chen.) The authors are with the Department of Physics, Xiamen University, Xiamen 361005, China (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LED.2018.2789920
Fig. 1. The schematic diagram of a graphene-based thermionic solar cell.
co-existing and complementing thermionic and photoelectric emission from an illuminated monolayer graphene sheet. Yuan et al. [10] proposed the prototype TEC with a backgated graphene anode, in which the maximum efficiency is 6.7 times higher than that of a TEC with a tungsten anode. However, performance researches and design manufactures of solar-driven graphene-based TECs have not been investigated. In this Letter, a new model of the thermionic solar cell (TSC) using graphene as the cathode is proposed. The effects of some key parameters on the performance of the TSC are analyzed in detail. The optimally operating conditions of the TSC are determined. The present theoretical work will be helpful to develop high efficient solar cells. II. M ODEL D ESCRIPTION TSCs are one class of concentrated solar devices that convert sunlight into electricity at sufficiently high temperatures. Figure 1 shows a TSC composed of a solar concentrator, an absorber, and a TEC, where the TEC consists of a graphene cathode at temperature TC , a metal plate anode at temperature TA , and a nanoscale vacuum gap between the two parallel planar electrodes. In Fig. 1, qC denotes the concentrated solar radiation flux, qR stands for the reflected solar radiation of the absorber, qL is the heat flow via radiation from the absorber to the environment, and qH is the heat flow from the absorber to the TSC. U (TA − TE ) and μσ (TA4 − TE4 ) are, respectively, the heat flows through the heat conduction and blackbody radiation between the anode and the environment, U is the heat transfer coefficient, μ is the surface emissivity, and σ is the Stefan-Boltzmann constant. JC and JA are the electric currents emitted by the cathode and anode, respectively, and closely rely on the electrode temperatures and material parameters. σ TC4 and σ TA4 are, respectively, the full spectrum blackbody emissions at the cathode and anode, and R is the load resistance.
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IEEE ELECTRON DEVICE LETTERS, VOL. 39, NO. 3, MARCH 2018
III. W ORKING P RINCIPLE AND E FFICIENCY OF THE TSC Serving as the intermediate body between the solar concentrator and the thermionic emission device, the absorber absorbs the concentrated broad solar spectrum via a photonic crystal and changes it to heat, which is transferred to the cathode and converted to a narrower thermal spectrum [12]. When the cathode absorbs the heat qH from the absorber and is heated to high temperatures, electrons rapidly thermalize to the equilibrium thermal distribution and diffuse throughout the cathode. Electrons gaining sufficient energy can overcome the potential barrier of the graphene near the cathode surface and escape from the cathode into the vacuum. Electrons across the vacuum gap condense at the anode, and return to the cathode through an external load. Generally, the electric current in bulk materials (i.e., metals) with parabolic energy dispersion (E k ∝ k 2 ) obeys the Richadson-Dushman law and the current-temperature scaling relation J ∝ T 2 is a signature of the parabolic energy dispersion of the transport electrons. For monolayer graphene, electrons behave like the massless Dirac fermion and possess a linear energy dispersion (E k ∝ k). Hence, the revised electric current of the graphene is given by J ∝ T 3 [8]. Consequently, the graphene-based TEC generates the net electric current, which is given by [8] and [11] J = JC − JA =
− EF φA ] − γ AC TA2 exp[− ], (1) kB TC kB TA
φC β AC TC3 exp[−
where β = qkB3 /π3 v F2 , q is the elementary charge, m is the electron mass, kB is the Boltzmann constant, is the reduced Planck constant, v F stands for the velocity of massless Dirac fermions in the graphene, AC are the surface areas of the cathode and anode, φC and φA are, respectively, the work functions of the graphene and metals, E F is the Fermi level, and γ = 4πqmkB3 / h 3 is the Richardson constant. According to Fig. 1 and the first law of thermodynamics, the energy balance equation of the absorber, cathode, and anode can be, respectively, expressed as qC = qH + qR + qL ,
(2)
qH − JC AC (φC + 2kB TC ) + JA AC (φC + 2kB TA ) − (1 − μ)AC σ (TC4 − TA4 ) = 0,
(3)
and JC AC (φA +2kB TC ) − JA AC (φA +2kB TA ) +μAC σ (TC4 − TA4 ) − U AC (TA − TE ) − μAC σ (TA4 − TE4 ) = 0.
(4)
(3) and (4) include the irreversible losses of the cathode and anode, heat transfer losses from the anode to the environment, and thermal radiation losses between the cathode and the anode. In addition, qC , qL , and qR can be, respectively, calculated by [12] ∞ (λ)dλ, (5) qC = ηA C AA ∞ 0 [F(TC ) − F(TE )]dλ, (6) qL = A A 0
and
qR = ηA C AA
∞
R(λ)(λ)dλ.
(7)
0
∞ In (5)-(7), 0 (λ)dλ represents the solar constant as well as a measure of flux density, and (λ) denotes the solar irradiance, which is the power per unit area received from the sun in the form of electromagnetic radiation. (λ) varies with the wavelength and the range of the corresponding radiation wavelength λ of the AM1.5 solar spectrum is 280-4000nm [13]. ηA is the optical efficiency of the concentrator, C is the concentration factor, AA is the front surface 2πhc2 ε(λ) areas of the absorber, F(T ) = λ5 exp(hc/λk 5 , ε(λ) denotes B T )−λ the spectral emissivity of the absorber surface, which equals the spectral absorption coefficient according to Kirchhoff law, and R(λ) = 1 − ε(λ) stands for the reflection coefficient. Substituting (3)-(7) into (2), we can obtain the expression of an absorber-electrode area ratio AA /AC ≡ α. Based on the above analyses, the efficiency and power density of the TSC can be, respectively, expressed as P ∞ (8) η= C AA 0 (λ)dλ and P = J V = (JC − JA )(φC − φA ),
(9)
where V = φC − φA is the voltage output. It should be pointed out that the performance of the TEC is closely dependent on the solar concentrating factor, temperature difference between the cathode and the anode, work function, and heat transfer coefficient of the anode material. In addition, the operating temperatures of the cathode and anode for the TSC are closely dependent on other parameters, which can be determined through (1)-(4). Note that the structural defects in the graphene strongly affect the properties of the electrical transport and significantly lower the mobility of electrons, and consequently, deteriorate the performance of graphene-based devices [14], [15]. Here we consider the thermionic electron emission along the direction that is perpendicular to the monolayer graphene plane. Thus, the effect of defects or decreasing mobility on the electrons transferring in the graphene can be negligible theoretically. IV. R ESULTS AND D ISCUSSION Using (1)-(9), one can generate the three-dimensional projective graphs of TC , TA , j , and η varying with α and V , as shown in Fig. 2, where j = J/AC . It is seen from Figs. 2(a)-(c) that TC monotonically increases with the increase of α and V , while both TA and j monotonically increase with α and decrease with V . Because the heat exchange between the whole system and the heat sink is finite, the more solar heat flux can be harvested by the absorber as α increases, and consequently, TC and TA simultaneously increase. Fig. 2(d) shows that η is not a monotonic function of α and V , and has the maximum value 21% when α = 0.24 and V = 2.01V. One can further obtain the curves of several parameters varying with C, as shown in Fig. 3, where αopt , Vopt, jopt, TC,opt, and TA,opt are the optimal values of α, V , j , TC , and TA
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the more the input energy, the higher the electrode temperature, and the larger the electric current. For the different values of C, we can calculate the maximum efficiencies and the optimum values of some important parameters at the maximum efficiency, which are listed in Table I. These figures and data obtained here will be helpful to engineers to design and operate graphene-based TSCs. V. C ONCLUSION In conclusion, we have established a model of the graphenebased TES including various irreversible losses. Through energy balance equations, the operating temperatures of the cathode and anode are determined and the formulas of the efficiency and power output are derived. The optimally operating conditions of some significant parameters are determined. The selection criteria of the parametric design are given. R EFERENCES Fig. 2. Three-dimensional projective graphs of (a) TC , (b) TA , (c) j, and (d) η as functions of α and V, where C = 800, ηA = 0.95, φC = 3eV, EF = 0.8eV, U = 100Wcm-2 K-1 , μ = 0.1, and TE = 300K are chosen.
Fig. 3. Curves of several parameters such as (a) ηmax and αopt , (b) Vopt and jopt , and (c) TC,opt and TA,opt varying with C, where the values of other parameters are the same as those used in Fig. 2. TABLE I T HE M AXIMUM E FFICIENCY AND O PTIMUM VALUES OF S OME PARAMETERS U NDER D IFFERENT C ONCENTRATION FACTORS
at the maximum efficiency ηmax , respectively. It is seen from Fig. 3(a) that when C increases, the maximum efficiency ηmax monotonically increases, while αopt monotonically decreases. Figure 3(b) shows that Vopt is not a monotonic function of C and has the maximum, while jopt monotonically increases with C. In Fig. 3(c), both TC,opt and TA,opt monotonically increases with the increases of C. The cause may be simply explained as follows. The larger the concentration factor is,
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