1. Introduction

Global warming and energy crisis are long-term issues resulted from prevalent use of fossil fuels. In developing the carbon-free, high-efficiency and low-cost renewable energy harvesting and recycling technology, thermophotovoltaic (TPV) energy conversion system is considered as a promising alternative to mitigate the current crisis [1]. A TPV system that generates electric power directly from absorbed thermal radiation can reap benefits because it can reuse the wasted heat in industry. In addition, a TPV system has no moving parts, leading to low noise and high portability [2].

Major challenges in practical use of a TPV system are relatively low power throughput and energy conversion efficiency because the source temperature in many industrial processes is considerably lower than that of the sun. In order to resolve these issues, the near-field thermal radiation has been employed by several groups [2–7]. Considering that the radiative heat transfer between two objects can exceed the blackbody limit through tunnelling of evanescent waves when distance between them is smaller than the characteristic wavelength of thermal radiation [8, 9], the source can transfer more radiative energy to the TPV cell in certain temperature difference, leading to increased power throughput. Whale and Cravalho [2] considered a Drude-model-based fictitious source and InGaAs cell to analyze the performance of TPV cell, while Pan et al. [3] calculated the near-field radiation between two dielectric materials for a near-field TPV system. Narayanaswamy and Chen [4] studied the effect of surface phonon-polaritons on the near-field radiative transfer for increasing the conversion efficiency of a TPV system. Later, the performance of TPV system consisting of tungsten and GaSb TPV cell was analyzed by Laroche et al. [5] by assuming the quantum efficiency to be 100% (i.e., all electron-hole pairs generated from TPV system can contribute to the photogeneration current). However, because generated electron-hole pairs can experience the recombination as they are transported in the TPV cell, the assumption of 100% quantum efficiency can lead to overestimation of the performance of a TPV system. Park et al. [6] considered this minority carrier transport and conducted more rigorous analysis on the performance of TPV system. Later, Franceur et al. [7] developed a theoretical model for coupled near-field thermal radiation, charge and heat transport problems to ensure required cooling for maintaining near-field TPV performance.

The conversion efficiency of a TPV system can be defined as ratio of electric power generated from a TPV cell to the absorbed radiative power. Given that low-energy photons (i.e., energy lower than bandgap energy of a TPV cell) cannot generate the electron-hole pair and the excessive energy absorbed from high-energy photons is eventually dissipated in the form of heat, the spectral radiative heat transfer in frequency slightly higher than bandgap energy is preferred for achieving high conversion efficiency. Recently, spectral tuning of the thermal radiative heat flux using gold reflector [10] or hyperbolic metamaterial [11] has been investigated to increase the conversion efficiency.

In order to achieve both enhanced power throughput and conversion efficiency, recent studies focused on the use of graphene for TPV application [12–14]. Because graphene, a two-dimensional lattice of carbon atoms with exceptionally high crystal and electronic quality [15, 16], can support surface plasmon in the near-infrared (IR) spectral region [17, 18], it can enhance the near-field thermal radiation by matching the surface plasmon frequencies of the source and the receiver [19–23]. For example, Ilic et al. [12] analyzed the performance of InSb TPV cell with suspended graphene source, and Messina et al. [13] investigated the effect of graphene on matching frequencies of hBN source and InSb TPV cell. However, all these analyses on the performance of graphene-assisted TPV system are based on the assumption of 100% quantum efficiency, resulting in inevitable overestimation of the performance.

In this work, we solve the near-field thermal radiation together with the minority carrier transport model used in Park et al. [6] in a semi-analytic manner to conduct more realistic performance analysis of the graphene-assisted near-field TPV system. In the proposed TPV system, doped Si is employed as the source because its optical property can be readily tuned by changing the doping concentration, and InSb is selected as a TPV cell for low temperature applications (∼ 500 K) because of its low bandgap energy (0.17 eV). In order to enhance the near-field thermal radiation between the source and the TPV cell, monolayer of graphene is employed to the cell side so that surface plasmon can play a critical role in heat transfer.

2. Theoretical modeling

2.1. Near-field thermal radiation

Figure 1 depicts the proposed near-field TPV system consisting of a doped-Si source at T₁ = 500 K and an InSb TPV cell coated with monolayer of graphene at T₂ = 300 K, separated by vacuum gap width d. The doping concentration of phosphorous-doped Si, N_Si, varies from 10²⁰ to 10²¹ cm⁻³, an its dielectric function ε₁(ω, T₁), where ω is angular frequency, is calculated from Basu et al. [24] with the assumption of complete ionization at 500 K. InSb has the bandgap energy of 0.17 eV, and its dielectric function at 300 K is calculated from ${\epsilon }_2(\omega )={\left(n+i\frac{\alpha (\omega )}{2k_0}\right)}^2$ [12, 13, 25], where n is the refractive index and α(ω) = 0 for ω < ω_g and $\alpha (\omega )=(0.7\times {10}^6\hspace{0.17em}{\text{m}}^{-1})\sqrt{(\omega -{\omega }_g)/{\omega }_g}$ for ω > ω_g. Here, k₀ = ω/c₀ with c₀ being the speed of light in vacuum, and ω_g is the angular frequency corresponding to the bandgap energy (i.e., ω_g = 2.583 × 10¹⁴ rad/s). In the present study, ε₂(ω) is assumed to be independent of the doping concentration of InSb, and the value of refractive index is taken from [26] near λ_g = 7.29 μm (i.e., the wavelength corresponding to the bandgap energy). Therefore, the TPV cell consisting of two quasi-neutral p- and n-doped InSb can be treated as a homogeneous isotropic medium, which has commonly done in several previous works [6, 7, 10].

 

Fig. 1 Schematic of a graphene-assisted Si-InSb thermophotovoltaic device in three-dimensional view.

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The net spectral heat flux from the semi-infinite doped-Si source (i.e., medium 1) to the semi-infinite TPV cell (i.e., medium 2) can be calculated from [9, 27]:

Because the TPV system generates the electric power from the absorbed radiation inside the TPV cell only, the absorption by the graphene layer should be excluded in the TPV performance analysis. It is important to understand that Eq. (1) contains the factor of $T_{12}{T}_{12}^{*}$, where T₁₂ is the transmission coefficient for the electromagnetic wave from the doped-Si source (medium 1) to the InSb cell (medium 2) passing through the graphene layer. Thus, the absorption by the graphene layer is not included. However, the effect of graphene is fully accounted for when calculating the Fresnel reflection and transmission coefficients at the vacuum- graphene/InSb interface in Eq. (2) [12, 13, 23, 28, 29]. Consequently, Eq. (1) represents the net spectral radiative heat flux absorbed only by the TPV cell, whose optical absorption is tuned by the graphene layer. Finally, the total net heat flux q″_net from the Si source to the InSb cell can be calculated as $q_{\mathit{net}}^{″}={\int }_0^{\infty }d\omega \hspace{0.17em}\hspace{0.17em}{q}_{\omega ,\mathit{net}}^{″}={\int }_0^{\infty }d\omega \hspace{0.17em}{\int }_0^{\infty }{S}_{\beta ,\omega }(\beta ,\omega )d\beta$ or $q_{\mathit{net}}^{″}={\int }_0^{\infty }d\lambda \hspace{0.17em}{q}_{\lambda ,\mathit{net}}^{″}={\int }_0^{\infty }d\lambda \hspace{0.17em}{\int }_0^{\infty }{S}_{\beta ,\lambda }(\beta ,\lambda )d\beta$.

2.2. TPV performance

As mentioned earlier, the TPV cell consisting of two quasi-neutral p- and n-doped InSb is treated as a homogeneous isotropic medium. Therefore, the net radiative heat flux Q(z) inside the TPV cell is simply [30]

With the solution of Eq. (6), the photocurrent density flowing from the n-region to the p-region (i.e., −z direction) can be obtained from the derivative of the minority carrier concentration at the edge of the depletion region [33]:

The total current density is required to calculate the maximal power output that can be derived by the ideal diode equation [34]: $\left|J\right|=\left|\left|J_{\mathit{ph}}\right|-\left|J_s\right|\times \left\{\text{exp}\left(\frac{eV}{k_{B}T}\right)-1\right\}\right|$, where $\left|J_s\right|=e{n}_i^2\left(\frac{\sqrt{D_e/{\tau }_e}}{N_A}+\frac{\sqrt{D_h/{\tau }_h}}{N_D}\right)$ is the magnitude of saturation current density with n_i being the intrinsic carrier concentration, and V is the bias voltage of the TPV system. The maximum value of electric power can then be calculated by [5, 6]:

In the calculation, the impurity concentrations of InSb TPV cell are set as N_A = 10¹⁹ cm⁻³ (p-region) and N_D = 10¹⁹ cm⁻³ (n -region). The properties of the InSb are adopted from the works of González-Cuevas et al. [35] and Francoeur [7]. The diffusion coefficients in the p-and n- region are D_e = 186 cm²/s and D_h = 5.21 cm²/s, respectively. The relaxation times at each region are τ_e = 1.45 ns and τ_h = 1.81 ns, respectively. The width of p-region is set to be a = 400 nm, and the n-region is thicker than the hole diffusion length L_h = 970 nm. The depletion region is estimated to be L_dp = b − a = 10 nm at the initial impurity condition. The surface recombination velocity is taken as u_p = 10⁴ m/s [36].

3. Results and discussion

Figure 2 shows the calculated spectral radiative heat flux between the doped-Si source and the graphene-coated InSb cell while varying the chemical potential (μ) of graphene. It can be clearly seen that higher chemical potential yields higher spectral heat flux, regardless of N_Si and d values. Because the excessive energy absorbed from high-energy photons is eventually dissipated in the form of heat, the spectral radiative heat transfer in slightly lower wavelengths than λ_g is preferred for achieving high conversion efficiency. The maximum enhancement achieved by applying monolayer graphene occurs at slightly lower wavelength than λ_g, reaping the benefits of increasing the conversion efficiency. For N_Si = 1 × 10²⁰ cm⁻³ in Fig. 2(a), we can obtain five times higher heat flux with graphene of μ = 1.0 eV compared to the case without graphene layer at d = 10 nm. Because presence of the graphene layer greatly increases the net heat flux in the vicinity of λ_g, graphene of μ = 1.0 eV can significantly enhance both the power throughput (EF_P = 30.4) and the conversion efficiency (EF_η = 6.1). On the other hand, for higher N_Si values in Figs. 2(b) and 2(c), although graphene increases the spectral heat flux to some extent, the enhancement occurs rather in a broad wavelength range of 3.5 μm < λ < λ_g. Consequently, the enhancement in the conversion efficiency is less than that of the case for N_Si = 1 × 10²⁰ cm⁻³, as can be read from Table 1. If the vacuum gap increases to 50 nm as shown in Fig. 2(d), graphene of μ = 1.0 eV still enhances the conversion efficiency by 11%, but its effect is not as considerable as the case of d = 10 nm.

 

Fig. 2 Spectral radiative heat flux between the doped-Si source and the InSb cell: (a) d = 10 nm and N_Si = 1 × 10²⁰ cm⁻³; (b) d = 10 nm and N_Si = 5 × 10²⁰ cm⁻³; (c) d = 10 nm and N_Si = 1 × 10²¹ cm⁻³; and (d) d = 50 nm and N_Si = 1 × 10²⁰ cm⁻³. In the figure, λ_g indicates the wavelength corresponding to the bandgap energy of InSb.

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Table 1. The enhancement factor compared to the case without graphene layer in terms of the conversion efficiency (EF_η) or the power throughput (EF_P).

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In order to elucidate the effect of graphene, Fig. 3 shows the contour of S_β,λ (β, λ) together with the surface plasmon polariton (SPP) dispersion curves that are calculated by finding poles of the transmission coefficient for p polarization in Eq. (2). As shown in Fig. 3(a), the SPP at the vacuum-InSb interface is not even close to the wavelength regions in which heat transfer dominantly occurs. When graphene of μ = 0.5 eV is coated on the InSb cell, however, it shifts the SPP at the vacuum-graphene/InSb interface to the shorter wavelength region so that the SPP at the vacuum-Si interface can interface with that at the vacuum-graphene/InSb near 8 μm. Because the InSb cell rarely absorbs the radiation when λ > λ_g, the spectral heat flux is mainly enhanced along the SPP dispersion curve at the vacuum-graphene/InSb interface due to evanescent waves with $\beta >200\times \frac{2\pi }{{\lambda }_g}$ in 6 μm < λ < λ_g (i.e., refer to Fig. 3(b)). If graphene’s chemical potential increases to 1 eV, then the SPP dispersion curve at the vacuum-graphene/InSb interface increases faster with respect to β, as can be seen from Fig. 3(c). In other words, as graphene’s μ increases, the SPP at the vacuum-graphene/InSb interface can be excited near λ_g by evanescent waves with relatively lower β values (i.e., $\beta >50\times \frac{2\pi }{{\lambda }_g}$). Therefore, the resulting heat transfer enhancement becomes larger than the case of μ = 0.5 eV because the evanescent waves with smaller β values are more favorable to the photon tunneling.

 

Fig. 3 Contour of S_β,λ (β, λ) in logarithmic scale: (a)–(c) d = 10 nm and N_Si = 1 × 10²⁰ cm⁻³; (d)–(f) d = 10 nm and N_Si = 5 × 10²⁰ cm⁻³; and (g)–(i) d = 50 nm and N_Si = 1 × 10²⁰ cm⁻³. For simplicity, the parallel wavevector component β is normalized by bandgap wavelength (λ_g = 7.29 μm). Surface plasmon dispersion curves are also overlaid.

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For N_Si = 5 × 10²⁰ in Figs. 3(d)–3(f), the SPP dispersion curve at the vacuum-Si interface appears when λ is much less than λ_g, resulting in rather broadband absorption over 3 < λ < 7 μm. However, similar to Fig. 3(c), the graphene of μ = 1.0 eV in Fig. 3(f) makes the SPP dispersion curves at the vacuum-Si interface and at the vacuum-graphene/InSb interface get closer, resulting in larger S_β,λ (β, λ) values along both dispersion curves.

If d increases to 50 nm as shown in Figs. 3(g)–3(i), although SPPs are excited at the vacuum-graphene/InSb interface, evanescent waves associated with SPPs do not actively participate in the photon tunneling. Graphene of μ = 1.0 eV has certain effects on enhancing the radiative heat flux at d = 50 nm because it’s SPPs are associated with smaller β at λ_g than that of 0.5 eV; however, the heat transfer enhancement is not as significant as in Fig. 3(c).

The magnitude of spectral photocurrent density is plotted in Fig. 4 for selected N_Si and d values. The spectral photocurrent density exhibits similar wavelength-dependency with the net spectral heat flux shown in Fig. 2. In addition, the graphene changes the spectral photocurrent density according to its chemical potential, μ, in the similar manner as in the spectral heat flux. The difference between the spectral photocurrent density and the spectral heat flux lies at wavelengths longer or far shorter than λ_g. Because InSb cell cannot generate electron-hole pairs by absorbing photons of λ > λ_g, there is no spectral photocurrent density for λ > λ_g. On the other hand, in the short wavelength region, the excessive energy of photons cannot generate extra electron-hole pairs and will eventually be dissipated as heat generation inside the TPV cell. Therefore, the spectral photocurrent is relatively small when λ > λ_g or 2 < λ < 4 μm, resulting in low conversion efficiency.

 

Fig. 4 Spectral photocurrent density generated in the InSb cell: (a) d = 10 nm and N_Si = 1 × 10²⁰ cm⁻³; (b) d = 10 nm and N_Si = 5 × 10²⁰ cm⁻³; (c) d = 10 nm and N_Si = 1 × 10²¹ cm⁻³; (d) d = 50 nm and N_Si = 1 × 10²⁰ cm⁻³.

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The power throughput of the graphene-assisted TPV system is plotted in Fig. 5(a) with respect to the vacuum gap width when N_Si = 1 × 10²⁰ cm⁻³. Monolayer of graphene enhances the power throughput about 30 times than the case without graphene for d = 10 nm, but its effect diminishes quickly as d increases to 50 nm. This is because SPPs at the vacuum-graphene/InSb interface do not contribute to the photon tunneling at larger vacuum gap widths as noted by Figs. 3(g)–3(i). Similarly, the effect of graphene of μ = 0.5 eV decreases faster than graphene of μ = 1.0 eV because graphene of lower μ supports SPPs at larger β values with which the photon tunneling hardly occurs.

 

Fig. 5 Effect of the vacuum gap width on the TPV performance: (a) power throughput; and (b) conversion efficiency.

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Figure 5(b) shows the conversion efficiency with respect to the vacuum gap width. In contrast to previous works [6, 7, 10] that show nearly constant conversion efficiencies in 10 nm < d < 10 μm, the graphene-assisted TPV system exhibits monotonically decreasing conversion efficiency with respect to the vacuum gap width. Concerning the near-field thermal radiation, as the vacuum gap width decreases, evanescent waves with larger β values start to participate in heat transfer. As mentioned earlier, the radiative heat transfer inside the TPV cell is proportional to exp{−2ℑ(k_2z)z}, and the corresponding radiation penetration depth is {2ℑ(k_2z)}⁻¹ [6]. When β ≫ ω/c₀, $k_{2z}=\sqrt{k_2^2-{\beta }^2}$ can be approximated as iβ. In such cases, the radiation penetration depth is simply 1/(2β); thus, the larger β is, the shorter penetration depth is, resulting in the smaller quantum efficiency because most of the incident radiation is absorbed near the surface (i.e., far away from the junction). In our case, however, large portion (∼ 90%) of the radiative heat transfer is carried by evanescent waves with β < 5ω/c₀ especially when d ≥ 50 nm because of the relative low T₁ as well as the mismatch between plasmon frequencies of doped Si and InSb cell. In addition, the quantum efficiency of the proposed TPV system does not increase with the vacuum gap because of its negligible β-dependence, which will be further discussed later. As a result, the resulting conversion efficiency monotonically decreases with respect to the vacuum gap. Similar to the case of power throughput in Fig. 5(a), graphene rarely affects the conversion efficiency when d ≥ 50 nm.

It becomes clear now that in order for a near-field TPV system to take benefit of monolayer graphene even for d ≥ 50 nm, graphene’s chemical potential should be higher than 1.0 eV so that graphene can support SPPs at much smaller β values in the vicinity of λ_g. This is because the radiative heat transfer is dominated by evanescent waves with β < 5ω/c₀ when d ≥ 50 nm. For example, if we consider doped Si with N_Si = 1.2 × 10²⁰ cm⁻³ and graphene with μ = 2.0 eV, the resulting EF_P = 28.5 and EF_η = 4.8 at d = 10 nm, and EF_P = 2.1 and EF_η = 2.0 at d = 50 nm. Although monolayer graphene with μ > 1.0 eV has rarely been fabricated so far, multilayer graphene with moderate chemical potential can be used instead [37].

Figure 6 depicts the quantum efficiency of the InSb cell for selected N_Si and d values. In general, the quantum efficiency becomes lower at shorter wavelengths and at smaller vacuum gap widths in which the corresponding radiation penetration depth is smaller. However, the InSb cell without the graphene layer has higher quantum efficiency at shorter wavelengths and smaller vacuum gaps, due to the fact that contribution of evanescent waves with β < 5ω/c₀ is substantial at shorter wavelengths and larger vacuum gap widths. Given that the radiation penetration depths of such evanescent waves are comparable to the total thickness of p- and n-regions, the smaller penetration depth is, the more photons can be absorbed, leading to higher quantum efficiency. In the similar manner, at larger vacuum gap widths, radiative heat transfer occurs dominantly by evanescent waves with β < 5ω/c₀ so that most of photons are absorbed beyond the hole diffusion length, reducing the quantum efficiency.

 

Fig. 6 Quantum efficiency, η_q: (a) d = 10 nm and N_Si = 1 × 10²⁰ cm⁻³; (b) d = 10 nm and N_Si = 5 × 10²⁰ cm⁻³; (c) d = 10 nm and N_Si = 1 × 10²¹ cm⁻³; (d) d = 50 nm and N_Si = 1 × 10²⁰ cm⁻³.

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When graphene is coated on the InSb cell, graphene can affect the contribution by evanescent waves whose penetration depth is smaller than the thickness of p-region. Considering that graphene with higher chemical potential enhances the heat transfer through evanescent waves with smaller β values (i.e., with larger penetration depths; refer to Fig. 3), graphene can increase the quantum efficiency around λ_g, as shown in Figs. 6(a)–6(d). Recalling that the net spectral heat flux is also enhanced near λ_g, graphene can enhance both the net spectral heat flux and the quantum efficiency, which in turn leads to the increased power throughput with high conversion efficiency. As clearly seen from Fig. 6, quantum efficiency, η_q < 80% in most cases; thus, considering the minority carrier transport inside the TPV cell is very important to properly estimate the TPV system performance.

 

Fig. 7 Effect of the thickness of p-region on the TPV performance: (a) power throughput; (b) conversion efficiency; and (c) photocurrent density.

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It was proposed that decreasing the thickness of the p-region can improve the quantum efficiency [6,10] by reducing the probability of recombination in the p-region. Consistently, when we decrease the thickness of p-region, it certainly increases both the power throughput and the conversion efficiency in 10 < d < 40 nm. For 40 < d < 100 nm, however, p-region with thickness of 1.6 ∼ 2.0 μm yields the maximum power throughput and the conversion efficiency, as can be seen from Figs. 7(a) and 7(b). This can be explained based on Fig. 7(c) in which |J_e| and |J_h| are plotted together. Because |J_dp| ≪ |J_h|, it is not shown in the figure. For 40 < d < 100 nm, evanescent waves with β < 5ω/c₀ are dominantly contribute to the heat transfer and the corresponding radiation penetration depths become comparable to the total thickness of p- and n-regions (i.e., few micrometers). In this d range, |J_e| has the higher value for thicker p-region as more photons are absorbed in the p-region, but |J_h| always decreases as the p-region becomes thicker because less photons are absorbed in the n-region. On the other hand, if the thickness of p-region becomes too small, then the whole p- and n-regions cannot absorb all the photons from the source. Therefore, there exists an optimal thickness of p-region for the maximum power throughput and the conversion efficiency. For the considered graphene-assisted InSb cell, the optimal p-region thickness is found to be 1.6 ∼ 2.0 μm. With these p-region thicknesses, we can obtain further enhancement in the power throughput and the conversion efficiency for large vacuum gap width, although the enhancement factor is not as significant as for the smaller vacuum gap cases. It can be readily seen that solving minority carrier transport is very important to get physical insight on the proper p-region thickness for a certain vacuum gap width.

4. Concluding remark

In the present work, we proposed a graphene-assisted near-field TPV system composed of a doped-Si source and InSb cell for low temperature applications. The performance of the proposed TPV system was theoretically analyzed by solving the near-field thermal radiation with the minority carrier transport in semi-analytic manner. In order to tune the surface plasmon frequency, monolayer graphene was applied to the TPV cell side and then SPPs at the vacuum-graphene/InSb interface can interact with those at the vacuum-Si interface. It was found that the graphene layer (μ = 1.0 eV) can enhance the power throughput by 30 times and the conversion efficiency by 6.1 times for the doped-Si source (N_Si = 1 × 10²⁰ cm⁻³) at d = 10 nm. However, such an enhancement diminishes quickly as the vacuum gap increases. We also showed that the InSb cell with thicker p-region is advantageous for the larger vacuum gaps. In the previous studies proposing the near-field TPV system for the tungsten source [6, 7, 10] at 2000 K, the conversion efficiency was around 20 ∼ 30% when the corresponding Carnot efficiency was 85%. It is worthwhile to mention that we achieved the maximum conversion efficiency of 19.4% under the Carnot efficiency of 40% at the source temperature of 500 K. Therefore, monolayer graphene played significant impact on further enhancing the efficiency of near-field TPV system.