ε2(ω)=−1 denotes the FD distribution of electrons in the reservoir at chemical potential μ and temperature T, the dispersion relation E(k→)=ℏ2(kx2+ky2+kz2)/2m∗, and v(kx)=ℏkx/m∗ indicates the velocity of an electron moving along direction x.By adopting the energy space, Eq. (9) is rewritten as3636. M. O'Dwyer, “ Solid-state refrigeration and power generation using semiconductor,” Ph.D. thesis ( University of Wollongong, 2007). Je=e2πℏ∫0∞n(μ,T)ξ(Ex)dEx(10)with n(μ,T)=m∗kBTπℏ2log [1+exp (−Ex−μkBT)].(11)By using Eqs. (10) and (11), the net electric current density through the ESC is given by Jnet,e=e2πℏ∫0∞[n(Efe,TH)−n(Efn,TC)]ξ(Ex)dEx.(12)From the first law of thermodynamics, each electron leaving the absorber carries away energy Ex+kTH−Efe. An electron from the electrode traveling through the ESC will dump energy Ex+kTC−Efe to the absorber. By using Eq. (12), the net heat flux density of the absorber is expressed as Qnet,e=12πℏ∫0∞[(Ex+kTH−Efe)n(Efe,TH)−(Ex+kTC−Efe)n(Efn,TC)]ξ(Ex)dEx.(13)Note that Jnet,h=Jnet,e and Qnet,h=Qnet,e, because it has been assumed that the structures of the conduction and valence bands are symmetry.The conservation of particles requires On the other hand, the conservation of energy leads to Qnet,e=Qnet,h=QEvan+QProp.(15)As a result, the efficiency and the power output of the HNTD are, respectively, expressed as and In the following discussion, Eg=0.2 eV, TS=1800 K, TC=300 K, δE=0.001 eV, d=10 nm, and m∗=0.1me with me being the mass of free electron. Unless otherwise specified, these parameter values are adopted. Figure 3(a) presents the electric current Jnet,e as a function of V for a given extraction energy ΔE. In the regime of small V, Jnet,e is independent of V, leading to a constant value of Jnet,e. It is owing to the fact that the energy difference Δμ in the absorber lays in the value suppressing the radiative recombination effect. Under this condition, most carriers are extracted through the ESCs. In the regime of large V, a drastic drop in Jnet,e appears, because a large number of photons are lost due to the radiative recombination. Figure 3 also shows that the short-circuit current JSC monotonically decreases with increasing ΔE, because fewer carriers are distributed at higher energy levels. However, the open-circuit voltage VOC is a non-monotonic function of ΔE. For the extraction energy ΔE less than 0.65 eV, the energy extracted per collected electron–hole pair is small, resulting in a limited open-circuit voltage VOC. For an extraction energy ΔE above 0.65 eV, the available energy in carriers is not sufficient for all carriers to be extracted, resulting in limited JSC and VOC. By increasing ΔE from 0.25 to 0.75 eV, an optimal value of ΔE is found that maximizes VOC.Figure 3(b) reveals the efficiency η as a function of V for a given extraction energy ΔE. In Fig. 3(b), η increases with the increase in V in the small-V regime (V<Vm), but it decreases with the increase in V in the large-V regime (V>Vm), where Vm is the voltage at the maximum efficiency. In the former, the power significantly increases as V increases [Fig. 3(c)]. In the latter, a surge of excited electrons in the absorber are recombined. This rate of radiative recombination exceeds the growth of V, such that η diminishes. Comparing with Figs. 3(c) and 3(d), one observes that the power output P decreases faster than the net photon energy Qin in the region of V>Vm. Therefore, both P and η reach zero at the open-circuit voltage. At the same time, η can be maximized by optimally carefully choosing ΔE. At ΔE=0.55eV, η approaches 54%, which is obviously larger than the efficiency obtained by the NFTPV device with an intermediate band in the absorber,1717. W. Shen, J. Xiao, Y. Wang, S. Su, J. Guo, and J. Chen, J. Appl. Phys. 128, 035105 (2020). https://doi.org/10.1063/5.0010965 a graphene-covered cell,2626. R. Messina and P. Ben-Abdallah, Sci. Rep. 3, 1383 (2013). https://doi.org/10.1038/srep01383 or a GaSb-based cell.3737. M. Laroche, R. Carminati, and J.-J. Greffet, J. Appl. Phys. 100, 063704 (2006). https://doi.org/10.1063/1.2234560 As a result, hot carrier-based NFTPVs is capable of dramatically enhancing the efficiency of heat conversion into electricity.The existing studies of HCSC generate electricity by converting energy from concentrated solar irradiation. In order to further show the advantage of combining a near-field thermal emitter and a hot-carrier solar cell, it would be more straightforward to compare the performances of NFTPV devices powered by thermal radiations of different intensity. Thus, Fig. 4 plots the electric current efficiency η varying with the voltage for different values of the distance d between the emitter and the absorber. It is observed that the maximal η decreases by increasing d, because the near-field radiative heat transfer is reduced.

In summary, based on the detailed balance principle and the assumptions of ideal photon absorption and extraction processes, the characteristics of a HNTD with ESCs are analyzed deeply. The electric current and heat flux densities through the ESCs are calculated. The obtained results show that one of the main energy dissipation mechanisms to decrease the efficiency of HNTDs comes from the thermalization loss in electrodes, which can be reduced by optimizing the extraction energy levels of the ESCs. These results are helpful for further understanding the performance of HNTDs and will lay the foundation for developing realistic NFTPV devices.

This work has been supported by the Fundamental Research Fund for the Central Universities (No. 20720210020) of China, the Natural Science Foundation of Fujian Province (No. 2020J05148), and the Doctoral Research Foundation of Jimei University (No. ZQ2018005).

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Junyi Wang: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Youlin Wang: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Project administration (equal); Validation (equal); Writing – review & editing (equal). Xiaohang Chen: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Jincan Chen: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Shanhe Su: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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