Dirac semimetal-assisted near-field radiative thermal rectifier and thermostat based on phase transition of vanadium dioxide

Shuang Wen; Yuhang Zhang; Yuhang Zhang; Yicheng Ma; Zhiqiang Sun

doi:10.1364/OE.496766

1. Introduction

The heat transfer quantity can exceed several orders of magnitude over the black body limit when the feature size of the radiation heat transfer system is smaller than the peak wavelength determined by Wien's displacement law [1]. In the near-field thermal radiation (NFTR) the evanescent wave replaces the propagating wave as the key to the heat transfer process. This significant enhancement of radiation heat transfer is expected to be used in new energy conversion technologies, and thermal management of nano components, such as electroluminescent cooling [2,3], subwavelength imaging [4,5], thermophotovoltaics [6–8], to name a few. Recently, NFTR is mainly focused on active regulation of heat transfer quantity.

In order to achieve active regulation, the radiative thermal switch and transistor, which are composed of the black phosphorus or graphene covered with silicon carbide, SiO₂, and VO₂, have become research hotspots in the field of NFTR [9–12]. The study of NFTR began with a simple two parallel plates structure. The radiative heat transfer quantity between two parallel plates is calculated accurately by Polder and van Hove for the first time [13]. Since then, a rapid progress has been seen in the field of NFTR, no matter in theoretical or experimental studies.

The research on active control of the NFTR can be roughly divided into two-body and multi-body structures. The research on NFTR of two-body structures is earlier and deeper. Francoeur et al. [14] deduced the expression of near-field radiative heat flux between two finite thickness plates. Biehs et al. [15] used the scattering matrix method to study the NFTR between the two parallel plates composed of multilayer hyperbolic metamaterials. Various types of surface electromagnetic modes can be excited under near-field mechanisms by utilizing materials such as metals [16], polar dielectric materials [17,18], hyperbolic materials [19,20], and two-dimensional materials [21,22]. As for these materials, active control of NFTR can be achieved through reasonable structural design. Yang et al. [23] investigated the effect of magnetic polaritons (MPs) inside the silicon carbide (SiC) grating microstructures on the NFTR. Zhang et al. [24] studied the NFTR between two suspended monolayer black phosphorus (BP) sheets, and investigated the effect of the number of layers, electron density of BP and the mechanical rotation on the NFTR. In addition, a magnetic field is a common method for achieving active thermal regulation and applied to the NFTR system. Moncada-Villa et al. [25] demonstrated the NFTR of doped semiconductors medium can be severely affected by the application of a static magnetic field. Similarly, Song et al. [26] regulated near-field thermal radiation between two multilayer hyperbolic metamaterial plates by utilizing an external magnetic field successfully. As a phase transition material, the vanadium dioxide (VO₂) is introduced in NFTR to broaden the application range due to its unique properties. Zwol et al. [27] researched the NFTR between VO₂ thin films. The results indicated that the near-field radiative heat flux can be regulated broadly. Yang et al. [28] proposed the concept of thermal rectifier based on the characteristic of VO₂. In order to achieve active regulation and enhancement of the heat transfer quantity, the graphene is introduced in NFTR because it is possible to achieve electromagnetic modulation from mid-infrared to terahertz wavelengths [29]. Ilic et al. [30] first studied the role of SPPs of graphene in the NFTR by using the structure of two parallel graphene films. Messina et al. [31] achieved enhancement of the NFTR between polar materials by covering graphene films on polar material substrates. Zhang and Chang et al. [32,33] analyzed the graphene-assisted NFTR configurations by calculating both the near-field dispersion relations and Rabi frequencies. He et al. [34] realized the magnetoplasmonic manipulation of NFTR by adding an external magnetic field on the twisted graphene gratings. Zheng et al. [35] reported a graphene-assisted near-field radiative thermal rectifier and achieved a larger rectification factor under the same conditions compared to the VO₂-SiO₂ thermal rectifier.

In practical applications, the simple two-body structure cannot meet the demand of the heat transfer of complex equipment obviously. Therefore, the NFTR of the multi-body structures has attracted more and more attentions. Ben Abdallah et al. [36] studied the NFTR between simple multi-body system composed of small interacting body. Choubdar et al. [37] demonstrated the heat flux can be switched effectively by changing the relative direction of ellipsoidal nanoparticles, especially the intermediate nanoparticles. Zheng et al. [38] realized flexible control of the heat flux by changing the thickness and position of the suspended film between two parallel plates. Ott et al. [39] realized the thermal rectification of NFTR between two particles by using the nonreciprocal surface waves excited by the magneto-optical material substrate. Xu et al. [40] built an effective multilayer model to approach the NFTR between random rough surfaces of SiC. Chen and Dong et al. [41,42] investigated the NFTR between two or more SiC/Au nanoparticles inside a cavity configuration composed of two semi-infinite SiC plates. Latella et al. [43] studied the near-field heat engine of a three-body structure and demonstrated that the energy conversion process driven by three-body photon tunneling imparts superior thermodynamic performance to the three-body structure heat engine compared to the two-body structure. When phase transition materials (VO₂) and two-dimensional materials (graphene) are applied in NFTR, the multi-body structure can achieve more functions. Song et al. [44] studied the NFTR effect of a three-body structure composed of graphene/SiC core-shell nanoparticles. Zhang et al. [45] analyzed the effect of the separation distances, number of layers and chemical potentials on NFTR in graphene/vacuum multilayers. He et al. [46] studied the steady-state temperature distribution of multilayer graphene systems and observed a unique multi-body enhancement phenomenon. Ben Abdallah et al. [47] designed a three-body structure based on the VO₂ and SiO₂ plates, which realized the transistor-like effect in NFTR. The thermal transistors have three key elements: the source, the drain, and the gate, which is similar to the electronic counterparts. The effective modulation of heat transfer from the source to the drain can be achieved by manipulating the temperature of the gate. Recently, Li et al. [48] reported an experimental demonstration of thermal transistor effect and an amplification parameter of heat flux over 20 is reached in the experiment. Such structures not only enable heat flux modulation and amplification but also facilitate the design of various functional devices, such as near-field thermal storage devices, thermal photon logic operators, and near-field thermostats [49]. For instance, Ito et al. [50] designed a near-field thermal memory and studied the radiative heat transfer hysteresis between a VO₂ film and a fused silica substrate under steady-state conditions. Ben Abdallah et al. [51] introduced the concept of thermal radiation logic gates based on phase-change materials, and discussed the implementation of NOT, OR, and AND gates operating in the near-field regime in detail. He et al. [52] coated the graphene on the simple three-body parallel plate structure, and a near-field thermostat was designed.

In the existing research, graphene is one of the key materials for achieving active regulation of NFTR. However, graphene, as an ultra-thin two-dimensional material, lacks the physical space that could be manipulated and controlled. Moreover, graphene is vulnerable to interference from the external dielectric environment, and the optical absorbance of single-layer graphene is weak [53]. In order to overcome the defects of graphene, researchers pay attention to a new topological quantum material called Dirac semimetal. This kind of material has similar energy band structure and linear dispersion relationship with graphene, and the dielectric function can also be adjusted by regulating Fermi level. Meanwhile, the structure of the topological quantum material is stable and easy to prepare [54]. Uchida et al. [55] introduced a precise growth technique for Dirac semimetal Cd₃As₂ thin films. The research indicated the Cd₃As₂ thin films with a thickness between 12 and 100 nm can be prepared using precise growth technique. Moreover, mechanical stripping and selenization technique can be employed to produce semi-metallic films at the atomic level [56,57]. Guo et al. [58] realized the active regulation of the Fermi level of Cd₃As₂ material successfully through the doping of Mn elements. In addition, similar to the graphene, the Fermi level of Dirac semimetal can be controlled by using a grid voltage control method [59,60]. The Fermi level was closer to the Dirac point by increasing the doping of impurities and applying a higher bias voltage. Dirac semimetal, as a substitute material of graphene, is applied in the field of optics widely. Kotov et al. [61] gave the calculation method of conductivity and dielectric function of Dirac semimetal through theoretical derivation. The research results proved that Dirac semimetal can support SPPs which is similar to graphene. Hu et al. [62] utilized Dirac semimetal and VO₂ to realize the dynamic adjustment of resonance frequency and absorption intensity of optical absorption peak. At present, research on the Dirac semimetal in the NFTR fields is limited. Xu et al. [63] coated Dirac semimetal films on the SiO₂ plates, and studied the variation of energy transmission coefficient with the thickness of Dirac semimetal, Fermi energy level and degeneracy factor. However, to the best of author’s knowledge, there is no investigation about modulating functions about Dirac semimetal in the two-body and three-body systems.

In order to further explore the performance of Dirac semimetal in the active regulation of NFTR, this paper utilized the Dirac semimetal, VO₂, and SiO₂ to design the two-body thermal rectifier and three-body thermostat, respectively. The paper is structured as follows. In section 2, the dielectric function and other parameters of materials is given, and calculation method of two-body and three-body parallel plate structure is introduced. In section 3, the NFTR of two-body and three-body structures is analyzed in detailed. Meanwhile, the influence of different parameters on the energy transmission coefficient, net radiative heat flux, and equilibrium temperature is studied, and the interaction forms of various electromagnetic surface modes are explored. Finally, the conclusion is given in section 4.

2. Model introduction

2.1 Heat transfer structure

The configuration of the near-field radiative thermal rectifier is showed in Fig. 1. The DSTR is consisted of a VO₂ plate covered with Dirac semimetal and a SiO₂ plate, in which the plates are semi-infinite. As shown in Fig. 1, the two plates are separated by a vacuum gap d and the thickness of Dirac semimetal film is t. VO₂ behaves like an insulator and exhibits anisotropy, when the temperature of VO₂ is lower than phase change temperature 341 K. On the contrary, VO₂ behaves like metal and exhibits isotropy.

Fig. 1. Heat transfer model of two-body structure thermal rectifier. Media 1 to 4 represent VO₂ plate, Dirac semimetal film, vacuum gap and SiO₂ plate, respectively. The temperature of the VO₂ and SiO₂ plate is set as 300 K and 353 K, and the scenario is defined as forward heat transfer. The heat transfer quantity of NFTR is expressed as Φ_F. The temperature of VO₂ and SiO₂ plates is set as 300 K and 353 K, and the scenario is defined as forward heat transfer. The heat transfer quantity of NFTR is expressed as Φ_R.

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On the basis of the two-body structure thermal rectifier and inspired by the idea of thermal transistors [49], a near-field thermostat based on SiO₂-VO₂-SiO₂ three-body structure covered with Dirac semimetal film is designed, as shown in Fig. 2. The structure is composed of three parallel plates. The left is a semi-infinite SiO₂ plate covered with a Dirac semimetal film with the thickness t₁, and the temperature is fixed at T₁. The middle is a VO₂ plate with the thickness δ. The right is a semi-infinite SiO₂ plate covered with a Dirac semimetal film with the thickness t₂, and the temperature is fixed at T₃ (T₁> T₃). The middle plate is separated from the left and right plates by a vacuum gap d.

Fig. 2. Heat transfer model of three-body thermostat. Media 1-3 represent the SiO₂ plate on the left (400 K), VO₂ plate, and SiO₂ plate on the right (300 K), and the two Dirac semimetal films are represented by 4 and 5. The heat flux released by the high temperature plate 1 is represented as Φ₁, and the heat flux absorbed by low temperature plate 3 is represented as Φ₃.

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2.2 Dielectric function calculation model

Determining the physical properties of materials is the premise for accurate calculation of near-field radiation heat transfer, and the basis for guiding near-field thermal radiation regulation. The dielectric function of Dirac semimetal can be written as [61,63]

(1)$${\varepsilon _\textrm{D}} = {\varepsilon _b} - \frac{{g\alpha c}}{{6\pi {v_F}}}\left[ {{{\left( {\frac{{2{E_F}}}{{\hbar \Omega }}} \right)}^2} + \frac{1}{3}{{\left( {\frac{{2\pi {k_B}T}}{{\hbar \Omega }}} \right)}^2} + i\pi G\left( {\frac{{\hbar \Omega }}{2}} \right) - 8\int_0^{{E_c}} {\frac{{G(\xi )- G\left( {\frac{{\hbar \Omega }}{2}} \right)}}{{{{({\hbar \Omega } )}^2} - 4{\xi^2}}}\xi d\xi } } \right]$$

where, ε_b represents the background dielectric function, g denotes the degenerate factor of 3D Dirac point; α refers to the fine structure constant; c indicates the speed of light; v_F corresponds to the Fermi velocity; E_F means the Fermi level; k_B is the Boltzmann constant; T is the temperature of Dirac semimetal; $\hbar$ is the Planck’s constant divided by 2π; Ω=ω+iτ⁻¹, where ${\tau ^{ - 1}} = {{ev_F^2} / {\mu {E_F}}}$ is the relaxation time and µ indicates the carrier mobility; Fermi Dirac distribution function G(ξ)=sinh(ξ/k_BT)/[cosh(ξ/k_BT)+cosh(E_F/k_BT)]; E_c means the cut-off energy. In present study, the AlCuFe quasicrystal, which is a typical Dirac semimetal material, is employed to analyze the performance of the DSTR and DST. The degeneracy factor, carrier mobility, and cut-off energy of the AlCuFe quasicrystal are 40, 6.42m²/($\textrm{V} \cdot \textrm{s}$), and 3E_F, respectively [63,64].

The crystal structure of VO₂ will be transformed near 341 K, and the dielectric function will change accordingly. VO₂ is in the insulating state when the temperature is lower than 341 K, and the dielectric function shows anisotropic characteristics. Assuming that the optical axis is the z-axis and perpendicular to the surface, the dielectric function of VO₂ can be expressed as a tensor [28]

(2)$$\overline{\overline \varepsilon } = \left[ {\begin{array}{{ccc}} {{\varepsilon_ \bot }}&0&0\\ 0&{{\varepsilon_ \bot }}&0\\ 0&0&{{\varepsilon_\parallel }} \end{array}} \right]$$

where the ordinary dielectric function ε_⊥ and extraordinary dielectric function ε_‖ can be calculated by Lorentz model, as following [65]

(3)$$\varepsilon (\omega = {\varepsilon _\infty } + \sum\limits_{j = 1}^N {\left[ {\frac{{({{S_j}\omega_j^2} )}}{{({\omega_j^2 - i{\gamma_j}\omega - {\omega^2}} )}}} \right]}$$

where ω indicates the angular frequency; ε_∞ denotes the high frequency constant; S_j means the phonon intensity; ω_j represents the phonon frequency; γ_j refers to the damping coefficient; the subscript j represents the jth optical phonon frequency.

VO₂ is in the metallic state when the temperature is higher than 341 K, and the dielectric function can be described by Drude model as [65]

(4)$$\varepsilon (\omega ={-} \frac{{{\varepsilon _\infty }\omega _p^2}}{{{\omega ^2} + i{\omega _c}\omega }}$$

where ω_p is the plasma frequency, ω_c is the collision frequency. The relative dielectric function of SiO₂ adopts Palik's data [66].

2.3 Calculation model of near-field thermal radiation

For the DSTR, the net radiative heat flux can be written as [13,67]

(5)$$\Phi = \int_0^\infty {\frac{{d\omega }}{{2\pi }}\varphi (\omega )}$$

where φ(ω) is the spectral radiative heat flux, which can be expressed as [35,67]

(6)$$\varphi (\omega ) = \frac{{\hbar \omega {n_{12}}}}{{4{\pi ^2}}}\int_0^\infty {\xi (\omega ,\kappa )\kappa d\kappa }$$

where n₁₂(ω)=n₁(ω)-n₂(ω) is the difference for the average energy of the harmonic oscillators at different temperatures, in which n_i(ω) = [exp(ħω/k_BT_i)-1]^-1; $\kappa$ denotes the component of the wave vector, which is parallel to the interface; ξ(ω,κ) refers to the energy transmission coefficient. Since the dielectric function of VO₂ in the insulating state is anisotropic, the contribution of the propagating wave (κ<ω/c) to the NFTR can be expressed as [28,35,68]

(7)$${\xi _{\textrm{prop}}}(\omega ,\kappa ) = \frac{{({1 - {{|{R_{321}^s} |}^2}} )({1 - {{|{r_{34}^s} |}^2}} )}}{{{{|{1 - R_{321}^sr_{34}^s{e^{2ik_{z,3}^sd}}} |}^2}}} + \frac{{({1 - {{|{R_{321}^p} |}^2}} )({1 - {{|{r_{34}^p} |}^2}} )}}{{{{|{1 - R_{321}^pr_{34}^p{e^{2ik_{z,3}^pd}}} |}^2}}}$$

The contribution of the evanescent wave (κ>ω/c) to the NFTR is expressed as [28,35,68]

(8)$${\xi _{\textrm{evan}}}(\omega ,\kappa ) = \frac{{4{\mathop{\rm Im}\nolimits} ({R_{321}^s} ){\mathop{\rm Im}\nolimits} ({r_{34}^s} ){e^{ - 2{\mathop{\rm Im}\nolimits} ({k_{z,3}^s} )d}}}}{{{{|{1 - R_{321}^sr_{34}^s{e^{2ik_{z,3}^sd}}} |}^2}}} + \frac{{4{\mathop{\rm Im}\nolimits} ({R_{321}^p} ){\mathop{\rm Im}\nolimits} ({r_{34}^p} ){e^{ - 2{\mathop{\rm Im}\nolimits} ({k_{z,3}^p} )d}}}}{{{{|{1 - R_{321}^pr_{34}^p{e^{2ik_{z,3}^pd}}} |}^2}}}$$

where d refers to the distance of vacuum gap between two plates; ${\mathop{\rm Im}\nolimits}$ indicates to the imaginary part; Rs 321 and Rp 321 indicate the reflection coefficients under s and p polarizations when the electromagnetic wave enters into VO₂ plate covered by Dirac semimetal film from vacuum; rs 34 and rp 34 represent the reflection coefficients under s and p polarizations when the electromagnetic wave enters into the SiO₂ plate from vacuum.

Rs 321 and Rp 321 can be expressed respectively as [63]

(9)$$R_{321}^s = \frac{{\left( {1 - \frac{{k_{z,1}^s}}{{k_{z,3}^s}}} \right)\cos k_{z,2}^st + i\left( {\frac{{k_{z,2}^s}}{{k_{z,3}^s}} - \frac{{k_{z,1}^s}}{{k_{z,2}^s}}} \right)\sin k_{z,2}^st}}{{\left( {1 + \frac{{k_{z,1}^s}}{{k_{z,3}^s}}} \right)\cos k_{z,2}^st - i\left( {\frac{{k_{z,2}^s}}{{k_{z,3}^s}} + \frac{{k_{z,1}^s}}{{k_{z,2}^s}}} \right)\sin k_{z,2}^st}}$$

(10)$$R_{321}^p = \frac{{\left( {1 - \frac{{k_{z,1}^p}}{{{\varepsilon_1}k_{z,3}^p}}} \right)\cos k_{z,2}^pt + i\left( {\frac{{k_{z,2}^p}}{{{\varepsilon_2}k_{z,3}^p}} - \frac{{{\varepsilon_2}k_{z,1}^p}}{{{\varepsilon_1}k_{z,2}^p}}} \right)\sin k_{z,2}^pt}}{{\left( {1 + \frac{{k_{z,1}^p}}{{{\varepsilon_1}k_{z,3}^p}}} \right)\cos k_{z,2}^pt - i\left( {\frac{{k_{z,2}^p}}{{{\varepsilon_2}k_{z,3}^p}} + \frac{{{\varepsilon_2}k_{z,1}^p}}{{{\varepsilon_1}k_{z,2}^p}}} \right)\sin k_{z,2}^pt}}$$

rs 34 and rp 34 can be expressed respectively as [35,68]

(11)$$r_{34}^s = \frac{{k_{z,3}^s - k_{z,4}^s}}{{k_{z,3}^s + k_{z,4}^s}}$$

(12)$$r_{34}^p = \frac{{{\varepsilon _4}k_{z,3}^p - {\varepsilon _3}k_{z,4}^p}}{{{\varepsilon _4}k_{z,3}^p + {\varepsilon _3}k_{z,4}^p}}$$

where ε_i is the dielectric function of medium i; ks z,i and kp z,i are the vertical component of the wave vector in medium i under s and p polarizations, ks z,i and kp z,i can be expressed as [35,68]

(13)$$k_{z,i}^s = \sqrt {{\varepsilon _{ \bot ,i}}{{({{\omega / c}} )}^2} - {\kappa ^2}}$$

(14)$$k_{z,i}^p = \sqrt {{\varepsilon _{ \bot ,i}}{{({{\omega / c}} )}^2} - ({{{{\varepsilon_{ \bot ,i}}{\kappa^2}} / {{\varepsilon_{{\parallel} ,i}}}}} )}$$

where ε_⊥,2 and ε_‖,2 are the components of dielectric function perpendicular to the optical axis and parallel to the optical axis respectively. Equations (13) and (14) can be employed to compute the vertical component of the wave vector when VO₂ is in the metallic state.

For the DST, the net radiative heat flux absorbed by plate 3 can be written as [13,69,70]

(15)$${\Phi _3} = \int_0^\infty {\frac{{d\omega }}{{2\pi }}{\varphi _3}(\omega )}$$

where φ₃(ω) is the spectral radiative heat flux, which can be written as [13,71,72]

(16)$${\varphi _3}(\omega )\textrm{ = }\hbar \omega \sum\limits_{j = s,p} {\int {\frac{{{d^2}\kappa }}{{{{({2\pi } )}^2}}}[{{n_{12}}(\omega )\xi_j^{1 - 2}({\omega ,\kappa } )+ {n_{23}}(\omega )\xi_j^{2 - 3}({\omega ,\kappa } )} ]} }$$

where j = s, p represents s and p polarizations, respectively; n_ij(ω)=n_i(ω)-n_j(ω)refers to the difference for the average energy of the harmonic oscillators at different temperatures, where n_i(ω) = [exp(ħω/k_BT_i)-1]^-1; ξ1-2 j(ω,κ) and ξ2-3 j(ω,κ) denote the energy transmission coefficients between plates 1 and 2 and between plates 2 and 3 in the case of j polarization, respectively.

ξ1-2 j(ω,κ) and ξ2-3 j(ω,κ) can be expressed as [52,71]

(17)$$\xi _j^{1 - 2}({\omega ,\kappa } )\textrm{ = }\left\{ {\begin{array}{{c}} {\frac{{{{|{\tau_2^j} |}^2}({1 - {{|{\rho_1^j} |}^2}} )({1 - {{|{\rho_3^j} |}^2}} )}}{{{{|{1 - \rho_{12}^j\rho_3^j{e^{2i{k_{z,0}}d}}} |}^2}{{|{1 - \rho_1^j\rho_2^j{e^{2ik_{z,0}^dd}}} |}^2}}},\textrm{ }\kappa < \frac{\omega }{c}}\\ {\frac{{4{{|{\tau_2^j} |}^2}{\mathop{\rm Im}\nolimits} ({\rho_1^j} ){\mathop{\rm Im}\nolimits} ({\rho_3^j} ){e^{ - 4{\mathop{\rm Im}\nolimits} ({{k_{z,0}}} )d}}}}{{{{|{1 - \rho_{12}^j\rho_3^j{e^{2i{k_{z,0}}d}}} |}^2}{{|{1 - \rho_1^j\rho_2^j{e^{2ik_{z,0}^dd}}} |}^2}}},\textrm{ }\kappa > \frac{\omega }{c}} \end{array}} \right.$$

(18)$$\xi _j^{2 - 3}({\omega ,\kappa } )\textrm{ = }\left\{ {\begin{array}{{c}} {\frac{{({1 - {{|{\rho_{12}^j} |}^2}} )({1 - {{|{\rho_3^j} |}^2}} )}}{{{{|{1 - \rho_{12}^j\rho_3^j{e^{2i{k_{z,0}}d}}} |}^2}}},\textrm{ }\kappa < \frac{\omega }{c}}\\ {\frac{{4{\mathop{\rm Im}\nolimits} ({\rho_{12}^j} ){\mathop{\rm Im}\nolimits} ({\rho_3^j} ){e^{ - 2{\mathop{\rm Im}\nolimits} ({{k_{z,0}}} )d}}}}{{{{|{1 - \rho_{12}^j\rho_3^j{e^{2i{k_{z,0}}d}}} |}^2}}},\textrm{ }\kappa > \frac{\omega }{c}} \end{array}} \right.$$

where τj 2 is the transmission coefficient of the plate 2 under j (j = s, p) polarization; ρj i is the reflection coefficient of the plate i (i = 1, 2, 3) under j polarization; ρj 12 is the whole reflection coefficient composed of plate 1 and plate 2 under j polarization.

τj 2, ρj i, and ρj 12 can be expressed as [46,69]

(19)$$\tau _2^j = \frac{{t_2^j\overline {t_2^j} {e^{ik_{z,2}^j{\delta _2}}}}}{{1 - {{({r_2^j} )}^2}{e^{ik_{z,2}^j{\delta _2}}}}}\quad \quad \rho _i^j = r_i^j\frac{{1 - {e^{ik_{z,i}^j{\delta _2}}}}}{{1 - {{({r_i^j} )}^2}{e^{ik_{z,i}^j{\delta _2}}}}}\quad \quad \rho _{12}^j = \rho _2^j + \frac{{\rho _1^j{{({\tau_2^j} )}^2}{e^{2i{k_{z,0}}d}}}}{{1 - \rho _1^j\rho _2^j{e^{2i{k_{z,0}}d}}}}$$

where δ_i is the thickness of plate i; rj i is the vacuum-medium Fresnel reflection coefficient of plate i in the case of j polarization; tj 2 is the vacuum-medium transmission coefficient of plate 2 in the case of j polarization; $\overline {t_2^j}$ is the medium-vacuum transmission coefficient of plate 2 in the case of j polarization.

For SiO₂ plates 1 and 3 covered by the Dirac semimetal film (marked as D) with thickness t, rj i(i = 1,3) can be expressed as [46,63,69]

(20)$$r_i^s = \frac{{\left( {1 - \frac{{{k_{z,i}}}}{{{k_{z,0}}}}} \right)\cos {k_{z,D}}t + i\left( {\frac{{{k_{z,D}}}}{{{k_{z,0}}}} - \frac{{{k_{z,i}}}}{{{k_{z,D}}}}} \right)\sin {k_{z,D}}t}}{{\left( {1 + \frac{{{k_{z,i}}}}{{{k_{z,0}}}}} \right)\cos {k_{z,D}}t - i\left( {\frac{{{k_{z,D}}}}{{{k_{z,0}}}} + \frac{{{k_{z,i}}}}{{{k_{z,D}}}}} \right)\sin {k_{z,D}}t}}$$

(21)$$r_i^p = \frac{{\left( {1 - \frac{{{k_{z,i}}}}{{{\varepsilon_i}{k_{z,0}}}}} \right)\cos {k_{z,D}}t + i\left( {\frac{{{k_{z,D}}}}{{{\varepsilon_D}{k_{z,0}}}} - \frac{{{\varepsilon_D}{k_{z,i}}}}{{{\varepsilon_i}{k_{z,D}}}}} \right)\sin {k_{z,D}}t}}{{\left( {1 + \frac{{{k_{z,i}}}}{{{\varepsilon_i}{k_{z,0}}}}} \right)\cos {k_{z,D}}t - i\left( {\frac{{{k_{z,D}}}}{{{\varepsilon_D}{k_{z,0}}}} + \frac{{{\varepsilon_D}{k_{z,i}}}}{{{\varepsilon_i}{k_{z,D}}}}} \right)\sin {k_{z,D}}t}}$$

where k_z_,_α is the vertical component of the wave vector in the medium α ($\alpha = 0,i,D$), in which 0, i, and D represent vacuum, SiO₂ and Dirac semimetal; ε_α is the relative dielectric function of the material α.

For the intermediate plate 2, rj 2 can be expressed as [35,46,69]

(22)$$r_2^s = \frac{{{k_{z,0}} - k_{z,2}^s}}{{{k_{z,0}} + k_{z,2}^s}},\quad \quad r_2^p = \frac{{{\varepsilon _{ \bot ,2}}{k_{z,0}} - k_{z,2}^p}}{{{\varepsilon _{ \bot ,2}}{k_{z,0}} + k_{z,2}^p}}$$

where ks z,2 and kp z,2 are the vertical component of the wave vector in the VO₂ under s and p polarizations, respectively; ε_⊥,2 is the component of dielectric function perpendicular to the optical axis in the VO₂.

tj 2 and $\overline {t_2^j}$ can be expressed as [46,52,69]

(23)$$t_2^s = \frac{{2{k_{z,0}}}}{{{k_{z,0}} + k_{z,2}^s}},\textrm{ }\quad \textrm{ }t_2^p = \frac{{2\sqrt {{\varepsilon _{ \bot ,2}}} {k_{z,0}}}}{{{\varepsilon _{ \bot ,2}}{k_{z,0}} + k_{z,2}^p}}$$

(24)$$\overline {t_2^s} = \frac{{2k_{z,2}^s}}{{{k_{z,0}} + k_{z,2}^s}},\textrm{ }\quad \textrm{ }\overline {t_2^p} = \frac{{2\sqrt {{\varepsilon _{ \bot ,2}}} k_{z,2}^p}}{{{\varepsilon _{ \bot ,2}}{k_{z,0}} + k_{z,2}^p}}$$

Similarly, the spectral radiative heat flux released by plate 1 with temperature T₁ can be written as [52,71,72]

(25)$${\varphi _1}(\omega )= \hbar \omega \sum\limits_{j = s,p} {\int {\frac{{{d^2}\kappa }}{{{{({2\pi } )}^2}}}[{{n_{23}}(\omega )\xi_j^{2 - 3,\ast }({\omega ,\kappa } )+ {n_{12}}(\omega )\xi_j^{1 - 2,\ast }({\omega ,\kappa } )} ]} }$$

The calculation of energy transmission coefficient only needs to exchange the subscript 1 and 3 in Eqs. (17) and (18).

3. Results and discussion

3.1 Properties of dielectric functions

The dielectric function of the material is the basis of calculation of the energy transmission coefficient and net radiative heat flux [24]. The real and imaginary parts of the relative dielectric function of Dirac semimetal, VO₂ and SiO₂ are shown in Figs. 3–5, respectively.

Fig. 3. The relative dielectric function of Dirac semimetal (a) real part; (b) imaginary part

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Fig. 4. The relative dielectric function of VO₂ (a) real part; (b) imaginary part

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Fig. 5. The relative dielectric function of SiO₂ (a) real part; (b) imaginary part

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Most of the electromagnetic surface modes are closely related to the dielectric function of the material in NFTR. Dirac semimetal supports surface plasmon polaritons (SPPs), and the electromagnetic mode can be effectively regulated by adjusting Fermi level. Figure 3 shows the relative dielectric functions of Dirac semimetal with different Fermi levels. The effect of SPPs on NFTR will be enhanced with the increase of the relative dielectric function. SiO₂ supports surface phonon polarizations (SPhPs). When the real part of the dielectric function is less than 0 (Re(ε) < 0), SPhPs of the vacuum-SiO₂ interface can be excited under the p-polarization. VO₂ supports SPhPs, type I and type II surface hyperbolic modes (HMsI and HMsII). When the real part of the ordinary dielectric function and extraordinary dielectric function are both less than 0 (Re(ε_⊥) < 0 and Re(ε_‖) < 0), SPhPs can be excited under p-polarization. HMsI can only be excited when Re(ε_⊥) > 0 and Re(ε_‖) < 0, however, HMsII can only be excited when Re(ε_⊥) < 0 and Re(ε_‖) > 0.

3.2 Discussion on two-body structure near-field thermal rectifier

Two-body parallel plate structure is one of the focuses in the NFTR. In the two-body structure, the variation trend of quantity of the heat transfer with various parameters can be observed directly, and the interaction of various electromagnetic surface modes can be explored effectively. Therefore, the NFTR between DSTR is explored at first. The forward and reverse heat transfer between two-body structure is investigated because VO₂ shows different properties in insulating and metallic state.

The forward and reverse energy transmission coefficients of DSTR under different combinations of the vacuum gap distance d, Fermi level E_F, and film thickness t are shown in Fig. 6. As shown in Fig. 6, the energy transmission coefficient decreases with the increase of the vacuum gap distance d. The energy transmission coefficient of forward heat transfer move towards the high frequently and low wave vector as the Fermi level E_F increases. Meanwhile, the energy transmission coefficient of reverse heat transfer is also enhanced slightly. With the increase of film thickness t, the energy transmission coefficient is shifted to direction of low wave vector furtherly (see Fig. 6(d)).

Fig. 6. Energy transmission coefficients of forward and reverse heat transfer of two-body structure under different parameter combinations. (a) d = 10 nm, E_F =0.15 eV, t = 5 nm; (b) d = 20 nm, E_F =0.15 eV, t = 5 nm; (c) d = 10 nm, E_F =0.18 eV, t = 5 nm; (d) d = 10 nm, E_F =0.15 eV, t = 10 nm. In Fig, the left side represents the energy transmission coefficient of forward heat transfer, and the right side represents the energy transmission coefficient of reverse heat transfer.

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With the vacuum gap distance d = 50 nm, Fermi level E_F= 0.25 eV, and film thickness t = 4 nm, meanwhile, the structure without Dirac semimetal film under the same vacuum gap, the forward and reverse spectral radiative heat flux are shown in Fig. 7. The rectification effect of the two-body structure can be significantly enhanced by utilizing the Dirac semimetal film (see Fig. 7). This enhancement is achieved by an additional heat transfer near ω=2.14 × 10¹⁴rad/s during forward heat transfer, while the reverse heat transfer is almost consistent with the non-covered structure. The forward net radiative heat flux of the DSTR, which is obtained by integral, is 2.22 × 10⁴ W/m². The forward net radiative heat flux of the structure without Dirac semimetal film is 1.47 × 10⁴ W/m². The forward net radiative heat flux is increased by 50% as the Dirac semimetal film is applied. On the contrary, the reverse net radiative heat flux of the DSTR is slightly smaller than that of the structure without Dirac semimetal film.

Fig. 7. Variation trends of the spectral radiative heat flux with different conditions. The black solid line and black dashed line represent the forward heat transfer scenario of the film-covered structure (d = 50 nm, E_F= 0.25 eV, and t = 4 nm) and non-covered structure, respectively, while the red solid line and red dashed line illustrates the reverse heat transfer scenario of the film-covered structure and non-covered structure, respectively.

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In order to illustrate rectification effect of DSTR, the global rectification factor is defined as M, which can be expressed as [35]

(26)$$M = \frac{{{\varphi _\textrm{F}}}}{{{\varphi _\textrm{R}}}}$$

where φ_F refers to the net radiative heat flux of forward heat transfer scenario; φ_R denotes the net radiative heat flux of reverse heat transfer scenario.

To offer an intuitive explanation of the rectification effect variation caused by the coverage of the Dirac semimetal film, the global rectification factor of the DSTR is shown in Fig. 8(a), where the Dirac semimetal film changes between 0-20 nm and Fermi level varies from 0.05-0.25 eV. In this part, the vacuum gap is set as 100 nm because it is attainable in existing near-field thermal radiation experimental research. In Fig. 8(a), the black contour line represents the global rectification factor of the structure without Dirac semimetal film, and the value is 2.45. It is evident that the global rectification factor of the DSTR is larger than that of the structure without Dirac semimetal film in the vast majority of cases (see Fig. 8(a)). However, as the film thickness and Fermi level increase, the surface electromagnetic mode is suppressed, resulting in a deterioration of the rectification effect of film-covered structure. The numerical experiments demonstrate that the rectification effect of the DSTR can be enhanced by adjusting the Fermi level and thickness of the Dirac semimetal film (Meanwhile, the rectification effect of the DSTR is larger than that of non-film structure). It offers valuable insights for optimizing the Dirac semimetal film thickness and the Fermi level in experimental studies.

Fig. 8. (a) Contour nephogram of the global rectification factor of DSTR as a function of film thickness and Fermi level. The black contour line represents the global rectification factor of non-coverd structure; (b) Contour nephogram of the global rectification factor of DSTR as a function of vacuum gap distance and Fermi level.

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In practical applications, the active regulation of the film thickness is challenging, but the vacuum gap is controllable within a certain range. Meanwhile, the Fermi level can be controlled using grid voltage. The influence of the vacuum gap distance d and Fermi level E_F on the global rectification factor is investigated, where the thickness of the Dirac semimetal film is 10 nm. Figure 8(b) shows the global rectification factor with vacuum gap distance between 50-200 nm and Fermi level between 0.05-0.25 eV. As shown in Fig. 8(b), the maximum global rectification factor is 4.4, when the Dirac semimetal film is existed. For the non-covered structure, as the vacuum gap varies between 50 nm and 200 nm, the global rectification factor decreases from 2.76 to 2.04. The optical Fermi level with different vacuum gap and film thickness can be determined by numerical experiment to help in selecting the best experimental parameter settings.

The variation of global rectification factor with temperature is studied, in which the value of the film thickness, vacuum gap distance, and Fermi level are 10 nm, 100 nm, and 0.18 eV. The reason is that the combination of parameters has good rectification effect and can be realized by existing experimental technology. To ensure that the VO₂ maintains in the insulating state under the forward heat transfer scenario and maintains in the metallic state under the reverse heat transfer scenario, the temperature of the low-temperature plate is fixed at 300 K while the temperature range of the high-temperature plate is varied from 350 K to 450 K. Therefore, the temperature difference varies from 50 K to 150 K. Figure 9 shows the variation of global rectification factor of the DSTR with temperature difference. As a comparison, the global rectification factor of the structure without Dirac semimetal film is also shown in Fig. 9. It can be observed that when the temperature difference increases, the decreasing trend of the global rectification factor in film-covered structure is smoother that in the non-covered structure.

Fig. 9. The variation of global rectification factor with temperature difference under specific combination of parameters. The black solid line represents the global rectification factor of film-covered structure, while the red line represents the global rectification factor of non-covered structure.

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3.3 Discussion on three-body structure near-field thermostat

Based on the research of two-body parallel plate structure, the middle plate is introduced to form a three-body parallel plate structure to expand the application value of NFTR furtherly (see Fig. 2). Based on the principle of the near-field thermostat [52], the temperature of the intermediate plate can be controlled by changing the parameters of the Dirac semimetal film. Furthermore, an independent near-field radiative thermal transistor-like effect without external heat flux can be realized by combining with the phase-change characteristics of VO₂.

The energy transmission coefficients of the three-body parallel plate structure are shown in Fig. 10, in which the Fermi level and thickness of the Dirac semimetal films on both sides are equal. In Fig. 10, the energy transmission coefficient from the intermediate plate to the low-temperature plate is significantly higher than that from the high-temperature plate to the intermediate plate whether in the insulating or metallic state. The primary reason for this phenomenon is that although the Fermi level and thickness of Dirac semimetal films are equal, the two films are at different temperatures. It leads to that the differences exist in the dielectric function. The symmetry of the energy transmission coefficient is affected. Meanwhile, compared to the insulating state, the distribution of the energy transmission coefficient of the metallic state is more concentrated, but the value is smaller (see Fig. 10). The phenomenon is consistent with the results in the two-body structure. This fact can be predicted that when VO₂ is in the metallic state, the net radiative heat flux is also less than that of the insulating state greatly in three-body structure.

Fig. 10. Energy transmission coefficient of three-body structure when E_F₁= E_F₂= 0.15 eV and t₁= t₂= 5 nm. (a) VO₂ in the insulating state; (b) VO₂ in the metallic state. In each figure, the left side represents the energy transmission coefficient between plate 1 and plate 2, and the right side represents the energy transmission coefficient between plate 2 and plate 3.

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The thermal equilibrium of the three-body structure is investigated to explain the working mechanism furtherly. When the net radiative heat flux Φ₁ released by plate 1 is equal to the net radiative heat flux Φ₃ absorbed by plate 3, the heat transfer reaches the steady state and plate 2 reaches the equilibrium temperature T_2,eq. At this time, the net radiative heat flux of the three-body structure is defined as the equilibrium heat flux Φ_eq. To illustrate the effect of the Dirac semimetal film on heat transfer in DST, a contour nephogram of the equilibrium temperature T_2,eq and equilibrium heat flux Φ_eq is calculated when the Fermi levels and film thickness range from 0.05-0.15 eV and 2-20 nm, and the thickness of the intermediate plate and vacuum gap are 30 nm and 60 nm as shown in Fig. 11. In addition, a contour line representing the equilibrium heat flux of non-film covered structure is existed in Fig. 11(b) to demonstrate the role of the Dirac semimetal film directly. It can be clearly observed that the Dirac semimetal film can enhance the heat transfer quantity of the DST in most cases. In Fig. 11(b), the maximum equilibrium heat flux is 2.62 × 10⁴ W/m², which is nearly double that in the structure without Dirac semimetal film (1.57 × 10⁴ W/m²).

Fig. 11. Contour nephograms of equilibrium temperature T_2,eq and equilibrium heat flux Φ_eq varying with Fermi level E_F and film thickness t at intermediate plate initial temperature T₂= 400 K. (a) equilibrium temperature T_2,eq; (b) equilibrium heat flux Φ_eq. The black solid line in (b) represents the equilibrium heat flux of non-film covered structure.

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The principle of the near-field thermostat effect is introduced in the part. Meanwhile, the realization approach of thermal transistor-like effect is discussed in DST. Figure 12 shows the equilibrium temperature T_2,eq and equilibrium heat flux Φ_eq change with the Fermi level, when the initial temperature of the intermediate plate is 300 K and thickness of the Dirac semimetal film is 2 nm. As shown in Fig. 12(a), the equilibrium temperature of the intermediate plate can be effectively controlled within a range of 325-371 K, as the Fermi level ranges from 0.05 eV to 0.15 eV. Notably, this temperature includes the phase transition temperature of VO₂ (the phase transition temperature of VO₂ is 341 K). This fact illustrates the DST can realize the thermostat effect. The Fig. 12(b) reveals the significant difference in the equilibrium heat flux at different Fermi level. The difference is mainly caused by the phase transition properties of VO₂. As shown in Fig. 12(b), the structure demonstrates a remarkable thermal transistor-like effect, where the maximum heat flux near 5.5 × 10⁴ W/m² and the minimum heat flux is around 1.9 × 10⁴ W/m². This substantial difference of 3 times highlights the outstanding thermal modulation capabilities of the DST.

Fig. 12. Contour nephograms of equilibrium temperature T_2,eq and equilibrium heat flux Φ_eq varying with Fermi level E_F at intermediate plate initial temperature T₂= 300 K. (a) equilibrium temperature T_2,eq; (b) equilibrium heat flux Φ_eq. In Fig. 10, black solid line in (a) represents the phase transition temperature contour of VO₂, which corresponds to the black solid line in (b).

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When the initial temperature of the intermediate plate changes from 300 K to 400 K, the equilibrium temperatures T_2,eq and equilibrium heat flux Φ_eq of the intermediate plate under different Fermi level are shown in Fig. 13(a) and (b), respectively. Combined with the Fig. 12, the interesting phenomenon that the initial temperature of the intermediate plate will affect the final equilibrium temperature can be found in this study. In order to explain the phenomenon and explore the regulation mechanism of equilibrium heat flux, the heat flux released by plate 1 and absorbed by plate 3 with different Fermi level are investigated in detail. The change process of the intermediate plate temperature is also analyzed. The thickness of Dirac semimetal films on both sides is set as 2 nm, and Fermi level of Dirac semimetal film covered on the low-temperature SiO₂ plate is 0.15 eV. The Fermi level of Dirac semimetal film covered on the high-temperature SiO₂ plate is set as 0.05 eV, 0.08 eV and 0.12 eV, respectively. As the intermediate plate temperature changes between 300 K and 400 K, the heat flux released by plate 1 and absorbed by plate 3 are shown in Fig. 14. Figure 14 also depicts the results of non-covered structure for comparison purposes.

Fig. 13. Contour nephograms of equilibrium temperature T_2,eq and equilibrium heat flux Φ_eq varying with Fermi level E_F at intermediate plate initial temperature T₂= 400 K. (a) equilibrium temperature T_2,eq; (b) equilibrium heat flux Φ_eq. Auxiliary lines are the same as those in Fig. 12.

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Fig. 14. Heat flux released by plate 1 and absorbed by plate 3 under different conditions. The solid line represents the heat flux released by plate 1, and the dotted line represents the heat flux absorbed by plate 3. The black line combination corresponds to case 1, the blue line combination corresponds to case 2, and the red line combination corresponds to case 3.

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In Fig. 14, the four intersection points might be the equilibrium temperature T₂,eq of the intermediate plate 2 under the thermal equilibrium state of the system. As shown in Fig. 14, when the Fermi level of Dirac semimetal covered on high-temperature SiO₂ plate is 0.05 eV and 0.12 eV, the equilibrium temperature of the intermediate plate is fixed at 325 K or 352 K (points 1 and 4). The equilibrium temperature is located in the region of insulating and metallic state of VO₂, respectively. For the structure without Dirac semimetal film, the result is similar to the case of the Fermi level of 0.05 eV. An interesting phenomenon that two different intersection points are existed in absorption and release heat flux in the whole temperature range from 300 K to 400 K occurs, when the Fermi level of Dirac semimetal covered on high-temperature SiO₂ plate is 0.07 eV. The corresponding temperature of intersection point 2 is located in the insulating state region of VO₂, and the corresponding temperature of intersection point 3 is located in the metallic state region. When the initial temperature T₂ of the intermediate plate is lower than the phase change temperature 341 K, the equilibrium temperature T_2,eq is equal to 333 K (point 2). The equilibrium temperature T_2,eq is equal to 346 K (point 3) if the initial temperature T₂ is higher than the phase change temperature 341 K. This is the reason that the equilibrium temperature of the intermediate plate is affected by the initial temperature.

In order to explore the role of Dirac semimetal in the three-body structure near-field thermostat. The coupling effect between the SPPs supported by Dirac semimetal, SPhPs supported by SiO₂, and SPhPs, HMsI, and HMsII supported by VO₂ is investigated in detail. The p-polarized energy transmission coefficients (${\xi_{2p}^{1-}}+{\xi_{3p}^{2-}}/2$ of the three-body structure are calculated for different Fermi levels of Dirac semimetal. It can be seen obviously from Fig. 15(a) that a new mode of energy transmission coefficient appears in the low-frequency region due to the addition of Dirac semimetal. The new mode of energy transmission coefficient is caused by the SPPs supported by Dirac semi metal. Meanwhile, the energy transmission coefficient increases as with the increasement of the Fermi level significantly. As shown in Fig. 15(b), the Fermi level of Dirac semimetal films has no obvious effect on the energy transmission coefficient of the three-body structure. This is due to the fact that when the intermediate plate VO₂ is in the metallic state, VO₂ cannot support any electromagnetic surface mode. In addition, the SPhPs supported by SiO₂, the SPPs supported by Dirac semimetal, and the interaction between them are suppressed greatly.

Fig. 15. The p-polarized energy transmission coefficients (${\xi_{2p}^{1-}}+{\xi_{3p}^{2-}}/2$ of the 5 nm film-covered three-body structure under different Fermi levels of Dirac semimetal. (a) intermediate plate VO₂ is in insulating state; (b) intermediate plate VO₂ is in metallic state. The four pictures from left to right show the situation without Dirac semimetal film coverage and when the Fermi levels of Dirac semimetal film are 0.05 eV, 0.10 eV and 0.15 eV respectively.

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4. Conclusions

This paper focuses on the active regulation of NFTR. We report the Dirac semimetal-assisted DSTR and DST, respectively. The DSTR is made of a Dirac semimetal-covered vanadium dioxide VO₂ plate and SiO₂ plate separated by a vacuum gap. The left and right sides of DST are consisted of the SiO₂ covered with Dirac semimetal, and the intermediate plate is the VO₂. The following conclusions are obtained:

1. In the NFTR, the Dirac semimetal supports the SPPs, which can interact with SPhPs supported by SiO₂, and HMsI and HMsII supported by VO₂.
2. In the two-body structure, the net radiative heat flux of VO₂ in the insulating state is significantly higher than that of VO₂ plate in the metallic state. The thermal rectification can be implemented. The NFTR of the two-body structure can be adjusted by changing the Fermi level E_F, thickness of the Dirac semimetal film t, and vacuum gap distance d.
3. In the three-body structure, the equilibrium temperature of the intermediate plate can be adjusted to make it slightly higher and lower than the phase transition temperature based on the principle of a near field thermostat. The equilibrium heat flux will step with the switching of VO₂ between the metallic and insulating state, so as to realize the near-field thermal transistor-like effect.

Funding

National Natural Science Foundation of China (52106125); Science and Technology Program of Hunan Province (2021RC4006).

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Dirac semimetal-assisted near-field radiative thermal rectifier and thermostat based on phase transition of vanadium dioxide

Abstract

1. Introduction

2. Model introduction

2.1 Heat transfer structure

2.2 Dielectric function calculation model

2.3 Calculation model of near-field thermal radiation

3. Results and discussion

3.1 Properties of dielectric functions

3.2 Discussion on two-body structure near-field thermal rectifier

3.3 Discussion on three-body structure near-field thermostat

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (15)

Equations (26)

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