Magnetic Field Effect of Near-Field Radiative Heat Transfer for SiC Nanowires/Plates

Abstract

The SiC micro/nano-scale structure has advantages for enhancing nonreciprocal absorptance for photovoltaic use due to the magneto optical effect. In this work, we demonstrate the near-field radiative transfer between two aligned SiC nanowires/plates under different magnetic field intensities, in which Lorentz-Drude equations of the dielectric constant tensor are proposed to describe the dielectric constant as a magnetic field applied on the SiC structure. The magnetic field strength is qualified in this study. Using local effective medium theory and the fluctuation-dissipation theorem, we evaluate the near-field radiation between SiC nanowires with different filling ratios and gap distances under an external magnetic field. Compared to the near-field heat flux between two SiC plates, the one between SiC nanowires can be enhanced with magnetic field intensity, a high filling ratio, and a small gap distance. The electric field intensity is also presented for understanding light coupling, propagation, and absorption nature of SiC grating under variable incidence angles and magnetic field strengths. This relative study is useful for thermal radiative design in optical instruments.

1. Introduction

When the distance between two materials is less than the thermal characteristic wavelength, also known as the characteristic wavelength of thermal radiation, evanescent waves play a dominant role in radiative heat transfer. The energy transfer can be further enhanced and Plank’s theory of thermal radiation is no longer valid, which is called near-field radiative heat transfer [1,2,3,4]. At nanometer distances, near-field radiative heat transfer can be several orders of magnitude greater than that between two far field blackbodies, especially when surface plasmon polaritons (SPPs) or surface phonon polaritons (SPhPs) are excited [5,6,7,8,9]. Due to the advent of near field radiative heat transfer and its wide potential applications in micro- and nano-scale thermophotovoltaic (TPV) cells, thermal imaging and local thermal management, near-field radiative heat transfer had attracted significant attention [10,11,12,13,14,15,16,17].

Some researchers have found that applying an external magnetic field to a semiconductor causes some changes in its optical properties. Under the action of a magnetic field, the atoms, or ions within the intrinsic magnetic moment material, produce magnetic induction phenomenon, resulting in the orderly arrangement of the magnetic moment, which affects the transmission of light in its internal features, called the magneto-optical effect. This effect can break the time reversal invariance of the quantum mechanics of materials, and the Kirchhoff’s law is no longer valid [18,19,20]. Wu et al. [21] discovered the near-field radiative characteristics of magneto dielectric uniaxial anisotropic media (MDUAM) that have both magnetic and electric anisotropy. However, the above works all focused on near-field radiative heat transfer (NFRHT) for homogeneous isotropic media. Then, Song et al. [22] studied the NFRHT between graphene and anisoropic magneto-dielectric hyperbolic metamaterials (AMDHMs) according to the fluctuational dissipation theorem. They found that extraordinary propagating modes enable the total NFRHT flux to exceed that between graphene and SiC nanowires by several fold. Although the above research considered the impact of magnetic permeability, the effect of the magnetic field was lacking the MDUAM and AMDHMs were all ideal models, and the magnetic field strength was not quantified.

SiC is a widely used semiconductor that can be manufactured as optical control elements, such as wavelength-selective photodetectors, photovoltaic cells, and absorbers. Few studies of NFRHT for SiC have been conducted, and most studies investigated the SPhP coupling of SiC . Basu et al. [23] theoretically investigated the improvement of near-field radiative transfer between SiC plates due to strong SPhPs coupling. Yue et al. [24] studied the effect of magnetic polaritons (MPs) inside the SiC grating microstructures on near-field radiative transfer. The presented MPs greatly affected the spectral heat flux beside the SPhPs. However, the above works all focused on near-field radiative transfer for reciprocal materials, lacking studies on effect of non-reciprocal materials as magneto-optical materials. Hence, we investigated the non-reciprocal effect between magneto-optical materials under near-field in order to explore new phenomena.

In a previous study, the optical characteristics of the magneto-optical materials, such as SiC, InSb, and HgCdTe, was examined as an external magnetic field was applied, and a detailed description of the dielectric constant was verified and provided [25,26]. In this work, we theoretically calculate the near field radiative transfer based on SiC nanowires with different filling ratios and gap distances under an external magnetic field by using fluctuation-dissipation theorem and local effective medium theory (EMT), in which Lorentz-Drude equations of dielectric constant tensor are proposed to describe the dielectric constant as a magnetic field is applied toon the SiC. The influence of magnetic field intensity on the near field thermal radiation for SiC nanowires and plates is quantified discussed in detail. We also analyzed the influence of magnetic field intensity on the near field thermal radiation of SiC nanowires/plates.

2. Theoretical Formulation

2.1. Geometry and Radiative Properties of SiC Nanowires

The emitter and receiver temperatures were respectively set as T₁ = 300 K and T₂ = 299 K in this work. From Figure 1, we can see that the vacuum gap distance is denoted as d. Note that the maximum f for aligned SiC nanowires is 0.3 due to using the approximation Maxwell Garnet method, which is suitable for a dilute system. The dielectric constant of SiC is described by the Lorentz-Drude model as: [27]

$${\epsilon }_{\left(\omega \right)}={\epsilon }_{\infty }\left(1+\frac{{\omega }_p^2}{{\omega }_T^2-{\omega }^2-i\gamma \omega }\right)$$

(1)

where ε_∞ is the high-frequency limit permittivity (ε_∞ = 6.38), ω_T is the angular frequency of the incident wave (ω_T = 1.50 × 10¹⁴ rad/s), γ is the collision frequency of free electrons (γ = 1/τ, τ is relaxation time) and γ = 4 × 10¹¹ rad/s, and ${\omega }_p=\sqrt{N{e}^2/{\epsilon }_0{m}^{\ast }}$ is the plasma frequency in rad/s.

In this paper, only the magnetization vector M parallel to the Z axis is considered. When the magnetic field H is perpendicular to the plane of incidence (i.e., p-polarized or Transverse Magnetic (TM) waves), medium 2 cannot be simply treated as an isotropic medium because the electric field and the optical axis form an acute angle. Hence, for anisotropic materials, the permittivity tensor under a magnetic field can be expressed as:

$$\overline{\epsilon }=\left(\begin{array}{ccc}{\epsilon }_{xx}& {\epsilon }_{xy}& 0\\ -{\epsilon }_{xy}& {\epsilon }_{yy}& 0\\ 0& 0& {\epsilon }_{zz}\end{array}\right)$$

(2)

To simplify the complexity of the computation, a unitary matrix transformation U is used to diagonalize the permittivity, as shown below, by treating the anisotropic dielectric tensor as being plane symmetric. The approximate dielectric constant can be expressed as:

$${\overline{\epsilon }}_D{=\mathrm{U}}^{\ast }\cdot \overline{\epsilon }\cdot \mathrm{U}=\left(\begin{array}{ccc}{\epsilon }_{xx}& 0& 0\\ 0& {\epsilon }_{yy}& 0\\ 0& 0& {\epsilon }_{zz}\end{array}\right)=\left(\begin{array}{ccc}{n}_{xx}^2& 0& 0\\ 0& n_{yy}^2& 0\\ 0& 0& n_{zz}^2\end{array}\right)$$

(3)

So, the dielectric constant tensor of SiC can be expressed as [24]:

$$\overline{\epsilon }=\left(\begin{array}{ccc}{\epsilon }_{\infty }\left(1-\frac{{\omega }_p^2\left(\omega +\gamma i\right)}{\omega \left[{\left(\omega +\gamma i\right)}^2-{\omega }_c^2\right]-{\omega }_T^2\left(\omega +\gamma i\right)}\right)& \frac{i{\omega }_p^2{\omega }_c}{\omega \left[{\left(\omega +\gamma i\right)}^2-{\omega }_c^2\right]-{\omega }_T^2\left(\omega +\gamma i\right)}& 0\\ -\frac{i{\omega }_p^2{\omega }_c}{\omega \left[{\left(\omega +\gamma i\right)}^2-{\omega }_c^2\right]-{\omega }_T^2\left(\omega +\gamma i\right)}& {\epsilon }_{\infty }\left(1-\frac{{\omega }_p^2\left(\omega +\gamma i\right)}{\omega \left[{\left(\omega +\gamma i\right)}^2-{\omega }_c^2\right]-{\omega }_T^2\left(\omega +\gamma i\right)}\right)& 0\\ 0& 0& {\epsilon }_{\infty }\left(1-\frac{{\omega }_p^2}{{\omega }_T^2-{\omega }^2-i\gamma \omega }\right)\end{array}\right)$$

(4)

The doping concentration can be expressed as $N=a{T}^{3/2}\exp \left(-E_g/2kT\right)$, a is a coefficient related material, the Boltzmann constant k = 8.625 × 10⁻⁵ eV/k, E_g = 2.2 eV is the energy gap in eV, T is the temperature which the structure operates, e = 1.6 × 10¹⁹ C is the elementary charge, and ε₀ = 8.85 × 10⁻¹² is the free-space permittivity in c²/m²N. The cyclotron frequency follows ω_c = eB/m*, where B refers to the applied field intensity. So the doping concentration was calculated as N = 3.7 × 10²³ m⁻³.

In Figure 2, the dielectric constants of SiC are plotted as a real part and imaginary part, respectively for B = 0 T and B = 3 T. From Figure 2, the results of the Lorentz-Drude model for both real and imaginary parts agree well with data reported by Palik [27]. The real part increases as the angular frequency increases, which shows the usual dispersion characteristics, and decreases as the angular frequency increases, showing a unusual dispersion characteristics. The imaginary part showed a similar trend. Hence, the Lorentz-Drude model is suitable to calculate the dielectric constant of SiC. As the magnetic field was applied, the real and imaginary part had obvious oscillation as shown in Figure 2b, as the magnetic field affected the optical constant of SiC.

In general, the application of a magnetic field somewhat affects the optical constant of SiC, especially in the x and y directions.

2.2. Effective Medium Theory

Local effective medium theory (EMT) was used to obtain the anisotropic dielectric function and was combined with fluctuation electrodynamics to calculate the near-field radiative heat transfer coefficient due to its simplicity and low computational demand. In the Maxwell-Garnett approximation, the effective properties of a composite medium are obtained by treating one constituent of the composite as the host and all other constituents as embedded fillers that are not in contact with one another [28].

For SiC nanowire arrays, we set the optical axis of the structure along the Z axis, and obtained the effective dielectric functions:

$${\epsilon }_{x,y}=\frac{{\overline{\epsilon }}_{x,y}\left(1+f\right)+\left(1-f\right)}{{\overline{\epsilon }}_{x,y}\left(1-f\right)+\left(1+f\right)}$$

(5)

$${\epsilon }_z={\overline{\epsilon }}_{z}f+1-f$$

(6)

EMT is a homogenization method for characterizing the optical properties of an inhomogeneous medium with different material constituents based on the field average method. In Equations (5) and (6), f means the volume filling ratio, which has been previously explained in detail [27].

Form Figure 3, the hyperbolic dispersions of SiC nanowires under magnetic field B = 3 T were supported in two narrow bands between 1.50 × 10¹⁴ rad/s and 1.69 × 10¹⁴ rad/s (blue shaded region, Type I), and between 1.79 × 10¹⁴ rad/s and 1.83 × 10¹⁴ rad/s (yellow shaded region, Type II). The heat transfer coefficient at temperature T between two semi-infinite plates can be expressed as:

$$H=\frac{1}{8{\pi }^3}{\displaystyle {\int }_0^{\infty }\mathsf{\Theta }\left(\omega ,T\right)}d\omega {\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{\infty }\xi \left(\omega ,\beta ,\varphi \right)\beta d}}\beta d\varphi$$

(7)

where $\mathsf{\Theta }\left(\omega ,T\right)=\frac{\partial }{\partial T}\left(\hslash \omega /(\exp \left(\hslash \omega /k_{B}T\right)-1)\right)$ is the average energy of a Planck’s oscillator, $\hslash$ is the reduced Planck constant, $\beta$ is the transverse wavevector (the magnitude of the wavevector component in the $x-y$ plane), $\varphi$ is the Azimuth of a wavevector, and $\xi \left(\omega ,\beta ,\varphi \right)$ is the energy transmission coefficient, which can be defined as:

$$\xi \left(\omega ,\beta ,\varphi \right)=\{\begin{array}{c}\mathrm{Tr}\left[\left({\mathrm{I}-\mathrm{R}}_2^{\ast }{\mathrm{R}}_2\right)\mathrm{D}\left({\mathrm{I}-\mathrm{R}}_1{\mathrm{R}}_1^{\ast }\right){\mathrm{D}}^{\ast }\right],\\ \mathrm{Tr}\left[\left({\mathrm{R}}_2^{\ast }{-\mathrm{R}}_2\right)\mathrm{D}\left({\mathrm{R}}_1{-\mathrm{R}}_1^{\ast }\right){\mathrm{D}}^{\ast }\right]e^{-2\left|{\mathrm{k}}_{y0}\right|d},\end{array}\begin{array}{c}\beta <k_0\\ \beta >k_0\end{array}$$

(8)

Here, the matrix D is expressed as $\mathrm{D}={\left(\mathrm{I}-{\mathrm{R}}_1{\mathrm{R}}_2{e}^{2i{k}_{y0}d}\right)}^{-1}$, I is a 2 × 2 unit matrix, R is the matrix related to Fresnel’s reflection coefficients and

$${\mathrm{R}}_{1,2}=\left[\begin{array}{cc}{r}_{ss}^{\left(1,2\right)}& r_{sp}^{\left(1,2\right)}\\ r_{ps}^{\left(1,2\right)}& r_{pp}^{\left(1,2\right)}\end{array}\right]$$

(9)

where j is for s or p polarization, and γ is Fresnel coefficient, which can be expressed as follows:

$$r_j=\{\begin{array}{c}\frac{k_{y0}-\sqrt{k_0^2{\epsilon }_x-{\beta }^2}}{k_{y0}+\sqrt{k_0^2{\epsilon }_x-{\beta }^2}},j=s\\ \frac{{\epsilon }_x{k}_{y0}-\sqrt{k_0^2{\epsilon }_x-{\beta }^2{\epsilon }_x/{\epsilon }_y}}{{\epsilon }_x{k}_{y0}+\sqrt{k_0^2{\epsilon }_x-{\beta }^2{\epsilon }_x/{\epsilon }_y}},j=p\end{array}$$

(10)

In order to check the reliability of the above model for near-thermal radiation, the heat transfer coefficient was calculated and compared to that obtained by Liu et al. [29] for doped Si nanowires and nanoholes, which is shown in Figure 4. The doping concentration was set to N = 10²⁰ cm⁻³.

From Figure 4, the results for the two nanostructures agree well. Hence, we assumed that the above model in this work is reliable. Note that no magnetic field was applied in this case. The calculated heat transfer coefficient between doped Si nanowires with f = 0.05 was 6.25 W/m²K and that between doped Si nanoholes with f = 0.3 was 3.83 W/m²K, which are basically the same values reported in the literature.

3. Results and Discussion

3.1. Spectral Heat Fluxes and Transmission Coefficient

From Figure 5, for the plate case, two peaks were observed at frequencies around 1.50 × 10¹⁴ rad/s and 1.81 × 10¹⁴ rad/s under B = 3 T. Compared to the plate case, a significantly enhanced spectral heat flux between 1.50 × 10¹⁴ rad/s and 1.70 × 10¹⁴ rad/s was observed, which is known as hyperbolic dispersion (Type I, blue region). Between 1.70 × 10¹⁴ rad/s and 1.90 × 10¹⁴ rad/s, the peak of spectral heat flux blueshifted toward increasing angular frequency, because the optical constants change due to different material structures. The linear shape of the spectral heat flux of the nanowires was obviously higher than that of the plates, producing violent shocks under the external magnetic field (Type II, yellow region).

Figure 5 shows that compared to hyperbolic dispersion type II, SiC nanowires exhibit large heat transfer flux in the hyperbolic dispersion type I region. Compared to the SiC plates, the spectral heat flux of the nanowires structure is more intense, especially in the hyperbolic dispersion regions.

Figure 6 displays the contour plots of the transmission coefficient for both SiC nanowires and SiC plates under the magnetic field B = 3 T at d = 10 nm. Compared to SiC plates, the transmission coefficient of SiC nanowires is higher at very large β values and in the hyperbolic band. This can allow photons on the surface of SiC nanowires to tunnel through the vacuum gap, which is why the heat fluxes of SiC nanowires is obviously higher than that of the plates in the hyperbolic band. Evanescent waves are the main surface mode of structure in this case. For SiC nanowires, because of their dielectric anisotropy, they support the hyperbolic band for p polarization. Therefore, in the hyperbolic regions, the nanowires can have a larger transmission coefficient than that of SiC plates.

3.2. Effect of Structure Parameters

3.2.1. Filling Ratio

The filling ratio $f$ is defined based on the volume fraction of SiC. The heat transfer coefficients for SiC nanowires under different magnetic field intensities at d = 10 nm were simulated as plotted in Figure 7. Note that the maximum filling ratio for the nanowires was 0.3.

From Figure 7, we found that the heat transfer coefficient of the nanowire structure obviously increased with increasing magnetic field. This is because the magnetic field leads to a significant increase in the photon tunneling phenomenon of the material structure. The SiC plates are not shown here because they remained constant and slightly changed as the external magnetic field strength increased.

3.2.2. Gap Distance

Figure 8 shows the heat transfer coefficient between SiC nanowires at several different vacuum gap distances. For SiC plates, the heat transfer coefficient of the plate under different magnetic field intensities did not change significantly at all gap distances. Hence, it is not shown here. This occurred because the dielectric constant of the SiC plates under the action of an external magnetic field had no obvious change, as shown in Figure 3.

For f = 0.2, SiC nanowires can yield a super-Planckian behavior when the gap distance is below 3 μm, whereas the heat transfer coefficient decreases with increasing gap distance. The heat transfer coefficient of the structure under a magnetic field was significantly higher than that of non-magnetic field when d was below 100 nm. The enhanced amplitude increased with increasing magnetic field intensity and decreasing gap distance. Evanescent waves were the main surface mode of the structure in this case. Therefore, the external magnetic field can significantly enhance evanescent waves in near field thermal radiation.

4. Conclusions

In this paper, we determined the radiative transfer properties of SiC nanowires under different external magnetic field intensities, such as spectral heat fluxes, transmission coefficient, and heat transfer coefficient, and theoretically investigated the effect of an external magnetic field on the SiC nanowires on near field thermal radiative transfer. Compared to SiC plates, the transmission coefficient of SiC nanowires was higher at very large lateral wavevector values and in the hyperbolic band. In general, the external magnetic field tunes the near field radiation properties of the subwavelength magneto-optical materials. Under the appropriate parameters, external magnetic fields obviously enhance the evanescent waves in near field thermal radiation. In summary, the external magnetic field’s effect on SiC nanowires in near field heat radiation has important applications like thermophotovoltaic energy conversion and radiation-based thermal management.