# Magnetic Field Effect of Near-Field Radiative Heat Transfer for SiC Nanowires/Plates
## 2 Theoretical Formulation
### 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:  
\[\varepsilon_{x,y}=\frac{\overline{\varepsilon}_{x,y}(1+f)+(1-f)}{\overline{ \varepsilon}_{x,y}(1-f)+(1+f)} \tag{5}\]  
\[\varepsilon_{z}=\overline{\varepsilon}_{z}f+1-f \tag{6}\]  
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 \(\times\) 10\({}^{14}\) rad/s and 1.69 \(\times\) 10\({}^{14}\) rad/s (blue shaded region, Type I), and between 1.79 \(\times\) 10\({}^{14}\) rad/s and 1.83 \(\times\) 10\({}^{14}\) rad/s (yellow shaded region, Type II). The heat transfer coefficient at temperature \(T\) between two semi-infinite plates can be expressed as:  
\[H=\frac{1}{8\pi^{3}}\int_{0}^{\infty}\Theta(\omega,T)d\omega\int_{0}^{2\pi} \int_{0}^{\infty}\xi(\omega,\beta,\phi)\beta d\beta d\phi \tag{7}\]  
where \(\Theta(\omega,T)=\frac{\partial}{\partial T}(\hbar\omega/(\exp(\hbar\omega/k_ {B}T)-1))\) is the average energy of a [PERSON]'s oscillator, \(\hbar\) is the reduced Planck constant, \(\beta\) is the transverse wavevector (the magnitude of the wavevector component  
Figure 2: Equivalent refractive index \(n\) of SiC under magnetic field (**a**) \(B=0\) T, (**b**) \(B=3\) T.  
in the \(x-y\) plane), \(\phi\) is the Azimuth of a wavevector, and \(\xi(\omega,\beta,\phi)\) is the energy transmission coefficient, which can be defined as:  
\[\xi(\omega,\beta,\phi)=\left\{\begin{array}{cc}\mathrm{Tr}[(\mathrm{I}- \mathrm{R}_{2}^{*}\mathrm{R}_{2})\mathrm{D}(\mathrm{I}-\mathrm{R}_{1}\mathrm{R }_{1}^{*})\mathrm{D}^{*}],&\beta<k_{0}\\ \mathrm{Tr}[(\mathrm{R}_{2}^{*}-\mathrm{R}_{2})\mathrm{D}(\mathrm{R}_{1}- \mathrm{R}_{1}^{*})\mathrm{D}^{*}]e^{-2|\mathrm{k}_{\phi 0}|d},&\beta>k_{0}\end{array}\right. \tag{8}\]  
Here, the matrix \(\mathrm{D}\) is expressed as \(\mathrm{D}=\left(\mathrm{I}-\mathrm{R}_{1}\mathrm{R}_{2}e^{2 ik_{\phi 0}d} \right)^{-1}\), \(\mathrm{I}\) is a \(2\times 2\) unit matrix, \(\mathrm{R}\) is the matrix related to [PERSON]'s reflection coefficients and  
\[\mathrm{R}_{1,2}=\left[\begin{array}{cc}r_{ss}^{(1,2)}&r_{sp}^{(1,2)}\\ r_{ps}^{(1,2)}&r_{pp}^{(1,2)}\end{array}\right] \tag{9}\]  
where \(j\) is for \(s\) or \(p\) polarization, and \(\gamma\) is Fresnel coefficient, which can be expressed as follows:  
\[r_{j}=\left\{\begin{array}{cc}\frac{k_{\phi 0}-\sqrt{k_{0}^{2}\varepsilon_{x}- \beta^{2}}}{k_{\phi 0}+\sqrt{k_{0