# Magnetic Field Effect of Near-Field Radiative Heat Transfer for SiC Nanowires/Plates
## 2 Theoretical Formulation  
### Geometry and Radiative Properties of SiC Nanowires  
The emitter and receiver temperatures were respectively set as \(T_{1}\) = 300 K and \(T_{2}\) = 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]  
\[\varepsilon_{(\omega)}=\varepsilon_{\infty}\Bigg{(}1+\frac{\omega_{p}^{2}}{ \omega_{T}^{2}-\omega^{2}-i\gamma\omega}\Bigg{)} \tag{1}\]  
where \(\varepsilon_{\infty}\) is the high-frequency limit permittivity (\(\varepsilon_{\infty}\) = 6.38), \(\omega_{T}\) is the angular frequency of the incident wave (\(\omega_{T}\) = 1.50 \(\times\) 10\({}^{14}\) rad/s), \(\gamma\) is the collision frequency of free electrons (\(\gamma\) = 1/\(\tau\), \(\tau\) is relaxation time) and \(\gamma\) = 4 \(\times\) 10\({}^{11}\) rad/s, and \(\omega_{p}=\sqrt{Ne^{2}/\varepsilon_{0}m^{*}}\) 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:  
\[\overline{\varepsilon}=\left( \begin{array}{ccc}\varepsilon_{xx}&\varepsilon_{ xy}&0\\ -\varepsilon_{xy}&\varepsilon_{yy}&0\\ 0&0&\varepsilon_{zz}\end{array} \right) \tag{2}\]  
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:  
\[\overline{\varepsilon}_{D}\mathrm{=U^{*}\cdot\overline{\varepsilon}\cdot U= \left( \begin{array}{ccc}\varepsilon_{xx}&0&0\\ 0&\varepsilon_{yy}&0\\ 0&0&\varepsilon_{zz}\end{array} \right)=\left( \begin{array}{ccc}n_{xx}^{2}&0 &0\\ 0&n_{yy}^{2}&0\\ 0&0&n_{zz}^{2}\end{array} \right) \tag{3}\]  
So, the dielectric constant tensor of SiC can be expressed as [24]:  
\[\overline{\varepsilon}=\left( \begin{array}{ccc}\varepsilon_{\infty}\left( 1-\frac{\omega_{z}^{2}(\omega+\gamma\varepsilon)}{\omega[(\omega+\gamma \varepsilon)^{-}\omega_{z}^{2}]-\omega_{z}^{2}(\omega+\gamma\varepsilon)} \right)&\frac{\omega_{z}^{2}\omega_{z}}{\omega[(\omega+\gamma\varepsilon)^{- }\omega_{z}^{2}]-\omega_{z}^{2}(\omega+\gamma\varepsilon)}&0\\ -\frac{\omega_{z}\omega_{z}}{\omega[(\omega+\gamma\varepsilon)^{-}\omega_{z}^ {2}]-\omega_{z}^{2}(\omega+\gamma\varepsilon)}&\varepsilon_{\infty}\left(1- \frac{\omega_{z}^{2}(\omega+\gamma\varepsilon)}{\omega[(\omega+\gamma \varepsilon)^{-}-\omega_{z}^{2}]-\omega_{z}^{2}(\omega+\gamma\varepsilon)} \right)&0\\ 0&0&\varepsilon_{\infty}\left(1-\frac{\omega_{z}^{2}}{\omega_{z}^{2}-\omega ^{2}-\gamma\varepsilon\omega}\right)\end{array} \right) \tag{4}\]  
The doping concentration can be expressed as \(N=aT^{3/2}\exp\left(-E_{g}/2 kT\right)\), \(a\) is a coefficient related material, the Boltzmann constant \(k=8.625\times 10^{-5}\) eV/\(k\), \(E_{g}=2.2\) eV is the energy gap in eV, \(T\) is the temperature which the structure operates, \(e=1.6\times 10^{19}\) C is the elementary charge, and \(\varepsilon_{0}=8.85\times 10^{-12}\) is the free-space permittivity in c\({}^{2}\)/m\({}^{2}\)N. The cyclotron frequency follows \(\omega_{c}=eB/m^{*}\), where \(B\) refers to the applied field intensity. So the doping concentration was calculated as \(N=3.7\times 10^{23}\) m\({}^{-3}\).