# Numerical Study of an Ultra-Broadband All-Silicon Terahertz Absorber
## 2 Structure and Design  
Figure 1 shows the proposed ultra-broadband THz metamaterial absorber with a magnified cubic unit cell. We can observe that the absorber consists of all-silicon diamond metamaterial layers placed on the base of the cubic substrate. The height of the metamaterial layer and the structural base are denoted as \(t\) and \(h\), respectively, while the period of the absorber is denoted as \(p\) and the length of the diamond is \(a\). These parameters are optimized as follows: \(p=170\)\(\upmu\)m, \(h=250\)\(\upmu\)m, \(t=60\)\(\upmu\)m, \(a=75\sqrt{2}\)\(\upmu\)m. The FDTD Solutions software produced by the Canadian Lumerical Solutions company (Vancouver, BC, Canada) was used to obtain the reflectance (R) and transmittance (T) of the structure at different geometrical parameters. This software adopts the finite difference time domain (FDTD) method. This algorithm uses Maxwell equations to analyze the interaction between electromagnetic waves and structures with subwavelength dimensions. Subsequently, the absorption (A) was calculated using the equation \(A=1-R-T\). We can design and explain the working principle for the investigated structures. The silicon used in this study is modeled by the Drude response model [1]:  
\[\varepsilon(w)=\varepsilon_{\infty}-\frac{{w_{p}}^{2}}{w^{2}+jw\gamma} \tag{1}\]  
in which \(\varepsilon_{\infty}=11.68\) is the intrinsic permittivity of Si, and \(f\) is the electromagnetic frequency. The plasma frequency \(W_{P}\) and the damping rate \(\gamma\) are related with the free carrier concentration \(N\) and the free carrier mobility \(\mu\), respectively. The equations follow  
\[{w_{p}}^{2}=\frac{Ne^{2}}{\varepsilon_{0}m^{*}} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **References** & **Structure** & **Waveband (THz)** & **Peak Absorption** & **Above 90\%** \\ \hline \multirow{4}{*}{**Dielectric-metal**} & [1] & Cu-SiO\({}_{2}\)-Si & 0.2–2 & 98\% & 0.08 THz \\  & [4] & Continuous & & & \\  & graphene-SiO\({}_{2}\)-Au & 0–2 & 98\% & 1 THz \\  & [29] & VO\({}_{2}\)-SiO\({}_{2}\)-Au & 0.2–2 & Nearly 100\% & 0.7 THz \\  & [30] & Cross-shaped & & & \\  & graphene-SiO\({}_{2}\)-Au & 1–8 & Nearly 100\% & 1.13 THz \\ \hline \multirow{4}{*}{**All-dielectric**} & [31] & Graphene & 0–5 & Nearly 100\% & 0.4 THz \\  & [27] & Al-silicon circular & 0–2 & Nearly 100\% & 0.36 THz \\  & [27] & groove & & & \\  & [32] & All-silicon & 0.5–2 & 99\% & 0.4 THz \\  & [33] & V-groove & & & \\  & [33] & All-silicon & & 0.2–1.5 & 93.80\% & 0.05 THz \\  & & hypersurface & & & \\  & This study & Diamond & 0.2–2 & Nearly 100\% & 1.3 THz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison with other absorbers. Figure 1: Schematic of the proposed terahertz metamaterial absorber. \[\gamma = \frac{e}{\mu m^{*}} \tag{3}\] \[\delta = Ne\mu \tag{4}\]  
where \(e\) is the electron charge, \(m^{*}\) is the effective mass of free carriers and has a value of 0.37 \(m_{0}\), \(m_{0}\) is the mass of the electron, \(\delta\) is the conductivity, and \(\varepsilon_{0}\) is the permittivity of free space. The empirical parameter of \(\mu\) is chosen to have a value of 386 cm\({}^{2}\) V\({}^{-1}\) S\({}^{-1}\). Its carrier density was derived from the conductivity of silicon wafers available on the market, which is approximately equal to \(0.03\times 10^{18}\) cm\({}^{-3}\). Finally, we calculated the plasma frequency value of \(1.6042\times 10^{13}\) rad/s and the collision frequency value of \(1.9573\times 10^{12}\) l/s. To gain a better understanding of the absorption performance, we also analyzed the impedance matching mechanism. Herein, an effective medium theory (EMT) is proposed. The absorber proposed in this study was considered to be homogeneous.