# Ultrathin and Electrically Tunable Metamaterial with Nearly Perfect Absorption in Mid-Infrared
## 3 Results and Discussion  
According to coupled-mode theory (CMT) [28; 29; 31; 32], the optical properties of the MIM structures are determined essentially by two competing loss channels, namely, the radiation loss of the resonance cavity and intrinsic loss due to material absorption. The tunable property of the MIM structure is therefore realizable by changing either of these two channels. From CMT theory, the reflection coefficient (r) of our system can be written as [29],  
\[\tau(\omega)=-1+\frac{2/\tau_{\mathrm{r}}}{-\mathrm{i}(\omega-\omega_{0})+1/ \tau_{\mathrm{r}}+1/\tau_{\mathrm{a}}}, \tag{1}\]  
where and \(1/\tau_{\mathrm{a}}\) are the decay rates of radiation and absorption, respectively. Furthermore, the radiative and absorptive quality factors (Q-value) of the system are defined as \(\mathrm{Q_{r}}=\omega_{0}\tau_{\mathrm{r}}/2\) and \(\mathrm{Q_{a}}=\omega_{0}\tau_{\mathrm{a}}/2\), which determine the physical properties of the system. From Equation (1), \(\mathrm{r}(\omega_{0})=\frac{\mathrm{Q_{a}}/\mathrm{Q_{r}}-1}{\mathrm{Q_{a}} /\mathrm{Q_{r}}+1}\) when frequency is at its resonance (\(\omega=\omega_{0}\)), which shows that \(\mathrm{Q_{a}}/\mathrm{Q_{r}}\) is the physical quantity to decide the minimum reflection or maximum absorption. Based on Equation (1) and the definitions of \(\mathrm{Q_{a}}\) and \(\mathrm{Q_{r}}\) we mentioned above, \(\mathrm{Q_{a}}\) and \(\mathrm{Q_{r}}\) of the system with specific structural parameters can be obtained through numerical fitting of our simulation or experimental results [29]. Figure 2a depicts the phase  
Figure 1: (**a**) The schematic of the graphene metamaterial perfect absorber and the unit cell of the metamaterial arrays. The tunable reflectance can be achieved by back-gating of graphene between the Si\({}_{3}\)N\({}_{4}\) membrane and the metasurface. (**b**) The SEM image of the metamaterial structure topping the absorber, with L = 1.25 \(\upmu\)m and W = 0.3 \(\upmu\)m. (**c**) Reflectance spectra for the absorbers in (**a**) with different metamaterial structures. (**d**) Simulated results for the same structures in (**c**). diagram of Q\({}_{\text{a}}\)/Q\({}_{\text{r}}\) as a function of the geometrical parameter W and the material loss tan\(\delta\). tan\(\delta\) defines the dielectric loss angular tangent of the system, which can indicate the inherent dissipation (such as heat) of electromagnetic energy in the dielectric layer [33]. When Q\({}_{\text{r}}>\) Q\({}_{\text{a}}\), the resonator is overdamped and the variation of phase only occupies a small range, less than 180\({}^{\circ}\). The solid line shows the perfect absorption condition with Q\({}_{\text{r}}\) = Q\({}_{\text{a}}\). Interestingly, this line shows a simple and approximately linear dependence in the region with small W (\(\leq\)0.6 \(\upmu\)m) and tan\(\delta\) (\(\leq\)0.08) This implies that the loss channel of the resonant mode may depend on specific parameters, W or tan\(\delta\). In Figure 2b,c, Q\({}_{\text{r}}\) and Q\({}_{\text{a}}\) are displayed as functions of W or tan\(\delta\) when the other one is fixed. It is obvious that Q\({}_{\text{r}}\) is affected mainly by W (\(\leq\)0.6 \(\upmu\)m), while Q\({}_{\text{a}}\) is affected by tan\(\delta\). This appears to be physically reasonable, since the non-radiative loss leads to material absorption and therefore depends on the imaginary part of its dielectric permittivity, while the radiation loss relies on the dipole oscillation strength of the resonant mode and hence increases with W. When W \(>\) 0.6 \(\upmu\)m, the field distribution becomes dramatically different from the case with small W, which prevents the radiative loss rate from further increase. The above scenario provides useful hints for the realization of tunable infrared devices with nearly perfect absorption. Experimentally, we examined the tuning performance through back-gating in our compact MIM structure with fixed geometrical parameters (L = 1.25 \(\upmu\)m and W = 0.3 \(\upmu\)m). The Raman spectra of the graphene we used are shown in Figure 3a.