# Implementation of Atomically Thick Graphene and Its Derivatives in Electromagnetic Absorbers
## 2 Graphene-Based Quarter-Wavelength Absorbers
### Graphene and Its Derivatives-Based Dallenbach Absorbers  
Unlike the Salisbury and Jaumann screens that require resistive layer and low loss/lossless dielectric spacer over a metallic ground, the Dallenbach absorber utilizes a lossy homogeneous material with complex relative permittivity \(\epsilon_{r}=\epsilon^{\prime}-j\epsilon^{\prime\prime}\) and permeability \(\mu_{r}=\mu^{\prime}-j\mu^{\prime\prime}\) above the metallic layer [34]. The EM wave absorption of Dallenbach layer can also be illustrated with the transmission line theory. The corresponding input impedance for normal incidence is derived as  
Figure 2: (**a**) schematic of a graphene capacitor with ionic liquid (or gel) and two graphene electrodes; (**b**) image of a graphene capacitor; (**c**) measured reflection spectra of the graphene capacitor based tunable Salisbury screen under various bias voltages, reprinted with permission from [39]; (**d**) schematic of ionic liquid based graphene Salisbury screen; (**e**) image of the flexible graphene capacitor based tunable Salisbury screen coated on the surface of a cylindrical object; (**f**) measured reflection spectra of the flexible Salisbury screen under various bias voltages, reprinted with permission from [40]; (**g**) schematic of the graphene Salisbury screen with ionic gel based top gate; (**h**) measured (dotted line) and simulated (solid line) normalized differential reflectance \(\Delta R/R\) vs. frequency under various bias voltages, where \(R\) is the reflectance of the device when \(E_{F}=0\) and \(\Delta R\) is the reflectance variation from \(R\) when \(E_{F}\
eq 0\), reprinted with permission from [41]. \[Z_{in}=Z_{0}\sqrt{\frac{\mu_{r}}{\epsilon_{r}}}\tanh\bigg{[}j\left(\frac{2\pi fd}{c} \sqrt{\mu_{r}\epsilon_{r}}\right)\bigg{]}. \tag{10}\]  
Hence, the absorption and return loss can also be calculated with Equations (8) and (9). In addition to the wave cancellation at the interface when \(d=M\cdot\lambda/4\), the Dallenbach absorber also exhibits magnetic and/or electrical conduction loss that enhance the EM wave absorption performance such as peak absorption rate and effective absorption bandwidth. The corresponding magnetic and electrical loss can be determined by the magnetic (\(\delta_{m}=\mu^{\prime\prime}/\mu^{\prime}\)) and electrical (\(\epsilon_{m}=\epsilon^{\prime\prime}/\epsilon^{\prime}\)) loss tangent. It is noted that both \(\epsilon_{r}\) and \(\mu_{r}\) play important roles in the EM wave absorption. In order to characterize the matching between \(\epsilon_{r}\) and \(\mu_{r}\), a factor \(|\Delta|\) is proposed [43]  
\[|\Delta|=|\sinh^{2}\left(Kfd\right)-M|, \tag{11}\]  
where  
\[K=\frac{4\pi\sqrt{\mu_{r}^{\prime}\epsilon_{r}^{\prime}}\sin\bigg{(}\frac{ \delta_{e}+\delta_{m}}{2}\bigg{)}}{c\cos\left(\delta_{e}\right)\cos\left( \delta_{m}\right)}, \tag{12}\]  
and  
\[M=\frac{4\mu_{r}^{\prime}\epsilon_{r}^{\prime}\cos\left(\delta_{e}\right) \cos\left(\delta_{m}\right)}{\left(\mu_{r}^{\prime}\cos\left(\delta_{e} \right)-\epsilon_{r}^{\prime}\cos\left(\delta_{m}\right)\right)^{2}+\left[ \tan\left(\frac{\delta_{m}-\delta_{e}}{2}\right)\right]^{2}\left(\mu_{r}^{ \prime}\cos\left(\delta_{e}\right)+\epsilon_{r}^{\prime}\cos\left(\delta_{m} \right)\right)^{2}}. \tag{13}\]  
For matched impedance, \(|\Delta|\) approaches zero and excellent absorption is achieved. When \(|\Delta|\) is away from zero, poor impedance matching is observed and the absorption performance is degraded. To achieve a Dallenbach absorber, graphene and its derivatives such as rGO are often mixed with non-magnetic (i.e., \(\mu^{\prime}\approx 0\) and \(\mu^{\prime\prime}\approx 1\)) [44; 45; 46] or magnetic [47] materials to form lossy composites. In 2011, [PERSON] et al. investigated the microwave absorbing properties of rGO/poly-(ethylene oxide) (PEO) composites based Dallenbach layer [44]. Due to the high electrical conduction loss and other losses result from dielectric relaxation, interface scattering, etc.