## 3 Material Characteristic and Numerical Analysis  
To accurately model the designed absorber, the characteristic of the carbon-loaded ABS materials is discussed first. There is previous literature about the carbon-loaded materials for microwave application [64], although the dielectric characteristic of the ABS materials used here is not available. In [62], the same material is used to design a low-cost terminal, and the dielectric constant of the materials at X-band is measured, which is not enough to model the material. To give a comprehensive description of the material, the permittivity measurement system, 85070D Dielectric Probe Kit, provided by Keysight Technologies (California, USA), is used to obtain the data of permittivity. A cube sample with dimensions of a 20 \(\times\) 20 \(\times\) 20 mm\({}^{3}\) is printed using the printer, MakerBot Replicator 2X (Brooklyn, NY, USA), based on Fused Deposition Modeling (FDM) technology. The printer has a nozzle with a diameter of 0.4 mm and the resolution can reach 0.05 mm along the \(Z\) axis and 0.1 mm in the XY plane. The printed cube sample and the setup of the permittivity measurement system is shown in Figure 2b. In the measurement system, an open-ended coaxial cable is used as the probe, and the probe is connected to a vector network analyzer (VNA). Before the measurement, a calibration process is carried out to avoid the effect of external noise. The air, a shorted circuit termination and the pure water at room temperature are used as the reference. When the sample is measured, the S-parameter of the probe is measured and then the dielectric permittivity of the material is calculated using measured S-parameter data. The measured real part and the imaginary part of the permittivity are shown in Figure 2c. Limited by the operating frequency of the VNA, we only measured the permittivity from 3 to 6 GHz, and the data corresponding to the rest of the band of interest is predicted using the fitting algorithm, based on general dispersion models integrated into CST Microwave Studio. The predicted data of the permittivity from 3 to 15 GHz are shown in Figure 2c. Additionally, the measured data of the permittivity reported are plotted in the figure. Observing the figure, one can see that there is a good agreement between the data measured in [62] and the predicted data obtained by the fitting algorithm, which can further verify the validity of predicted data. It is also worth noting that, at a low frequency band, i.e., 3-9 GHz, a decreasing trend can apparently be found for the real part of the permittivity. As we all know, for the all-dielectric resonator, the resonating bandwidth suffers narrowband, as the electrical size of the resonator varies with frequency. However, if the permittivity decreases with frequency, this issue may be eliminated, as the electrical length of the structure varies little over a wide bandwidth. This characteristic provides the possibility to design the resonator with a broad operating bandwidth. This will be discussed in detail later. Concerning the data, we also can see that the imaginary part of the permittivity of the materials is maintained at a high level over the whole band, indicating that the dielectric loss of the material is quite high. After the characteristics of the material are modeled, the numerical analysis, utilizing full wave simulation, is carried out to investigate the working principle of the designed absorber. In the simulation, the element is arranged in the _xoy_ plane, and the incident wave is assumed to propagate from the \(+z\) axis. The unit cell boundary is assigned both in the \(x\) and \(y\) axes. For a transmission structure, the absorptivity of the structure can be defined as \(A(\omega)=1-R(\omega)-T(\omega)\), where \(R(\omega)\) and \(T(\omega)\) are reflectivity and transmissivity, respectively, which can be calculated using the S-parameters. As the absorber designed here has a metal backplate, the absorption can be calculated with the simplified expression \(A(\omega)=1-R(\omega)\).