# Study on the Absorbing Properties of V-Doped MoS2
## 3 Result and Discussion
### The Absorbing Properties of VMS and VMSA
After Maxwell-Wagner effect correction, the Cole-Cole diagram of the actual material is deviated from the standard semicircle, but an arc and/or a long straight line with a positive slope, where the arc represents the relaxation process, and the long straight line represents the conduction loss. As illustrated in Figure 3c-f, all four samples display distinct positive-slope linear segments, confirming the presence of conduction losses. Notably, while the VMS-40 sample exhibits multiple relaxation arcs suggesting complex polarization mechanisms, the annealed VMSA-40 shows only a single arc, demonstrating that thermal treatment significantly alters the dielectric relaxation behavior. Based on the transmission line theorem [24], the RL performance of the sample can be calculated using Equations (5)-(9). In these equations, \(Z_{in}\) represents the incident impedance of MAMs, \(Z_{0}\) denotes the impedance of free space, d is the thickness of the absorber, c is the speed of light, and f is the frequency. RL is commonly used to describe the intensity of EM wave absorption by materials and serves as a key metric for characterizing their absorbing properties [25; 26; 27; 28; 29; 30; 31]. A negative RL value indicates that lower reflection loss  
Figure 3: (**a**) The \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) with different filling ratios (**b**) The \(\tan(\delta\varepsilon)\) with different filling ratios (**c–f**) Cole–Cole curves with different filling ratios. corresponds to higher absorption loss, suggesting that the material has better attenuation of the EM wave. \[\varepsilon_{r}=\varepsilon^{\prime}-je^{\prime\prime} \tag{5}\]  
\[\mu_{r}=\mu^{\prime}-j\mu^{\prime\prime} \tag{6}\]  
\[Z_{in}=Z_{0}\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}}\tanh\left[j\frac{2\pi fd}{c} \sqrt{\mu_{r}\varepsilon_{r}}\right] \tag{7}\]  
\[RL=20\log\left|\frac{Z_{in}-Z_{0}}{Z_{in}+Z_{0}}\right| \tag{8}\]  
\[\alpha=\frac{\sqrt{2}\pi f}{c}\sqrt{(\mu^{\prime\prime}e^{\prime\prime}-\mu^{ \prime}e^{\prime})+\sqrt{(\mu^{\prime\prime}e^{\prime\prime}-\mu^{\prime}e^{ \prime})^{2}+(\mu^{\prime}e^{\prime\prime}+\mu^{\prime\prime}e^{\prime})^{2}}} \tag{9}\]  
Figure 4 presents the three-dimensional reflection loss (RL) color maps for the four samples at various frequencies and coating thicknesses. As shown in Figure 4a-d, the RL\({}_{min}\) of the VMS-30, VMS-40, VMS-50, and VMSA-40 samples, at thicknesses of 5.10 mm, 1.78 mm, 1.38 mm, and 1.45 mm, respectively, are \(-28.17\) dB, \(-32.24\) dB, \(-7.40\) dB, and \(-9.64\) dB. The VMS-30 and VMS-40 samples exhibit RL\({}_{min}\) values below \(-10\) dB, indicating that their absorption efficiency of EM waves exceeds 90%, with their absorption bands located in the 12-18 GHz range, which falls within the Ku band. Notably, the VMS-40 sample demonstrates the best wave absorption performance, achieving an RL\({}_{min}\) of \(-32.24\) dB, corresponding to an absorption efficiency of 99.9%. In contrast, RL\({}_{min}\) of VMS-50 and VMSA-40 samples is greater than \(-10\) dB, so these two samples are obviously difficult to meet the actual needs. Another important factor in evaluating the absorption performance is the maximum EAB, so we calculated the EAB for the first two samples. As shown in Figure 5a,b, the EAB of the VMS-30 and VMS-40 samples are 2.4 GHz and 4.4 GHz, with corresponding  
Figure 4: (**a**–**d**) Three -dimensional figure of reflection loss of VMS-30, VMS-40, VMS-50, and VMSA-40. coating thicknesses of 5.46 mm and 1.43 mm. To compare the comprehensive absorbing properties of the four samples more intuitively, we summarized their absorption performance parameters in Table 1. According to Table 1, the RL\({}_{min}\) of the VMS-30 sample is \(-28.17\) dB, achieved at a layer thickness of 5.10 mm and a frequency of 18.00 GHz. The VMS-40 sample exhibits the best maximum RL value of \(-32.24\) dB, at a layer thickness of 1.78 mm and a frequency of 12.20 GHz.