## 3 Results and Discussion
### EM Wave Absorption Properties
The values of \(\upmu^{\prime}\) and \(\upmu^{\prime\prime}\) are generally lesser in magnitude as compared with the dielectric loss and can be seen to decrease as frequency increases. This abrupt decrease in permeability values of ferromagnetic materials at microwave frequency have been reported in the literature [28; 29; 30]. This can be explained by the fact that the skin depth for the propagation of the EM field into the materials at these frequencies is generally less than the sizes of the fundamental ferromagnetic domains. Thus, the domain mechanism important for microwave magnetization will breakdown, since the applied field ceases to be effective throughout the volume of a domain [31]. Noticeably, negative values of \(\upmu^{\prime}\) are observed as frequency increases. This occurrence could be associated with the conversion of magnetic to electric energy [12]. The \(\upmu^{\prime}\) and \(\upmu^{\prime\prime}\) were observed to decay towards unity and zero, respectively, as the frequency approaches 12 GHz and beyond, such behavior can be attributed to zero magnetic properties at high frequency [32]. The magnitude of the losses or energy dissipation can be estimated by evaluating the dielectric and magnetic loss tangents (tan\(\delta_{\epsilon}\) and tan\(\delta_{\mu}\)). Figure 7e,f present the dielectric loss and magnetic loss tangent respectively. General observation shows that the \(\tan\delta_{\mu}\) values are generally smaller compared to the \(\tan\delta_{\epsilon}\). Hence, the dielectric loss is the dominating attenuation mechanism of all samples presented in this study. To analyse the dielectric loss mechanism of the samples, the Debye dipolar relaxation was employed. The complex permittivity according to this model is as follows:  
\[\epsilon_{\mathrm{r}}=\epsilon^{\prime}-\mathrm{j}\epsilon^{\prime\prime}= \varepsilon_{\infty}+\frac{\epsilon_{\mathrm{s}}-\varepsilon_{\infty}}{1+ \mathrm{j}\omega\tau} \tag{12}\]  
where \(\tau\), \(\epsilon_{\mathrm{s}}\), \(\omega=2\pi\)f, \(\varepsilon_{\infty}\), denotes the relaxation time, static permittivity, angular frequency, and relative permittivity at the maximum frequency. The \(\epsilon^{\prime}\) and \(\epsilon^{\prime\prime}\) can then be described as  
\[\epsilon^{\prime}=\varepsilon_{\infty}+\frac{\epsilon_{\mathrm{s}}- \varepsilon_{\infty}}{1+\omega^{2}\tau^{2}} \tag{13}\]  
\[\epsilon^{\prime\prime}=\omega\tau+\frac{\epsilon_{\mathrm{s}}-\varepsilon_{ \infty}}{1+\omega^{2}\tau^{2}} \tag{14}\]  
Figure 7: (**a**) Real (\(\epsilon^{\prime}\)) and (**b**) imaginary (\(\epsilon^{\prime\prime}\)) parts of permittivity, (**c**) real (\(\mu^{\prime}\)) and (**d**) imaginary (\(\mu^{\prime\prime}\)) parts of permeability, (**e**) dielectric (tan\(\delta_{\epsilon}\)) and (**f**) magnetic (tan\(\delta_{\mu}\)) loss tangents of all samples at 8.2–18 GHz frequency. With the above equations, the correlation between \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime}\) can be expressed as  
\[\left(\varepsilon^{\prime}-\frac{\varepsilon_{\mathrm{s}}+\varepsilon_{\infty}}{ 2}\right)^{2}+\left(\varepsilon^{\prime\prime}\right)^{2}=\left(\frac{ \varepsilon_{\mathrm{s}}-\varepsilon_{\infty}}{2}\right)^{2} \tag{15}\]  
This model establishes the relationship between (\(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\)), which is termed the Cole-Cole semicircle. It implies that the plot of \(\varepsilon^{\prime\prime}\) against \(\varepsilon^{\prime}\) will exist as a semicircle indicating a polarization process [25]. Figure 8a-d presents the Cole-Cole plots of all samples in the frequency range of 8.2-18 GHz. The plots show obvious several semi-circles, which may be attributed to the nano size of the C-Fe\({}_{3}\)O\({}_{4}\). Meanwhile, the existence of carboxyl and hydroxyl functional groups in the sample as confirmed by the FTIR and EDS analysis could provide additional polarization centers resulting in interfacial polarization that contributes to enhance absorption of EM wave.