## 3 Results and Discussion
### Complex Conductivity  
The complex conductivity calculations are based on Equations (7) and (8). In order to calculate the complex \(\hat{\sigma}(\omega)\) of our graphene nanocomposite samples, we need to know the complex dependences of \(\hat{\varepsilon}(\omega)\) for not only the nanocomposite, but also for the neat polymer matrix. Therefore, we additionally calculated the \(\hat{\varepsilon}(\omega)\) spectrum for each neat copolymer sample, using Equations (1)-(6), with the reference corresponding to the THz transient spectrum measured in dry nitrogen. The resulting real and imaginary parts of the graphene nanocomposite complex conductivity are plotted as symbols for both the thin (open circles) and thick (open triangles) in Figure 8. To calculate error bars, we analyzed the data collected in different sets of measurements performed at different spots on our samples. Next, our \(\hat{\sigma}(\omega)\) spectra were fitted using the Drude-Smith model [38] (solid lines in Figure 8a), and what we call the Drude-tunneling model presented in [39] (solid lines in Figure 8b). The Drude-Smith model is given by  
\[\sigma(\omega)=\frac{\varepsilon_{0}\omega_{p}^{2}\tau}{(1-i\omega\tau)} \left[1+\sum_{m=1}^{\infty}\frac{c_{m}}{(1-i\omega\tau)}\right], \tag{11}\]  
where \(\omega_{p}\) is the plasma frequency, \(\tau\) represents the carrier collision time, and the \(c_{m}\) coefficient is the so-called scattering parameter. A first-order approximation is conventionally used, i.e., the terms \(m>1\) are truncated. When \(c_{m}=0\), Equation (11) simplifies to the classical Drude model (the term in front of the square bracket), while \(c_{1}=-1\) means that carriers are completely backscattered. This extension of the classic Drude model was introduced by [PERSON] [38] and is usually used to describe complex conductivity spectra in the THz range of materials with nanofillers [40], such as nanoparticles [41; 42], carbon nanotubes [43], and even monolayer graphene [43], when it is difficult to explain results using the simple, homogeneous-medium Drude model. The model presented in [39] is based on a concept that in non-homogeneous, granular materials carrier transport consists of both, intragrain scattering and intergrain tunneling, and the corresponding \(\sigma_{eff}(\omega)\) should be described, in general, as a superposition of both processes and given by as  
\[\frac{1}{\sigma_{eff}(\omega)}=\frac{f}{\sigma_{f}(\omega)}+\frac{1-f}{\sigma _{t}(\omega)}, \tag{12}\]  
Figure 8: Experimental complex conductivity (\(\sigma_{1}\) and \(\sigma_{2}\)) spectra for both thin (open circles) and thick (open triangles), 1-wt% graphene nanocomposite samples. Solid lines correspond to the fits based (**a**) the Drude–Smith and (**b**) Drude-tunneling models. Experimental data points for the thick sample were trimmed according to sample bandwidth (see Figure 5b). where \(\sigma_{f}(\omega)\) is the free carrier conductivity given by the Drude law (\(m=0\) in Equation (11)), \(\sigma_{t}(\omega)\) is the tunneling conductivity given in [39], and \(f\) is the spectral weight of intragrain transport. The latter explains why the authors in [39] call this approach the free (intra-grain) and tunneling (inter-grain; grain boundary) carrier transport model. Analyzing Figure 8a, we observe that we obtained an excellent fit using, as it is a standard procedure, only the first term (\(m=1\)) in the Smith expansion (Equation (11)) and the fitting parameters were \(c_{1}=-0.997\), \(\omega_{\rm p}=72.4\times 10^{12}\)\(1/\)s, and \(\tau=12\) fs for the thin nanocomposite sample, and \(c_{1}=-0.998\), \(\omega_{\rm p}=41.9\times 10^{12}\)\(1/\)s, and \(\tau=21\) fs for the thick one (in both cases, the \(\omega_{\rm p}\tau\) product is approximately the same and equal to 0.87). We observe that in both cases, the \(c_{1}\) values are very close to \(-1\), indicating a nearly complete backscattering mechanism of carriers inside our graphene-polymer samples.