## 3 Results and Discussion
### THz Conductivity and Permittivity
The similar behavior of conductivity of such materials was reported in several works [1; 2; 3], where the DC measurements were performed. Based on Figure 5 it can be concluded that for complex permittivity spectra the similar behavior is observed as for complex sheet conductivity ones. In 0.2-0.8 THz frequency range, the real part of permittivity has very large negative values, which is also in a good enough agreement with the classical Drude model for metals/semimetals. This model was applied simultaneously to both conductivity and permittivity spectra (to increase the approximation accuracy) in the next form:  
\[\hat{\sigma}(\omega)=\frac{\sigma_{0}}{1-i\omega\tau}=\frac{\sigma_{0}}{1+ \omega^{2}\tau^{2}}+i\omega\tau\frac{\sigma_{0}}{1+\omega^{2}\tau^{2}}, \tag{13}\]  
\[\hat{\varepsilon}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{ 2}+i\gamma\omega}, \tag{14}\]  
Figure 5: The dynamical complex permittivity spectra of Bi\({}_{1-x}\)Sb\({}_{x}\) films in 0.2–0.8 THz frequency range for the cases of mica (**a**,**b**) and PI (**c**,**d**) substrates and for 150 nm (**a**,**c**) and 70 nm (**b**,**d**) film thicknesses. Different colors represent different Sb concentrations. The solid and dashed lines represent the real and imaginary parts of permittivity, respectively. where \(\omega=2\pi f\) is the angular frequency of THz radiation, \(\tau\) is the relaxation time of charge carriers which can be approximately estimated (using Equation (13) after splitting \(\hat{\sigma}(\omega)\) into real and imaginary parts and their equalization through \(\sigma_{0}\)) as  
\[\tau=\frac{1}{\omega}\frac{Im[\hat{\sigma}(\omega)]}{Re[\hat{ \sigma}(\omega)]}, \tag{15}\] \[Re[\hat{\sigma}(\omega)]=\frac{\sigma_{0}}{1+\omega^{2}\tau^{2}},\] \[Im[\hat{\sigma}(\omega)]=\omega\tau\frac{\sigma_{0}}{1+\omega^{2 }\tau^{2}}\]  
with small dispersion (\(\tau\) can vary in process of THz wave transmission, because the THz pulse duration is about 10 times higher than the relaxation time value),  
\[\sigma_{0}=n(\omega)e^{2}\tau/m^{*} \tag{16}\]  
is the DC film conductivity, where \(n(\omega)\) is a concentration of the charge carriers (see below), \(e\) is the electron charge, \(m^{*}\) is the effective mass (the only one free fitting parameter), \(\varepsilon_{\infty}=100\) (high-frequency dielectric constant) for all the films, \(\gamma=1/\tau\) is the damping rate,  
Figure 4:0  
is the square of plasma frequency, \(\varepsilon_{0}\) is the permittivity of free space. Due to the fact that the film is irradiated with a THz wave, the additional charge carriers are generated besides the static carrier concentration:  
Figure 4:1  
where \(n_{0}\) is the carrier concentration without the external influence (obtained from galvanomagnetic measurements), \(R(\omega)\) is the film reflection coefficient (estimated using the Fresnel equation for a normal irradiation from Equations (11) and (12)), \(\alpha(\omega)\) is the absorption constant (from a complex refractive index, root of Equation (12)), \(W(\omega)\) is the intensity spectrum of THz emitter which is normalized through the introduction of some constant value \(C\), and \(\hbar\omega\) is the photon energy. Thus, the complete Drude model was used to fit both experimental conductivity and permittivity data. This model takes into account the complex dependency of carrier concentration on THz field intensity in semimetallic Bi\({}_{1-x}\)Sb\({}_{x}\) films. The results of data fitting are presented in Figure 6 for various Sb concentrations \(x\) (plasma frequency and relaxation time, DC conductivity and effective carrier masses). As seen in Figure 6 the film thickness and the substrate material strongly affect the values of all the presented parameters. The nonlinear behavior of the curves is defined by a complex dependency of Bi\({}_{1-x}\)Sb\({}_{x}\) band structure on antimony concentration \(x\). The DC conductivity is higher for films on mica substrate due to a higher relaxation time (lower damping rate, associated with a greater crystallite size).