## 3 Typical Examples of THz Conductivity in Nanomaterials  
For the purposes of this focused review, we will consider and discuss only a few selected examples of the THz conductivity, such as popular nano- semiconductors, oxides, and metals, rather than a comprehensive and extensive review. An extensive review of a large collection of materials is beyond the scope of this focused review. Note that the following experimental data are all provided by pulsed sources based on terahertz time-domain spectroscopy (THz-TDS) and measured at room temperature. There are a number of excellent reviews on this topics [1; 27] and, therefore, we do not discuss THz techniques in this article. Open circles and triangles in Figure 3 show the experimental results of real and imaginary parts of conductivity, respectively, in nanocrystalline Si films (average grain size is 15-25 nm) [8]. To extract the complex conductivity we use the relationship between the dielectric permittivity and conductivity given as:  
\[\sigma*(\omega)=-i\omega\varepsilon_{0}\varepsilon*(\omega) \tag{5}\]  
where \(\varepsilon*(\omega)+\varepsilon_{\infty}=\varepsilon_{\mathrm{R}}(\omega)+i \varepsilon_{\mathrm{I}}(\omega)\), \(\varepsilon_{0}\) is the absolute permittivity, \(\varepsilon_{\mathrm{R}}(\omega)\) and \(\varepsilon_{\mathrm{I}}(\omega)\) are the real and imaginary part of dielectric constants, respectively, and \(\varepsilon_{\infty}\) is the background dielectric constant. Note again that a time dependence of the form \(\exp(-i\omega t)\) is used here. Optical pumping spectroscopy has been used for producing enough free carriers (electrons) at room temperature, and, hence, we do not need \(\varepsilon_{\infty}\) for calculation as a physical parameter [1]. The negative value of the imaginary part of conductivity is obtained. Solid lines in Figure 3 show the best fit of the SSFTC model to the experimental data. Adjusted physical parameters are all reasonable and are listed in Table 1. We now recognize that the SSFTC model predicts a similar behavior of the Lorentz resonance as shown in Figure 1 (curves (c) and (d)). It is important to emphasize that experimental data are only available within a certain narrow frequency range. In the present case, a peak is predicted by the model outside the experimental frequency window. Fitting within the data range leads to a peak outside the data range, but there is no way of validating this prediction without extending the experimental data range to cover the peak. However, as will be shown for nanogold particles (Figure 7), a peak is indeed found in the observed frequency range. We, therefore, discuss a detailed comparison between the Lorentz resonance and the SSFTC prediction in the final part of this section. Figure 3: Complex conductivity in nanocrystalline Si films. Open circles and triangles are experimental results of the real and imaginary parts of conductivity, respectively; data extracted from [8]. Solid lines are conductivity predicted from the SSFTC model. Open circles and triangles in Figure 4 show the real and imaginary parts of conductivity in the crystalline phase of Ge\({}_{2}\)Sb\({}_{2}\)Te\({}_{5}\) (from the [PERSON] group at the Academy of Sciences of the Czech Republic in Pargue), which is the most useful phase-change material, developed for digital versatile disk (DVD) [28]. As the crystalline phase of phase-change materials is a degenerate semiconductor (witha high density of free carriers) and, hence, the optical-pumping (photocarrier) technique, as in nanocrystalline Si (see Figure 3) is not required. In this case, background conductivity (imaginary), \(\sigma_{\infty}=\omega\varepsilon_{0}\varepsilon_{\infty}\), is taken into account [1,2]. Solid lines show the real and imaginary parts of conductivity predicted by the SSFTC model, and the physical parameters from the fitting are listed in Table 1. Note that features obtained here are very similar to those obtained for other phase-change materials [14]. Let us examine the THz conductivity in so-called oxide metals and semiconductors.