## 2 VO2 Materials  
To describe the properties of VO2 thin films in incomplete phase transition, we need to use the effective medium theory (EMT). At present, the most commonly used are Maxwell-Garnett EMT and Bruggeman EMT. Because the metal particles in composite media are non-uniform distribution and the volume fraction of metal components is more than 20% during phase transition, Maxwell-Garnett EMT is not suitable for this condition [31]. In THz band, the IMT characteristics of VO2 can be described by Bruggeman EMT. The dielectric function \(\varepsilon_{C}\) can be expressed as  
\[\varepsilon_{C}=\frac{1}{4}\Bigg{\{}\ +\sqrt{\left[\varepsilon_{D}(2-3f_{v})+ \varepsilon_{M}(3f_{v}-1)\right]^{2}+8\varepsilon_{D}\varepsilon_{M}}\ \Bigg{\}} \tag{1}\]  
where \(f_{v}\) is the volume fraction of the metal region, \(\varepsilon_{D}\) and \(\varepsilon_{M}\) are dielectric functions of VO2 thin films in insulation and metal phase, respectively. The functional relationship between the volume fraction \(f_{v}\) and temperature \(T\) follows Boltzmann function,  
\[f_{v}=f_{\text{max}}\Bigg{(}1-\frac{1}{1+\exp[(T-T_{0})/\Delta T]}\Bigg{)} \tag{2}\]  
where \(T_{0}\) is the phase transition temperature, with heating and cooling points 68 \({}^{\circ}\)C and 64 \({}^{\circ}\)C, respectively; \(\Delta T\) = 2 \({}^{\circ}\)C is the transition width, and \(f_{\text{max}}\) is the maximum volume fraction of the metal component in VO2 at the highest temperature. Experimental results show that \(\Delta T\) = 2 \({}^{\circ}\)C and \(f_{\text{max}}\) = 0.95 [23]. For the insulator component in VO2, it can be regarded as a dielectric with dielectric constant \(\varepsilon_{D}\) = 9 [23]. The dielectric function \(\varepsilon_{M}\) of metal components in VO2 can be determined by Drude model. However, the traditional Drude model cannot accurately describe the dielectric function of VO2 in IMT process. Therefore, to make the Drude model better represent IMT characteristic of VO2, a simple generalization was derived based on the impulse response approach and Poisson statistics [32; 33], and the negative parameter b is introduced [23; 34],  
\[\varepsilon_{m}=\varepsilon_{\infty}-\frac{{\omega_{p}}^{2}}{\omega^{2}+i \omega\Gamma}\Bigg{(}1+\frac{ib\Gamma}{\omega+i\Gamma}\Bigg{)} \tag{3}\]  
where \(\omega=2\pi f\) is the circular frequency of THz wave, \(\varepsilon_{\infty}=\varepsilon_{D}\) = 9 is the high-frequency limit dielectric constant of VO2, \(\omega_{p}\) is the plasma frequency, which is expressed as,  
\[\omega_{p}=\sqrt{N\varepsilon^{2}/\varepsilon_{0}m^{*}} \tag{4}\]  
where \(N=1.3\times 10^{22}\) cm\({}^{-3}\) is the carrier concentration in the medium, \(m^{*}=2m_{\text{e}}\) is the effective mass, \(m_{\text{e}}=9.1094\times 10^{-31}\) kg is the electronic mass and \(\varepsilon_{0}\) is the vacuum dielectric constant. \(\Gamma=1/\tau\) is the scattering rate, where \(\tau=2.27\times 10^{-15}\) s is the carrier collision time. The complex permittivity can be expressed as a summation of real and imaginary parts,  
\[\varepsilon(\omega)=\varepsilon_{1}(\omega)+\frac{4\pi i}{\omega}\sigma_{1}(\omega) \tag{5}\]  
where \(\varepsilon_{1}(\omega)\) is the real dielectric function, and \(\sigma_{1}(\omega)\) is the frequency dependent conductivity.