## 3 Results and Discussion
### Determination of the Interface States of MIS Capacitors with Different Polymer Gate Insulators  
To investigate the influence of the polymer gate insulator on the interface-state density in P3 HT MIS capacitors, we analyzed the impedance characteristics by using the conductance method suggested by [PERSON] and [PERSON] [19; 20]. In order to extract the interface states from the conductance, three equivalent circuit models that incorporate the interface states with a single energy level, continuum energy levels, and continuum interface states affected by band-bending fluctuations were proposed. Hereafter, we refer to these as the single-level model, the continuum model, and the statistical model, respectively. Figure 6a illustrates an equivalent circuit of an MIS capacitor that incorporates the interface states with the single energy level in the depletion mode. Here, \(C_{t}\) and \(G_{t}\) represent the capacitance and conductance associated with interface states, respectively. The simplified version of Figure 6a is shown in Figure 6b. When the equivalent circuit shown in Figure 6b is converted into that shown in Figure 6c, the equivalent parallel conductance (\(G_{p}\)) divided by \(\omega\) (loss) is given by \begin{table}
\begin{tabular}{c c} \hline
**Annealing Cycle** & \(N_{A}\) **(cm\({}^{-3}\))** \\ \hline Before annealing & \(1.6\times 10^{16}\) \\
1 st & \(2.8\times 10^{15}\) \\
2 nd & \(2.0\times 10^{15}\) \\
3 rd & \(1.4\times 10^{15}\) \\
4 th & \(1.4\times 10^{15}\) \\
5 th & \(1.4\times 10^{15}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Variation of the dopant concentration (\(N_{A}\)) for annealing cycle in the ITO/PI/P3 HT/Au MIS capacitor. Figure 5: (a) Gate voltage dependences of the capacitance (C) and (b) \(1/C^{2}\)-gate voltage plots in the ITO/PI/P3 HT/Au MIS capacitor without the peripheral regions measured at 100 Hz after each thermal annealing cycle. \[\frac{G_{p}}{\omega}=\frac{qD_{it}\omega\tau_{p}}{1+\omega^{2}\tau_{p}{}^{2}}, \tag{3}\]  
where \(D_{it}\) and \(\tau_{p}\) are the density of interface states and the time constant of the interface states, respectively. Figure 6d shows an equivalent circuit with the continuum interface states, which is represented by \(G_{t}-C_{t}\) branches parallel to the depletion layer. In the case of the continuum model, the loss can be expressed as  
\[\frac{G_{p}}{\omega}=\frac{qD_{it}}{2\omega\tau_{p}}\ln\Bigl{(}1+\omega^{2} \tau_{p}{}^{2}\Bigr{)}. \tag{4}\]  
In the statistical model, the influence of built-in charges and/or charged interface states was also assumed [20; 28; 29]. It is assumed that a random spatial distribution of discrete interface charges results in a spatial distribution of band bending. The probability that the band bending is \(v_{s}\) is expressed as  
\[P(v_{s})=\Bigl{(}2\pi\sigma_{s}{}^{2}\Bigr{)}^{-1/2}\exp\Biggl{[}-\frac{(v_{s }-v_{s})^{2}}{2\sigma_{s}{}^{2}}\Biggr{]}, \tag{5}\]  
where \(\sigma_{s}{}^{2}\) is the variance of band bending, expressed in terms of \(kT/q\), and \(\langle v_{S}\rangle\) is the mean value of the band bending. In this case, Equation (4) can be rewritten as follows  
\[\frac{G_{p}}{\omega}=\frac{q}{2}\int_{-\infty}^{\infty}\frac{D_{it}}{\omega \tau_{p}}\ln\Bigl{(}1+\omega^{2}\tau_{p}{}^{2}\Bigr{)}P(v_{s})dv_{s}. \tag{6}\]  
From the above expressions, we obtain  
\[\frac{G_{p}}{\omega}=\frac{qD_{it}\bigl{(}2\pi\sigma_{s}{}^{2}\bigr{)}^{-1/2} }{2\omega\tau_{p}}\int_{-\infty}^{\infty}\exp\biggl{(}-\frac{\eta^{2}}{2\sigma _{s}{}^{2}}\biggr{)}\exp(-\eta)\ln(1+\omega^{2}\tau_{p}{}^{2}\exp 2\eta)d\eta, \tag{7}\]  
where \(\eta=v_{s}-\langle v_{s}\rangle\). The density of the interface states can be determined by fitting Equations (3), (4), or (7) to the frequency dependence of loss (\(G_{p}/\omega-f\)). First, we examined the applicability of the three models for the P3 HT MIS capacitors with polymer gate insulators.