Figure 1

(Color online) The scheme shows a semiconductor where the Fermi level can be controlled with external potential. The semiconductor has both extended states (e.g., conduction-band levels) and localized (band-gap) levels. The Fermi-level marks approximately the separation between occupied and empty localized levels, which applies in an exponential distribution of traps. The equivalent circuit for two specific levels (one occupied, another one empty) is shown. For each level, the resistance corresponds to the thermal release of electron to the extended levels, and the capacitance corresponds to the variation of equilibrium occupancy of the level under variation of Fermi level, i.e., a chemical capacitance. The extended states also possess a chemical capacitance, though this is not shown. The characteristic frequencies of the equivalent circuits of occupied and empty levels have different properties, as discussed in the text, and are denoted as “deep” and “shallow,” respectively.Reuse & Permissions

Figure 2

(Color online) (a) Energy diagram showing the thermal occupation at temperature $T=300\text{K}$ of an exponential DOS in the band gap. $E_F$ is the Fermi level of electrons and $E_c$ is the conduction-band energy. The clear (green) shaded region indicates the band-gap states that are occupied with electrons, and the dark (blue) shaded region indicates empty levels, as determined by the Fermi–Dirac distribution function (red dashed line). (b) The trap frequencies as a function of energy. The parameters used in the calculation: $T_0=800\text{K}$, $N_c={10}^{20}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, and $\beta ={10}^{-10}{\text{cm}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 3

(a) The equivalent circuit of the conduction-band states (represented by capacitance $C_0$) and the distribution of traps, each energy interval $dE$ being associated with a $RC$ circuit. (b) The trap impedance is represented by $Z_{\text{traps}}$. (c) Reduction of the impedance of an exponential distribution of traps $Z_{\text{traps}}$ to a much simpler equivalent circuit, valid in the high-frequency domain, as is demonstrated quantitatively in the text. The sum of individual $RC$ elements, each corresponding to a narrow energy interval, over all the shallow levels (above the Fermi level) collapses into a CPE element.Reuse & Permissions

Figure 4

(Color online) Representation of the real and imaginary parts of the frequency-dependent capacitance $C_{\text{traps}}^{\ast }$ for an exponential distribution of traps as a function of the frequency. Also shown are the approximations at low frequency $\left(C_{\text{deep}}^{\ast }\right)$, an $RC$ circuit, and high frequency $\left(C_{\text{shallow}}^{\ast }\right)$, a CPE element. The square point marks the deep trap frequency ${\omega }_{t0}$. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.4\text{eV}$, $E_c=1\text{eV}$, $T_0=800\text{K}$ $\left(\alpha =0.375\right)$, $N_c={10}^{20}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, and $\beta ={10}^{-10}{\text{m}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 5

(Color online) (a) Representation of the capacitance $C_{\text{traps}}^{\ast }$ and (b) correspondent impedance in the complex plot for an exponential distribution of traps. The square point marks the deep trap frequency ${\omega }_{t0}$. In (a) are indicated in dashed lines the CPE approximation at high frequency (green line) and the series circuit, $R_{\text{deep}}{C}_{\text{deep}0}$, at low frequency (brown line). The vertical line in (b) shows the value of low-frequency resistance. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.4\text{eV}$, $E_c=1\text{eV}$, $T_0=800\text{K}$ $\left(\alpha =0.375\right)$, $N_c={10}^{20}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, and $\beta ={10}^{-10}{\text{cm}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 6

(Color online) The thick red lines show the dc value of the (a) capacitance and (b) resistance for an exponential distribution of traps as a function of the Fermi level. In blue dashed lines are shown the dc values for the capacitance and resistance of the deep levels, according to the approximations discussed in the main text. The parameters used in the calculation: $T=300\text{K}$, $E_c=1\text{eV}$, $T_0=800\text{K}$ $\left(\alpha =0.375\right)$, $N_c={10}^{20}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, and $\beta ={10}^{-10}{\text{cm}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 7

(Color online) Representation of the impedance $Z_{\text{traps}}=1/\left(i\omega C_{\text{traps}}^{\ast }\right)$ for an exponential distribution of traps. (a)–(c) show successive enlargement of the axis. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.4\text{eV}$, $E_c=1\text{eV}$, $T_0=800\text{K}$ $\left(\alpha =0.375\right)$, $N_c={10}^{20}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, and $\beta ={10}^{-14}{\text{cm}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 8

(Color online) (a) Schematic representation of a porous semiconductor film, formed by connected nanoparticles deposited over a conducting substrate (shown at the left). (b) Electronic processes in the porous film, when it is immersed in a redox electrolyte. Electrons injected from the substrate diffuse by displacement in the extended states. They have a probability of being trapped and further released by localized states in the band gap. In addition, electrons in the conduction band have a chance to be captured by the oxidized form of the redox ions, which are indicted below around the redox potential level. The Fermi level is shown inclined, indicating a transient situation in which macroscopic diffusion of electrons occurs toward the right direction.Reuse & Permissions

Figure 9

(a) Transmission line representation of the diffusion impedance with the distributed diffusion resistance $r_n$ and the general transverse element ${\zeta }_n$. [(b)–(f)] Specific models are indicated, depending on the local electronic processes coupled with diffusion.Reuse & Permissions

Figure 10

(Color online) Representation of diffusion-recombination impedance of a film of thickness $L$. (a) Diffusion in the absence of recombination. The square point marks the angular frequency of turnover, related to ${\omega }_d=100\text{rad}{\text{s}}^{-1}$. [(b),(c) and (d)] Diffusion-recombination impedance, showing two cases with different values of lifetime, as indicated. The yellow round point marks the effective lifetime frequency ${\tau }_0^{-1}$, with the values ${\tau }_0={10}^{-1}\text{s}$ and ${\tau }_0={10}^{-3}\text{s}$. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.75\text{eV}$, $E_c=1\text{eV}$, $N_c={10}^{18}{\text{cm}}^{-3}$, $D_0={10}^{-4}{\text{cm}}^2{\text{s}}^{-1}$, $L={10}^{-3}\text{cm}$, and $A=1{\text{cm}}^2$.Reuse & Permissions

Figure 11

(Color online) Representation of the impedance of diffusion in a film of thickness $L$, with an exponential distribution of traps. [(a)–(c)] Impedance spectra in the complex plane. The gray thin line is the impedance of diffusion without traps. The yellow square points mark the diffusion frequency ${\omega }_{dn}=\left(C_0/C_{\text{traps}}\right){\omega }_{d0}$, the cutoff frequency ${\omega }_{tm}$, and the deep traps frequency ${\omega }_{t0}$. (d) The real part of the impedance as a function of the frequency. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.75\text{eV}$, $E_c=1\text{eV}$, $T_0=1000\text{K}$ $\left(\alpha =0.33\right)$, $N_c={10}^{18}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, $D_0={10}^{-4}{\text{cm}}^2{\text{s}}^{-1}$, $L={10}^{-3}\text{cm}$, and $A=1{\text{cm}}^2$.Reuse & Permissions

Figure 12

(Color online) Representation of the impedance of diffusion in a film of thickness $L$, with an exponential distribution of traps. (a) and (b) show the impedance in the complex plane. The gray thin line is the impedance of diffusion without traps. The yellow square point marks the cutoff frequency ${\omega }_{tm}$. (c) shows the real part of the impedance as a function of the frequency. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.75\text{eV}$, $E_c=1\text{eV}$, $T_0=1000\text{K}$ $\left(\alpha =0.33\right)$, $N_c={10}^{18}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, $D_0={10}^{-4}{\text{cm}}^2{\text{s}}^{-1}$, $L={10}^{-3}\text{cm}$, and $A=1{\text{cm}}^2$.Reuse & Permissions

Figure 13

(Color online) Repesentation of impedance of a film of thickness $L$, with an exponential distribution of traps with recombination from the conduction band. The green square point marks the deep trap frequency ${\omega }_{t0}$, and the blue round point marks the effective lifetime frequency ${\tau }_n^{-1}$, with the values (a) ${\tau }_0=1\text{s}$ and ${\tau }_n=362\text{s}$, (b) ${\tau }_0=0.01\text{s}$ and ${\tau }_n=3.62\text{s}$, and (c) ${\tau }_0=0.001\text{s}$ and ${\tau }_n=0.362\text{s}$. In (c) in green dashed line the CPE approximation at high frequency is indicated. The parameters used in the calculation: $T=300\text{K}$, $E_F=0.4\text{eV}$, $E_c=1\text{eV}$, $T_0=800\text{K}$ $\left(\alpha =0.375\right)$, $N_c={10}^{20}{\text{cm}}^{-3}$, $N_t={10}^{20}{\text{cm}}^{-3}$, and $\beta ={10}^{-10}{\text{cm}}^2{\text{s}}^{-1}$.Reuse & Permissions

Figure 14

(Color online) Representation of (a) capacitance and (b) impedance in the complex plane, for single and double trap models, as indicated. Parameters: $C_{t1}={10}^{-3}\text{F}$, $R_{t1}={10}^3\Omega$, ${\omega }_{t1}=1\text{rad}{\text{s}}^{-1}$, $C_{t1}=5\times {10}^{-3}\text{F}$, $R_{t1}=2\times {10}^3\Omega$, ${\omega }_{t1}=0.1\text{rad}{\text{s}}^{-1}$, and $C_{\text{tot}}=C_{t1}+C_{t2}$.Reuse & Permissions