Assuming only the space-charge capacitance that varies with the changing electrode potential, the reverse-bias dependence of the inverse square capacitance of the space-charge region in the semiconductor is given by the Mott-Schottky relation:
\[\frac{1}{C^2}=\frac{2}{A^2\epsilon\epsilon_0 q N_D}(E-E_\textrm{fb}-\frac{kT}{q})\]where $C$ is the relative permittivity of Si (11.7), $q$ is the elementary charge, $N_D$ is the free charge carrier density in the semiconductor, $E$ is the potential difference between the semiconductor and the redox potential of the solution, $E_\textrm{fb}$ is the flat band potential, $k$ is the Boltzmann’s constant and $T$ is temperature (298 K).
Mott-Schottky is an electrochemical impedance spectroscopy (EIS) technique that can be difficult to perform and interpret if the system is not ideal. When the measurement is successful, it is able to determine both the $E_\textrm{fb}$ and the free charge carrier density (donors or acceptors, $N_D$) of the photoelectrode. The Mott-Schottky plot will possess a negative slope for p-type materials and a positive slope for n-type materials.
The primary disadvantage of M–S analysis is that the C sc of a photoelectrode can be very complicated to measure depending on the sample composition and mor- phology. The ideal sample is a single-crystal material of high crystalline quality with moderate doping grown on an ohmic contact. Single-frequency measurements of the C sc of an ideal sample are fairly straightforward. However, measurements for samples which are not ideal may require a full evaluation of the frequency range over many decades, followed by fitting of the data to an appropriate, but often complicated equivalent circuit model
The $E_\textrm{fb}$ can be determined by taking the value of the intercept between the extrapolated linear region of the $C^{−2}$ with the $x$-axis in the Mott-Schottky plot.
The barrier height ($\phi_b$) was calculated using the Schottky’s relation: \(\phi_b=E_\textrm{fb}+V_n\)
Where $V_n$ is the difference between the potential of the conduction band edge and the Fermi level, and was obtained by using the following relationship:
\[V_n=kT\,\textrm{ln}(\frac{N_C}{N_D})\]The density of conduction band states ($N_C$) was calculated by
\[N_C=2(\frac{2\pi m^*_ekT}{h^2})^{3/2}\]where $m^*_e$ is the effective mass of electron and $h$ is the Planck’s constant.
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