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The Mott–Schottky equation relates the capacitance to the applied voltage across a semiconductor-electrolyte junction.

${\displaystyle {\frac {1}{C^{2}}}={\frac {2}{\epsilon \epsilon _{0}A^{2}eN_{d}}}(V-V_{fb}-{\frac {k_{B}T}{e}})}$

where ${\displaystyle C}$ is the differential capacitance ${\displaystyle {\frac {\partial {Q}}{\partial {V}}}}$, ${\displaystyle \epsilon }$ is the dielectric constant of the semiconductor, ${\displaystyle \epsilon _{0}}$ is the permittivity of free space, ${\displaystyle A}$ is the area such that the depletion region volume is ${\displaystyle wA}$, ${\displaystyle e}$ is the elementary charge, ${\displaystyle N_{d}}$ is the density of dopants, ${\displaystyle V}$ is the applied potential, ${\displaystyle V_{fb}}$ is the flat band potential, ${\displaystyle k_{B}}$ is the Boltzmann constant, and T is the absolute temperature.

This theory predicts that a Mott–Schottky plot will be linear. The doping density ${\displaystyle N_{d}}$ can be derived from the slope of the plot (provided the area and dielectric constant are known). The flatband potential can be determined as well; absent the temperature term, the plot would cross the ${\displaystyle V}$-axis at the flatband potential.

Derivation

Under an applied potential ${\displaystyle V}$, the width of the depletion region is

${\displaystyle w=({\frac {2\epsilon \epsilon _{0}}{eN_{d}}}(V-V_{fb}))^{\frac {1}{2}}}$

Using the abrupt approximation, all charge carriers except the ionized dopants have left the depletion region, so the charge density in the depletion region is ${\displaystyle eN_{d}}$, and the total charge of the depletion region, compensated by opposite charge nearby in the electrolyte, is

${\displaystyle Q=eN_{d}Aw=eN_{d}A({\frac {2\epsilon \epsilon _{0}}{eN_{d}}}(V-V_{fb}))^{\frac {1}{2}}}$

Thus, the differential capacitance is

${\displaystyle C={\frac {\partial {Q}}{\partial {V}}}=eN_{d}A{\frac {1}{2}}({\frac {2\epsilon \epsilon _{0}}{eN_{d}}})^{\frac {1}{2}}(V-V_{fb})^{-{\frac {1}{2}}}=A({\frac {eN_{d}\epsilon \epsilon _{0}}{2(V-V_{fb})}})^{\frac {1}{2}}}$

which is equivalent to the Mott-Schottky equation, save for the temperature term. In fact the temperature term arises from a more careful analysis, which takes statistical mechanics into account by abandoning the abrupt approximation and solving the Poisson–Boltzmann equation for the charge density in the depletion region.

References

1. Gelderman, K. (2007). "Flat-Band Potential of a Semiconductor: Using the Mott–Schottky Equation". Journal of Chemical Education. 84 (4): 685. Bibcode:2007JChEd..84..685G. doi:10.1021/ed084p685.
2. ^ Grundmann, Marius (2010). "Section 20.2.2". The Physics of Semiconductors. Springer. ISBN 978-3-642-13883-6.

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