Mott–Schottky equation

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

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

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

This theory predicts that a Mott–Schottky plot will be linear. The doping density $N_{d}$ 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 $V$-axis at the flatband potential.

Derivation

Under an applied potential $V$, the width of the depletion region is

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

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 $eN_{d}$, and the total charge of the depletion region, compensated by opposite charge nearby in the electrolyte, is

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

Thus, the differential capacitance is

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

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.