Mott–Schottky equation

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

$$\frac{1}{C^2} = \frac{2}{\epsilon \epsilon_0 A^2 e N_d} (V - V_{fb} - \frac{k_B T}{e})$$

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 $$w A$$, $$e$$ 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

$$w = (\frac{2 \epsilon \epsilon_0}{e N_d} ( V - V_{fb} ) )^\frac{1}{2}$$

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 $$e N_d$$, and the total charge of the depletion region, compensated by opposite charge nearby in the electrolyte, is

$$Q = e N_d A w = e N_d A (\frac{2 \epsilon \epsilon_0}{e N_d} ( V - V_{fb} ) )^\frac{1}{2}$$

Thus, the differential capacitance is

$$C = \frac{\partial{Q}}{\partial{V}} = e N_d A \frac{1}{2}(\frac{2 \epsilon \epsilon_0}{e N_d})^\frac{1}{2} ( V - V_{fb} )^{-\frac{1}{2}} = A (\frac{e N_d \epsilon \epsilon_0}{2(V - V_{fb})})^\frac{1}{2}$$

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.