Mott–Schottky equation

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

$\frac{1}{C^2}=\frac{2}{ϵ{ϵ}_0{A}^{2}e{N}_d}(V-V_{fb}-\frac{k_{B}T}{e})$

where $C$ is the differential capacitance $\frac{\partial Q}{\partial V}$, $ϵ$ is the dielectric constant of the semiconductor, ${ϵ}_0$ 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

$w=(\frac{2ϵ{ϵ}_0}{e{N}_d}(V-V_{fb}){)}^{\frac{1}{2}}$

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

$Q=e{N}_{d}Aw=e{N}_{d}A(\frac{2ϵ{ϵ}_0}{e{N}_d}(V-V_{fb}){)}^{\frac{1}{2}}$

Thus, the differential capacitance is

$C=\frac{\partial Q}{\partial V}=e{N}_{d}A\frac{1}{2}(\frac{2ϵ{ϵ}_0}{e{N}_d}{)}^{\frac{1}{2}}(V-V_{fb}{)}^{-\frac{1}{2}}=A(\frac{e{N}_dϵ{ϵ}_0}{2(V-V_{fb})}{)}^{\frac{1}{2}}$

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.