Mott–Schottky Equation

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

$\frac = \frac (V - V_ - \frac)$

where $C$ is the differential capacitance $\frac$, $\epsilon$ 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, $V_$ 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

$w = (\frac ( V - V_ ) )^\frac$

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

$Q = e N_d A w = e N_d A (\frac ( V - V_ ) )^\frac$

Thus, the differential capacitance is

$C = \frac = e N_d A \frac(\frac)^\frac ( V - V_ )^ = A (\frac)^\frac$

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

{{DEFAULTSORT:Mott-Schottky equation
Equations