Mott–Schottky equation

The Mott–Schottky equation relates the capacitance to the applied voltage across a semiconductor-electrolyte junction[1].

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

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 flatband 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[2]

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

Using the abrupt approximation[2], 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[2].

References

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