Mott–Schottky equation explained

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

(1)/(C ^{2}) = (2)/(\epsilon \epsilon _{0} A ^{2} e N _{d}) (V - V _{fb} - (k _{B} T)/(e) )

where

C is the differential capacitance (\partial{Q)/() }, \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 isw = ( (2 \epsilon \epsilon _{0})/(e N _{d}) ( V - V _{fb} ) ) ^{(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

e N _{d} , and the total charge of the depletion region, compensated by opposite charge nearby in the electrolyte, isQ = e N _{d} A w = e N _{d} A ( (2 \epsilon \epsilon _{0})/(e N _{d}) ( V - V _{fb} ) ) ^{(1)/(2)}

Thus, the differential capacitance is

C = (\partial{Q)/() } = 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.^[2]

Notes and References

1. Gelderman . K. . Flat-Band Potential of a Semiconductor: Using the Mott–Schottky Equation . . 84 . 685 . 2007 . 4 . 10.1021/ed084p685. 2007JChEd..84..685G .
2. Book: Grundmann, Marius . The Physics of Semiconductors . Section 20.2.2 . Springer . 2010 . 978-3-642-13883-6.