Mott–Schottky equation
The Mott–Schottky equation relates the capacitance to the applied voltage across a semiconductor-electrolyte junction.[1] [math]\displaystyle{ \frac{1}{C^2} = \frac{2}{\epsilon \epsilon_0 A^2 e N_d} (V - V_{fb} - \frac{k_B T}{e}) }[/math]
where [math]\displaystyle{ C }[/math] is the differential capacitance [math]\displaystyle{ \frac{\partial{Q}}{\partial{V}} }[/math], [math]\displaystyle{ \epsilon }[/math] is the dielectric constant of the semiconductor, [math]\displaystyle{ \epsilon_0 }[/math] is the permittivity of free space, [math]\displaystyle{ A }[/math] is the area such that the depletion region volume is [math]\displaystyle{ w A }[/math], [math]\displaystyle{ e }[/math] is the elementary charge, [math]\displaystyle{ N_d }[/math] is the density of dopants, [math]\displaystyle{ V }[/math] is the applied potential, [math]\displaystyle{ V_{fb} }[/math] is the flat band potential, [math]\displaystyle{ k_B }[/math] is the Boltzmann constant, and T is the absolute temperature.
This theory predicts that a Mott–Schottky plot will be linear. The doping density [math]\displaystyle{ N_d }[/math] 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 [math]\displaystyle{ V }[/math]-axis at the flatband potential.
Derivation
Under an applied potential [math]\displaystyle{ V }[/math], the width of the depletion region is[2]
[math]\displaystyle{ w = (\frac{2 \epsilon \epsilon_0}{e N_d} ( V - V_{fb} ) )^\frac{1}{2} }[/math]
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 [math]\displaystyle{ e N_d }[/math], and the total charge of the depletion region, compensated by opposite charge nearby in the electrolyte, is
[math]\displaystyle{ Q = e N_d A w = e N_d A (\frac{2 \epsilon \epsilon_0}{e N_d} ( V - V_{fb} ) )^\frac{1}{2} }[/math]
Thus, the differential capacitance is
[math]\displaystyle{ 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} }[/math]
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
- ↑ Gelderman, K. (2007). "Flat-Band Potential of a Semiconductor: Using the Mott–Schottky Equation". Journal of Chemical Education 84 (4): 685. doi:10.1021/ed084p685. Bibcode: 2007JChEd..84..685G.
- ↑ 2.0 2.1 2.2 Grundmann, Marius (2010). The Physics of Semiconductors. Springer. ISBN 978-3-642-13883-6.
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