The Mott–Schottky equation relates the

capacitance Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related ...

to the applied

voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...

across a

semiconductor A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...

electrolyte An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...

junction Junction may refer to: Arts and entertainment * ''Junction'' (2012 film), an American film * ''Junction'' (2024 film), an American film * ''Jjunction'', a 2002 Indian film * ''Junction'' (album), a 1976 album by Andrew Cyrille * Junction (E ...

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

where

C

is the differential capacitance

\frac

\epsilon

is the

dielectric constant The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...

of the semiconductor,

\epsilon_0

is the

permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...

A

is the area such that the

depletion region In semiconductor physics, the depletion region, also called depletion layer, depletion zone, junction region, space charge region, or space charge layer, is an insulating region within a conductive, doped semiconductor material where the mobil ...

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 Flat or flats may refer to: Architecture * Apartment, known as a flat in the United Kingdom, Ireland, and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), a two-dimens ...

k_B

is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...

, and T is the

absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...

. This theory predicts that a

Mott–Schottky plot In Photoelectrochemistry#Semiconductor electrochemistry, semiconductor electrochemistry, a Mott–Schottky plot describes the reciprocal of the square of capacitance (1/C^2) versus the potential difference between bulk semiconductor and bulk elect ...

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 In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

into account by abandoning the abrupt approximation and solving the

Poisson–Boltzmann equation The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules ...

for the charge density in the depletion region.

References

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