Schottky diode current

The current across a metal-semiconductor junction is mainly due to majority carriers. Three distinctly different mechanisms exist: diffusion of carriers from the
semiconductor into the metal, thermionic emission of carriers across the Schottky barrier and quantum-mechanical tunneling through the barrier. The diffusion theory
assumes that the driving force is distributed over the length of the depletion layer. The thermionic emission theory on the other hand postulates that only energetic
carriers, those, which have an energy equal to or larger than the conduction band energy at the metal-semiconductor interface, contribute to the current flow.
Quantum-mechanical tunneling through the barrier takes into account the wave-nature of the electrons, allowing them to penetrate through thin barriers. In a given
junction, a combination of all three mechanisms could exist. However, typically one finds that only one current mechanism dominates.

The analysis reveals that the diffusion and thermionic emission currents can be written in the following form:

.

This expression states that the current is the product of the electronic charge, q, a velocity, v, and the density of available carriers in the semiconductor located
next to the interface. The velocity equals the mobility multiplied with the field at the interface for the diffusion current and the Richardson velocity for the thermionic emission current. The minus one term ensures that the current is zero if no voltage is applied as in thermal equilibrium any motion of carriers is balanced by a motion of carriers in the opposite direction.

The tunneling current is of a similar form, namely:

where vR is the Richardson velocity and n is the density of carriers in the semiconductor. The tunneling probability term, Q, is added since the total current depends
on the carrier flux arriving at the tunnel barrier multiplied with the probability, Q, that they tunnel through the barrier.

Diffusion current
This analysis assumes that the depletion layer is large compared to the mean free path, so that the concepts of drift and diffusion are valid. The resulting current
density equals:

The current therefore depends exponentially on the applied voltage, Va, and the barrier height, fB. The prefactor can more easily be understood if one rewrites it
as a function of the electric field at the metal-semiconductor interface,

_max:

Thermionic emission
The thermionic emission theory assumes that electrons, with an energy larger than the top of the barrier, will cross the barrier provided they move towards the
barrier. The actual shape of the barrier is hereby ignored. The current can be expressed as:

where

is the Richardson constant and fB is the Schottky barrier height.

The expression for the current due to thermionic emission can also be written as a function of the average velocity with which the electrons at the interface approach
the barrier. This velocity is referred to as the Richardson velocity given by: