# 3.3.3 Field emission - Fowler-Nordheim tunneling

Field emission - also called Fowler-Nordheim tunneling - is the process whereby electrons tunnel through a barrier in the presence of a high electric field. This quantum mechanical tunneling process is an important mechanism for thin barriers as those in metal-semiconduictor junctions on highly-doped semiconductors.

We derive of the tunnel probability from the time independent Schrödinger equation:

[3.1.44]

which can be rewritten as

[3.1.45]

Assuming that V(x)-E is independent of position in a section between x and x+dx this equation can be solved yielding:

[3.1.46]

The minus sign is chosen since we assume the particle to move from left to right.For a slowly varying potential the amplitude of the wave function at x = L can be related to the wave function at x = 0 :

[3.1.48]

This equation is referred to as the WKB (Wigner, Kramers, Brillouin) approximation. From this the tunneling probability, Q, can be calculated for a triangular barrier for which V(x)-E = qfB (1- )

[3.1.49]

the tunneling probability then becomes

[3.1.50]

where the electric field equals E = fB/L.

The tunneling current is obtained from the product of the carrier charge, velocity and density. The velocity equals the Richardson velocity, the velocity with which on average the carriers approach the barrier while the carrier density equals the density of available electrons multiplied with the tunneling probability, yielding:

The tunneling current therefore depends exponentially on the barrier height to the 3/2 power.

3.3.2 ¬ ­ ® 3.4