In practice you may want to consider a set of discrete trap levels or a continuum of trap states (with a density of states gt(
Et)). Both of these approaches can be accommodated by defining
Nt, the density of traps per unit volume at a particular trap energy,
Et. For a continuum of trap states,
Nt is given by:
For the trap energy Et, carriers in the conduction band or valence band with energy
E make the following contributions to the total recombination or generation rate per unit volume for each process:
where Nt is the number of traps per unit volume,
ft is the trap electron occupancy (between 0 and 1),
gc(
E) is the conduction band density of states,
gv(
E) is the valence band density of states, and
cec(
E),
cee(
E),
chc(
E), and
che(
E) are rate constants. The net rate of capture for electrons and holes in the energy interval d
E can be written as:
In thermal equilibrium the occupancy of the electron traps ft is determined by Fermi-Dirac statistics,
fte=1/(1+exp[(
Et-
Ef)/(
kBT)]/g
D) (where
gD is the degeneracy factor). The above equations can be simplified to yield:
Equation 3-82 applies even away from equilibrium. Substituting this equation back into
Equation 3-81, rearranging and integrating, gives the total rate of electron (
re) or hole (
rh) capture to traps at the specified energy,
Et:
Introducing the constants Cn and
Cp, which represent the average capture probability of an electron over the band:
Equation 3-83 and
Equation 3-84 define the electron and hole recombination rates associated with the trap energy level.
Equation 3-87 determines the occupancy of the traps at the level
Et. For a continuous distribution of traps the equivalent expression is:
The total recombination rate for electrons and holes is given by integrating Equation 3-83 over all the distributed traps and summing over the distinct discrete traps (denoted by the superscript
i), giving the following result:
The total charge density, Q, that results from the traps is given by:
Equation 3-89 can be rewritten in the form:
where E0 is an energy within the band gap referred to as the neutral level.
E0 is chosen such that: