An ionization coefficient ratio k is defined such that divided by and an equation for the excess noise factor is given that relates average multiplication M with k.
(1)
McIntyre [1] made the assumption that the ionization coefficient only depended on the local electric field and did not consider the ‘dead space effect’.
Nonlocal ionization coefficients
Dead space is nonlocal effect. A nonlocal ionization coefficient n(z) is defined by Marsland [2] such that is the probability that a carrier startin8 with no kinetic energy at z = 0 will impact ionize in the interval (z, z + dz).
The ionization path length PDF, can be defined such that is the probability that a carrier will impact ionize for the first time in the given interval.
The probability that a carrier will travel to z without ionizing is called the survival probability .
The survival probability can be related to the ionization path length PDF:
The nonlocal ionization coefficient n(z) is related to the PDF :
A model for ionization PDF
This behaviour can be described by the following expression where 1 is the length of dead space region and a and b are the constants governing the slope of the rise and fall of h(z):
(4)
The above equation has been fitted to h(z) calculated using Monte Carlo techniques [3] for electrons in GaAs at a field of 3 x 10 7 using the following parameters;
Jacob et al. [3] have also calculated the ionization path length PDF for electrons at higher field of . Again this result can be fitted using the following parameters; :
and l=0.0415{\mu m}^{-1}\):