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208 Fiber Optic Communications

CB

Initial state of
an electron

Final state of
two electrons in
CB and one
hole in VB

VB

Figure 5.15 Schematic representation of impact ionization in a direct band-gap semiconductor. Note: CB = conduction
band and VB = valence band.

process terminates when all the free carriers are swept out of the high-electric-field region. In the end, one ini-
tial electron or hole generates M extra e–h pairs, where M is called the multiplication gain of the photodetector.
A generalized theory regarding the impact ionization phenomena in semiconductor materials has been
developed by Baraff [20] in terms of threshold ionization energy (E ), optical phonon scattering energy (E ),
r
i
and carrier mean free path () limited due to optical phonon scattering. However, Baraff’s expression for the
impact ionization could be evaluated only numerically. A simple expression for the ionization parameters
of the charged carrier in a semiconductor as a function of an electric field and a lattice temperature has been
developed by Okuto and Crowell [17, 19]. They expressed the ionization parameters in terms of a power-series
expansion of the functions of an electric field (F), optical phonon energy, carrier mean free path due to optical
phonon scattering, and threshold energy for ionization. By fitting Baraff’s numerical results for and versus
electric field at low field values and imposing energy conservation conditions, Okuto and Crowell obtained a
semi-analytical expression for ionization coefficients. The expression is given by
√
⎧ √ [ ] 2 ⎫
( ) 1.14 √ ( ) 1.14 [ ] 2
qF ⎪ E ie;h √ E ie;h E ie;h ⎪
; = exp 0.217 − √ 0.217 + , (5.33)
E ⎨ E E qF ⎬
ie;h re;h re;h e;h
⎪ ⎪
⎩ ⎭
where E denotes the threshold energy of electron and hole ionization, and E represent the mean free
ie;h e;h re;h
path and average energy loss of carriers per collision, respectively, due to optical phonons. The temperature
dependence of and in the above relation comes from the temperature dependence of E ie;h , , and E re;h
e;h
and is given below (E is for the case in InP, as an example)
g
3.63 × 10 −4 2
E (T)= 1.421 − T , (5.34)
g
(T + 162)
E ieo
E (T)= E (T) ⋅ ,
ie g
E (300)
g
E iho
E (T)= E (T) ⋅ ,
ih
g
E (300)
g