144 CHAPTER 4. OPTICAL RECEIVERS
remains constant at every point inside the multiplication region. If we replaceihin Eq.
(4.2.3) byI−ie, we obtain
die/dx=(αe−αh)ie+αhI. (4.2.6)In general,αeandαharexdependent if the electric field across the gain region is
nonuniform. The analysis is considerably simplified if we assume a uniform electric
field and treatαeandαhas constants. We also assume thatαe>αh. The avalanche
process is initiated by electrons that enter the gain region of thicknessdatx=0. By
using the conditionih(d)=0 (only electrons cross the boundary to enter then-region),
the boundary condition for Eq. (4.2.6) isie(d)=I. By integrating this equation, the
multiplication factordefined asM=ie(d)/ie( 0 )is given by
M=
1 −kA
exp[−( 1 −kA)αed]−kA, (4.2.7)
wherekA=αh/αe. The APD gain is quite sensitive to the ratio of the impact-ionization
coefficients. Whenαh=0 so that only electrons participate in the avalanche process,
M=exp(αed), and the APD gain increases exponentially withd. On the other hand,
whenαh=αe, so thatkA=1 in Eq. (4.2.7),M=( 1 −αed)−^1. The APD gain then
becomes infinite forαed=1, a condition known as theavalanche breakdown.Al-
though higher APD gain can be realized with a smaller gain region whenαeandαhare
comparable, the performance is better in practice for APDs in which eitherαeαhor
αhαeso that the avalanche process is dominated by only one type of charge carrier.
The reason behind this requirement is discussed in Section 4.4, where issues related to
the receiver noise are considered.
Because of the current gain, the responsivity of an APD is enhanced by the multi-
plication factorMand is given by
RAPD=MR=M(ηq/hν), (4.2.8)where Eq. (4.1.3) was used. It should be mentioned that the avalanche process in APDs
is intrinsically noisy and results in a gain factor that fluctuates around an average value.
The quantityMin Eq. (4.2.8) refers to the average APD gain. The noise characteristics
of APDs are considered in Section 4.4.
The intrinsic bandwidth of an APD depends on the multiplication factorM. This
is easily understood by noting that the transit timeτtrfor an APD is no longer given
by Eq. (4.2.1) but increases considerably simply because generation and collection of
secondary electron–hole pairs take additional time. The APD gain decreases at high
frequencies because of such an increase in the transit time and limits the bandwidth.
The decrease inM(ω)can be written as [24]
M(ω)=M 0 [ 1 +(ωτeM 0 )^2 ]−^1 /^2 , (4.2.9)whereM 0 =M( 0 )is the low-frequency gain andτeis the effective transit time that
depends on the ionization coefficient ratiokA=αh/αe. For the caseαh<αe,τe=
cAkAτtr, wherecAis a constant (cA∼1). Assuming thatτRCτe, the APD bandwidth is
given approximately by∆f=( 2 πτeM 0 )−^1. This relation shows thetrade-offbetween