84 CHAPTER 3. OPTICAL TRANSMITTERS
rate, andRtot≡Rrr+Rnris the total recombination rate. It is customary to introduce
the recombination timesτrrandτnrusingRrr=N/τrrandRnr=N/τnr, whereNis the
carrier density. The internal quantum efficiency is then given by
ηint=τnr
τrr+τnr. (3.1.17)
The radiative and nonradiative recombination times vary from semiconductor to
semiconductor. In general,τrrandτnrare comparable for direct-bandgap semicon-
ductors, whereasτnris a small fraction (∼ 10 −^5 )ofτrrfor semiconductors with an
indirect bandgap. A semiconductor is said to have a direct bandgap if the conduction-
band minimum and the valence-band maximum occur for the same value of the elec-
tron wave vector (see Fig. 3.2). The probability of radiative recombination is large in
such semiconductors, since it is easy to conserve both energy and momentum during
electron–hole recombination. By contrast, indirect-bandgap semiconductors require
the assistance of a phonon for conserving momentum during electron–hole recombina-
tion. This feature reduces the probability of radiative recombination and increasesτrr
considerably compared withτnrin such semiconductors. As evident from Eq. (3.1.17),
ηint1 under such conditions. Typically,ηint∼ 10 −^5 for Si and Ge, the two semicon-
ductors commonly used for electronic devices. Both are not suitable for optical sources
because of their indirect bandgap. For direct-bandgap semiconductors such as GaAs
and InP,ηint≈ 0 .5 and approaches 1 when stimulated emission dominates.
The radiative recombination rate can be written asRrr=Rspon+Rstimwhen radia-
tive recombination occurs through spontaneous as well as stimulated emission. For
LEDs,Rstimis negligible compared withRspon, andRrrin Eq. (3.1.16) is replaced with
Rspon. Typically,RsponandRnrare comparable in magnitude, resulting in an internal
quantum efficiency of about 50%. However,ηintapproaches 100% for semiconductor
lasers as stimulated emission begins to dominate with an increase in the output power.
It is useful to define a quantity known as thecarrier lifetimeτcsuch that it rep-
resents the total recombination time of charged carriers in the absence of stimulated
recombination. It is defined by the relation
Rspon+Rnr=N/τc, (3.1.18)whereNis the carrier density. IfRsponandRnrvary linearly withN,τcbecomes a
constant. In practice, both of them increase nonlinearly withNsuch thatRspon+Rnr=
AnrN+BN^2 +CN^3 , whereAnris the nonradiative coefficient due to recombination at
defects or traps,Bis the spontaneous radiative recombination coefficient, andCis the
Auger coefficient. The carrier lifetime then becomesNdependent and is obtained by
usingτ−c^1 =Anr+BN+CN^2. In spite of itsNdependence, the concept of carrier
lifetimeτcis quite useful in practice.
3.1.4 Semiconductor Materials
Almost any semiconductor with a direct bandgap can be used to make ap–nhomojunc-
tion capable of emitting light through spontaneous emission. The choice is, however,
considerably limited in the case of heterostructure devices because their performance