200 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES
There are several important limits of the spontaneous rate:
i.Inthecasewheretheelectronandholedensitiesnandparesmall (non degenerate case),
the Fermi functions have a Boltzmann form(exp(−E/kBT)). The recombination rate is
found to be
Rspon=
1
2 τo
(
2 π^2 m∗r
kBTm∗em∗h
) 3 / 2
np (4.9.7)
The rate of photon emission depends upon the product of the electron and hole densities.
If we define the lifetime of a single electron injected into a lightly doped(p=Na≤
1017 cm−^3 )p-type region with hole densityp, it would be given from equation 4.9.7 by
Rspon
n
=
1
τr
=
1
2 τo
(
2 π^2 m∗r
kBTm∗em∗h
) 3 / 2
p (4.9.8)
The timeτrin this regime is very long (hundreds of nanoseconds), as shown in figure 4.35,
and becomes smaller aspincreases.
ii.Inthecasewhereelectronsareinjectedintoaheavilydopedp-region(orholesareinjected
intoaheavilydopedn-region),thefunctionfh(fe)canbeassumedtobeunity. The
spontaneous emission rate is
Rspon∼
1
τo
(
m∗r
m∗h
) 3 / 2
n (4.9.9)
for electron concentrationninjected into a heavily dopedp-type region and
Rspon∼
1
τo
(
m∗r
m∗e
) 3 / 2
p (4.9.10)
for hole injection into a heavily dopedn-type region.
The minority carrier lifetimes (i.e.,n/Rspon) play a very important role not only in LEDs
but also in diodes and bipolar devices. In this regime the lifetime of a single electron (hole)
is independent of the holes (electrons) present since there is always a unity probability that
the electron (hole) will find a hole (electron). The lifetime is now essentiallyτo,asshown
in figure 4.35.
iii.Anotherimportantregimeisthatofhighinjection,wheren=pissohighthatonecan
assumefe=fh= 1 intheintegralforthespontaneousemissionrate. The spontaneous
emission rate is
Rspon∼
n
τo
∼
p
τo
(4.9.11)
and the radiative lifetime (n/Rspon=p/Rspon)isτo.
