Page 131 - Photonics Essentials an introduction with experiments

Light-Emitting Diodes

Light-Emitting Diodes 125

ing rate – radiative recombination rate – nonradiative recombination
rate, or

dN J N – n D
= – B(NP – n i ) – (6.15)
2
dt qd n–r
where:
dN/dt = the rate of change in the electron density at the p-n
junction
B(NP – n i ) = the net change in electron–hole concentration due to
2
radiative recombination
(N – n D )/ n–r = the change in the carrier concentration due to nonra-
diative recombination.
In these expressions, the majority carrier doping density, n D , and the
intrinsic carrier concentration, n i , are written in lower case to remind
us that these quantities remain constant during the LED operation.
The optical output comes from the net electron–hole recombination
rate. The output power can be written as

P out = N photons/sec × photon
2
= ext · volume · B(NP – n i )· (6.16)
In this expression, ext is the external quantum efficiency and is the
fraction of photons created that actually are emitted in free space. Be-
cause of the large index of refraction of semiconductor materials, we re-
call from Section 6.4 that most of the photons emitted are trapped by
total internal reflection inside the LED. A typical value for ext is 0.02.
In steady state, dN/dt = 0 and

J N – n D
2
= B(NP – n i ) + (6.17)
qd n–r
We would like to rewrite this equation in terms of N – n D . This will al-
low us to combine the two terms on the right-hand side of the equa-
tion, and to compare the recombination rates for radiant recombina-
tion to those for nonradiant recombination. This comparison gives the
internal quantum efficiency.
We can use the charge neutrality condition to write

NP = N(p 0 + P) and P = N = N – n D
n i 2 = N – n D
= N
n D
2
2
= N – n D N + (~n i ) (6.18)
2
Note that B(NP – n i ) = BN(N – n D ).

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