Page 138 - Photonics Essentials an introduction with experiments

Light-Emitting Diodes

132 Photonic Devices

In the low-injection limit, the rise time, step , is equal to the recom-
bination time, r :

1 1 1 1
= B(N 1 + P 1 + N) + = Bn D + =
step n–r n–r r
Basically, the level of injection remains low enough that the recom-
bination processes in the diode are still governed by the doping levels,
which, of course, are not dependent on time or current.
The output power of the light-emitting diode during the step input
can be calculated using Eq. (6.15):
2
2
P out (t) = ext VB[NP – n i ] = [(N 1 + N)(P 1 + N) – n i ]
2
= ext VB[N 1 N + N 1 P 1 – n i ] (6.32)
where smaller terms have been neglected. This result can be separat-
ed into two terms, one representing the diode behavior before the cur-
rent pulse and the other being the transient response:

P out (t) = ext VB[N 1 P 1 – n i ] + ext VB(N 1 N)
2
= ext V int + ext V N (6.33)

1
J 1

qd r–r
step
= I 1 + ext [(I 2 – I 1 )(1 – e –t/ step )]
r–r
= [(I 2 – I 1 )(1 – e –t/ step ) + I 1 ] (6.34)
Note that
1 1 1 1
= B(N 1 + P 1 + N) + = Bn D + =
step n–r n–r r
is identical to the total recombination rate determined earlier in Eq.
6.21 for an LED operating in steady state. This result is obtained be-
cause the current does not induce an excess carrier concentration as
large as the carrier concentration from the doping. Recombination dy-
namics are therefore determined by the doping and not by the smaller
amount of nonequilibrium carriers induced by current injection.

Case 2. The High-Injection Limit. In this analysis we will consider the
case where the excess carrier density, injected during the current
pulse, is comparable to the equilibrium carrier density introduced by
doping. We start from the same expression for the response time:
1 1
= B(N 1 + P 1 + N) +
step n–r

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