Page 139 - Photonics Essentials an introduction with experiments

Light-Emitting Diodes

Light-Emitting Diodes 133

which is no longer constant. The rate equation is
d J 2 – J 1 N
N = – B N(N 1 + P 1 + N) –
dt qd n–r
Since we cannot solve the rate equation explicitly, we will develop
an expression for the rise time of the LED in response to a current
pulse. To carry out this analysis, we will focus on the variables that
are changing with time. To simplify the rate equation, we will assume
that the current at time t = 0 is also 0.

J 1 = 0
Define a relaxation time, r–0 :

1
= B(N 1 + P 1 )
r–0
The simplified rate equation is expressed as
d J 2 N N
N = – – – B N 2 (6.35)
dt qd r–0 n–r
Note from the rate equation that the presence of a quadratic term
means that the transient behavior of the diode during turn-on will not
be the same as its behavior during turn-off. That is, the rise time will
no longer equal the fall time.
In the heavy-injection limit, the excess carrier density N = N –
n D N. We assume that the LED has been turned on at current
density J 2 . After the LED has reached steady state, we apply a small
ac modulation around the steady-state current. The LED bandwidth
can be determined for this small modulation in the approximation
that d/dt N = 0. The rate equation under these conditions is ex-
pressed as
d J 2 N
N = – B(N + P + N) N – = 0 (6.36)
dt qd n–r
N P N (6.36)
We will further assume that the LED is a good device so that nonra-
diative recombination is negligible. This means that n–r r–r .
J N
2
– 3BN – = 0
qd n–r
J 3 3 2
1
2
2
3BN = (BN) = (6.37)
qd B B ac
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