Page 137 - Photonics Essentials an introduction with experiments

Light-Emitting Diodes

Light-Emitting Diodes 131

Equation (6.30) shows the basic physics of the situation. In a good
LED, we can neglect the nonradiative term compared to the radiative
term. Thus, the transient response time, step , is inversely proportion-
al to the carrier concentration. This could be the carrier concentration
due to doping or induced by the current pulse.
The rate equation can now be expressed as

d J 2 – J 1 N
N = – (6.31)
dt qd step
This is a simple (in appearance!) differential equation in N. Howev-
er, since 1/ step also depends on N, a closed-form solution will be pos-
sible only under special circumstances. If N N 1 , we can treat step
as a constant. This condition corresponds to the limit of high doping
or low injection. Then Eq. (6.31) can be solved analytically. Otherwise,
only a numerical solution is possible.

Case 1. Low-Injection Limit
In the low-injection limit, step is treated as a constant. Eq. (6.31) is a
first-order differential equation with a driving term N/ step . The so-
lution is written as

N(t) = Ae –t/ step + Be –t/ step + C
where A, B, and C are constants to be determined by the boundary
conditions.

Boundary conditions

1. At time t = 0, N = 0.
2. At time t = , N = [(J 2 – J 1 )/qd]· step .

Applying boundary condition (1), A + B + C = 0.
Applying boundary condition (2), it follows that B = 0, and

(J 2 – J 1 )
C = · step = –A
qd
So the particular solution is expressed:
(J 2 – J 1 ) (J 2 – J 1 )
N(t) = – · step e –t/ step + · step
qd qd
(J 2 – J 1 )
= · step (1 – e –t/ step )
qd

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