Page 136 - Photonics Essentials an introduction with experiments

Light-Emitting Diodes

130 Photonic Devices

rise time. In the second case, known as the low-injection limit or the
high-doping limit, the rate equation can be solved explicitly to give
the time response of the LED to a current pulse, as well as the rise
time. We will treat this second case first.
We will analyze the response of the LED to a step increase in the
drive current. At time t 0, the LED is operating at steady state at
current density J 2 . First let us look at the steady state current before
and after the current step:
N 1 – n D
2
J(t = 0) = J 1 = qd B(N 1 P 1 – n i ) –
n–r
At t > 0, the current density is raised to J 2 , where it remains:
N 2 – n D
J(t = ) = J 2 = qd B(N 2 P 2 – n i ) –
2
n–r
The key to the transient analysis is the excess current density,
which is now a function of time:

N(t) = N(t) – N 1 = P(t) – P 1
The rate equation for t > 0 can be written as

d d d J 2 N – n D
N(t) = [N 1 + N(t)] = N(t) = – B(NP – n i )– (6.28)
2
dt dt dt qd n–r
Next, we substitute for N and P:
d J 2 N 1 + N – n D
2
N(t) = – B[(N 1 + N)(P 1 + P) – n i ] –
dt qd n–r
In this last equation, we can identify the current density before the
current pulse was applied:
d J 2 N 1 – n D N
2
N = – B[N 1 P 1 – n i ] – – B[ NP 1 + PN 1 – P N] –
dt qd n–r n–r
N
J 2 – J 1
= – B N[P 1 + N 1 + P] –
qd n–r
J 2 – J 1 1
= – N B(P 1 + N 1 + N) + (6.29)
qd n–r
As we have done before, we recognize in the second term a relaxation
time. This is the transient response term that we are looking for:
1 1
B(P 1 + N 1 + N) + (6.30)
step n–r

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