Page 135 - Photonics Essentials an introduction with experiments

Light-Emitting Diodes

Light-Emitting Diodes 129

1
= BN 1.41 × 10 sec –1
9
r–r
1.41 × 10 –9
int = = 0.74
1.41 × 10 –9 + 0.5 × 10 –9
1.24
Power = 0.1 · 0.74 · · 50 mA = 3.4 mW
1.35

In the example above we found the steady-state output after an in-
crease in the current from 0 to 50 mA. In the next section, we will con-
sider the dynamic response of the LED to this step. The step response
time, or the rise time, step , can be used to determine the modulation
bandwidth directly:
1
Bandwidth = (6.27)
· step
Summarizing some of the important results so far:

N = P (Excess electron concentration = excess hole concen-
tration)
2
B(NP – n i ) = BN(N – n D )
(Radiative recombination rate)
BN = 1/ r–r
P out = (I/q) (Optical power is proportional to electric current)

6.8 Rise Time of the Light-Emitting Diode
The basic signal in a digital optical communications link is a pulse of
photons. The pulse is created by increasing the bias current from
some initial value I 1 to I 2 and then from I 2 to I 1 . Although the current
through the LED can be increased as fast as the RC time constant of
the diode will permit, there is an additional delay associated with the
appearance of additional light emission. The pulse of photons appears
only after the excess carrier density starts to recombine. We will de-
scribe this transition using the rate equation. The current pulse cre-
ates excess carriers, just as we saw in the steady-state analysis in the
previous section. There are only two possibilities: either this excess
carrier density is greater than the majority carrier density created by
doping, or it is not. In the first case, called the high-injection or low-
doping limit, it is not possible to solve the rate equation explicitly. Nu-
merical modeling can be used to map out the LED time response. De-
spite this difficulty, we can still determine a functional form for the

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