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### Modulation Bandwidth of LED

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We derive the expression for modulation bandwidth of an LED in this lecture. We start by
writing the rate equation, which describes the rate at which the carrier density is
generated/converted to photons.

The rate at which the carriers are injected is I/q, where I is the injected current (in Amperes) and
q represents the charge of an electron. Injected current is also represented in terms of current
density J, defined as the current per unit area (measured in A/m2

). The rate at which the carrier
density reduces is given by n/τc, where τc represents the carrier lifetime, considering both the
radiative and non-radiative recombination. Thus, the rate equation for the carrier density
(measured per unit volume) is given as
dn
dt =
I
q

n
τ௖
=
J
qd −
n
τ௖
where d represents the thickness of the recombination region.
In the steady-state operating conditions, when a direct current I is applied to the junction, dn/dt
should be zero and hence the steady-state carrier density (n) is given by,

I
q
=
n
τ௖
=
J
qd
n =
Jτ௖
qd

In case of intensity modulation, the applied current is not a constant and it changes depending
on the bits that need to be transmitted. In order to find the largest speed with which the LED can
respond, we carry out a harmonic analysis where the response of the system is analysed at a
given frequency. Using Fourier analysis, the response of the system for any periodic function can
then be estimated from the harmonic response.
For carrying out the harmonic analysis, consider the input current applied to the pn junction as, I
= Ib+Imexp(jωmt) where Ib is the DC value and Im is the amplitude of the modulating current with
frequency ωm. We assume that the response of the carrier density is also sinusoidal of the form,
n =nb + nmexp(jωmt). Substituting I and n in the rate equation, the modulation amplitude of
carrier density can be derived as,

n௠ =
I௠/q
jω௠ + 1/τ௖

The frequency response of the system, represented as H(jω) can be calculated as, nm(ω)/nm(0,
where nm(ω)is the carrier density when the system is excited at frequency ω and nm (0)is the
H(jω) =
1
jω௠τ௖ + 1
The magnitude of frequency response can be found as,
|H(jω)| =
1
ඥ(ω௠τ௖)
ଶ + 1

The 3- dB bandwidth defined as the frequency for which the magnitude response reduces to half
its peak value can be found as, ωm= √3/τ௖

or, f௠ =
√ଷ
ଶగఛ೎
. Thus, carrier recombination time –that includes both the Radiative and non-radiative lifetimes. Larger recombination time would imply that the carrier density cannot respond as fast as the input modulation speed and so this results in a reduction in the bandwidth of the device.