Page 145 - Fiber Optic Communications Fund

126 Fiber Optic Communications

to the active region, but also photons, which increases the interaction among them, and the efficiency of light
generation in a double heterostructure is much higher than in the devices using homojunctions.

3.8.2 Radiative and Non-Radiative Recombination
When a PN junction is forward-biased, electrons and holes recombine to produce light. This is called radiative
recombination. In a semiconductor, electrons and holes can also recombine non-radiatively. In this case, the
energy difference is released as lattice vibrations or given to another electron or hole to increase its kinetic
energy [14–16]. This type of recombination is called non-radiative recombination. In a practical light source,
we like to maximize the radiative recombination by reducing the energy loss due to non-radiative recombi-
nation. Therefore, it is useful to define the internal quantum efficiency of a light source as
R rr R rr
int = = , (3.105)
R tot R + R nr
rr
where R is the radiative recombination rate, R is the non-radiative recombination rate, and R tot is the total
rr
nr
recombination rate. Radiative recombination occurs in two different ways: (i) spontaneous emission and (ii)
stimulated emission,
R = R spont + R stim . (3.106)
rr
For direct band-gap materials, the radiative recombination rate could be larger than the non-radiative rate
since the conservation of energy as well as momentum can be achieved when an electron makes a transition
from the conduction band to the valence band emitting a photon. In contrast, for indirect band-gap materi-
als, such as Si and Ge, the electron–hole recombination is mostly non–radiative and, therefore, the internal
quantum efficiency is quite small. Typically, n is of the order of 10 −5 for Si and Ge.
int

3.8.3 Laser Rate Equations

In Section 3.6 we developed the rate equations for an atomic system with two levels. In the atomic system, the
interaction takes place among the photons, the atoms in the excited level, and in the ground level. Similarly,
in the semiconductor laser diode, the interaction is between the electrons in the conduction band, holes in the
valence band, and photons. Therefore, Eqs. (3.89) and (3.90) may be used to describe the time rate of change
of electrons and photons in a cavity with N being replaced by the electron density N ,
2 e
dN
e
= R pump + R stim + R + R nr
sp
dt
N e
= R pump − GN − , (3.107)
ph
e
dN ph
= R stim + R + R loss
sp
dt
N ph
= GN + R − . (3.108)
ph sp

ph
Here, ≡ 21 represents the lifetime of electrons associated with spontaneous emission and non-radiative
e
transition. In Section 3.3, we found that G = g. This result was derived under the assumption that the light
is a plane wave. But in a double-heterojunction laser, the active region has a slightly higher refractive index