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Lasers 123

where
m eff,1 m eff,2
m = (3.104)
r
m eff,1 + m eff,2
is the reduced effective electron mass [13]. For indirect band-gap materials, the momenta of electrons in
the conduction band and in the valence band are different. Typically, the difference in momenta is much
larger than the photon momentum and, therefore, the momentum can not be conserved in the electron–photon
interaction unless the photon emission is mediated through a phonon. A phonon refers to the quantized lattice
vibration or sound wave. If the momentum of the phonon is equal to the difference ℏ(k − k ), the chance of
1
2
photon emission as an electron jumps from conduction band to valence band increases. In other words, in the
absence of phonon mediation, the event that an electron makes a transition from conduction band to valence
band emitting a photon is less likely to happen. Therefore, indirect band-gap materials such as silicon and
germanium are not used for making lasers, while direct band-gap materials such as GaAs and InP (and their
mixtures) are used for the construction of lasers. However, silicon can be used in photo-detectors. As we will
discuss in Chapter 5, an electron in the valence band jumps to the conduction band by absorbing a photon.
We may expect that such an event is less likely to happen in silicon because of the momentum mismatch in
the electron–photon interaction. But the crystal lattice vibrations (crystal momentum) provide the necessary
momentum so that the momentum is conserved during the photon-absorption process. In contrast, during the
photon-emission process, phonon mediation is harder to come by since the free electrons in the conduction
band are not bound to atoms, and, therefore, they do not vibrate within the crystal structure [10].

Example 3.6
In a direct band-gap material, an electron in the conduction band having a crystal momentum of
7.84 × 10 −26 Kg⋅ m/s makes a transition to the valence band emitting an electromagnetic wave of wavelength
0.8 μm. Calculate the band-gap energy. Assume that the effective mass of an electron in the conduction
band is 0.07m and that in the valence band is 0.5m, where m is the electron rest mass. Assume parabolic
conduction and valence band.

Solution:
The reduced mass m is related to the effective masses by Eq. (3.104),
r
m eff,1 m eff,2
m = .
r
m eff,1 + m eff,2
m eff,1 = 0.07m, m eff,2 = 0.5m, electron mass m = 9.109 × 10 −31 kg,
0.07 × 0.5 −31 −32
m = × 9.109 × 10 = 5.59 × 10 kg.
r
0.07 + 0.5
The electron momentum is
ℏk = 7.84 × 10 −26 kg ⋅ m/s.
1
The photon energy is
c 1.054 × 10 −34 × 2 × 3 × 10 8
ℏ = ℏ2 =
0.8 × 10 −6
−19
= 2.48 × 10 J.