Page 168 - Electrical Properties of Materials

150 Semiconductors

of wavelength for a thin Si sample is plotted in Fig. 8.18(c). The transmission
may be seen to vary much more gradually with the wavelength.
How does an electron make a transition when excited by a photon? We have
not yet studied this problem in any detail. All I have said so far is that if an elec-
tron receives the appropriate amount of energy, it can be excited to a state of
higher energy. This is not true in general because, as in macroscopic collision
processes, not only energy but momentum, as well, should be conserved.
There is no way out now. If we wish to explain the excitation of electrons
by light in an indirect-gap semiconductor, as shown in Fig. 8.19(b), we have
to introduce phonons and we have to consider the momentum of our quantized
particles.
Phonons are the quantum-mechanical equivalents of lattice vibrations. For
a wave of frequency ω the energy of the phonon is ω, analogous to the energy
of a photon. What can we say about momentum? In Chapter 3 we talked about
the momentum operator, and subsequently we showed that the energy of a free
2 2
electron is equal to E = k /2m, where k comes from the solution of the wave
equation. It is the equivalent of the wavenumber of classical waves.
The momentum of a free electron is k. When an electron is in a lattice,
its energy and momentum are no longer related to each other by the simple
quadratic expression. The relationship is then given by the E–k curve, but the
momentum is still k. The momenta of photons and phonons are given again
by k, but now k = ω/v, where v is the velocity of the wave. For an electro-
–1
8
magnetic wave it is v = c =3 × 10 ms . The velocity of a lattice wave,
which may also be called an acoustic wave or, more commonly, a sound wave,
is smaller by four or five orders of magnitude. Hence, for the same frequency
the momentum of a phonon is much higher than that of a photon.
Let us next do a simple calculation for the momentum of an optical wave.
14
With a wavelength of λ =1 μm, the corresponding frequency is 3 × 10 Hz,
6
–1
It may be seen that the energy of the energy is 1.24 eV, and k photon =6.28 × 10 m . In our simple models of
the photon is comparable with the Chapter 7 the zone boundary came to π/a. With a = 0.3 nm, this means that
–1
gap energies of semiconductors, the range of the electronic value of k extends from k =0 to k =10 10 m .On
but its k value is relatively small. this scale k photon is at the origin to a very good approximation. Hence, when
we talk about electron excitation in a direct-gap semiconductor, we can regard
the transition as practically vertical. The picture of a two-particle interaction is
permissible: a photon gives all its energy to an electron.
In order to excite an electron in an indirect-gap semiconductor, the photon
still has enough energy, but its momentum is insufficient. It needs the good
services of a third type of particle, which can provide the missing momentum.
These particles are phonons, which are always present owing to the finite
temperature of the solid. They can provide high enough momentum to sat-
isfy momentum conservation. Nevertheless, this is a three-particle interaction
between an electron, a photon, and a phonon, which is much less likely than
a two-particle interaction. Hence, as the wavelength decreases, the transmis-
sion will not suddenly increase as in Fig. 8.18(b). It will instead gently rise
and reach saturation when the frequency is large enough [f v in Fig. 8.19(b)] to
affect direct transitions between the valence and conduction bands. Whether
or not a material is a direct-gap material is of increasing importance in the
development of optical semiconductor devices—semiconductor lasers are all
direct-gap materials.