The document discusses the optical properties of solids, focusing on interband absorption in indirect gap semiconductors like silicon and germanium. It explains that indirect transitions require both photon and phonon interactions, resulting in a lower transition rate compared to direct transitions. The document also includes comparisons of absorption rates between silicon and GaAs, highlighting the differences in their band gap properties.

1. Optical Properties of Solids
Presented By: Dr. Adnan Khalil
1

2. Books Recommended:
2
1. Optical properties of solids by Mark Fox, 2nd edition
(Oxford University Press, 2010)
2. Solid State Physics II: Optical Properties of Solids by M. S.
Dresselhaus
3. Optical Properties of Solids by Dick Bedeaux
4. Non Linear Optics, theory, numerical Modeling and
applications by Partha P. Banerjee . 1st edition (CRC
Press; 2003)
5. Optical Properties of Solids by Frederick Wooten (Elsevier
Science, 2013)

3. Interband absorption
3
Band edge absorption in indirect gap semiconductors
In the previous sections, we have been concentrating on direct interband transitions. As
it happens, several of the most important semiconductors have indirect band gaps, most
notably silicon and germanium. Indirect gap semiconductors have their conduction band
minimum away from the Brillouin zone centre, as shown schematically in Fig. 3.2(b).
Transitions at the band edge must therefore involve a large change in the electron wave
vector. Optical frequency photons only have a very small k vector, and it is not possible
to make this transition by absorption of a photon alone: the transition must involve a
phonon to conserve momentum.

4. Interband absorption
4
Band edge absorption in indirect gap semiconductors
Consider an indirect transition that excites an electron in the valence band in state (Ei,
ki) to a state (Ef , kf ) in the conduction band. The photon energy is ω, while the
phonon involved has energy Ω and wave vector q. Conservation of energy demands
that:
Ef = Ei + ω ± Ω,
while conservation of momentum requires that:
kf = ki ± q
The ± factors allow for the possibility of phonon absorption or emission, with the + sign
corresponding to absorption, and the − sign to emission.

5. Interband absorption
5
Band edge absorption in indirect gap semiconductors
Before considering the shape of the band edge absorption spectrum, we can first make
a general point. Indirect transitions involve both photons and phonons. In quantum-
mechanical terms, this is a second-order process: a photon must be destroyed, and a
phonon must be either created or destroyed. This contrasts with direct transitions which
are firstorder processes because no phonons are involved. The transition rate for
indirect absorption is therefore much smaller than for direct absorption.

6. Interband absorption
6
Band edge absorption in indirect gap semiconductors
The smaller transition rate for indirect processes is clearly shown by the data given in
Fig., which compares the band edge absorption of silicon and GaAs. Silicon has an
indirect band gap at 1.12 eV, while GaAs has a direct gap at 1.42 eV. We see that the
absorption rises much faster with frequency in the direct gap material, and soon exceeds
the indirect material even though its band gap is larger. The absorption of GaAs is
roughly an order of magnitude larger than that of silicon for energies greater than ∼ 1.43
eV.

7. Interband absorption
7
Band edge absorption in indirect gap semiconductors
The derivation of the quantum-mechanical transition rate for an indirect gap
semiconductor is beyond the scope of this book. such a calculation give the following
result:
This shows that we expect the absorption to have a threshold close to Eg, but not
exactly at Eg. The difference is ∓Ω, depending on whether the phonon is absorbed or
emitted. Note that the frequency dependence is different to that for direct gap
semiconductors given in eqn 3.25. This provides a convenient way to determine whether
the band gap is direct or not.

8. Interband absorption
8
Band edge absorption in indirect gap semiconductors
Indirect absorption has been thoroughly studied in materials like germanium. The band
structure of germanium is shown in Fig. 3.10. The overall shape of the band dispersion
is fairly similar to that of GaAs given in Fig. 3.4. This is hardly surprising, given that
gallium and arsenic lie on either side of germanium in the periodic table, so that GaAs
and Ge are approximately isoelectronic materials. There is, however, one very important
qualitative difference: the lowest conduction band minimum of germanium occurs at the
L point, where k = (π/a)(1, 1, 1), and not at k = 0. This makes germanium an indirect
semiconductor with a
band gap of 0.66 eV.

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