Page 36 - Photonics Essentials an introduction with experiments

Electrons and Photons

30 Introductory Concepts

at room temperature, there are not very many occupied states in the
conduction band compared to the occupied states in the valence band.
Under most conditions, Boltzmann statistics can be used, as we have
done in Eqs. 2.1–2.5, to calculate the number of states in the conduc-
tion band that are occupied by electrons, or the number of empty sites
in the valence band. These are called holes.

2.7 Summary
The behavior of electrons in semiconductors at equilibrium is ruled by
the Boltzmann distribution under almost all circumstances. The Boltz-
mann distribution says that the probability of finding an electron with
energy E a decreases exponentially as E a increases. The three funda-
mental energy excitations in semiconductors are electrons, photons,
and phonons. We treat the indivisible units of these excitations as par-
ticles. Each particle has a wavelength that is proportional to the recip-
rocal of its momentum. Each particle obeys the two basic laws of con-
servation of energy and momentum. These two laws are the foundation
that determines all the possibilities that photonics has to offer.
The map of allowed electron states is called a band structure. For
semiconductors like GaAs and Si, the electron states are generally
filled up to and including the valence band states or the bonding
states. This is followed by an energy gap that results because there is
an energy difference between the bonding and the antibonding, or
conduction band states. If the highest energy valence band state oc-
curs at the same momentum as the lowest energy conduction band
state, the material is called a direct band gap semiconductor. GaAs
and InP are examples of direct band gap semiconductors. If the mini-
mum energy of the conduction band occurs at a different momentum
than the maximum energy on the valence band, then the material is
known as an indirect band gap semiconductor. Si and Ge are exam-
ples of indirect band gap materials.
The thermal energy available from the environment can act to
break bonding states. This action creates vacancies in the occupation
of the valence band called holes, because the electrons that main-
tained those bonds are absent. The liberated electrons are now in an-
tibonding states in the conduction band. The Boltzmann distribution
is used to keep track of the number of electron states that are occu-
pied in the conduction band as a function of temperature.

Bibliography

C. Cercignani, Ludwig Boltzmann, The Man Who Trusted Atoms, New York,
Oxford Univeristy Press, 1998. Boltzmann’s ideas about the direction of

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