Page 38 - Photonics Essentials an introduction with experiments

Electrons and Photons

32 Introductory Concepts

Problems
2.1 A p-n junction is a metallurgical junction between two materials
having different numbers of free electrons in their respective
conduction bands. At equilibrium, Boltzmann statistics can be
applied. Use this information to determine the energy difference
in electron volts between the conduction bands on each side of
the junction if the n-side has 10 18 cm –3 free electrons and on the
4
p-side there are 10 cm –3 electrons. Assume that the junction is
at room temperature.
2.2 We know that a photon cannot interact with a free electron be-
cause simultaneous conservation of energy and momentum is
not possible. That is, their energy band structures do not inter-
sect. In a collision between an electron, a photon, and a phonon,
an interaction is possible. This can happen in a solid like Si or
GaAs.
a. Calculate the wavelength, the frequency, and the energy of the
phonon in silicon that will allow a 1 eV photon to transfer all its
energy to an electron. Assume that the electron is initially at
rest (E = 0) (that is, T = 0). The velocity of sound in silicon is
3
about 8.5 × 10 meters per second at room temperature.
b. What is the final energy of the electron?
c. If the collision takes place in silicon at room temperature,
what is the likely initial energy of the electron?
2.3 Electrons in a semiconductor have the full electronic charge q,
but often their mass appears to be different from the free elec-
tron mass. In GaAs, for example, the effective mass of an elec-
tron is equal to 0.065 the value of the free electron mass. The
size of the effective mass depends on both the structure and the
crystalline potential of the semiconductor. Given this informa-
tion:
a. Calculate the de Broglie wavelength of a conduction band
electron in GaAs, assuming a kinetic energy equal to the ther-
mal energy at room temperature.
b. The wavelength corresponds to how many unit cells of the
crystal?
c. In three dimensions, estimate how many atoms could be
found in a sphere the diameter of which is equal to a de
Broglie wavelength in GaAs.

2.4 Show from first principles that the energy of a photon can be cal-
culated from its wavelength by the following relationship:

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