Page 24 - Photonics Essentials an introduction with experiments

Electrons and Photons

18 Introductory Concepts

Exercise 2.5
For each broken bond in a perfect crystal of silicon, an electron is pro-
moted from the valence band to the conduction band. Using Boltz-
mann statistics you can write:

n antibonding
= e – E/kT
n bonding
At room temperature, we will approximate kT by 0.025 eV,

n antibonding
= e –1/0.025 = e –40
10 24 atoms/cm 3
n antibonding e –40 ·10 24 = ?? (2.24)
This is an interesting number. Take the log of both sides:
log 10 (n antibonding ) 24 – 40log 10 (e) = 24 – (40)(0.4) = 24 – 16 = 8
8
n antibonding 10 bonds/cm 3

This back of the envelope estimate shows that on the average a
semiconductor whose band gap (= antibonding – bonding energies) = 1
8
3
eV will have about 10 broken bonds per cm . A more detailed calcula-
tion for silicon based on the same principles gives ~10 10 cm –3 broken
bonds at room temperature.
When the bond is broken, the electron is promoted from the valence
band, or bonding orbitals to the conduction band or antibonding or-
bitals. Another name of the conduction band is simply the set of unoc-
cupied levels that are closest in energy to the valence band levels.
When the bonds are not broken, they act like springs that hold the
atoms in the crystal at the right distance from each other. These
springs vibrate as a way of storing the thermal energy of the crystal.
The vibrational energy of each atom = – kT for each degree of freedom,
1
2
3
or – kT. So the average vibrational energy at room temperature is
2
about 40 meV. These vibrations have a frequency and a wavelength
that are related by the speed of sound:
v s = f
5
3
The speed of sound in solid materials is about 10 cm/sec = 10 m/sec.

Exercise 2.6
What is the ratio of v s to the speed of light?

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