Page 25 - Photonics Essentials an introduction with experiments

Electrons and Photons

Electrons and Photons 19

5
v s /c ~ 10 /10 10 ~ 10 –5
So for the same frequency f (= same energy),

s
= ____________?

What is the frequency of a 40 meV vibration?
–3
E (40 × 10 )(1.6 × 10 –19 )
f = = = 9 × 10 12 = 10 13 Hz (2.27)
h 6.6 × 10 –34
What is the wavelength ?
5
= v s /f = 10 /10 13 = 10 –8 cm
Well, this is only a few times larger than the lattice parameter of Si.
Does this make sense?

The lower limit on the wavelength is the interatomic distance
which is about 0.12 × 10 –8 cm in silicon. So lattice vibrations have a
wavelength that is an integral multiple of the lattice parameter.
These vibrational quanta are called phonons. They are important be-
cause they allow the semiconductor to reach equilibrium.
To summarize our story so far:

Wavelength of a 1 eV electron = 12 Å
Wavelength of a 1 eV photon = 1240 nm
(only true around 1 eV!)
= 1000 × electron
So, what is the wavelength of a 1 eV phonon? The answer is, a 1 eV
phonon does not exist. It cannot exist because its wavelength would
be much smaller than the separation between atoms, and the phonon
represents vibrations of atoms. However, the wavelength of a 40 m eV
phonon is about the same as that for the 1 eV electron.
Since momentum = h/ , at room temperature, the momentum of a
typical phonon is similar to the momentum of 1 eV electron.
As electrons move around in the semiconductor, they need to con-
serve energy and momentum. In this never ending struggle, the
phonon acts as a source of momentum that contributes very little en-
ergy, whereas the photon can contribute energy with very little mo-
mentum. As the electron interacts with light, the electric field, etc.,
both phonons and photons interact with the electron so that both en-
ergy and momentum are conserved.

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