Page 29 - Photonics Essentials an introduction with experiments

Electrons and Photons

Electrons and Photons 23

the cat out the bag so to speak, for which he was awarded the Nobel
Prize in 1929. He claimed credit in his thesis for “the first plausible
physical explanation for the condition of stable orbits as proposed by
Bohr and Sommerfeld.”
I find that the most interesting part of de Broglie’s reasoning to be
the notion that because quantization exists, there must be an associ-
ated wave behavior.

2.6 Changing Places: How Electrons Behave
in Solids
The energy momentum relationship for an electron is the same as the
energy momentum relationship for a baseball. But, because the elec-
tron has a wavelength, we can represent its behavior by a wavefunc-
tion:
(k, x) = A sin(kx)
A semiconductor crystal is a periodic arrangement of atoms. The peri-
odicity applies to all the physical properties of the crystal. This means
that the allowed values for energy and momentum have to be period-
ic, too:
A sin(kx) = A sin[k(x + a)], where a = the period of the crystal lattice
= A sin kx cos ka – A cos kx sin ka
This is true if
ka = 2

or
2
k =
a
At these special k values, everything looks the same. Since every-
thing looks the same, we just keep the central zone that has the
unique information between k = – /a and k = /a. This is called the
Brillouin zone. Brillouin was a classmate of de Broglie.
The diagram in Fig. 2.7 has its characteristic shape because of the
periodicity, or to use a more general term, the symmetry of the crys-
tal. There are two essential components of the energy–momentum re-
lationship in crystals of real materials: symmetry and chemistry. The
component added by chemistry is the potential added by the atoms
that make up the crystal. Si atoms have a different potential from Ge
atoms, and the energy–momentum relationship for Si is slightly dif-
ferent from that for Ge.

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