Physics for Information Science: Unit I: Electrical Properties of Materials
Brillouin Zones
Definition, Explanation, Proof, Equation, Formula
Brillouin Zones are the boundaries that are marked by the values of propagation wave vector k in which the electrons can have allowed energy values without diffraction.
BRILLOUIN ZONES
Definition: Brillouin Zones are the boundaries that are marked by the values of propagation wave vector k in which the electrons can have allowed energy values without diffraction. Since k is a vector, it has different values along different directions.
Explanation: The relation between the wave vector and the energy of
the electron in a constant potential field can be got as follows.
We know, in a length of potential box ‘Ɩ' the energy of the electron in a constant potential field
............(1)
Also,we
know wave vector
............(2)
(or) 
............(3)
Substituting
equation (3) in (1), we get
............(4)
A plot is made between the total energy 'E' and the wave vector k, for various values of k with n = ±1, ±2,... (since k = n π /Ɩ)
(i.e) for n = ± 1k=± π/Ɩ
for
n= ± 2k = ± 2π/
for
n= ±3k = ± 3π/
For
the above values of k, the curve is obtained as shown in Fig. 1.21, which is in
the form of a parabola with discontinuities.
From the figure it can be seen that the energy of the electron increase continuously from zero to ± π/Ɩ then the electron meets the wall and is reflected.
This
range of allowed energy values in the region between to is called First Brillouin Zone.
The second allowed energy values consists of two parts; one from π/l to 2π/Ɩ and another for-π/Ɩ to -2π/Ɩ is called Second Brillouin Zone.
Similarly
there will be 3rd, 4th etc, Brillouin Zones for various
ranges of k values.
Also
from Fig. 1.21 we can see that each Brillouin Zone is separated by breakup of
energy values known as Forbidden Zone
(or) Energy Gap.
Conclusion
Therefore,
we can conclude that the electron can go from one brillouin zone to the other
only if it is supplied with an energy equal to forbidden gap energy. Thus the
forbidden gap is the one which decides whether the solid is a conductor (or)
insulator (or) semiconductor.
Physics for Information Science: Unit I: Electrical Properties of Materials : Tag: : Definition, Explanation, Proof, Equation, Formula - Brillouin Zones
Physics for Information Science: Unit I: Electrical Properties of Materials
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