Page 186 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Applications of the Bose-Einstein Distributions
The normal vibrations of the crystal lattice as described in Chapter 2 may
Lattice 5.3 Applications of the Bose-Einstein Distributions
Vibrations be described as set of linear non-interacting harmonic oscillators when
neglecting the nonlinear interaction between the lattice atoms. Each of
these oscillators has a specific frequency ω and each of the phonons
i
i
contributes with an energy hω . With n phonons present frequency ω
i i i
we have a contribution of E = n hω to the energy contained in that lat-
i i i
tice vibration mode.
Photons For photons, the situation looks alike and the only constraint on the parti-
β
tion function is given by the parameter , the temperature, which deter-
mines the average energy of the system. There is no prescribed number
i
of particles for a single sate in either cases of photons or phonons. Any
number of photons or phonons may occupy a given energy level. For this
case the partition function reads
∞
Z = ∑ exp – ( βΣ n E[ i i i k ∏ ∑ exp – ( βE n )
] ) =
i i
k i n i = 0 (5.34)
1
= ∏ --------------------
–
–
βE i
i 1 e
From (5.34) we can calculate the average number of particles with
energy E
i
∂
1 lnZ e – βE j 1
n = – ---------------- = -------------------- = ------------------ (5.35)
j
∂
β E – βE j βE j
–
j 1 e e – 1
Planck which is the well known Planck distribution for photons. It gives the sta-
Distribution tistical occupation number of a specific energy level for a photonic sys-
tem. Consider a piece of material at a specific temperature, i.e., with
β
given . Then (5.35) describes the intensity distribution of the irradiated
electromagnetic spectrum with respect to the frequency (note that the
energy of the electromagnetic wave is proportional to its frequency.) Of
course this is a non-equilibrium situation. Strictly, we also have to take

Semiconductors for Micro and Nanosystem Technology 183