Page 187 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
into account the power absorbed from the environment and calculate the
balance in order to determine the heat radiation. Nevertheless, (5.35)
reveals the statistical nature of heat radiation from a material.
The description of phonons via (5.35) is valid only with some restric-
tions. At low temperature only waves with large wavelengths compared
to the interatomic spacing are excited. In this case the lattice can be
treated as a continuum. At short wavelengths the spectrum of frequencies
shows a maximum value ω , which is independent of the shape of the
max
crystal. In general the spectrum is quite difficult and must be calculated
or measured in detail to give realistic distributions at higher frequencies.
Nevertheless in a wide range of applications (5.35) is a good approxima-
tion.

For the case of photons and phonons, the partition function does not
depend on the constraint of the number of particles N . Therefore, the
derivative ln Z with respect to N vanishes and thus photon or phonon
statistics are special cases of the Bose-Einstein statistics with µ = . 0

5.4 Electron Distribution Functions

The distribution functions for electrons in the periodic lattice of a semi-
conductor follow Fermi-Dirac statistics. To distribute a certain number of
electrons in the conduction band of a semi-conductor, we have to know
the shape of the band with respect to the momentum of the electron, as
explained in Section 3.3.3.

5.4.1 Intrinsic Semiconductors
An intrisic semiconductor is a pure crystal, where the valence band is
completely occupied and the conduction band is completely empty. This
is the whole truth for a purely classical description. Such a classical
model does not allow electrons to occupy the valence band. We know,

184 Semiconductors for Micro and Nanosystem Technology