Page 242 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Electron
where τ
is the phonon lifetime or relaxation time, which in turn is the
p
the inverse of the phonon scattering rate.
Clearly, (7.5a) predicts that κ is temperature dependent, with three
potential sources. We do not expect the phonon velocity v to show a
p
strong temperature dependence for the solid state, since its value should
be proportional to the root of the material density. In Section 2.4.2 we
have seen that the specific heat at high temperatures tends to a constant
value c . We are thus left with the phonon lifetime as only temperature-
v
dependent influence at high temperatures. No simple theoretical explana-
tion of these scattering processes is possible, and experimental evidence
suggest that κ ∝ T – β for 1 ≤ β ≤ 2 . This makes sense, since we expect
that as the number of generated phonons increase due to the raised tem-
perature, so does the number of inter-phonon scattering events, and hence
the thermal conductivity should decrease with raised temperature.

7.2 Electron-Electron

For this section we assume that all electron-electron interaction of
ground state electrons has already been considered in calculating the
occupied ground states, i.e., the valence band structure. Therefore, we
only consider the interaction of excited electrons. The density of excited
electrons is assumed to be this low that a single electron description is
justified. Remember that this single electron description allows the
description of a completely occupied valence band missing one electron
as a single defect electron or hole in the same manner as for a single elec-
tron in the conduction band.

In the light of the above simplifying assumptions we calculate the transi-
tion rates between two electronic states as they enter the Boltzmann
transport equation (6.2) by means of the Fermi golden rule (3.82). For the
electron-electron interaction this yields a nonlinear term in the Boltz-
mann transport equation. For this section we only want to describe the

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