Page 239 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
ner that expresses it as a three, respectively a four phonon interaction. For
the three phonon interaction, four different actions are possible from this
description:
• Absorption of a phonon, with two photons created;
• Absorption of two phonons, with one phonon created;
• Absorption of three phonons (energy balance violation);
• Creation of three phonons (energy balance violation);

τ
The lifetime is then defined as the inverse of the probability that a par-
ticular phonon state with wavevector k will, through one of the inter-
j
j
actions above, disappear. The probability is formed by summing over all
possible interactions to destroy a phonon given all possible distributions
among the phonon states, minus all possible interactions to create a
phonon given all possible distributions among the phonon states. In anal-
ogy to other “particles”, given the existence of a phonon lifetime τ , the
p
phonons are now also endowed with a mean free path l .
p

7.1.2 Heat Transport

Thermal Because the phonons transport energy through the crystal, and are gener-
Conductivity ated by raising the temperature of a crystal, we expect that the macro-

scopic laws of heat conduction will be predicted by the microscopic
lattice phonon model, at least for electrically insulating materials in
which heat transport is not dominated by freely moving energetic elec-
trons. Macroscopically, heat conduction in a solid is observed to be gov-
erned by the Fourier law

⋅
q = – κ ∇ T (7.1)
– 2
q
which relates the heat flux density in Wm to the negative tempera-
– 1
ture gradient ∇– T in Km through the material’s thermal conductivity
–
1
κ
(tensor) in Wm K – 1 .

236 Semiconductors for Micro and Nanosystem Technology