Page 240 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Phonon-Phonon
The harmonic model for the lattice predicts that once a lattice wave is set
into motion, it continues forever, i.e., there is no resistance to energy
transport. We know from observations that this is not true. Already the
fact that a crystal is limited in extent introduces reflections and distur-
bances to the perfectly homogeneous lattice model. However, all “parti-
cles” travelling through a real lattice also experience scattering centers,
i.e., phonons interacting with each other, and this is what ultimately
causes finite thermal resistances. Since a full derivation of the heat con-
ductivity goes well beyond the level of this text (again, the interested
reader should consult Madelung [7.10]), we instead look at an alternative
formulation [7.2].

Our model assumes that we have a small, one-dimensional temperature
gradient along the negative x-axis, as illustrated in Figure 7.2. For the

Scattering radius = l
Centre y
dθ
y x x
l p
θ θ
x
0

a) b)

Figure 7.2. (a) Phonons arrive at point x from a sphere of scattering centers with radius
0
equal to the mean free path length. (b) Nomenclature for the solid angle integration.

point x along the axis, our approach will be to estimate the 1D phonon
0
energy flux q that arrives due to the scattered phonons. For a phonon
x
mean free path of l , we may assume that all phonons arriving at x
p 0
come from the surface of a sphere of radius l . These phonons will have
p
θ
on average the velocity v . If we call the angle that a point of phonon
p
Semiconductors for Micro and Nanosystem Technology 237