Page 238 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Phonon-Phonon
diagram. The dark blocks indicate interactions between similar particles,
i.e., phonons with phonons. The lighter blocks indicate interactions
between different particle families. We do not restrict ourselves only to
quantum phenomena, but also include effects that are comfortably
described by a more classical model. At the end, we look at effects that
arise due to material inhomogeneities, i.e., when we depart from the
“infinite”, perfect silicon crystal.

7.1 Phonon-Phonon

The harmonic potential is in fact a truncated series expansion model (see
(2.13)) of the true crystal inter-ion potential, and as such is not capable of
explaining all lattice phenomena satisfactorily. If we include further
terms of the series, we say that the potential is anharmonic.

An- This has a host of consequences: the phonon frequencies become shifted
harmonicity from their harmonic frequencies; anharmonic phonons interact with each

other; a phonon in a particular state has a finite lifetime before it disap-
pears, and this fact is due to an anharmonic phonon-phonon interaction.
We will briefly discuss two important further consequences in the sec-
tions to follow: the lattice thermal conductivity and the expansion of the
lattice with temperature, both of which cannot be explained with the har-
monic potential. But first we take a closer look at the concepts of interac-
tion and lifetime.

7.1.1 Phonon Lifetimes

Phonon The theory required for an adequate description of phonon lifetimes is
Lifetimes fairly involved, going beyond the level and goals of this text. Instead, the
interested reader is referred to Madelung [7.10]. Here we provide only a
brief description of the phenomena. Each of the third and fourth terms of
the series expansion of the interatomic potential can be written in a man-

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