Fundamentals of the Physics of Solids / 12-The Quantum Theory of Lattice Vibrations

12.4 Anharmonicity

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for each vibrational mode, which specifies the volume dependence of frequency for the mode in question. The Grüneisen relation (12.4.11) is satisfied if the parameter γ is defined as the weighted average of the γλ(q):

γ =	qλ qλ cV λ(q) .	(12.4.19)
	γλ(q)cV λ(q)

Collision (decay and merger) processes among phonons play an important role in rendering the thermal conductivity of nonmetallic solids finite. If the sample is not in thermal equilibrium but the temperature is di erent at the two ends, then a larger number of phonons are generated thermally at the highertemperature end, and their motion toward the lower-temperature side may give rise to a heat current. This will be discussed in more detail in Chapter 24; only a simple picture is presented below.

We shall apply the results of the kinetic theory of gases to a gas of phonons. If the phonon mean free path Λ is finite because of the collisions among phonons, then the thermal conductivity is given by

λph = 31 ΛvcV ,

(12.4.20)

where v is the velocity of phonons, and cV is the specific heat per unit volume. At high temperatures, where the Dulong–Petit law applies, the temperature dependence of thermal conductivity is governed by the temperature dependence of the phonon mean free path. Since the frequency of collision increases with the number of thermally generated phonons, it is plausible to assume that the mean free path is inversely proportional to the number of phonons. Above the Debye temperature the number of phonons

nλ(q) =	1		≈	kBT		(12.4.21)
	e ωλ (q)/kBT − 1			ωλ(q)
is proportional to T , so in this temperature range
Λ 1/T	and	λph 1/T .				(12.4.22)

At room temperatures the typical value of the mean free path ranges from 10−6 to 10−7 cm. At lower temperatures it can be substantially larger: around 20 K it is 10−2–10−3 cm. At such low temperatures, where only long-wavelength (low-energy) phonons are generated thermally in significant numbers, one has to take into account that collisions do not reduce the heat current (energy current) of phonons as long as only normal scattering processes exist. This is because the dispersion relation of long-wavelength acoustic phonons is linear, and so momentum conservation goes hand in hand with energy conservation in the decays of such phonons. The dissipation of the heat current requires umklapp processes.

428 12 The Quantum Theory of Lattice Vibrations

Further Reading

1.T. A. Bak, Phonons and Phonon Interactions, W. A. Benjamin, New York (1964).

2.M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford Classic Texts in the Physical Sciences, Oxford University Press, Oxford (1998).

3.P. Brüesch, Phonons: Theory and Experiments, Springer Series in SolidState Sciences, Springer-Verlag, Berlin (1982).

4.A. A. Maradudin, E. W. Montroll, G. H. Weiss, I. P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Second Edition, in: Solid State Physics, Supplement 3, Academic Press, New York (1971).

5.J. A. Reissland, The Physics of Phonons, John Wiley & Sons Ltd., London (1973).