Question

Problem 6.15 Prove that the temperature dependence of the low-temperature specific heat of phonons depends on the dimensionality of the material.

Answer 1

Problem:6.15- Lattice vibrational motion can also produce traveling waves in which localized regions of vibratory atomic motion travel through the lattice. Examples of such traveling waves are sound moving through the air, or seismic waves that start at the epicenter of an earthquake, and travel thousands of miles to reach a seismograph detector that records the earthquake event several minutes later.

Localized traveling waves of atomic vibrations in solids, called phonons, are quantized with the energy ђω = hv, wherev= ω/2π is the frequency of vibration of the wave. Phonons play an important role in the physics of the solid state. From the quantum mechanics point of view, we know that the energy levels in the harmonic oscillator are quantized. Similarly, the lattice vibrations are quantized. This quantum of vibration is called as phonon analogous to photon which is the quantum of light energy. Phonons play an important role in the physics of the solid state. The allowed energy levels in the harmonic oscillator are

where, n is the quantum number. A normal vibration mode in a crystal of frequency ω is given by .

If the energy of this mode is given by

,

we can say that this mode is occupied by n phonons of energy ħω. The term ½ħω is the zero point energy of the mode. Phonons govern the thermal properties in semiconductors and insulators. Their influence on thermal, electrical, optical and other properties of bulk materials is well known. As we know that the density of states D(E) of conduction electrons are strongly affected by the dimensionality of a material, phonons also have a density of states D(PH) which depends on the dimensionality, and like its electronic counterpart, it influences some properties of solids. In specific, the specific heat of a solid is the amount of heat that must be added to it to raise its temperature by one degree Celsius. The main contribution to this heat is the amount that excites lattice vibrations, and this depends on the phonon density of states D(PH). The interactions of phonons are unavoidably altered by the effects of dimensional confinement on phonon modes of nanostructures. These effects are similar to that of electrons confined in a quantum well. The dimensional confinement of phonons results in restrictions in the phase space of the phonon wave vectors, due to which it is certain that carrier-phonon interactions in nanostructures would be modified by the phonon confinement.