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Thermal Conductivity - Coggle Diagram
Thermal Conductivity
Thermal Conductivity
In general, heat (i.e., energy) can be transmitted through a crystal by phonons, photons, free electrons or holes, and electron-hole pairs.
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A single phonon cannot be used to describe a deviation from equilibrium in one region of the crystal.
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We can describe a temperature variation localized in space within a region small compared to sample dimensions but large compared to a unit cell dimension (i.e., ∆k must be small compared to the Brillouin zone dimensions).
In a perfectly harmonic crystal such a wave packet of phonons travels unaltered and thus the thermal conductivity is infinite.
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Phonon Mean Free Path
The details of thermal conduction by phonons are best approached via a macroscopically defined mean free path.
The thermal conductivity 𝜅 is defined as the constant of proportionality between a temperature gradient ∇T and the rate of energy flow per unit area Q as in
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The phonons must diffuse through the sample, suffering frequent collisions, from the higher temperature to the lower temperature end (in the latter case Q depends only on the temperature difference, ΔΤ, between the ends).
However, for any distribution of phonons, we may define a nominal mean free path.
Anharmonic effects
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There is a similar equation for each of the Ν atoms. The two terms come from the two springs that are attached to the nth atom.
Nevertheless, the forces are linear in the displacements.
In this case the normal mode decomposition of the atomic motion is exact and phonons cannot interact (collide) with another, so in a perfect crystal there would be no phonon scattering and no thermal resistivity (the thermal conductivity would be infinite).
However, the quadratic term is just the first nonzero term in a Taylor expansion of the potential energy about the equilibrium position.
These anharmonic terms couple one phonon to another (phonon-phonon interaction), causing phonon "collisions" and a finite mean free path and, by several different processes, establishing equilibrium.
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