A crystal carrying a heat current has a phonon distribution displaced from equilibrium

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Solid State Physics, Ch Anharmonicity: expansion, phonon lifetimes and thermal conductivity

Problem

A crystal carrying a heat current has a phonon distribution displaced from equilibrium, so that the total P≡∑q,sℏq nqs\mathbf{P}\equiv\sum_{\mathbf{q},s}\hbar\mathbf{q}\,n_{\mathbf{q}s}, the sum running over every mode in the first Brillouin zone, is not zero. A three-phonon splitting event removes one phonon (q1,s1)(\mathbf{q}_1,s_1) and creates two, (q2,s2)(\mathbf{q}_2,s_2) and (q3,s3)(\mathbf{q}_3,s_3), subject to ℏω1=ℏω2+ℏω3,q1=q2+q3+G,\hbar\omega_1=\hbar\omega_2+\hbar\omega_3,\qquad \mathbf{q}_1=\mathbf{q}_2+\mathbf{q}_3+\mathbf{G}, with G\mathbf{G} a reciprocal lattice vector, written as G=0\mathbf{G}=0 for a normal process. ((a)) Write down the change ΔP\Delta\mathbf{P} produced by one such splitting event in terms of q1,q2,q3\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3, and show, using the conservation condition above, that if G=0\mathbf{G}=0 then ΔP=0\Delta\mathbf{P}=0 exactly for that event, however many such events occur throughout the crystal. ((b)) Explain why this means a crystal in which only normal (G=0\mathbf{G}=0) three-phonon processes occurred would still have an infinite thermal conductivity, even though its phonons do decay with a finite lifetime. State what property an event needs, in terms of G\mathbf{G}, in order to relax P\mathbf{P} and so give a finite κ\kappa.

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