Silicon's measured lattice thermal conductivity at is

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

Condensed Matter Physics, Ch Thermal expansion and lattice thermal conductivity

Problem

Silicon's measured lattice thermal conductivity at 300 K300\,\mathrm{K} is κ=150 W m−1K−1\kappa = 150\,\mathrm{W\,m^{-1}K^{-1}}, its branch-averaged sound speed is v=6400 m s−1v = 6400\,\mathrm{m\,s^{-1}}, and its molar heat capacity is CV=20.0 J mol−1K−1C_V = 20.0\,\mathrm{J\,mol^{-1}K^{-1}} (Chapter 6). One calculation, extracting the mean free path from κ=13cVvℓ\kappa = \tfrac13c_Vv\ell, writes: "ℓ=3κ/(CVv)=3(150)/[(20.0)(6400)]=3.52×10−3 m\ell = 3\kappa/(C_Vv) = 3(150)/[(20.0)(6400)] = 3.52\times10^{-3}\,\mathrm{m}, so the phonon mean free path in silicon is about 3.5 mm3.5\,\mathrm{mm}." Something has gone wrong: a millimetre-scale mean free path would already exceed the thickness of an ordinary silicon wafer. Find the error and give the correct mean free path in nanometres and in lattice constants (silicon's lattice constant is a=5.43 A˚a = 5.43\,\mathrm{\mathring{A}}).

Browse the problem library