Wavevector k for phonons the dispersion is a plot of

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Unformatted text preview: n ( 0 ) < 2 F + (0) υ x + 0 < n ( L ) < F + (0) υ x 1c) Explain why the simple application of Fick’s Law often leads to incorrect results. Solution: It is very common to apply the boundary condition, n ( L ) = 0 , but this assumes diffusive transport. It is not Fick’s Law that is wrong, it is the boundary condition. It is more physical to impose boundary conditions from the fluxes incident from the two sides, not the carrier densities on the two sides. ECE- 656 3 Fall 2013 Mark Lundstrom 10/19/13 2) For electrons, the bandstructure is a plot of energy, E k , vs. wavevector, k . For phonons, the dispersion is a plot of phonon energy, ω ( q ) , vs. phonon wavector, q . () For electrons, we often approximate the bandstructure with simple, parabolic bands, 2k 2 Ek = 2 m* For phonons, we can sometimes approximate the phonon dispersion with the Debye approximation, () ω = υ D q , where υ D is the Debye velocity (an average of the longitudinal and transverse acoustive celocities.) 2a) Compute the density- of- states, D ph ( ω ) , for phonons in the Debye model. Solution: Equate the DOS in q- space to energy space: 1 N dq = D ph ( ω ) d ( ω ) Ωq ( ) 1 1 N q dq = 3 × 3 4π q 2 dq = D ph ( ω ) d ( ω ) Ω 8π (i) (ii) Note that there is no factor of 2 for spin in this case, but we have a factor of three because of the three polarizations, longitudinal and two transverse acoustic phonons. From the d...
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