Question

Consider the phonon dispersion relation for a 10cm long GaAs crystal, shown in the figure below.

5.1 x 1013 rad/s - - - - - - - - - - - - - - - - 3.2 x 1013 rad/s o(4) ידידידידידידי == סןEדידידידידידידי 0 E/G Wavevector, k

(a) How can you obtain the phonon speed from the dispersion relation? What is the speed of the phonons with the shortest wavelength? (2 marks) (b) Which is more effective at carrying energy across a solid: long wavelength or short wave- length phonons? Why? (2 marks) (c) What is the wavelength of the fundamental mode of this crystal? (1 mark) (d) In the long wavelength limit, how many optical phonons would be created if 8.08 × 10−21 J of heat was transferred to the crystal? (2 marks) (e) Does the long wavelength limit correspond to low or high sample temperatures? Explain. (1 mark) (f) In the long wavelenth limit, would this sample of GaAs be transparent to light having a frequency of 5.1 × 1013 rad/s? How about 3.2 × 1013 rad/s? (1 mark) Explain.

Answer 1

Ansa relation is given by the the - as the dispersion I equation is I where, 'k' in the Apring constant as a wave Vector and 'k' then, The particle / phonon Given by the equation speed is The equation - du dk d o snode with respect to ware vector 2 Then the sout boot 4 sin ka [eth ) estar al is felha aig person he has 4 Sin ka mm 2 2 x 8 Sinka 4 coska x 4a. mime mim mim,

From Schrodinger equation ka 2me, do 'k' in the wave e lo city vector Here Then, 2 ME So that, ettere, Pis High 'E io low Pin Low when 'E' in High where, Ein the energy. So that Case "shart wavelength is suitable ! (c) To find the evare length a we know kata where 'c so the speed of light Also , k = 21 Thun 2 211 where he is given by equation ④