Numerical evidence for the bottleneck frequency of quasidiffusive acoustic phonons

Abstract

A kinetic equation on quasidiffusion of phonons was recently analyzed by Esipov and he predicted the existence of a bottleneck frequency ( BN ) which separates the phonons decaying from those diffusing to a detector. We have solved numerically the kinetic equation and obtained the temporal evolution of phonon concentration excited at the center of a spherical sample. We have also performed Monte Carlo simulations of phonon propagation in the same geometry. At a time much later than the ballistic arrival time of phonons, both sets of results exhibit a sharp peak in the phonon concentration around the predicted BN . With Monte Carlo simulations we have also confirmed the same relaxation rate for the phonons of frequencies Ͻ BN . ͓S0163-1829͑97͒01645-7͔Propagation of high-frequency acoustic phonons in nonmetallic crystals at low temperatures is governed by the ͑1͒ focusing effect due to the anisotropy of the lattice, 1,2 ͑2͒ elastic scattering due to foreign and isotopic impurities, 3 and ͑3͒ anharmonic decay via three-phonon interaction. [4][5][6] The propagation in the regime where the latter two effects are dominant is called quasidiffusion. 7-10 Because of a highlyfrequency-dependent decay rate of phonons A Ϫ1 ϭA 5 (A is a constant depending on phonon polarization and is the frequency͒, there exists a characteristic frequency (t) at an elapsed time t determined by A ( )ϭt. This frequency gives the length l of space expansion at a time t as lϳv͓ A ( ) I ( )͔ 1/2 ϳt 9/10 , where v is the Debye velocity and I Ϫ1 ϭB 4 (B is a constant͒ is the elastic scattering rate. 7,8 Thus the average phonon distribution in a sample spreads more slowly than in ballistic propagation but faster than in normal diffusion. More recent observations show that the characteristic behavior of quasidiffusive phonons appears in the time trace of the detected phonon signal as an exponentially decaying tail at a time much later than the ballistic time of flight t b through the sample. [11][12][13] Experimentally, this exponential behavior has been observed for several semiconducting samples of slab geometry such as silicon, germanium, and GaAs with a photoexcitation technique at a low input power level. 13 It is critical for observing the exponential tails to remove liquid helium from the surface where the phonons are excited. Otherwise, the high-frequency phonons excited at the sample surface are lost into the helium bath and the quasidiffusive tail originating from these phonons is hardly observed. The quasidiffusion is also important in analyzing the phonon signal produced by high-energy particles in a crystal. [14][15][16][17][18] The applicability of a quasidiffusive model is seen by comparing the experimental heat pulses with Monte Carlo simulations. [19][20][21] Originally a simple one-branch model was proposed, where three-phonon polarizations were approximated by a single mode with the Debye velocity. 19 The Monte Carlo simulations based on this model can reproduce the exponential tail in the time-of-flight spectrum of phonons arrivi...

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