Lifetimes of High Frequency Phonons

We discuss the inhibition of anharmonic spontaneous multiphonon decay by energymomentum conservation and the reduction of isotope scattering by the sharing of ionic motion to permit a frequency window for long-lived transverse acoustic phonons. l.~ntroduction Recent high frequency phonon experiments have revealed anomalously long phonon lifetimes at low temperatures.' Indeed, for phonons generated by e-h recombination in GaAs mean free paths ok a few mm have been observed for near zone boundary acoustic phonons. Since elementary theories of isotope scattering and of two-phonon decay lead to scattering rates that grow as w4 and 0' respectively, the experimental results require some mechanism or mechanisms that inhibit such scattering processes at least over some frequency window near the zone boundary. 2.&~&meous Phonon D e w In an isotropic solid transverse phonons can not decay into two other phonons because of energy and momentum conservation.' ~ a r i s ~ recognized that the same conclusion would be likely to apply to an anisotropic crystal, but that conservation of energy and momentum would be model dependent. He demonstrated the result for a face-centered cubic crystal with central forces between nearest neighbor atoms. Lax, Hu and Narayanamurti5 proved the following theorem: A phonon can not decay by anharmonic processes of any order into a set of phonons each of whose phase velocities is higher than that of the initial phonon. This result is applicable in the presence of frequency and angular dispersion in an anisotropic crystal of arbitrary symmetry. It is applicable to U (umklapp) as well as N (normal) processes. The above theorem does not completely preclude the possibility that a transverse acoustic phonon can decay by anharmonic processes, but as found in Maris's example, only modes with small wave-vector q can decay. The reason for this is that the phase velocity of a phonon will normally decrease with increasing q making it more difficult to find phonons of lower phase velocity to decay into. 3.lsotope Scattering To explain anomalously long-lived phonons, we must also demonstrate that isotope scattering, at least over some frequency "windown, is weaker than expected from the usual theoretical treatments of monatomic lattices. We have suggested5 that this weakness is to be expected in multi-atomic lattices when not all of the constituents possess isotopes. If only one atom (say Ga) has isotopes, there is a reduction factor of the form lup(q')* up(q ) I Z where up (q ) is a (normalized) amplitude for Ga in a mode of type t with propagation vector q. Here t and q refer to the initial phonon state and t' and q' refer to the final state. For the case of GaAs, which has cubic site as well as space group symmetry, the initial and final state amplitudes can be disentangled and the decay rate l / ~ , ( q ) associated with a single scattering center in a lattice of N sites is given by * M. Lax is also at Physics Dept., City College of New York, New York 10031. **W. Weber is also at Kernforschungszentrum Karlsruhe, FRG, Inst. f. Annew. Kern~hvs. I. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981649