Group velocities of phonons in superlattices

We theoretically study the magnitude of the group velocities of acoustic phonons in periodic superlattices. The group velocities in a perfect, periodic (infinite) superlattice are calculated from the dispersion relations and those in a finite superlattice are defined from the motion of phonon wave packets with the help of the stationary-phase approximation. In a frequency band, the group velocities in the infinite and finite superlattices coincide with each other. In a frequency gap, their magnitude in the finite superlattice becomes much larger than that in the band region, and increases as the periodicity N increases. This N dependence is qualitatively different depending on whether the gap in the corresponding infinite superlattice is due to the intramode or intermode Bragg reflection. To explain these unusual behaviors of phonon group velocities, we develop analytical calculations based on the transfer-matrix method. Numerical examples are given for a GaAs/AlAs superlattice sandwiched between GaAs substrate layer and GaAs cap layer.