Propagation of acoustic phonon solitons in nonmetallic crystals

We consider the theory of the propagation of acoustic-phonon solitons in nonmetallic crystals. For a soliton with a high strain amplitude, the width is small and so the soliton contains Fourier components of high frequency. These components are attenuated as a result of spontaneous anharmonic decay processes and, in addition, in some crystals there will be scattering due to the variation in mass of the different isotopes. At nonzero temperature there will be additional attenuation due to anharmonic interactions with thermal phonons. These attenuation mechanisms result in a steady decrease in the amplitude and velocity of a soliton, and we calculated attenuation constants for several nonmetallic crystals. We derive expressions for the rate of decay of the soliton amplitude and test these results by comparison with the results of numerical simulations.

Figure 1

Plot of the functions $F_{\mathrm{an}}$, $F_{\mathrm{iso}},$and $F_H$ defined in Eqs. (22), (30), and (35), respectively. The dashed curve shows the strain profile of the soliton.

Figure 2

The calculated strain amplitude $A$ of a soliton propagating in the [100] direction in MgO, Ge, Si, GaAs, and along the $c$ axis of Al${}_2$O${}_3$ as a function of time. The initial strain amplitude is ${10}^{-3}$ and the temperature is 0 K. The soliton is attenuated by anharmonic decay and isotope scattering. The solid curves are from the analytical calculations described in the text and the open circles for MgO are the results of a numerical simulation.

Figure 3

The calculated strain amplitude $A$ of a soliton propagating in the [100] direction in Ge as a function of time. The initial strain amplitude is 10${}^{-3}$ and the temperature is 300 K. The soliton is attenuated by the Herring process. The solid curve is from the analytical calculations described in the text and the dotted line is the result of numerical simulations.

Figure 4

The calculated strain amplitude $A$ of a soliton propagating in the [100] direction in Si as a function of time. The initial strain amplitude is 10${}^{-3}$ and the temperature is 300 K. The solid curves are from the analytical calculations described in the text for the Herring process, the dotted line is the result of numerical simulations for the Herring process, and the dashed curve is the result of a numerical simulation that includes the damping due to both the Herring and Akhiezer processes. The inset shows the frequency dependence of the decay rate of the phonons in Si due to the Akhiezer and Herring processes in Si.

Figure 5

The shape of a strain pulse propagating in the [100] direction in Ge. The initial strain amplitude is ${10}^{-3}$ and the temperature is 300 K. The horizontal axis measures distance relative to the position $x=\mathit{ct}$ that a pulse travelling at the speed of sound would reach at time $t$. The soliton is attenuated by the Herring process only. The inset shows the details of the strain distribution at the last four times.