Decay of the Vibrational States

in Glasses

Calculation of the decay rates of vibrational modes in amorphous silicon can be found in J. Fabian and P. B.  Allen, “Anharmonic decay of vibrational modes in a-Si”, Phys. Rev. Lett. 77, 3839 (1996). The relevance of the three-locon interactions for vibrational relaxation is also explained on the one-dimensional example of a disordered atomic chain in J. Fabian, “Decay of localized vibrational states in glasses: a one-dimensional example”, Phys. Rev. B 55, R3328 (1997).

Some experiments have suggested that vibrational states in glasses have extremely large relaxation times, up to microseconds. It also seemed that the relaxation times were longer for vibrational modes with higher frequencies. This contradicts the conventional wisdom of phonon physics: Phonons in crystals decay on picosecond time scales and the higher the frequency, the shorter is the relaxation time. Picosecond time scales follow from the amount of anharmonicity present in the interatomic forces, while the faster decay of higher frequency modes follows from the fact that at higher frequencies there are more possibilities (larger phase space) for the decay of the modes into two (or more) other modes. Since there is no obvious reason for why this argument would fail in glasses (not everybody agreed with that--see below), the interpretation of the experimental findings (but not the findings themselves!) were under suspicion and we set out to perform what turned out to be the first realistic calculation of vibrational relaxation times in a glass--amorphous silicon. The result of our calculation is clear: the experiment was misinterpreted. Vibrational states in glasses should decay on picosecond time scales, with increasing intensity at higher frequencies. The only difference between crystals and glasses is in the details of the decay mechanism itself. In crystals momentum conservation severely restricts the phase space for decay products. In glasses, where the majority of modes are diffusons which have no well defined momentum, this restriction no longer holds. Does that mean that one should expect much faster decay in glasses? No, because the resulting decay matrix elements are much larger in crystals than in glasses (due to the random polarization pattern of the modes in glasses), compensating for the loss of phase space.

It was not only the experimental interpretation which was rejected by our calculation. Some theoretical models claimed to understand the experimental findings in terms of “slow relaxational dynamics of vibrations in glasses.” The “fracton model” in particular claimed to explain why vibrations at higher frequencies should have smaller relaxation rates than modes at lower frequencies. The fracton model assumed that the majority of vibrational states in glasses are locons. The higher the frequency, the higher is the degree of localization of vibrational states. It follows that the modes at higher frequencies decay mostly into two other modes which are also localized. As the probability for finding three locons at the same place in a glass is “negligibly” small, such events can be (see below) completely neglected. It then follows that the higher is the frequency, the smaller is the decay rate. The simplicity of this reasoning did not call for a more careful analysis. In fact, this argument is false. We showed by a scaling argument and by an explicit numerical calculation on a one-dimensional atomic disordered chain that three-locon processes cannot be neglected. Such processes are as (or even more) important than all other processes. The reason is that although the probability of three locons meeting at one place is small, the resulting matrix element coming from the overlap is huge, compensating for the small probability. As a result, we showed that the fracton model too did not explain the experiment. The correct interpretation of the experiment is still not known, but most probably the experimental results are influenced by trapping electrons in localized states (traps). Such electrons would recombine over a long time period by emitting phonons which are then measured as if they decayed on the same time scale as the trapped electrons

Calculated decay rates (in meV) of the vibrational modes in an amorphous silicon model (of 216 atoms) as a function of frequency (in meV) at 300 K (top) and 10 K (bottom). The decay rates are typically 1 meV, which translates into picoseconds for vibrational lifetimes. Perturbation theory is valid since the calculated decay rates are always smaller then the corresponding mode frequencies. The decay intensity nicely follows the joint-density-of-states (JDOS) curve (dashed), which measures the available phase space for (combinational) decay at the given frequency. At higher temperatures also differential decay occurs (a mode decays into one mode with a higher and another one with a lower frequency).