Joe Feldman (NRL), Jaroslav
Fabian (formerly a PhD student at Stony Brook, now assistant professor
of physics at the Institute for Theoretical Physics of Karl-Franzens University
in Graz, Austria) and I have been studying the lattice vibrations of amorphous
silicon using a realistic model of the atomic coordinates and the interatomic
information is on Jaro Fabian's site.
The picture above shows the vibrational densities of states of both
crystalline and amorphous silicon. There is surprisingly little change
when crystalline order is lost. However, the nature of the vibrational
normal modes does change. The plot also shows the ``bond stretching character''
of each mode. Crystalline silicon has vibrations which can be classified
by wavevector and by branch. Transverse branches involve atom motions perpendicular
to the direction of propagation. These vibrations do not stretch the bonds
very much. Longitudinal branches cause large stretches of bonds. In the
low-frequency part of the spectrum, the crystalline modes clearly separate
into these very different categories. The vibrational normal modes of amorphous
silicon completely lose this differentiation. At a given frequency, all
modes have the same ``bond stretching character.''
The next picture shows for the amorphous case how the vibrational amplitude
of a given normal mode changes with distance as you go away from the atom
with the largest vibrational displacement (for that branch.) Modes with
frequency less than 72 meV extend throughout the material. The amplitude
falls initially, but saturates at a constant value. Modes with frequency
higher than 72 meV are localized. Their amplitudes fall off exponentially
with distance. The frequency 72 meV which divides these two categories
is called the mobility edge.
The picture above shows thermal conductivity measured on films of amorphous silicon in various laboratories around the world. In the temperature interval 10K < T < 20K the conductivity shows a "plateau" where it is nearly independent of temperature. Our theory shows that this plateau arises as a crossover from the low T regime where heat is carried by ballistically propagating long wavelength vibrational modes to the high T regime where heat is carried by modes which extend throughout the sample, but have no wavevector or group velocity. These modes have an intrinsically diffusive behavior. Wavepackets built from these modes spread diffusively (r^2 = 6Dt) with diffusivity D of order the Debye frequency times interatomic separation squared, or about 1 mm^2/s. These modes are not significantly populated until T > 20K, and then they begin to dominate the conductivity. We call these modes "diffusons."
4096 Atom Model
There are 12288 vibrational normal modes for a system with 4096 atoms.
Each normal mode is described by an eigenvector of length 12288.
Storage of these eigenvectors used to be quite expensive. Brian Davidson
had calculated them for us in 1996, but soon they were erased.
Recently, Bill Garber and Folkert Tangermann (Applied Math, Stony Brook)
recalculated them for us and we are storing them securely. Bill has
just calculated the inverse participation ratio for
these modes. Click here to see the logarithm of the participation ratio versus frequency for the
4096 atom model compared with the 1000 atom model.