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Lattice Vibrations of Disordered Solids. Thin Films
On 1 January 2001, it will be thirty years ago that Bob Zeller submitted his MS thesis entitled "Heat Conduction in Glasses" to the Cornell graduate school. The findings first reported there, combined with what we know now, can be summarized as follows:
"In all amorphous solids at temperatures below a few Kelvin, acoustic waves, regardless of their polarization (transverse or longitudinal) are attenuated such that the ratio of their wavelength l to their mean free path l is constant, to within a mall factor, perhaps 2 or 3, independent of the frequency, from 1011 Hz (the thermal phonon frequency at ~ 1K) to the lowest frequencies studied, which have been as low as 10-3 Hz. For all amorphous solids, regardless of their chemical composition, this ratio, l /l , ranges from ~ 10-3 to ~ 10-2, a mere one order of magnitude."
This statement is based on a review of all measurements of low temperature thermal conductivity and acoustic attenuation, approximately one hundred of them, which have been published since Zeller wrote his thesis. This review is just being readied for submission to the Proceedings of the National Academy of Sciences (PNAS - for short), with the title "Low Energy Lattice Vibrations of Amorphous Solids".
Two exceptions to this universal behavior have to be noted. In amorphous silicon and germanium, l /l depends on the way in which this material is prepared (it only exists in the form of thin films), and can be smaller than 10-3, by as much as a factor of ten, although the independence of frequency still seems to me maintained. In one case, however, in amorphous silicon films especially prepared at the Natural Energy Resources Laboratory (NREL) containing small (~ 1 at %) hydrogen, l /l has been found to be smaller than 10-5 (Xiao Liu et al., "Amorphous Solid without Low Energy Excitations," Phys. Rev. Lett. 78, 4418 (1997). Since these films are definitely structurally amorphous, the significance of this finding is that it has shown that the amorphous structure itself is not the cause for the acoustic attenuation common to all other amorphous solids. The other exception is that the same behavior found in all of those solids has also been established in a large number of crystalline solids, most of them chemically disordered, but also in a quasicrystalline solid ("Glass-like lattice vibrations in the quasicrystal Al72.1Pd20.7Mn72," E. Thompson et al., Phys. Rev. B 62, 11437 (2000)). These examples, also reviewed in the PNAS article, demonstrate again clearly that the amorphocity is irrelevant as far as the universal magnitude of the acoustic damping is concerned. So then, what is its physical origin? We really don't know the answer to this very basic question in condensed matter physics. In some recent papers, we have been trying to come up with a model. If you want to know, look at "Generation of Low-Energy Excitations in Silicon", Phys. Rev. Lett. 81, 3171 (1998), by Xiao Liu et al., and "Lattice Vibrations of Disordered Solids", Current Opinion in Solid State and Materials Science 4, 281 (1999), by R.O. Pohl et al.
The model for the physical nature of the defects leading to the anomalies described above, though not for their universality, is that all disordered solids contain tunneling defects with a wide spectrum of tunnel splittings. With some reasonable assumptions, this so-called Tunneling Model, suggested by P.W. Anderson, B.I. Halperin, and C.M. Varma (Phil. Mag. 25, (1972) and by W.A. Phillips, J. Low Temp. Phys. 7, 351 (1972), has successfully described most low temperature thermal and acoustic measurements. One direct consequence of such a picture is that at sufficiently low temperatures, when the thermal motion of the lattice becomes small enough, these defects will elastically interact with one another, eventually perhaps forming something resembling a spin glass. Estimates of the temperatures at which such interactions might occur have been around 10 mK. With the help of Gavin Lawes and Jeevak Parpia, EunJoo Thompson has measured the elastic constants (sound velocity and damping) of a bar of amorphous SiO2 above 6 mK, using a torsional oscillator. She found deviations from the predictions of the Tunneling Model below ~ 20 mK. One of them was that the speed of sound ceased to vary as rapidly as predicted by the Model, and the other was that the frequency response below 12 mK showed an erratic, irreproducible behavior. Both of these observations may be the result of tunnel-defect interactions ("Low Temperature Acoustic Properties of Amorphous Silica and the Tunneling Model," E. Thompson et al., Phys. Rev. Lett. 84, 4601 (2000)). She also searched for similar deviations in disordered crystals, but ran out of time (the trip around the world was too alluring). Therefore, a new postdoc, Robert Merithew, who recently came to Cornell after earning his Ph.D. at the University of Illinois in Urbana, has taken over where she left off. The exciting data are just beginning to come in!
For a number of years, we have been using so-called double-paddle-oscillators, harmonic oscillators with extremely high elastic quality, to investigate the elastic constants of thin films deposited onto those oscillators ("Internal Friction of Subnanometer a-SiO2" B.E. White Jr., Phys. Rev. Lett. 75, 4437 (1995)). They were also used to explore the a-Si and a-Ge films mentioned above, and were also applied for a study of amorphous water ice films. The idea behind the work on the latter was that we wanted to understand how amorphous ice, believed to be the major constituent of cometary ice, will survive for long times in the interplanetary space. The most remarkable result of this work was the finding that freshly condensed on a cold substrate, these amorphous films had very small shear moduli, which stiffened during annealing ("Annealing and Sublimation of Noble Gas and Water Ice Films", B.E. White Jr. et al., J. Low Temp. Phys. 111, 233 (1998)). The question was whether this effect was a special property of water ice, or was of a more general nature. In order to answer this question, Tom Metcalf has undertaken a similar study of argon and of neon films. Such films are never amorphous, which should simplify their understanding, and secondly, the interatomic forces are well known (Lennard-Jones potential). It appears at this time that the freshly condensed films are nearly as soft as the water ice films. New, and perhaps even more interesting, however, is the observation that the first layers, those deposited close to the silicon substrate, are much softer than the layers deposited later on, when the film grows thicker. Thus, the crucial parameter determining the stiffness of a freshly condensed film is its thickness. This observation may be important for the understanding of thin film properties ("Annealing of Quench-condensed Argon Films", T.H. Metcalf, to appear in J. Low Temp. Phys.).
With the double-paddle oscillator having played such an important role for our thin-film work in recent years, Chris Spiel decided that its various normal modes needed to be identified and their damping be understood - the latter being quite different for the different models. Working with Alan Zehnder in our Department of Theoretical and Applied Mechanics, he used a finite element method to predict the modes, which he then compared with the experiment. He was then able to show that the internal friction of the different modes is related to the restoring force required to keep the paddle in place ("Normal Modes of a Si (100) Double-Paddle Oscillator" C.L. Spiel et al., to appear in Rev. Sci. Instr). This is the first step towards the design of oscillators with minimal clamping losses, which is essential for the development of micro-electro-mechanical oscillators (MEMS'es), a project that has just been funded by Cornell's Center for Materials Research, and also by the Naval Research Lab.
Working in this group are Robert Merithew, postdoc, and Tom Metcalf, graduate, and, hopefully soon, another postdoc.
Last modified: November 10, 2000
Douglas E. Milton, Sr., firstname.lastname@example.org