7411
J. Phys. Chem. 1994,98, 7411-7413
High-Energy Vibrational Lifetimes in a-Si:H R. Orbach’ Department of Physics, University of California, Riverside, Riverside, California 92521 -0101
A. Jagannathan Loboratoire de Physique des Solides, Batiment 510. Universite de Paris-Sud,
Orsay 91405 Cedex, France
Received: April 15, 1994; In Final Form: May 23, 1994”
The lifetimes of high-energy lattice vibrational states of a-Si:H are calculated on the basis of vibrational localization for energies w > wc,where wcsignifies the mobility edge. Anharmonicity-induced localized vibrational state hopping, with the emission of an extended vibrational state (a phonon) is found to be the dominant decay mechanism. Because of the contribution of the same vertex to thermal transport via localized vibrational state hopping, thevibrational lifetimes can be expressed in terms of this hopping contribution to the thermal conductivity, with only wc as a n undetermined variable. At low temperatures, the high-energy vibrational lifetime is found to be proportional to ( W ~ / W )exp[ ~ ( ~ / w ~ ) ~ owhere / ~ ] , de is the “superlocalization” exponent appropriate to random vibrational networks at the hopping length scale, and D is the fractal dimension. Taking w, = 40 cm-1 for a-Si:H from the plateau temperature, and the ratio d e / D = 1.22 from estimates for percolating networks in d = 2 (there are no comparable calculations at the hopping length scale in d = 3), we find vibrational lifetimes of 1.1 ns (TO, 480 cm-l), 0.8 ps (LA, 300 cm-l), and 17 fs (TA, 150 cm-l). These values are not inconsistent with recent observations of Scholten et al.
I. Introduction The lifetime of high-energy vibrational states in a-Si:H have recently been measured in a remarkable series of experiments by Scholten et al.1 They observed anomalously long lifetimes (e.g. the TO vibration at 480 cm-1 lived for 70 ns) roughly 4 orders of magnitude longer than found for the similar vibration in crystalline silicon.2 It was suggested1 that this behavior was a consequence of ”... localization of phonons with hw 1 100 cm-I” in a-Si:H. Similar observationsof long lifetimes for high-energy vibrational excitations have also been observed in glasses, using time-resolved vibronic wide band ~pectroscopy.~ We had hypothesized some years ago4 that vibrational states in amorphous materials are localized above a “mobility edge” w,. This localization was suggested to be a consequence of the “network-like”atomic structure in amorphousmaterials for length scales less than E = 20-30 A and was deduced from the universal behavior of the thermal conductivity exhibited by amorphous materials.5 This concept has been made more quantitative by a recent analysis of S. R. Elliott.6 He ascribes the low-energy vibrational behavior exhibited by amorphous solids to ’... phonon scattering caused by intrinsic density fluctuation domains in the structure, within which short- and medium-range order is maintained and beyond which the material is structurally isotropic and homogeneous. Phonon localization occurs when the meanfree path is comparable to the size of the domains.” He sets this length scale equal to a radius length scale E = 17.1 A in vitreous silica, with an associated vibrational frequency of 40 cm-1. Quite independently, our own analysis’ of the thermal conductivity of vitreous silica led to a crossover length scale of =20 A, and an associated vibrational energy w, of 35 cm-I. We termed4.7w, a mobility edge, such that the vibrational states for w C w, are extended and, for w > w,, are localized. We stress that this is supposed to hold for all w > w,, and not for a narrow band of energies in the vicinity of w,, as has been hypothesized by Karpov et a1.8 Given this analysis, it is quite reasonable to ask for the consequences for vibrational lifetimes for w > w,. We explore here the case of w >> w,, appropriate for the experiments of Scholten e? al.1 They explored lifetimes of Abstract published in Advance ACS Abstracrs, July 1, 1994.
0022-3654/94/2098-7411%04.50/0
vibrations in a-Si:H with energies w = 480 cm-l (TO), w = 300 cm-I (LA), and w = 150 cm-’ (TA). We shall eventually set wc = 40 cm-l for a-Si:H, so that the experimentalconditions certainly satisfy the limit w >> 0,. This condition is important because it determineswhich anharmonicity-induceddecay process dominates for the high-energy lattice vibrations. Our previous analyseshave relied on the so-call “fracton“model9 as applied to amorphous material^.^ We noted7 that most such materials (with the exception of the silica aerogels) are certainly not fractal but neverthelesscould be modeled accordingly because of their common features. The fracton model has the great advantage that the vibrational states are well defined and the excitation spectrum and the length scaleof localization are known.9 We shall make use of this model for our explicit calculations but at the end set all of the relevant dimensions (with the exception of the ratio of the superlocalization exponent do to the fractal dimension D ) equal to the Euclidean dimension, d = 3. This simply means that our space will be Euclidean, with the exception that the localization will reflect the network character of the medium-range order for glasses.6 This will keep the calculations quantitative, and consistent with the vibrationalanalogueof Mott’s variable-range hopping model for electronic states.10 The fundamental assumption we make is that localized vibrational states decay for w >> w, via third-order anharmonic interactions. The localized vibration ‘hops” to another localized vibration site, with the energy difference taken up by theemission (or absorption) of an extended latticevibrational state (a phonon). The probability for three localized states interacting anharmonically is negligible, and the decay of the localized vibration into two extended states is forbidden for w > 2 wc. The latter process will contribute about equally to localized vibration hopping7 but only applies in the frequency range w, < w 2 w,. We can now make use of the analysis of ref 7 to calculate the vibrational lifetime T ( W ) for high-energy vibrations in amorphous systems. This is done in section 11. We use the results of ref 7 for the contribution of vibrational hopping to the thermal conductivity, K h o p ( T ) , to express T ( W ) in terms of K h o p ( T ) . Very recent thermal conductivity experiments of Cahill et al.11 for a-Si:H are introduced in section I11 to evaluate ~ ( w ) enabling , 0 1994 American Chemical Society
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7412 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
uroacn ana Jagannatnan . I
At low temperatures, kBT
