Fractons

Physica 140A (1986) 397-404 North-Holland, Amsterdam

FRACTONS S. ALEXANDER The Raeah Institute of Physics, The Hebrew University, Jerusalem, Israel and Department of Physics, University of California, Los Angeles, CA 90024, USA

The fracton description of vibrational excitations in the strong scattering regime, above the Ioffe-Regel limit is reviewed. The reasons why this is a plausible model for vibrations in amorphous materials are explained and some predictions discussed. It is shown how such a model arises in the context of scattering theory when the Ioffe-Regel length is large and a new self-consistencyargument suggesting that the relevant fracton dimension for such vibrations has a universal value of 4/3 is described.

1. The fraeton model The concept of fractons was introduced in the context of the study of the dynamics of self-similar fractal objectsX). When one tries to map a Laplacian or a second-order difference operator on a fractal geometry one finds anomalous behavior2). F o r the low frequency vibrations one finds x) a density of states N(o~) a: ~g-1

(1)

and a dispersion relation =

X ~. ~ - D / d ,

(2)

where D is the fractal (Haussdorf) dimension and d is a new intrinsic fracton (or spectral) dimension. The scaling form of these expressions follows from the self-similarity (dilation symmetry) of the fractal. The fact that d is, in general, different f r o m D results from the fact that quantities like the diffusion constant 3) or the elastic rigidity are not related to the scaling of the mass in the same way they would be in a Cartesian geometry. F o r m a n y purposes the relevant dimension is this intrinsic fracton dimension. Thus, e.g., the generalization of standard arguments shows that d is the relevant dimension for Anderson localization 4) so that, in the presence of disorder, all states are localized when d ~< 2 independent of the value of the fractal dimension ( D ) . Moreover, most connected fractals of interest are weakly connected and 0378-4371/86/$03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

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have a fracton dimension between one and two. In particular, d-= 1 for all configurations of linear polymers1'5). More interesting is the fact that d = 4 / 3 for percolation clusters generated in spaces of dimension d >/2, within experimental accuracy. Mostly one is of course interested in disordered fractals so that all eigenstates must be localized4). An interesting question is the meaning of the scaling length (X) in (2). The scaling is of course that one usually finds for wavelengths and that is indeed a correct interpretation for a folded linear chain (say a SAW) with no internal disorder (d = 1)1). In the other cases of interest the disorder is an intrinsic property of the geometry (e.g., for percolation or DLA clusters). We have conjectured 6'7) that on such fractals )t is a unique length describing both the wavelength and the localization length. One notes that the localization edge is at w = 0 on the fractal, so that the localization length also scales with a power of w. One must therefore have X, ~ ,o -x.

(3)

Comparing this with (2) one notes that x > D / d is obviously inconsistent because it implies At < X at small w. x < D/d only contradicts one's intuition because it implies that Xt(o~)/A(w) increases with a power of frequency. I will show below that a more careful argument 8) shows that x - - D / d implies -- 4 / 3 . If one accepts this conjecture (Xt : X for all w) one has very peculiar eigenstates described by a single length scale (= w-D/d). This has many implications. It should be contrasted with the situation near the Anderson localization edge where one has three length scales - the wavelength (~t), the scattering length (Xs) and the localization length (Xz), which are all different. Because of this special property we 1'6'7) gave these excitations a name and called them fractons. In a different terminology one can say that fractons are always at the Ioffe-Regel limit. They remain there at all frequencies. It is fairly straightforward to apply these ideas to a percolation model of quenched disorder6'7). The main new feature is that one has a crossover to phonons at the correlation length (~) and that one has to take into account a Stauffer 9) distribution of finite clusters to make sure that the overall density is uniform. The model one obtains has a lot of predictive power. One can for example calculate a density of states and the temperature dependence of the specific heatT'l°), the high-temperature thermal conductivity11), scattering form factors (S(q, to)), relaxation rate distributions 12) and inelastic scattering off fractons13). One is also forced 6) to introduce two-level systemsZ4). It is of course tempting to try to apply these results to real amorphous materials without worrying too much how one would map realistic microscopic

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models on this type of scaling model. One notes that percolation is, by far, the best understood model for quenched disorder and that the physical properties we are talking about seem, experimentally, to be surprisingly universal. One also notes that the only generally accepted universal feature of amorphous materials, the two-level systems14), has still, after many years, not been connected to microscopic models in any satisfactory way. The results are surprisingly, sometimes embarrassingly, encouraging. This is of course particularly true when they are not very sensitive to the detailed scaling indices which are not really accessible. Some of these results are far from obvious. Thus one predicts a crossover in the density of states to a convex frequency dependence with an excess density of state in the crossover region1°). The prediction of a plateau in the thermal conductivity when the vibrations become strongly localized is sort of built into the modelT). The prediction of a linear temperature dependence of this quantity 11) at high temperatures is much more striking and we had considerable difficulty in getting any sort of experimental results (new or old but all unpublished) on this regime. None of this can of course be regarded as conclusive proof of the validity of the scaling, fracton description of the high frequency vibrations of amorphous materials. Published results also usually have other explanations related to the specific properties of the materials involved. Thus the most common explanation of the universal plateau in the thermal conductivity in (transparent "window") glass is not localization but scattering from the torsional modes of SiO4 tetrahedra. This may certainly be the correct explanation combined with a series of material-specific explanations for all other materials for which such a plateau is observed. It is nevertheless surprising to read a recent didactic Phys. Rev. Letter 15) so enthusiastically acclaimed (apparently) by all referees that it was published within four weeks of submission, whose only point is that a convex density of states can arise in many ways and does not imply the validity of the fracton model (which is of course correct). The conclusion of this paper is (in a free translation from the German) that the only merit of the fracton model is that it is a "beautifully complicated" explanation of phenomena which have been well understood for a long time. The shopping list of options offered in ref. 15 is rather long, but as far as I have checked they are mostly valid mechanisms for changing a density of states- when applicable. The editor of Nature found this letter sufficiently important to merit an editorial 16) calling it "a gentle warning on fractal fashions". I may be forgiven for refraining from commenting in detail on this criticism in spite of the implied compliment that we have indeed managed to convince the scientific community that amorphous materials should be considered "fractal at short distances"16). We have certainly tried, but I must confess so far with very limited success.

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S. A L E X A N D E R

2. S c a t t e r i n g

and strong scattering

I would like to devote the rest of this paper to a discussion of the reasons why a fractal (fracton) description of this (relatively) high frequency vibrational mode in amorphous solids is reasonable, what it means and why it implies a fracton dimension d = 4/3. The last part of this discussion is very recent work8). It also contains the most important new results in this paper. Thus A. Aharony O. Entin-Wohlman, and R. Orbach should certainly be considered authors for the parts of this paper which are really new. I have not included their names above solely because I am not sure they would agree to accept responsibility for the preceding section of this paper. I want to consider the vibrational spectrum of amorphous materials. MacroscopicaUy such materials behave as isotropic solids. The low frequency modes are therefore phonons with a dispersion X(w) = C / w ,

(4)

where C is the velocity of sound and a density of states cod-1

N(w)

=

Ca .

(5)

At low frequencies the scattering is weak and can be described by a Rayleigh scattering 17) time 1

0~d + l

-- = - Ts

(6)

wIdR '

where we have introduced the Ioffe-Rege118'x9) frequency (WIR) for which 1 -

=

(7)

% The scattering time ('rs) in (5) introduces a second length scale (8)

Below 0~ia one is in the weak (elastic) scattering regime. The (Born) scattering matrix elements are small. One result is that the density of states is not modified by the scattering and is still given by its free form [eq. (5)]. Somewhere in this regime Anderson localization sets in so that the phonons become localized. This

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~1

has been discussed in great detail in a number of recent papers, particularly those by John, Sompolinsky and Stephen2°), by Akkerman and Maynard 21) and by Anderson2Z). It is useful to look at the structure of the eigenfunctions or the Green's function [G(0, r) before averaging]. There are three length scales involved with an inequality A < As ~< At,

(9)

where Al is the length scale associated with multiple scattering Anderson localization. One has free particle ballistic motion for r < As and anomalous diffusion in the range As < r < At. For scales large compared to Al one has normal diffusive behavior below the localization edge (w < WA) with a renormalized diffusion constant. Above wh the diffusion constant (D(w)) vanishes in this limit. Now all these three length scales decrease above wA. As decreases faster than A [eq. (8)] and they become equal at the Ioffe-Regel frequency (win). In considerhag the behavior of the localization length one has to remember that the scattering length (As), or equivalently the scattering Coupling parameter, is an ingredient in the multiple scattering theory. Thus on frequency scales for which As changes the critical behavior of At basically multiplies the noncritical frequency dependence of As [eq. (8)]. Thus At(w ) must approach both As(w ) and A(w). We shall claim that the three scales all coincide for w = w tR. A heuristic argument is to consider the Thouless localization length23) (A-r) which neglects the anomalous character of the diffusion between As and At. This should not matter when As -- At. One has

D(w) = N(w)Ad'

(10)

and, using D(w) = CAs(w ) and (5) and (8), one finds [ ¢..0 ~2/(d-2)

so that the three lengths become comparable at w ia. The eigenfunctions at tOlR thus have a single length scale, A -- As -- Ai, as we assumed for fractons. They are of course localized. One expects this single length scale feature to be a characteristic of the eigenfunctions at higher frequencies. It seems implausible that even stronger scattering can be renormalized so that A and As regain their separate identity.

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S. ALEXANDER

This assumes that the basic length scale for the disorder is a (<<)~m) - so that such a strong scattering regime has physical significance. The latter is not a trivial assumption. We note that the treatment of the disorder in refs. 20 and 21 implicitly assumes ~ IR = a. Fractons live in this strong scattering range. Recent experimental evidence 24) also shows that hip" >> a. We therefore consider the meaning of having a physical model for which ~ IR >> a in detail. The basic quantity one wants is clearly some local index of refraction or velocity of sound. One is interested in the averages relevant to an excitation of characteristic length I. A simple procedure would be to divide space into cells of size l. For each cell we determine the lowest frequency (with some reasonable boundary conditions). We then define a local (cell) velocity of sound (12) In a disordered system one will find a distribution with an average

(C 2) = (C2),

(13)

and a variance



1 = 7:,

(14)

where yl(w) is the scattering parameter governing the effect of the disorder for these waves. In particular the Ioffe-Regel condition [eq. (4)] is equivalent to some critical value T, = y * -- 1.

(15)

Consider now the statistical distribution. For a normal distribution one would have ( C 2 ) t = C 2,

(16)

independent of l and

]7,

(17)

where y2 is the variance at the atomic level (l = 1). This is only consistent with a large Ioffe-Regel length XI~ = t(-~*) >> 1,

(18)

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if 7 2 >> 1. While one cannot exclude this option it is rather difficult to think of (mechanically stable) microscopic models for which it would make sense. An obvious alternative is suggested by percolation, ( C 2 ) t cx l -x,

(19)

v, - v .

(20)

The meaning of eq. (19) is that the coupling of cells of size l into larger cells (of size bl) involves not only their internal structure (expressed by the C2) but also the coupling between the ceils. Eqs. (19) and (20) are a scaling description of the result if these couplings are sufficiently random. For a percolation model, for which only a fraction of the potentially possible bonds exists, this means that space is uniformly covered by clusters, of any size 1, but only a fraction (b o - d = b - a / ~ ) of these clusters is coupled into larger clusters (of size bl). Variations on this scenario are of course possible25). Finally I want to show that the consistency of this argument leads in the strong scattering Ioffe-Regel limit to a fracton spectral dimension of 4/3. I follow ref. 8. Consider vibrations on a fractal. We want to compute the scattering width in the Born approximation analogous to Rayleigh scattering [eq. (6)], assuming weak disorder at some small length scale. This requires the definition of an effective strain on the scale of the "wave length" (Xtr(~0))

1 -

,'r

(&)4 =

V2U(

)=

(21)

where q, is the vibrational amplitude and R(X) an effective distance on the fractal corresponding to a distance X in cartesian space. It can be seen26) that R(X) is the distance along the bonds, and therefore proportional to the resistance between two points on the fractal at a distance X in space when there is a unique connecting path. More generally R(X) scales like the point to point resistance (or force constant) one would need if one were to coarse grain the fractal to scale X. In both situations one finds 26) Rcx )t?,

~ = (2 - 4)(D/d-),

(22)

and using ep/R - [oY2R] -1 and eq. (2) one finds V z - ~6-4~ and finally, using

(1) in (21) 1/~" cx o~5-3'7.

(23)

This forces an effective "quantum" fractal dimension of 4/3 if one wants the scattering to remain marginally strong (or weak) and obey the Ioffe-Regel condition [eq. (4)] at all frequencies. The philosophy of this calculation is of

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s. ALEXANDER

course that o n e tries to take a c c o u n t of the strong disorder a n d multiple s c a t t e r i n g b y " c h a n n e l i n g " the v i b r a t i o n s o n a fractal o n which the scattering is (relatively) weak. I have tried to show what a scaling model for the strong scattering regime in a m o r p h o u s materials means, why it seems reasonable, a n d to discuss some implications.

Acknowledgement I w o u l d like to t h a n k H.E. Stanley for insisting o n m y writing this m a n u s c r i p t for these Proceedings.

References 1) 2) 3) 4) 5)

S. Alexander and R. Orbach, J. de Physique Lett. 43 (1982) L625. S. Alexander, Phys. Rev. B 27 (1983) 1541. Y. Gefen, A. Aharony and S. Alexander, Phys. Rev. Lett. 50 (1983) 77. R. Rammal and G. Toulouse, J. de Physique Lett. 44 (1983) L13. This well-known result was recently rediscovered in ref. 15. It was certainly not new in ref. 1, where it is used as a didactic illustration. 6) S. Alexander, Ann. Isr. Phys. Soc. 5 (1983) 144. 7) S. Alexander, C. Laermans, R. Orbach and H.M. Rosenberg, Phys. Rev. B 28 (1983) 4615. 8) A. Aharony, O. Entin-Wohlman, S. Alexander and R. Orbach, Phys. Rev. Lett., submitted. 9) D. Stauffer, Phys. Rept. 54 (1979) 3. 10) A. Aharony, S. Alexander, O. Entin-Wohlman and R. Orbach, Phys. Rev. B 31 (1985) 2565. 11) S. Alexander, O. Entin-Wohlman and R. Orbach, Phys. Rev. Lett., submitted; Phys. Rev. B 34 (1986) 2726. 12) S. Alexander, O. Entin-Wohlman and R. Orbach, J. de Phys. Lett. 46 (1985) L549, L555; Phys. Rev. B 32 (1985) 6447; 33 (1986) 3935. 13) O. Entin-Wohlman, S. Alexander and R. Orbach, Phys. Rev. B 32 (1985) 8007. 14) P.W. Anderson, B.I. Halperin, and C.M. Varma, Phil. Mag. 25 (1972) 1; W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. 15) J.A. Krumhausi, Phys. Rev. Left. 56 (1986) 2696. 16) J. Maddox, Nature 322 (1986) 303. 17) Lord Rayleigh, The Theory of Sound (Macmillan, London, 1986), vol. II. 18) A.F. Ioffe and A.R.-Regel, Prog. in Semiconductors 4 (1960) 237. 19) N.F. Mott, Phil. Mag. 13 (1974) 93. 20) S. John, H. Sompolinsky and MJ. Stephen, Phys. Rev. B 27 (1983) 5592. 21) E. Akkermans and R. Maynard, Phys. Rev. B 32 (1985) 7850. 22) P.W. Anderson, in T. Holstein Memorial Volume, R. Orbach, Ed. (Los Angeles, 1986). 23) D. Thouless, Phys. Repts. 13 (1974) 93. 24) J.E. Graebner, B. Golding and L.C. Allen, prepfint. 25) In particular viscoelasticeffects can become important. Limitations of time and space prevent me from exploring this here. 26) A. Aharony, Y. Gefen and Y. Kantor, J. Stat. Phys. 36 (1984) 795; Ann. Isr. Phys. Soc. 5 (1983) 301.