Tunneling-states model and the propagation characteristics of surface phonons in amorphous solids at low temperatures

Solid State Communications, Vol. 30, pp. 517—520. Pergamon Press Ltd. 1979. Printed in Great Britain. TUNNELING-STATES MODEL AND THE PROPAGATION CHARACTERISTICS OF SURFACE PHONONS IN AMORPHOUS SOLIDS AT LOW TEMPERATURES A. Tate, S. Tamura and T. Sakuma Department of Engineering Science, Hokkaido University, Sapporo 060, Japan (Received 4 December 1978 by A.A. Maradudin) The propagation characteristics of the surface phonon in an amorphous film are theoretically investigated at low temperatures based on the tunneling-states model. It is shown that the attenuation rate of the surface phonon is proportional to w tanh hw/2kBT and the relative velocity variation to In (T/T0) for the thickness of the amorphous film comparable to or larger than the surface phonon wavelength. Below 3 K, our result can well account for recent measurements of the relative velocity variation by Hartemann et al. 3/T and the relative variation of the surface AT VERY LOW TEMPERATURES amorphous varies as velocity w exhibit anomalous behavior in their specific heatsolids [1—3], phonon is proportional to w2(l IT 0 lIT). thermal conductivity [2—41and acoustic propagation Most recent measurements [5]of surface acoustic wave [3, 5—7]. These characteristics of amorphous solids are velocity for a layer obtained by ion-implanted quartz in common independent of their chemical compositions show essentially the same temperature dependence as and have been successfully explained by means of the that of the bulk phonon in the amorphous bulk materlow energy excitations associated with the two level jals. This discrepancy between the theory and the expersystems which are assumed to exist in the amorphous iment may be due to the difference in the film thickness solids. These two levels are made by the quantumwhich is experimentally chosen to be close to the penemechanical tunneling of atoms between two possible tration depth of the surface phonon. configurations localized at two nearly equal potential The purpose of the present paper is to investigate wells. Transitions of atoms between the two positions the surface phonon propagation in an amorphous film are possible by tunneling, whereby a phonon is absorbed at very low temperatures by means of the tunnelingor emitted resonantly. This model will be referred to as states model in the situation where the film thickness the tunneling-states model [8, 9] hereafter. a is comparable to or larger than the penetration depth In amorphous bulk materials, the tunneling-states of the surface phonon. The surface phonon traversing model can account for the excess specific heat varying the film with the 2 states in will the resonantly amorphous interact film through the tunneling deformation linearly with temperature T and the anomalous T dependence of the thermal conductivity at low tempera- potential coupling and suffer the attenuation as well as tures. As for the ultrasonic propagation, the model can in the amorphous bulk materials. also explain the ~2/T dependence of the attenuation In a solid extending for z > 0 with a stress-free rate for phonon energy hw small compared with kBT plane boundary at z = 0, we can expand the displacement and the sound velocity variation proportional to vector u(r, t) of the medium at a point r = (x, z) and at In (T/T 0), where T0 is the fiducial temperature for several a time t in terms of eigenwaves Uj(Z) e1k~in the homokinds of amorphous bulk materials. At higher temperageneous [111 and even in a layered half space as follows: tures, the finite life times of the two levels become / h \1/2 important and then a relaxation mechanism will dominu(r, t) = [ajuj(z) e~~~1t-4- h.c.], ate the resonant mechanism [9]. \2p(~)jS/ (1) The propagation characteristics of the surface where p is the mass density of the solid, J represents a phonon (the quantum of the surface acoustic wave) set of quantum numbers specifying the eigenmodes both have recently been studied by Nakayama [101 at low surface and bulk in character, S is an appropriate normaltemperatures using the tunneling-states model for the ization area, k and x are the wave and the coordinate amorphous film thin enough compared with the panevectors parallel to the surface (x—y plane) and aj and its tration depth of the surface phonon. The results of his Hermitian conjugate aj~jare annihilation and creation work are quite different from those of the bulk phonon operators of the J-mode phonon satisfying the ordinary in the amorphous materials, that is, the attenuation rate commutation relations of the Bose type. —

~I .~

517

)

518

SURFACE PHONONS IN AMORPHOUS SOLIDS AT LOW TEMPERATURES

x-y plane

1/2

u~(z)= _7(~)

{e_7’~~_ 1 ~

2e~u’Z],

(2)

where k = IkI, ~yand ,~ are decay constants and K is a constant needed for the normalization of the wave functions. We can now write down the interaction Hamiltonian between the jth tunneling state and the surface phonon;

Substrate

I

=

Z

~ [V~jb~1bpjai~ + h.c.],

(3)

j,k

Fig. I. The layered system consisting of the amorphous film with thickness a on the crystalline substrate. Surface phonon propagates parallel to the surface of the film (x—y plane). In the following, we shall be devoted to the propagation characteristics of the surface phonon. In the layered system consisting of an amorphous film and the crystalline substrate shown in Fig. 1, the solution for the surface acoustic wave is generally complicated since in order obtain the complete solution, we must solve the wavetoequation separately both in the film and the substrate and then form a linear combination of the waves to satisfy the boundary condition at the interface. When the elastic properties of both mateials are assumed to be isotropic, the Rayleigh-type wave can propagate in this layered system [12]. The most significant effect on the surface wave produced by a film on the otherwise free infinite surface of a substrate is the change of its velocity, which takes a value between those for the Rayleigh waves in the layer and in the substrate materlals depending upon the film thickness. Therefore, it should be noted that as far as the interaction of the surface acoustic wave in the film is concerned the surface wave can be approximated by the Rayleigh wave with appropriately modified values for the velocity and the decay parameters. In this paper, we wish only to discuss the qualitative characteristics of the interaction of the surface phonon with the two level systems. Accordingly, we assume that both the film and the substrate of the above system can be approximated to be isotropic in their elastic properties and the above mentioned Rayleigh-type wave exists as the surface acoustic wave in the layered system [13]. In the limiting case where the film thickness becomes much larger (smaller) than the penetration distance of the wave, it may be reduced to the Rayleigh wave in the material of the film (substrate). The explicit forms of the z-dependence of the surface phonon wave functions within the film are thus given by 2 2’vn e”’~’, ~ = x, y, k ~K / [e ~ I + ~2 u~(z)= ~~_l(k~’ —

Vol. 30, No.8

J

where a and 13 denote the upper and the lower levels of the two-level states, respectively, which can be obtained by diagonalizing the relevant tunneling states. b~and b~1creates a tunneling particle at the upper level and annihilates it at the lower level of the jth two-level state, respectively. Considering the deformation potential coupling between the localized tunneling states and the surface phonon, the vertex function ~ in equation (3) is given by 3__~1/2 (1 _72) e~7~zJ, (4) v~,= —D E~ (__hk 1 \2pKSwJ where D is the deformation potential coupling parameter. ~, = ~ e~iis the off-diagonal matrix element of the jth tunneling state due to the small overlap of the wavefunctions of the localized oscillators, where hw0 is a typical zeropoint energy in the wells and Xj is a par. ameter describing the extent of wave-function overlap between the states in the jth double well. E1 = -sJC + is the level splitting of the jth two-level system, here ej is a potential The depth state from theasymmetry. surface is denoted by of z1. the fth tunneling In the lowest order approximation of the perturbation theory,ofwe the attenuation face phonon thefind frequency w = ck asrate of the sur-

a

(

=

2ir ‘ç’ tanh E~ iV~pjI2&(E1 hC~), hc \2kBT /

(5)

~

where ~ means the sum over all tunneling states subjected t~othe interaction with the surface phonon and c is the velocity of the surface phonon. It is convenient to evaluate the sum over the possible tunneling states by averaging over the double-well poten. tial parameters e, X and the depth parameter z from the surface by using the following substitution ~

-÷5

de dX dz P(e, X)F(z).

(6)

Here, P(e, X) represents the probability distribution of the states for the varying values of z-direction. and X, and tunneling F(z) is their distribution function in the We assume that the tunneling states have a uniform

Vol. 30, No.8

SURFACE PHONONS IN AMORPHOUS SOLIDS AT LOW TEMPERATURES

distribution for the parameters e and X at low temperature and also for the spatial parameter z in the layer. With this assumption, we have for P(e, A) and F(z) F(, A)

=

PoO(X

—

(Fop 0,

=

a,

P

c

with Xmin = hni (~~~,0IE1) [8], and F(z)

fd~ c a(c~T)

—

(7)

Amin)

~

(8)

z > a,

—

519

a(oJ~T0) (10)

~2_~2

where P indicates the principal part of the integral. We assume that the expression (9) is valid for all frequencies below a certain cutoff frequency “~maxand that the relation hWm~~ k~Tholds. Substituting equation (9) in equation (10) and taking the large thickness of the film into consideration, we obtain the following expression for the relative velocity variation of the surface phonon. 2(1 —y22 ~ ~ (T/T i.~c noD 0). (11)

where P0 and F0 are constants. The attenuation rate of the surface phonon can be calculated from equation (5) together with equations (6)—(8). The result is given by c 2K7pC 2(l 72)2~) hw This logarithmic temperature dependence agrees well tanh 2kBT (9) with the results observed experimentally by Hartemann a = irnoD2K7pc3 (1 e_2~~) — —

— _________________

—

Here, n 0 = P0F0/S is the number of the tunneling states per unit volume and energy. In the cases where the film thickness a is much smaller or larger than the relevant penetration depth 1= (7k)’ of the surface phonon, we find for the attenuation rate of the surface phonon due to the resonant interaction with the tunneling states as 2tanh— hw a c~., 2kBT

for a very thin film satisfying a/I ~ L and a cx ~ tanh 2kBT for a thick film satisfying a/I> 1. It should be empha. sized that the former and the latter have the same frequency and temperature dependences as those in the very thin amorphous fIlms [10] and for the bulk wave in the amorphous materials [8, 9], respectively. The difference of the frequency dependence in the attenuation rates due to film thickness may be qualitatively understood as follows: the surface phonon penetrates the bulk material to a depth I cx c~f’.When the film thickness is larger than the penetration depth I, that is a/I> I the available number of the tunneling states which contribute to the attenuation of the surface phonon is the total number of those in the film multiplied by the factor i/a. Since the penetration depth I is proportional to w~’,the frequency dependence of the attenuation rate of the surface phonon in the thick amorphous film is weaker than that in the thin film by w. Next we consider the relative velocity variation of the surface phonon in an amorphous film through the resonant interaction with the tunneling states. The relative variation can be calculated using the surface phonon attenuation rate equation (9) by means of the wellknown Kramers—Kronig relation as ,

etal. [5]. It may be worthwhile to note that the temperature dependences of both the attenuation rate (9) and the velocity variation (11) are independent of the velocity and the decay parameters of the surface phonon. For the surface phonon with the penetration depth comparable to the film thickness, we can expect that the velocity variation mayform haveofthe dependence essentially of the In temperature (nT 0) as well, which is independent of the velocity and the decay parameters characteristic to the layered system. This is beacuse the term 27k0 which has been neglected to derive equation (11) for a thick amorphous film has a small contribution to e_ the integral of equation (10), since it is nonvanishing only for the restricted region of the frequency, i.e. w c/2-ya. Finally, we estimate the magnitude of the surface phonon attenuation rate. Fitting the expression for the relative velocity variation equation (II) to the experimental results obtained in [5],we can determine the value of the product n 2 and consequently evaluate the 0D magnitude of the attenuation rate of equation (9). The result shows that the surface phonon attenuation rate becomes hw a (1 x lO9sec cm~)wtanh 2kB T ~,

where we have roughly estimated its order of magnitude by taking the following set of parameters for vitreous silica, c = 3.4 x iO~cm sec~’,p = 2.2 g cm3, y = 0.81, K = 0.75, a = 0.74 pm and the frequency v around 1 GHz. For T = 2K and v = 1 GHz, the attenuation rate becomes a 0.7 dB cm’ and this gives almost the same magnitude of the attenuation rate as that of the bulk phonon in amorphous materials [6]. In conclusion, we have investigated theoretically the propagation characteristics of the surface phonon in an amorphous film with a finite thickness assuming the

520

SURFACE PHONONS IN AMORPHOUS SOLIDS AT LOW TEMPERATURES

possible existence of the tunneling states. Our results naturally agree with those obtained by Nakayama [10] for the film thickness thin enough compared with the penetration depth of the surface phonon. In the case of the thick amorphous film, however, the frequency and temperature dependences of the attenuation rate and the relative velocity variation of the surface phonon are

2.

3.

just the same as those [6, 8,9] of the bulk phonon at low temperature. The surface phonon attenuation rate behaves as w2/T for the phonon energy small compared with k~Tand the relative velocity variation is proportional to In (T/T 0). Below T = 3K, our theoretical result can well account for the relative velocity variation measurement in an ion-implanted quartz by Hartemann et aL [51. At higher temperatures a relaxation mechanism [9] may become important and further theoretical calculations involving all modes of phonons in a half space will be required.

~

Acknowledgments One of the authors (S.T.) expresses his thanks to the Sakkokai Foundation for financial support. This work was also supported by a Scientific Research Fund from the Ministry of Education.

8.

—

s. 6. 7.

9. 10. 11. 12.

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Vol. 30, No.8

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