Ultrasonic mean free path in a granular aluminum film

Solid State Communications,Vol. 17, pp. 653—656, 1975.

Pergamon Press.

Printed in Great Britain

ULTRASONIC MEAN FREE PATH IN A GRANULAR ALUMINUM FILM* M. Tachiki,t H. Salvo, Jr., D.A. Robinson~and M. Levy University of Wisconsin—Milwaukee, Milwaukee, Wisconsin 53201, U.S.A. (Received 3 February 1975 by A.G. Chynoweth)

The ultrasonic mean free path has been measured and compared to the electrical mean free path of a thin granular aluminum film. They have been found to differ by an order of magnitude which is believed to indicate that mean free path determined ultrasonically is for the Al metal while the one determined electrically is for the Al—Al2 03 matrix structure.

THE ELECTRON—PHONON contribution to the ultrasonic attenuation coefficient of a 300 A Al film has been measured using 2 GHz surface waves. The value obtained for the electron mean free path from these measurements will be compared to that obtained from the electrical conductivity of the same film. The result will indicate that the ultrasonic mean free path is about 10 times larger than the electrical mean free path. This was is consistent with that since the aluminum evaporated inthe the fact presence of I O~torr of oxygen, the Al film is composed of metal islands 50 A in diameter embedded in an Al 1 Thus 2 03 matrix. it appears that the surface phonons probe the metal islands themselves so that the mean free path is important for this process is the electron mean free path within the islands. On the other hand the electrical current has to travel across the islands and in this case the effective mean free path is determined by the tunneling from one island to another. Since the attenuation is linearly proportional to the mean free path, the contribution to the attenuation of the matrix

with its very small mean free path is very small cornpared to the contribution of the Al islands. Figure 1 shows a schematic of the experimental arrangement. The substrate is lithium niobate. The Al film was evaporated in the presence of 1 O~torr of oxygen between the two interdigital transducers, one of which was used as a transmitter and the other3 as a receiver. This delay line was then placed be in aobtained. He cryostat where temperatures of 0.4°Kcould Figure 2 shows a superposition of the attenuation in the normal and superconducting states of the Al film down to 0.74°K.The normal state was obtained by applying a magnetic field. Both superconducting and permalloy shieldings were used to reduce the earth’s magnetic field below fifty milligauss when the measurements were performed in the superconducting state of Al. The d.c. resistance of this film in the normal state was 100 &2/o. As may be seen from Fig. 2, the attneuation change experienced by the surface waves as the film went from the normal to the superconducting state was

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*

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Research supported by the U.S. Air Force Office of Scientific Research under Grant No. AFOSR 712079.

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0.45 dB which yields an attenuation coefficient of 0.33 dB/cm. It should be pointed out that this attenuation is being produced only by electron—phonon interaction in the Al film since when the Al film be. comes superconducting this part of the attenuation is

t Permanent Address: Tohoku University, Sendai, Japan.

eliminated as shown in Fig. 2. The problem now is to determine the attenuation coefficient of the Al

+

~ Permanent Address: Texas Instruments, Dallas, Texas U.S.A.

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653

654

ULTRASONIC MEAN FREE PATH IN A GRANULAR ALUMINUM FILM

Vol. 17, No.6

i8oxc~i Z

kdsgra$os

4

______________

I~~I

__

Al

( ~

q

_____

FIG. 3. Model LiNb used ~ for calculating the attenuation in

~

h

\F~ _ ___________ I In.

FIG 1. Schematic of system used for producing, propagating and detecting surface waves.

the Al film. h is theq thickness pagation direction is along x.of the Al film. The proand (2), ~ and ~c2are constants indicating the penetration depth of the surface wave and are given by Ki = ~ (3)

qjl

K2 =

—

(~2 \Vj

/

~2

(4)

where v1 and Vt are the velocities of longitudinal and transverse bulk waves, and ~is defined by the ratio of the surface wave approximated by velocity v to Vt and reasonably

V

c..J 9

0.874 + 1.12v 1+ii

(5)

being Poisson’s ratio for the substrate. In an isotropic 2 is also expressed as a function of Poisson’s ratio(vt/v,) as(l 2v)/(l v). Thex andy cornelastic body, ponents of the surface vibration induce compressive and shearing strains in the Al film. These strains cause i’

0.6

0.9

2

.5

18

2.1

24

2.7

30

33

I °K FIG. 2. Surface wave attenuation as a function of ternperature. The Al film held in normal state in upper curve. film. This may be done if we look at the following model. Surface waves propagate along the x axis in the coordinate system shown in Fig. 3. We assume that the substrate is an isotropic elastic body and also surface waves in the substrate are not affected by the presence of the Al thin film covering the substrate surface. The x andy components of the2 displacement vector of surface wave are expressed as KIZ e~Z] cos(qx wt), (1) S, AK 2+K~ 1 [e q S~ Aq [e’ciz 2K 2 1K2 + K~eS(2Z sin(qx—wt), (2) q where A is a constant, and q and w the wave number and frequency of the surface wave. In equations (1) —

—

—

—

—

—

—

energy dissipation of the surface wave to the conduction electron system in the Al film due to Pippard’s mechanism.3 When the electron mean free path is much shorter than the wave length of the surface wave, the mean rate of the compressive strain energy loss per unit surface area is given by 2 Nmv~c.,i4rh 2 (6) Q = 5x0 where h is the film thickness, v 0 the Fermi velocity of the conduction electrons, N the per unit volume, r the relaxationnumber time ofof theelectrons conduction electrons, x component of the surface wave amplitude 5x0 at z the = 0. The mean rate of the shearing strain energy loss is given by

J

=

2 4rh 1 Nmv 0w 52zO

(7)

where S~ 0is the z component of surface wave amplitude

Vol. 17, No.6

ULTRASONIC MEAN FREE PATH IN A GRANULAR ALUMINUM FILM

oo8r 0.o7r

I

From the electrical resistivity measurements we find 1 = v0r = l.3A using a = Ne2 rim. We would like to call this mean free path an effective mean free

I~

-

~o.o6r

path since the actual mean free path in the Al is probably 14A. However, this mean free path reflects the fact that the electrons are tunneling from one metal island to the other. So the number obtained for the mean free path is really a conversion of the tunneling probability to a mean free path and the equation for

oo5r~ 004f. 0.20 0.25 030 035 040 0.45 0.50 U .

I

I

I

I

I

I

the conductivity is our means of converting the measured electrical resistivity which is limited by tunneling into an effective mean free path. As is seen the two values for 1 differ by a factor of 10.8. This appears to be consistent with a model wherein the ultrasonic surface waves sample the aluminum islands and Al 2 03

FIG. 4. Graph of equation (11) of text. F(v) vs v. at z = 0. The attenuation coefficient of surface wave is given by

Q~+

(8)

matrix in parallel while the electrical current goes through them in series.

yE where E is the energy of the surface wave per unit surface area. It can be shown that the mean kinetic energy of a surface wave is equal to its potential energy. Thus the total surface energy is given by twice the kinetic 0 energy E p[((.~’~)2 ) + ((s~)2)]~ (9)

=f

where p is the density of the substrate and (A) denc~tes the time average of A. The value of E is calculated using equations (1), (2), and (9). Inserting the energy E and equations (6) and (7) into equation (8), we obtain Nmv~w2r a = ~ (qh)F(v), (10) pv 2ab .~.(l_l+2) FØ.’)

=

a 2

2

~2

a

4ab +a2 Xa+b)+

with a = \/iT~j~ and b = ~/1 (Vt/Vj)2 ~2 The calculated values of F(v) are shown in Fig. 4. —

The above results are obtained using free electron model parameters. An alternative way of analyzing this data is to use values for the parameters obtained 5 Hauser6 and Abeles7 a!. have found that Alinfilms from theetanomalous skin effect Al. may be characterized by an effective mean free path 1eff obtained from P1o = 1.6 x lO~~ ~2cm2.This would yield ‘eff = 5.3A for our film. The expression for a in equation 10 would then have to be modified to

=

1m2 v~.,2 1.78 x l0_23e2pv3~~

+j~~a2(l_j+_2)2

1 (1 +a2Xa+b)+(1 +a2 )2b +~(l ~2

655

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Using the measured attenuation coefficient and equation (10), we obtain the electron mean free path as 1 = 14.1 A. Here we have used N = 1.806 x 1 023cm’, m = free electron mass, v 0 = 2.02 x 108 cm/sec, p = 4.7, v = 3.226 x l0~cm/sec, and v = 0.309.~This mean free path is consistent with a mean free path which is given by diffuse scattering over the surface of a sphere of diametera = 50 A, in which case the average mean free path 7= 2a/3 = 33 A.

(

2a2b +a2)2

(11)

in order to include the anomalous skin effect results. This would yield an acoustically obtained 1eff = 57A. The ratio of these two values is 10.8 which is the same as the one obtained using the free electron values. Acknowledgements

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The authors would like to

acknowledge the invaluable help of Ernest Stern,. Richard Williamson, and Harry Smith of the Surface Wave Technology Group at Lincoln Labs, who designed and fabricated the surface wave delay line. The authors also wish to thank L. Testardi, Bell Labs, for several fruitful discussions.

656

ULTRASONIC MEAN FREE PATH IN A GRANULAR ALUMINUM FILM

Vol. 17, No.6

REFERENCES I.

HAUSER J.J.Phys. Rev. B3, 1611 (1971).

2.

VIKTOROV l.A., Rayleigh and Lamb Waves (1967).

3.

PIPPARDA.B.,PhiI. Mag. 46,1104(1955).

4. 5.

LI R.C.M.,Proceedings 1972 Ultrasonics Symposium, p. 263, October 1972 (Edited by DE KLERK J.) IEEE, New York (1972). OLSEN J.L., Electron Transport in Metals p. 84, Interscience, New York (1962).

6. 7.

HAUSERJ.J.,J. Low Temp. Phys. 7,335(1972). ABELES B., COHEN R.W., and STOWELL VLR.,Phys. Rev. Lett. 18,902(1967).

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Physical Theory and Applications Plenum Press, New York