Temperature- and pressure-dependence of the resistivity of MgxZn100−x metallic glasses

Journal of Non-Crystalline Solids 109 (1989) 267-276 North-Holland, Amsterdam

267

TEMPERATURE- AND P R E S S U R E - D E P E N D E N C E OF T H E RESISTIVITY OF Mgx Zn too- x METALLIC GLASSES A. DROST, G. FRITSCH a, W. D Y C K H O F F and E. L U S C H E R Physik Department, Technische Universitiit Miinchen, D-8046 Garching, FRG a Phys. lnstitut FB BauV/l, Universitiit der Bundeswehr Miinchen, D-8014 Neubiberg~ FRG Received 14 December 1987 Revised manuscript received 6 January 1989

The resistance of the metallic glass MgxZnl0o_ x (x = 67, 72, 77 at.%) has been measured as a function of temperature (2-250 K) and quasihydrostatic pressure (up to 10 GPa). We show that the behaviour of the resistivity can be understood on the basis of the Faber-Ziman theory for metallic glasses.

1. Introduction The electrical resistivity of amorphous metallic alloys with low resistivity (p = 50 /t12 cm) and without magnetic contributions can be successfully described by the F a b e r - Z i m a n theory [1]. At temperatures far below the Debye temperature O there may be additional contributions by quantum interference effects [2]. Besides the variation of the resistivity with temperature its dependence on hydrostatic pressure is another effective tool to check theory. While the temperature dependence of the resistivity is mainly given by the temperature dependence of the resistivity structure factor SP(K) in the F a b e r - Z i m a n theory [1], pressure changes in addition quantities such as scattering potential, Fermi energy and Debye temperature [3]. In this paper we present the results of resistance measurements as a function of temperature and quasihydrostatic pressure (up to 10 GPa) on amorphous MgxZnl00_ x. The data will be discussed within the framework of the F a b e r - Z i m a n theory.

2. Experimental details The high pressure apparatus used in our experiments has been described earlier [4]. It is based on 0022-3093/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

the principle of two opposed anvils as introduced by Bridgman. The pressure was determined in situ by recording the superconducting transition temperature of lead [5]. We measured the temperature with a calibrated carbon or platinum resistor. The resistance could be detected by using the common four-lead technique. The relative resolution of the resistance data is about 2 x 10 -5. In the case of pressure and temperature the accuracies are about 10% as well as less than 0.1 K below 50 K and 0.5 K above this value. The samples were produced in 1985 (March) by melt spinning (Mg 4N, Zn 4N) and stored at liquid nitrogen temperature. The amorphous state was confirmed by X-ray diffraction just before the measurement. The data were taken in 1986 during the period July to December.

3. Corrections To compare the measured resistance, R, with the resistivity, p, obtained from the theory, R has to be multiplied with a temperature- and pressuredependent geometrical factor

G(T, P ) = p(T, P)/R(T, P) =A(T, P)/I(T, P),

(1)

where A and l are the cross section and length of

268

A. Drost et al. / Resistivity of M g x Zn lo o _

metallic glasses

:,

56.0 • ..............................

••.....•...,

.. •

~

55.0

"•.•,.. "-... •'•',..••.

E tJ

54.0

~

••.,,••...

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./.7

a)

~-~-~.....

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,'o

o I

A I

IlK J

3'0 I

0

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so J

J

I

l

~00

1

I

1

200

T/K Fig. 1. The resistivity of Mg72Zn28 at P = 2.6 GPa, corrected without (a) and with (b) a temperature- and pressure-dependent geometrical factor. The insert shows the low temperature region. Here, both curves are shifted closer together in order to demonstrate their detailed shapes.

the sample, respectively. The factor G(T, P) as a function of temperature and pressure can be written as follows: G(T, P)=G(T0,

Po)* 1+

-~Ix(T,

,) dP).

~,(T, Po)dr (2)

Because of the lack of data of al(T, Po) and the amorphous phase we used available data for the single crystal elements and averaged them according to the volume proportion in the alloy [6,7]. For Mg72Zn28 the procedure yields the following numbers:

x(T, P) in

%(239 K) - 25.9 × 10 - 6 K - I , Here, G(To, Po) is the geometrical factor at room temperature, T0, and ambient pressure, Po; al(T, Po) is the linear thermal expansion coefficient at normal pressure and g(T, P) is the temperature- and pressure-dependent compressibility• These corrections are not negligible because changes in p with temperature, Ap/p I AT=293K= 5%, and those with pressure, Ap/p Jze=sGpa = 20%, for MgZn are only a factor of about 10 larger than the effects caused by the temperatureand p r e s s u r e - d e p e n d e n t geometrical factor

G(T, P).

1(0 K ) / l ( 2 9 3 K) - 1 = - 5 . 1 × 10 -3 ,

x(To, P0)

= 25.9 TPa -1,

dr/dP=

- 2 . 7 × 103 TPa -2 at

P=Po,

d K / d T = 5.6 × 10 -3 (TPa K) -1 at T-- TO. To justify the averaging procedure in the case of we compare G(T)/G(To) as measured in ref. [8] in the temperature range from 77 K to 300 K with the values as calculated using averaged data. We find differences smaller than 11% which constitutes an acceptable agreement in this tern-

al(T )

A. Drost et aL / Resistivity o f M g x Z n lo o _ ~ metallic glasses

269

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P/GPa 75

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Fig. 2. The resistivity of Mg67Zn33, Mg72Zn28 and Mg77Zn23 as a function of T and P.

,

r

,

,

300

270

.4. D r o s t et aL / Resistioity o f M g x Z n so o _ ~ m e t a l l i c glasses

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53.63 30 40 50 T/K Fig. 3. P and T d e p e n d e n c e of the resistivity of Mg67Zn33, Mg72Zn28 a n d Mg77Zn23 in the low-temperature region. T h e n u m b e r s at the ordinate refer to the resistivity at the m i n i m u m Pmin. 0

I0

20

A. Drost et al. / Resistivity of Mg:, Zn ~oo- x metallic glasses

perature range. This number has to be considered as a systematic error. Figure 1 illustrates the effect of correcting the data with a temperature-dependent geometrical factor G ( T ) . The overall shift is caused by the pressure dependence of the geometrical factor G ( P ) . The slope in the high temperature range gets reduced by about 10%. A value of a~(T) showing a systematic error of 15%, therefore, leads to a final error in the slope of less than 2%. In the low temperature regime (T < 30 K) the geometrical factor can be taken to be nearly constant because of the smallness of a l ( T ) (G(30 K ) / G ( 0 K) - 1 -- 2 × 10-5). Thus, the structure in R ( T ) below T < 30 K is not affected by the geometrical factor G ( T ) , as can be seen in the insert of fig. 1. In the case of compressibility there exist no experimental data for MgZn. By comparing the compressibility of an amorphous with a crystalline system it can be concluded that the compressibility of the amorphous alloy deviates less than 5% according to ref. [9] and less than 15% according to ref. [10] from the value of the compressibility for that crystalline metal being the main component in the alloy. The influence of the pressure dependence of the geometrical factor G ( P ) on the data will be discussed below. In addition, the averaged values may be compared with those obtained by Hafner in his "first principle" calculations [11]. He gets

(T0, t0) = 20.8 TPa al(300 K) = 20 × 10 - 6 K -1. These numbers deviate by about 20% from the values quoted above. However, since they depend to a certain extent on the theoretical procedure applied, we prefer to use the averaged values in the rest of this paper.

271

of the not precisely known geometrical factor G(To, Po)). In the range T > 175 K the resistivity is linear in temperature; the T C R = p -1 d p / d T turns out to be negative and is of the order of - 3 × 10 4 K-1. Increasing the Zn contents the TCR takes more negative values. Applying pressure the resistivity increases ( H = p-1 d p / d P = 0.04 G P a - 1) and the T dependence appears to be less steep in the high-temperature region. In fig. 3 the low temperature behaviour of the resistivity is shown. The curves are plotted in such a way that the distances between the minima are equidistant in the y-axes (the lowest curve belongs to the highest pressure). At T = 10 K a minimum exists. Its relative depth with respect to the value at 4 K is about (p(4 K) - P m i n ) / P m i n ~-~ 6 × 10 5 _+ 0.2%. This systematic error stems from the T-dependence of the geometrical factor. When increasing the Zn contents or the pressure the minimum shifts towards higher temperatures and its depth is more pronounced. Above rmi n the resistivity increases approximately as T 2. The increase is followed by a broad maximum at about T - - 4 0 K. Its relative heights (Pmax -- Pmin)/Pmin is about 5 × 1 0 - 4 + 4% (systematic error as above). Increasing the Zn concentration and applying pressure produces a decrease of the height of the maximum and a shift of Tma× to lower temperatures. Systematic errors in the compressibility drop out since does not depend on T in this temperature range. Our data are in general agreement with the less accurate ones obtained by Matsuda and Mizutani [10] on the same system. There is, however, a difference in the low temperature region: the increase of the depth of the minimum with increasing Zn concentration is not observed by these authors.

5. Discussion 4. Results

In fig. 2 data of the temperature and pressure dependence of the resistivity are shown in the full T and P ranges. One observes an increase of the resistivity when increasing the Zn contents (there may be a systematic error as large as 20% because

The resistivity of amorphous MgZn and its temperature dependence were calculated by several authors within the framework of the Faber-Ziman theory [11,13,14]. The results presented by Hafner [11] as well as by Cote and Meisel [13] are in very good agreement with experiment (even concerning the small structure at very low temper-

272

A. Drost et al.

/ Resistivity of M g ~ Z n ~oo _ ~ metallic glasses

atures). Following Cote and Meisel the phonon ineffectiveness is taken into account in our effort to understand the pressure dependence of the resistivity. The resistivity p is given by

D

1.2

p = e2hv~(N/V)

1 / "v° f.

UZ(K)SO(K)"

j.

C9

(3) Here, U(K) denotes an effective atomic scattering potential, kr the Fermi vector and vf the Fermi velocity. The other symbols have their usual meaning. The resistivity structure factor SP(K) is given by

f

f

Q.

I,.

tH

I,

°

I

. °°

A

~..""""

1.1 O..

1.0

S~(K)

---a ( K )

e -2"~'~

×

+

I

3h2K 2 0 MkO T

n(x)(n(x) dl2

×fTga(IK+ql).

0 +

1)F.( qqal]/ qd

(4)

a ( K ) describes the static structure factor, 2W(K) the Debye-Waller factor, n(x) is the Bose function with x = qO/%T, where ~9 is the Debye temperature and qd the Debye wave vector. F(ql) is the Pippard function, l being the electron mean free path.

F(y)=2(y*arctan(y) ,r y - arctan(y)

3) y "

(5)

In eq. (4) the resistivity structure factor S°(K) is given by a sum of an elastic and an inelastic scattering contribution. While the first term causes a negative TCR by reducing the elastic scattering through the Debye-Waller factor, the second one leads to a positive TCR. Following the arguments of Cote and Meisel, in MgZn alloys the elastic part dominates at very low temperatures ( T < 10 K) because of the reduction of the inelastic scattering by phonon ineffectiveness, so the TCR is negative in this temperature range. Above this temperature the behavior of the resistivity is determined by the inelastic scattering, the resistivity increases until T = 40 K, where the contribution of the elastic scattering again dominates and causes a negative TCR.

I

J

I

2

I

I

3 P/GPa

J

I

4

i

I

5

Fig. 4. The normalized residual resistivity p o ( P ) / o o ( O ) as a function of pressure. Mg67Zn33, ~ ; Mg72Zn28, zx; Mg77Zn23, D. Data taken from Hafner's plot in ref. [11] (H), as well as from Vassiliou [15] (V), and Fritsch et al. [3] (W), are also included. The dotted line shows o 0 ( P ) / p o ( O ) for Mg72Zn28 when not correcting the data with a pressure-dependent geometrical factor. In (W) a pressure independent compressibility was used.

5.1. Pressuredependenceof the residual resistioity The pressure dependence of the residual resistivity P0 is shown in fig. 4. This quantity is defined as the resistivity for T---, 0. Together with our results we also show data measured by Vassiliou [15] and Wildermuth [3] and values calculated by Hafner [11]. The slopes of the straight lines reproduced in fig. 4 turn out to depend on the choice of the compressibility x(P) when correcting the resistance data with a pressure-dependent geometrical factor G according to eq. (1). As can be seen from fig. 4 the pressure dependence of the resistivity, defined as ( 1 / O o ) d p / d P , changes approximately 10% from 0.031 GPa -~ to 0.028 GPa -1 at Mg72Zn28, when correcting the data with a pressure-dependent geometrical factor. Therefore, a systematic error in the compressibility below 20% leads to a corresponding one in ( 1 / 0 o ) d o / d P smaller than 2%.

-6

273

A. Drost et al. / Resistivity of M g x Z n too - ~ metallic glasses

In the following discussion we use the quantity /3, as defined below, instead of pressure P in order to characterize the volume change

/3 = (V(O)/V( P)) 1/3 = 1 + ½fpP~( P) dP >~1.

(6) Here, V(0) describes the volume at Po. The resistivity at T = 0 K is given by (see eqs. (3) and (4)) 12~r

Oo- eZhvZ(U/V) K

K

a(K)UZ(K).

1

U(B, K ) =

/3 -5 , a ( f l , K ) and -~Ef(/3)u(fl, K).

(8)

We assume that the main effect of a change in volume on a(K) is a scaling of the static structure factor a(fl, K)=a(1,

ilK).

(9)

After a transformation of the coordinates from flK to K the following expression can be derived:

p(1) 1 f o ' d ( ~ ) ( 2 ~ f ) 3a(l'K)u2(fl'flK) fl fold(~----~r)(2~f)3a(1, K)uZ(i, K) (10) u may be given for example by a screened pseudo-potential u(fl, K ) =

cos(R¢(fl)K)

1 + 7raoK2/(4kf(fl)) '

Alloy

p-1 d p / d P ( P < 5 GPa) ( G P a - 1)

p-i dp/dP ( P = 9 GPa) ( G P a - 1)

p(0, 0) (td2 cm)

Mg67Zn33 MgvzZn2s MgTvZn23

0.040 0.028 0.048

0.008 0.011 0.017

57 51 39

(7)

The following quantities show a dependence on pressure:

vff( N / V )

Table 1 Mgx Znl0o_ x: P-dependence of the resistivity and extrapolated resistivity p(0, 0) at P = 0 GPa and T = 0 K. d p / d P is linear below about 5 GPa and levels off at higher pressures. The systematic uncertainty from the correction due to the compressibility is 0.002 GPa-1, the statistical error amounts to about 10%. The p(0, 0) values are associated with a systematic error of about 20%

(11)

where a 0 denotes the Bohr radius (a 0 = 0.53 A). According to eqs. (8) and (11) Re is the quantity responsible for the scattering strength of the

potential. Calculations, using parameters appropriate to the system MgZn (a(K) taken from the hard sphere PY model with a packing fraction 71= 0.525 and a hard-sphere radius o = 2.63 ,~, 2 k e = 2.9 ,~-1 and Re(1 ) = - 0 . 6 7 7 ,~) show that R c must decrease with pressure but less so than Rc(fl) = Rc(1)/fl. This result seems to be correct, since the quantity R¢ is associated with the core radius of the scattering atom, which in turn should become smaller with rising pressure. According to eq. (11) the zero of the pseudo-potential u(fl, ilK) is given by Ko(fl) = 7r/(2R¢(fl)fl) < K0(1 ). Hence, the zero of the pseudo-potential is shifted to lower K values in the picture of scaled coordinates, whereas K p , the K coordinate of the peak of a(K), remains unchanged. If the zero of the pseudo-potential is below but close to the peak of a ( K ) as in the case of MgZn alloys, this yields an increase of the integral value in eq. (7) and therefore an increase of O. The levelling off of o(P) at higher pressures ( P > 5 GPa) remains unexplained at the moment. The relevant data are summarized in table 1. Systematic errors in /3 due to errors in t¢ do not change the conclusions stated above and in the following. The increase of resistivity with increasing Zn concentration may be explained by similar arguments. From the work of Matsuda et al. [12] it is known that increasing the Zn contents will increase both kf and K value of the peak in a(K). Therefore, a scale transformation similar to the one used in the case of pressure seems to be a good approximation to take into account the

A. Drost et al. / Resistivity of MgxZn joo x metallic glasses

274

changes of a(K) when varying the Zn concentration. Assuming that R c of Zn is larger than the one for Mg, the zero of the pseudo-potential, K0, shifts away from Kp to lower values with rising Zn contents. This effect produces an increase of p, a behaviour which is also observed in the case of pressure. We do not attempt here to discuss the concentration and pressure dependence quantitatively, since then one has to look into the details of various structure factors and pseudo-potentials.

-7.9

c

~

D

-8.1 o o

-8.3 o

5.2. Changes in the temperature dependence with pressure 5.2.1. High-temperature range For T > O, eq. (4) predicts a linear temperature dependence for the elastic as well as the inelastic part of the resistivity structure factor SP(K). The temperature-dependent part of SP(K) is given by [161

S~.( K )

+ql)].

x f-4-;a(l

I t c a n b e s h o w n t h a t a scale t r a n s f o r m a t i o n

(12) can

be performed on the inelastic contribution (second term in the brackets) to S~(K), too. Using this scaling, the following expression for the slope Pr of the resistivity normalized to the residual resistivity P0 can be derived: PT( J~ ) /Po ( ~ )

=

/~2-6"y

(13)

with p(T)= Po- pT(T) • The value '2' in the exponent of fl stems from the transformation of coordinates K ' = flK in the prefactor of eq. (12). The quantity T denotes the Griineisen parameter. It is defined by y=-dlnO/dlnV

5

P/GPa

10

Fig. 5. The normalized slope ln(pr/#o) of the resistivityin the high-temperature region. The solid line is calculated using eq. (13). Mg67Zn33, O; Mg72Zn28,zx; Mg77Zn23, n. bye temperature. A convenient value for y is between 1 and 2 [17]. Taking y = 1.5, then we have 2 - 6"t = - 7 . The Debye temperature itself drops out of eq. (13) because of normalisation. Together with the x ( P ) values, as listed above, the results can be obtained which are shown in fig. 5. The agreement between theory and experiment is probably to be improved by applying correct •(P, T ) and y ( P ) values in eq. (13). The correction of the experimental data by the pressure- and temperature-dependent geometrical factor is of no importance in fig. 5. The slopes p ( T ) are normalized by the residual resistivity #o- Hence, the influence of x(P) drops out. In addition, an incorrect linear thermal expansion coefficient produces only an overall shift of the data but does not change their pressure dependence.

5.2.2. Low-temperature region

p (1)/po(1) = ( l +½ fp~x(P) dP) 2-6v,

~,

of

For T < 30 K the temperature-dependent part SP(K) is approximately given by [16]

S~-(K) = 3h2K2 ] MkO a( K ) - - f 0 1 d ( ~ d ) ( q ~ ) n ( x ) 8

1

( )J0

q

q

2

(14)

and assumed to be pressure-independent. It accounts for the pressure dependence of the De-

- M kO a ( K ) f

,qd I .

(15)

A. Drost et al.

/ Resistivity

Again x = qO/(qdT). The first term in the brackets stems from an expansion of the D e b y e - W a l l e r factor (eq. (4)) and is related to the elastic contribution in SP(K). The second one takes into account the inelastic scattering. The prefactor in S t ( K ) reduces all relative slopes by a factor f12-3~ as a function of pressure. The function f ( T / O , qd l) is pressure dependent through the Debye temperature, O, and the product of the Debye wavevector, qd, and electron mean free path, L This function is shown in fig. 6 for several values of qal. The parameter can be estimated within the N F E model by

275

of M g x Zn ~oo- .~ metallic glasses

1.0004

Mg77Zn23 +.

1.0003

P/GPo

E o_

2

2.15 ..

t. 0002

•

Q.

...'

.

5./+-.-

'

., .

1.0001

qal = 7

krP '

where Z is per atom. Taking p appropriate The Debye

, ~

i ~

X+

~, %

the number of conduction electrons

i

+...

. i '"

",-

.

",:.."'

.i "

"."

'~,,-j'%"-.v.... -. v"

.....

'

..

lOI3

...

+"" ..,

..-""

~:<. • 'W%:. ".'-.' • :.+• ,, ........- .-

= 50/~12 cm, k r = 1.5 ~ - 1 and Z = 2 to amorphous MgZn yields qd I = 16. temperature can be determined from

.........

i'0 ....

T> K' ' '{o ....

Fig. 7. The resistivity, normalized to Pmin, of Mg77Zn23 in the low-temperature region for three pressures. For clarity the curve for P = 5.4 G P a is shifted upwards by 3.75 x 10 -5 and that for P = 2.15 G P a by 7.5 × 1 0 - 5 units.

15

iI

I0

iII

0

0-

o"

- ' ~

//

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13/

/

,k •

-5

/

\

10,

-1o

+

;,

'

o

T/O lO0 Fig. 6. The quantity f(T/O, qd1) contaimng phonon ineffectiveness as a function of the normalized temperature T/O with qal as parameter.

measurements of the specific heat at low temperatures [18,19]; its value is approximately 8 = 300 K. The scale for qd I may contain a systematic error due to the not exactly known absolute value of the geometrical factor G. The quantity qdl decreases with increasing pressure or Zn concentration, because both p and k f increase with pressure and Zn concentration, respectively. A lower value of qd l leads to an increase of phonon ineffectiveness and according to fig. 6 to a shift of the normalized minimum temperature, T,m J O , to higher temperatures and an increase of the depth of the minimum. This is in qualitative agreement with the experimental results concerning the dependence on both pressure and Zn concentration (see figs. 3 and 7). In quantitative respect, however, the expected variations should be larger than the changes determined. In principle, this discrepancy can be caused by a variety of reasons. First of all, magnetic impurities may produce a K o n d o effect.

276

A. Drost et al. / Resistivity o f M g x Z n lo o _ x metallic glasses

However, since we d o n o t observe a n y non-linearities in the H a l l resistivity as a function of the m a g n e t i c field [20], such an e x p l a n a t i o n is highly i m p r o b a b l e . Second, two-level systems [21] c o u l d c o n t r i b u t e to the resistivity. W h e n a n n e a l i n g the s a m p l e s one w o u l d expect a decrease in the density o f those systems and, hence, also a c h a n g e in their c o n t r i b u t i o n to p. W e d o n o t see such an effect. Finally, q u a n t u m corrections to the Boltzm a n n resistivity have been d e t e c t e d in the m a g n e toresistivity of M g Z n [20]. Therefore, the resistivity # will also be m o d i f i e d at t e m p e r a t u r e s b e l o w a b o u t 30 K due to the presence o f the C o u l o m b i n t e r a c t i o n as well as of the q u a n t u m interference p a r t s of this correction. I n o r d e r to allow a q u a n t i t a t i v e c o m p a r i s o n the pressure d e p e n d e n c e of the inelastic scattering time, of the electronic diffusivity, a n d of the C o u l o m b screening p a r a m e ter has to b e analysed. However, this p r o c e d u r e is b e y o n d the scope o f this p a p e r . C a l c u l a t i o n s using the p a r a m e t e r s f r o m m a g n e t o r e s i s t i v i t y d a t a [20] i n d i c a t e that the q u a n t u m corrections to the resistivity should be small in the t e m p e r a t u r e r a n g e a b o v e 4 K.

6. Summary T h e t e m p e r a t u r e - a n d p r e s s u r e - d e p e n d e n c e of the resistance has been m e a s u r e d for a m o r p h o u s M g x Z n l 0 o _ x ( x = 67, 72, 77 at.%). T h e d a t a were corrected by a temperature- and pressure-depend e n t g e o m e t r i c a l factor. T h e increase of the resistivity with rising p r e s s u r e c a n b e e x p l a i n e d b y an increased scattering strength within the F a b e r - Z i m a n theory. T h e decrease of the negative T C R with increasing p r e s s u r e is c a u s e d b y an increase o f the D e b y e t e m p e r a t u r e in this model. Changes in the b e h a v i o u r o f p at very low t e m p e r a t u r e s as a f u n c t i o n o f p r e s s u r e c a n be u n d e r s t o o d qualitativly within the F a b e r - Z i m a n t h e o r y b y t a k i n g i n t o a c c o u n t p h o n o n ineffectiveness. W e w o u l d like to t h a n k P. L~Sbl for p r e p a r i n g the samples, L. G o n z o for h e l p i n g with the m e a -

s u r e m e n t s a n d Prof. G . M . K a l v i u s for the p e r m i s sion to use e q u i p m e n t of his institute.

References [1] P.J. Cote and L.V. Meisel, in: Glassy Metals I, eds. H.J. Giintherodt and H. Beck, Topics in Applied Physics, Vol. 46 (Springer, Berlin, 1983) p. 141. [2] P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. [3] G. Fritsch, J. Willer, A. Wildermuth and E. Liischer, Physics of Solids under High Pressure, eds. J.S. Schilling and R.N. Shelton, (North-Holland, Amsterdam, 1980) p. 239. [4] J. Wilier and J. Moser, J. Phys. E 12 (1979) 886. [5] A. Eichler and J. Wittig, Z. Angew. Phys. 25 (1968) 319; A. Eiling and J.S. Schilling, J. Phys. F 11 (1981) 623. [6] Y.S. Touloukian and C.Y. Ho, eds., Thermophysical Properties of Matter, Vol. 12, Thermal Expansion IFI/Plenum, New York, Washington, 1975), p. 694. [7] K.H. Hellwege and A.M. Hellwege, eds., Landolt-B~rnstein, Group III, Vol. ll, Elastic Constants (Springer, Berlin, Heidelberg, New York, 1979) p. 873. [8] T. Fukunaga, T. Sugawara, F. Itoh and K. Suzuki, Z. Phys. Chemie 157 (1988) 79. [9] H.S. Chen and J.T. Krause, Script. Met. 11 (1977) 761. [10] C.P. Chou, L.A. Davies and M.G. Narasimhan, Scripta Met. 11 (1977) 417. [11] J. Hafner, J. Non-Cryst. Solids 69 (1985) 325. [12] T. Matsuda and U. Mizutani, J. Phys. F 12 (1982) 1877. [13] L.V. Meisel and P.J. Cote, Phys. Rev. B 27 (1983) 4617; P.J. Cote and L.V. Meisel, J. Non-Cryst. Solids 61 (1984) 1167; L.V. Meisel and P.J. Cote, Phys. Rev. B 17 (1978) 4652; L.V. Meisel and P.J. Cote, Phys. Rev. B 31 (1985) 4872. [14] J. Laakkonen and R.M. Nieminen, J. Phys. F 13 (1983) 2265. [15] J.K. Vassiliou, J. Appl. Phys. 59 (1986) 482. [16] L.V. Meisel and P.J. Cote, Phys. Rev. B 16 (1977) 2978. [17] N.W. Ashcroft and N.D. Mermin, in: Solid State Physics (Holt Saunders, London, 1981) p. 213. [18] U. Mizutani and T. Mizuguchi, J. Phys. F 11 (1981) 385. [19] R. van den Berg, S. Grondey, J. K~istner and H. v. L6hneysen, Sol. St. Commun. 47 (1983) 137. [20] F. Kiiss, "A. Schulte, P. LiSbl, E. Liischer and G. Fritsch, in: Amorphous and Liquid Metals, eds. E. Liischer, G. Fritsch and G. Jacucci, NATO ASI Series (Nijhoff, Dordrecht, 1987). [21] A.B. Kaiser, Phys. Stat. Sol. b 136 (1986) 313.