Z. Physik270, 195-202 (1974) 9 by Springer-Verlag 1974

Electrical Resistivity of Amorphous Alloys D. Korn, H. Pfeifle and G. Zibold Fachbereich Physik der Universit~it Konstanz Received April 30, 1974 Abstract. Amorphous alloys of Ga, Sn, Pb and Bi with Cu, Ag and Au are produced by

evaporation on a cold substrate. The residual resistivity, the temperature dependence of the resistivity, the transition temperature of superconductivity and the temperature of the amorphous-crystalline transformation are measured. We observe e.g. that the residual resistivity increases with the noble metal concentration, and that the temperature coefficient of the resistivity of the Au alloys is always negative. In these two respects amorphous alloys differ in behaviour from the corresponding liquid alloys. These observations can be correlated with the atomic energy levels of the free atoms.

I. Introduction

Amorphous alloys stand between crystalline and liquid alloys. With liquid alloys they have in common the disordered structure, whereas their solid state relates them with the crystalline alloys. Therefore, they are of interest for understanding the difference between the crystalline and the liquid metallic state. Since the temperature range, where the metallic amorphous state is stable, could be extended [1] up to 300 K, the application of numerous straightforward experimental techniques is possible. We report here measurements of the temperature dependence of the electrical resistivity of the amorphous alloys of Ga, Sn, Pb, and Bi with Cu, Ag, and Au. We find that the change in electrical resistivity with temperature has about the same order of magnitude in amorphous and crystalline alloys (10 -2 ~f~ cm K - l ) . However, the behaviour of the electrical resistivity of amorphous and crystalline alloys differs fundamentally: In the crystalline phase the resistivity of the mentioned alloys always shows a positive temperature coefficient and decreases with temperature to a con-

stant value, unless superconductivity appears. The resistivity of amorphous alloys, however, exhibits positive and negative temperature coefficients. The alloys with positive temperature coefficient show a transition to superconductivity and never a constant residual resistivity at low temperature. In alloys with negative temperature coefficient, the resistivity approaches a constant value at low temperature, followed by a transition to superconductivity in some alloys. A linear temperature dependence with both positive and negative coefficients is well known for liquid alloys. The difference in resistivity between crystalline and amorphous alloys is greater than that between liquid and amorphous alloys. This result reflects the importance of the periodicity of the lattice for electron transport. The periodicity masks the properties of the individual atoms which form the alloy. Since in amorphous alloys the structure is disordered and no periodicity exists, it can be expected that the influence of the atomic properties increases in the amorphous phase. That will be shown in Section III, 2.

196

Z. Physik 270 (1974)

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Ga-Au

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122

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-6

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68,6

61,9

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I

0

40

80

Sn-noble metal

I

omorphous

i

i

i

0

20

40

i

60K

i

160

Temperature

61,1

f

I

12OK

Fig. 2. Electrical resistivity of O a - A u alloys as a function of temperature. Annealing temperature Tt = 150 K i

80 249,5

Temperature Iz~cm

Fig. 1. Electrical resistivity of different S n - n o b l e metal alloys as a function of temperature after annealing at Tt = 80 K

II. Experimental Results

1. a) Sn-Noble Metal Alloys Recently we reported on measurements of the electrical resistivity of amorphous S n - C u alloys [1, 2]. The results can be characterized, with regard to rising Cu concentration, in the following way: c~) the residual resistivity increases; fl) the temperature coefficients are constant within certain temperature intervals and change from positive to negative; 7) the transition temperature of superconductivity decreases and is proportional to the positive temperature coefficient. To study the influence of different solutes on these parameters, we measured amorphous Sn + 20 at ~o Ag and S n + 2 0 a t % A u . The result is shown in Fig. 1 together with an earlier measurement of Sn + 25 at % Cu. Both curves for Ag and Cu alloys are similar in the magnitude of the residual resistivity, whereas the positive temperature coefficient is larger in S n - A g

~:~ 225 ._> .m_ t/) o# re"

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15

I

I

I

0

20

L0

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60K

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80

Temperature Fig. 3. Electrical resistivity of B i - A g alloys as a function of temperature. Annealing temperature about 80 K

D.

Korn

aL : Electrical Resistivity of Amorphous Alloys

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Tern peru ture Fig. 5. Electrical resistivity of P b + 5 0 a t % Au as a function of temperature. Annealing temperature Tt = 150 K LU

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114,3

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Temperature Fig. 4. Electrical resistivity of P b - A g alloys as a function of temperature. Annealing temperature Tt = 150 K

than in S n - C u . For the Cu alloy the slope of the curve decreases at about 60 K. A much higher residual resistivity is found in the Au alloy together with a negative temperature coefficient. The transition temperature of superconductivity decreases slightly going from Ag to Cu and to Au alloys. b) Ga - Au Alloys

Earlier measurements on G a - A g [3] showed a positive temperature coefficient. Measurements of Ga + x at ~o Au, where x-- 20, 49 and 70, are displayed in Fig. 2. All three curves are similar in shape, showing a negative temperature coefficient of about the same magnitude, but much higher than in the other alloys. The temperature coefficient decreases at low temperature. The residual resistivity increases with Au concentration whereas the transition temperature of superconductivity decreases. For Ga + 70 at % Au no superconductivity is observed above 1 K.

m

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80at% 100

Noble Metal Concentration Fig. 6. Residual resistivity of different alloys as a function of noble metal concentration. Annealing temperature Tt normally 8 0 K except G a - A u ( Tt = 150 K)

c) B i - Ag Alloys

The temperature dependence of the electrical resistivity has been investigated for pure amorphous Bi [4, 5] and an alloy of B i - A g [2]. In Fig. 3, the resistivity of amorphous Bi + x at % Ag, where x = 15, 51 and 70, is plotted as a function of temperature. In general, the dependence is similar to that of G a - A u in Fig. 3. All three curves show a negative temperature coefficient

198

Z. Physik 270 (1974)

Table 1 Alloy

d) P b - N o b l e Metal Alloys T~

p~,

[K]

[gflcm] [10-2~tflcmK-~] [K] [K]

Ap/AT

T~

T~

pos.

neg.

68.3 65.0 61.0 92.1 86.9

0.8 0.9 1.5 -

1.1 1.1

5.8 6.0 6.6 5.2 5.3

50 100 150 80 150 80 150 150 260 150

82.1 83.7 88.5 120.9 126.9 120.8 122.0 154.1 161.0 154.9

-

2.0 2.0 3.5 2.1 3.4 2.1 3.8 3.7 4.2 3.2

5.2 5.0 4.7 < 1.5 -

42 74 51 at ~ Ag 40 80 150 23i 70 at ~o Ag 80 150 210 300

192.0 190.2 227.8 225.8 215.9 212.6 249.5 239.7 240.9 250.2

-

1.8 2.2 2.7 2.5 2.3 2.2 1.8 1.4 1.1 0.8

5.1 120 5.3 1.4 320 1.5 3.2 3.6 < 1.8 ~340

117.5 114.2 132.4 127.5 131.4 128.9 124.7 116.6 126.4 122.3 115.0 164.2 191.2

1.2 1.6 0.4 0.7

-

4.8 4.9 3.1 3.6

-

0.1 0.2

-

0.1 0.1 -

0.1 0.7 0.9

Sn + 25 at ~ Cu

80 187 20 at ~o Ag 80 20 a t ~ Au 80 160

Ga+20 at ~ Au

49 at ~o Au 49 at ~ Au 70 at ~ Au 70 at ~o Au Bi + 15 a t ~ Ag

P b + 2 0 a t ~ Ag 51 a t ~ Ag 70 at ~o Ag

71 at ~ Ag

50 at ~ Au

80 150 80 i50 40 80 ,150 240 80 150 240 80 150

190 120-220 210 200

270 230 280 308

R e s u l t s o n a m o r p h o u s P b + x at % Ag, w h e r e x = 20, 51 a n d 71, are s h o w n in Fig. 4. O n l y a small c h a n g e of the r e s i d u a l resistivity with c o n c e n t r a t i o n is o b served. T h e t e m p e r a t u r e coefficient is positive a n d decreases with rising A g c o n c e n t r a t i o n . I n P b + 71 at A g the t e m p e r a t u r e coefficient starts with a s m a l l p o s i t i v e v a l u e at low t e m p e r a t u r e a n d b e c o m e s slightly n e g a t i v e a b o v e 60 K. T h e t r a n s i t i o n t e m p e r a t u r e of s u p e r c o n d u c t i v i t y decreases with A g c o n c e n t r a t i o n . I n Fig. 5 the resistivity of P b + 50 at % A u is s h o w n . I n c o m p a r i s o n to P b + 51 at 9/o Ag, the r e s i d u a l resistivity is higher, the t e m p e r a t u r e coefficient is n e g a t i v e a n d d e c r e a s i n g in m a g n i t u d e with t e m p e r a t u r e , a n d the t r a n s i t i o n t e m p e r a t u r e of s u p e r c o n d u c t i v i t y is lower. T h e r e s i d u a l resistivities of the different alloy systems, as a f u n c t i o n of n o b l e m e t a l c o n c e n t r a t i o n , are p l o t t e d in Fig. 6. T h e r e s i d u a l resistivity increases with n o b l e m e t a l c o n c e n t r a t i o n . O n l y in P b - A g a flat m a x i m u m of the r e s i d u a l resistivity is observed. T h e highest resistivity v a l u e s are f o u n d in B i - A g alloys; the lowest v a l u e s in S n - C u a n d S n - A g alloys. A d d i t i o n a l results are p r e s e n t e d in T a b l e 1.

2. Annealing Behaviour and Transformation Temperature 210 270 280

-

260

3.1 ~290 3.0

Tt = maximum annealing temperature. Pr~,: for alloys with positive temperature coefficient the residual resistivity Pre~ is deduced by extrapolating the linear part of the resistivity curve to zero K. For alloys with negative temperature coefficient Pres is taken at the maximum. Ap/A T= temperature coefficient of the resistivity taken from the linear part of the curves. T~= transition temperature of superconductivity. T. = transformation temperature amorphous-crystalline. - = quantity not observed in the temperature range 1.5 K to T. no sign = quantity not investigated. v a r y i n g little with c o n c e n t r a t i o n . T h e r e s i d u a l resistivity increases with rising A g c o n c e n t r a t i o n , a n d the t r a n s i t i o n t e m p e r a t u r e of s u p e r c o n d u c t i v i t y decreases.

A l l alloys h a v e b e e n p r o d u c e d in the s a m e m a n n e r [1]. F o u r types of a n n e a l i n g b e h a v i o u r c o u l d be d i s t i n g u i s h e d (Fig. 7 a - d ) . T h e g e n e r a l p r o c e d u r e of m e a s u r i n g is e x p l a i n e d for the case of P b + 2 0 a t ~ Ag w h i c h b e h a v e s typical for a w h o l e g r o u p of alloys. It r e p r e s e n t s the first type of a n n e a l i n g b e h a v i o u r . P b - A g is q u e n c h - c o n d e n s e d as a film of a b o u t 500 A at a b o u t 5 K (Fig. 7a). (500 A film t h i c k n e s s t o g e t h e r with a m e a n free p a t h of a b o u t 5 A in a m o r p h o u s alloys m e a n s t h a t b u l k effects are investigated.) D u r i n g a n n e a l i n g u p to a c e r t a i n t e m p e r a t u r e T , the r e s i d u a l resistivity decreases ( b r a n c h e s I-II). A f t e r w a r d s the resistivity is o b s e r v e d to 'be reversible b e l o w the a n n e a l i n g t e m p e r a t u r e T~ (e.g. b r a n c h III). O n l y m e a s u r e m e n t s d o n e at lower t e m p e r a t u r e t h a n Tt are p r e s e n t e d in II, 1. T h e t r a n s f o r m a t i o n t e m p e r a t u r e T, b e t w e e n the a m o r p h o u s a n d c r y s t a l l i n e state, is r e a c h e d at the p o i n t , where the resistivity d r o p s sharply. F o r P b + 2 0 at ~ A g we o b s e r v e T u = 2 1 0 K. A t this t e m p e r a t u r e , the a m o r p h o u s p h a s e t r a n s f o r m s i r r e v e r s i b l y i n t o the c r y s t a l l i n e phase. T h e a b r u p t decrease of the resistivity at T, is c h a r a c t e r i s t i c for the m e t a s t a b l e a m o r p h o u s state of m e t a l films [6]. T h e highest a n n e a l i n g t e m p e r a t u r e is 380 K in o u r e x p e r i m e n t s . A t this t e m p e r a t u r e all o u r m e t a l films are in the crystalline p h a s e ( b r a n c h IV).

D. K o r n et al. :

Electrical Resistivity of Amorphous Alloys

199

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I

100

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100

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Temperature

Temperature

Bi§

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300K

at% Ag ~cm

p~cm

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50 cr

or"

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~d 250 v,,

kJ

25

I

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I

0

100

200

I

300 K

I

I

I

I

L00

0

100

200

Temperature

I

300 K

I

400

Temperature

Fig. 7 a - d. Electrical resistivity of quench-condensed alloys as a function of temperature between 1.5 a n d 380 K. Sample thickness about 500 ,~

A similar behaviour is seen in Sn+25at% Cu, Sn + 20 at % Au, Pb + 51 at % Ag, and Pb + 71 at % Ag. Only in Sn+20at% Ag the transformation amorphous-crystalline extends over a wider temperature region. The second type is seen in the example Ga + 49 at % Au (Fig. 7b). The annealing behaviour is characterized

by an increase of resistivity within the amorphous phase. At the transformation temperature the resistivity drops sharply. After the transformation the resistivity shows the behaviour typical of crystalline alloys. The same holds for Ga with 20 or 70 at % Au. Type 3 is seen in Bi+51 at% Ag (Fig. 7c). Bi+ 15 at% Ag behaves similar. In the amorphous phase, the

200 resistivity diminishes with annealing temperature in a way comparable to that of type 1. At the transformation temperature T,, a sharp increase of the resistivity is observed, leading to the crystalline phase. The crystalline phase shows the behaviour typical of Bi films, a high resistivity and a negative temperature coefficient [7]. In all alloys under investigation, the transformation temperature increases with noble metal concentration. The observed values are summarized in Table 1. In S n + 8 0 a t ~ Cu and in Bi+51 or 70at~o Ag, the amorphous state can be stabilized to temperatures higher than 300 K. In S n - C u and in P b - A g alloys, the values of transformation temperature T, and melting temperature T,~ are, to a good approximation, proportional, with T,=0.4 T,,. This relationship is not well observed, however, in the B i - A g and G a - A u systems. S n + 18 at ~o Zn does not exist in the amorphous phase, as can be seen in Fig. 7 d. On annealing the resistivity of S n - Z n falls continually and the temperature dependence (not shown in detail) is typical of the crystalline phase. In addition, it was not possible to produce the amorphous state of A1+9.5 a t ~ Cu, T1 + 10 at ~o Ag nor of the alloys with high noble metal concentration S n + 9 0 a t ~ Cu and G a + 8 3 at~o Au. III. Discussion

1. Comparison with Theory The appropriate theory for the resistivity of liquid and amorphous alloys is the Faber-Ziman theory [8]. It explains successfully the resistivity and its temperature coefficient of liquid metals in terms of the partial interference function, which can be measured in diffraction experiments, and the pseudopotential of individual cores. Besides quantitative calculations it allows qualitative predictions about the change of resistivity with alloy concentration. Furthermore, it predicts the sign of the temperature coefficient of the resistivity as a function of alloy valency. As interference functions have not been measured for the amorphous alloys under investigation here, we confine our discussion to a qualitative comparison of theory and experiment. The discussion holds also for a comparison between the experimental results of liquid and amorphous alloys. The theory states [9] a maximum of the resistivity for the valency range

C Z o + ( 1 - c ) z l = l . 5 to 2 (c is the alloy concentration, and Zo, z 1 are the solvent and the solute valency). A negative temperature coef-

z. Physik 270 (1974) ficient should be observed in the same valency range. The above relation determines the alloy concentration range where resistivity maxima and sign changes of the temperature coefficient are to be expected. Experimental data and theoretical values are collected in the following table. Table 2. Noble metal concentration range in at ~o for resistivity maximum Pmax and negative temperature coefficient Ap/AT of the resistivity Alloy

Ga- Au Sn-Cu Sn - Ag Sn-Au Pb-Ag Pb-Au Bi -Ag

Theory Investi- Experiment Pmaxand gated neg. Ap/AT range Pm,~ 50 to 75 66 to 83 66 to 83 66 to 83 66 to 83 66 to 83 75 to 88

20 to 70 10 to 80 20 20 20 to 71 50 20 to 70

neg. Ap/AT

not observed 20 to 70 not observed 40 to 80 not observed 20 20
A possible agreement between theory and experiment exists for P b - A g alloys. In the other alloys, no maximum of the resistivity was observed; only an increase of resistivity with concentration (Fig. 6). As the concentration range in question is not covered totally by the measurements, the maximum might exist even at higher concentrations. However, no tendency towards a maximum in the resistivity is indicated in Fig. 6, and a rather abrupt change would have to occur in G a - A u in order to satisfy the theory. In addition, G a + 8 3 a t ~ Au could not be obtained in the amorphous phase. Therefore, the experimental range left is narrow. In S n - C u too, the upper limit of the range is nearly reached experimentally, and no maximum can be seen (Fig. 6). Sn + 90 at ~o Cu is not amorphous. The temperature coefficients of G a - A u and S n - Cu are negative already at low concentration, instead of being positive. At higher concentrations, the negative values agree with theory. In B i - A g , only negative temperature coefficients are found in a concentration region where positive values are expected from the theory. Bi alloys show the highest resistivity of all investigated alloys. This is consistent with values found for liquid and crystalline metals and reflects the tendency towards higher resistivity values with increasing valency [10].

2. Interpretation of the Results The striking facts in the behaviour of the amorphous noble metal alloys under consideration are: ~) for the Cu and Ag alloys the increase of the residual resistivity with rising noble metal concentration and

D. Korn et al.: Electrical Resistivity of A m o r p h o u s Alloys

the negative temperature coefficient occuring at high concentration together with the high residual resistivity; /3) for the Au alloys both a high residual resistivity increasing with concentration and a negative temperature coefficient throughout the whole concentration range. These facts suggest that negative temperature coefficients as well as high residual resistivities, are linked and are caused by the noble metals. The specific properties of the noble metals, which account for this effect, must vary from Cu to Au to explain the different concentration dependence. Noble metals are situated at the end of the row of the transition elements. Transition elements show a high residual resistivity in the solid phase which is due to the s-d mixing and the high density of d-states at the Fermi level. When we go from lower to higher atomic number, the d-states are shifted from above the Fermi level to a few electron volts below the Fermi level for the noble metals. However even in the noble metals the Fermi electrons show an appreciable d-character. A measure for the d-character of the conduction electrons is the energy difference between the top of the d-band and the Fermi energy [11] which amounts to 2eV for Cu, 4eV for Ag and 2.5 eV for Au. If we want to connect the d-character of the conduction electrons with the high residual resistivity of the alloys, we should expect at least a similar behaviour in Cu and Au. In terms of this interpretation, we should observe in Cu alloys even stronger effects than in Au alloys. Actually the opposite behaviour is observed. One reason for this might be that the band structure is derived for a periodic lattice. As band structures for these amorphous alloys are not yet known, we might look at the amorphous solid as an agglomeration of individual atoms. In this picture, the atomic terms of the free atom would be the starting point to explain the electronic properties. Noble metals show a d 1~ s1 electronic configuration in the S-ground state. The difference of the noble metals lies in the first excited state being a D-state for Au and Cu and a P-state for Ag. The excitation energies are 0.9, 1.1 and 2.9 eV respectively [12]. The D-states in Cu and Au correspond to a d 9 s 2 electron configuration in which one of the electrons is excited into a higher lying s-shell. Unfilled d-shells are characteristic for high resistivity, as known from the transition metals. Hence we expect a higher resistivity for Cu and Au alloys than for the corresponding Ag alloys. Moreover the temperature coefficient can be correlated to the atomic energy levels.

201 Table 3. Free atomic states of the major component Alloy

Residual resistivity

Temperature coefficient

Ground state

First excited state

B i + 1 5 a t ~ Ag Au + 30 at ~ Ga Ag + 29 at ~ Pb Pb + 20 at ~ Ag Cu + 20 at ~ Sn Ga + 20 at ~o Au S n + 2 0 a t ~ Au Sn + 25 at ~o Cu S n + 2 0 a t ~ Ag Ga + 10 at ~ Ag

high

neg. neg, small pos. neg. neg. neg. pos. pos. pos.

S S S P S P P P P P

D D P P D P P P P P

low

Table3 shows the amorphous metals with about 20 at~o addition of another component. The alloys are arranged with respect to decreasing values of their residual resistivity. The temperature coefficient is written together with the terms of the ground state and the first excited state. A negative temperature coefficient exists for these amorphous metals, which have a S-ground state and a D-excited state. Ag with S-ground and P-excited states shows only a small temperature coefficient. Those alloys of atoms with a P-ground state and a P-excited state have positive temperature coefficients unless Au is the second component (see III, 2,/3). Bi behaves like Au, having also a S-ground state and D-excited state. For pure amorphous Bi there exists a negative temperature coefficient, too [4, 5]. Bi and Au determine the temperature coefficients even at low concentration in contrast to Cu. This may be due to the fact that the 5d-electrons of Bi and Au extend spatially further than the 3 d-electrons of Cu. Probably different mechanisms are the cause of the positive and negative temperature coefficients. Preliminary measurements show a different influence of magnetic additions on the two types of temperature coefficients. Furthermore the temperature coefficients distinguish in their behaviour at low temperature. The negative temperature coefficient grows less when approaching T = 0 K. This may be due to the freezing in of lattice vibrations at low temperature. Correspondingly the structure factor approaches a constant value at low temperature. The structure factor has been measured for amorphous pure Bi films [5] but, as far as we can see, only on the irreversible branch. Measurements on the reversible branch are necessary in order to correlate the negative temperature coefficient with a possible decrease of the structure factor with rising temperature. According to our interpretation the atoms constituting the amorphous alloy, exist at least partly in an excited

202 state. M o r e o v e r the electrical resistivity of alloys, composed of atoms existing in the S-state, is lower than the electrical resistivity of an alloy constituted by atoms in the D-state. (It is k n o w n that not only crystalline but also liquid alloys with flled d-shells have a lower resistivity than those with unfilled d-shells). N o w let us consider an a m o r p h o u s alloy with atoms in the D-state. W h e n the p h o n o n s create an electric field in the a m o r p h o u s solid and polarize the atoms, then a mixing of adjacent states occurs. These are S-states for Bi, A u and Cu. The alloy with a t o m s in the S-state has a lower resistivity. Therefore the resistivity decreases with rising p h o n o n excitation and a negative t e m p e r a t u r e coefficient results. 3. A m o r p h o u s - C r y s t a l l i n e Transformation

The t e m p e r a t u r e T, of the amorphous-crystalline transformation, has been correlated to the melting temperature Tm in the expression T,=0.3 T,, [-13]. We found the relation Tu=0.4 T m and the agreement is good for S n - C u and P b - A g alloys and also for Bi - Ag and G a - Au in the low concentration range. The deviation of Tu from 0.4 T,, for G a + 70 at % Au is due to the fact that the melting t e m p e r a t u r e for this alloy is especially low. Such low T,, exist in regions with intermetallic c o m p o u n d s . Intermetallic compounds and phase boundaries do not exist in these a m o r p h o u s alloys. The properties of metallic a m o r phous alloys vary continually with concentration. Finally, we observe that the alloys, which exist in the a m o r p h o u s phase, consist of one c o m p o n e n t with a P - g r o u n d state and the other c o m p o n e n t with a D-state as first excited state. In S n - A g and P b - A g

Z. Physik 270 (1974) this condition is not fullfilled, but the D-states in Ag lie close above the P-states. Accordingly S n - Ag does not show the abrupt transformation to the crystalline phase. In S n - Z n , the Zn D-states are far away from the P-states. This alloy does not exist at all in the a m o r p h o u s phase. Numerous discussions with Prof. Dr. J. J/ickle are gratefully acknowledged.

References 1. Korn, D., Mtirer, W., Zibold, G.: J. Physique 35, C4-257 (1974) 2. Korn, D, Mtirer, W., Zibold, G.: Phys. Letters 47A, 117 (1974) 3. Korn, D., Miirer, W., Zibold, G.: Z. Physik 260, 351 (1973) 4. Bergmann, G.: Z. Physik 225, 430 (1969) 5. Komik, Yu.F., Belevtsev, B.I., Yatsuk, L.A.: Soviet Physics JETP 36, 1177 (1973) 6. Buckel, W.: Z. Physik 138, 136 (1954) 7. Buckel, W., Hilsch, R.: Z. Physik 138, 109 (1954) 8. Faber, T.E, Ziman, J.M.: Phil. Mag. 11, 153 (1965) 9. Faber, T.E.: Theory of Liquid Metals, p. 455, Cambridge: University Press 1972 10. Faber, T.E.: Physics of Metals I. Electrons, p. 296, Cambridge: University Press 1969 11. Friedel, J.: Physics of Metals I. Electrons, p. 358, Cambridge: University Press 1969 12. Moore, C. E. : Atomic Energy Levels, I, II, I II, Washington D. C. : Nat. Bur. of Stand. Circular 467, 1949 13. Mader, S.: J. Vac. Sci. TechnoL 2, 35 (1965) Prof. Dr. D. Korn cand. phys. H. Pfeifle Dipl.-Phys, G, Zibold Ph.D. Fachbereich Physik der Universit/it Konstanz D-7750 Konstanz Postfach 733 Federal Republic of Germany