Limits to weak localization in Ca70Mg30−xAlx

90

Materials Science and Engineering, A133 (1991 ) 90-93

Limits to weak localization in

Ca70Mg30_xAlx

David V. Baxter and A. N. Fadnis Department of Physics, Indiana University, Swain Hall W, Bloomington, IN 47405 (U.S.A.) A. Sahnoune Department of Physics, McGill University, 3600 University Street, Montreal, PQ, H3A 2T8 (Canada)

Abstract We use computer simulations of random flights, in combination with measurements of the temperature and magnetic field dependence of resistivity, to estimate the size of the total contribution of quantum interference to the resistivity in a series of Ca-Mg-A1 alloys. We find that this effect alone is insufficient to explain the negative temperature coefficient in this system, even though it describes the low temperature magnetoresistance better for this system than any other glass studied to date. The techniques presented should allow similar bounds to be estimated for the weak localization contribution to resistivity in other alloys.

1. Introduction

The success which the theories of weak localization (WL) and enhanced interelectron interactions have had in explaining the low temperature resistivity of metallic glasses has led several authors to suggest that these same theories may provide an explanation for phenomena observed at room temperature [1-3]. Specifically the long-standing problem of the Mooij correlation (i.e. the negative temperature coefficient in more resistive alloys) [1, 3], and the anomalously large resistivity of certain Ca-AI alloys [2] have been attributed to WL by various authors. It is the purpose of this work to show that measurements of low temperature magnetoresistance establish limits on the size of the effect that could be expected at room temperature. Weak localization may be viewed as a reduction in the mobility of the electrons arising from the quantum interference between clockwise and counter-clockwise trajectories around closed loops [4]. The size of the reduction may be changed by any outside influence that alters this interference. An increase in temperature limits the interference by increasing the inelastic scattering rate for the electrons, but an applied magnetic field can also destroy the interference by 0921-5093/91/$3.50

introducing an additional phase shift between the interfering paths. Within this semiclassical picture of the effect, the WL correction to the classical conductivity is [5] ha=

=-

2e2

--~

D

i

Wtdt N,

3e2Dz ~. Pi 27t2h13 i=1

(1) (2)

where Wt is the probability per unit volume for the electron to return near the origin at time t, assuming that it moves along a classical trajectory. In the long time limit ( r~ "> 1-)for which ( 1 ) is valid, this probability may be assumed constant. The times r and r~ are the electron's elastic scattering time and phase coherence time, respectively. Equation (2) follows from (1) if the classical trajectory is modelled by an isotropic random flight where Pi is the probability for return to within one step (of size l = VEt) of the origin on the i th step, and N¢ steps are taken within the time re. Figure 1 shows values for PN and S(N) = Zj~=NPj as determined from Monte Carlo simulation of just such a model. In a typical experiment r¢ is the shorter of the inelastic scattering time (zi), or the field dephas© Elsevier Sequoia/Printed in The Netherlands

91 1

~

...... 1 . . . . . . . . I

200

.......

'a''l

....

I ....

I ....

|

I ....

~

3

4

0.01 I00

0.0ol 0.0001 10 -5

0 10-5 10-7

....... I

.... I ....

....

10

100

1000

10000

I,,

1

2

B (T)

N

Fig. 1. Probability for returning to within one step of the origin on the Nth step, PN, and the integrated probability from N to infinity,S(N), for an isotropic random flight.

......

/'

! I

ing time (rB = h/4eDB). The important thing to note here is that the temperature and field dependence of o enter the theory the same way, through the phase coherence time re. For example the change in conductivity for an applied field B, at a given temperature would be given by (O(x) is 1 for x > 0 and 0 for x < 0 )

1000

,~,

500

x=

& x=

0

i

i

i

I 2

o(8,r)-

o(8=0,

i

i

,

i

I 4

t

i

b

i 6

c)

-2:r2hl [S ( Ns)-S( N~)]O(N~- Ns) (3)

Fig. 2. (a) Weak localization contribution to the magnetoconductivityat 20 K and 4.2 K for CaToAl30.(b) Temperature dependence of the conductivity for Ca70Als0 and Ca70MgT.sA122.s. The solid line is an extrapolation of the low temperature form of the inter-electron interaction contribution to the conductivity.

As a quantitative test of some of these ideas we consider some experimental results on a series of CaToMgs0_ xAl x alloys. It has been shown recently that the magnetoresistance of these alloys at low temperature is explained essentially perfectly by the predictions of W L (with a small contribution from electron interaction effects) [6]. Therefore this system represents an ideal one for testing the ability of the W L theory to explain the temperature dependence of resistivity above 4.2 K. Figure 2(a) shows the W L contribution to the magnetoresistance of Ca70A130 at 4.2 and 20 K as determined in ref. 6, while Fig. 2(b) shows the temperature dependence of the resistance in the same alloy. We note that Ti(4.2 K ) > ri(20 K ) > ~'B(4 T), since there is a measurable magnetoresistance at 4 T and 20 K. According to (1), therefore, the change in conductance due to W L in going from 4.2 K to 20 K in zero field must be less than the change seen due to a field of 4 T at a temperature of 4.2 K. From the data we see that 0(20 K ) - a(4.2) is 540 Sin-]. In contrast to this,

a comparison of the magnetoresistance measurements at 4.2 K and 20 K indicates that the W L contribution to the change in conductivity between these temperatures is only 126 S m - l . We conclude therefore that some other effect must contribute to, in fact must dominate, the negative change of resistance with increasing T in this alloy, even for temperatures below 20 K. This conclusion depends only on the assumption that the field and temperature dependence of a both enter through the phase coherence time r~; an assumption which is at the very heart of the W L theory. We also point out that for temperatures between 10 K and 20 K, a combination of W L and electron interaction effects is still unable to explain the temperature dependence quantitatively (compare Fig. 2(b) and Table 1). Similar results are summarized in Table 1 for two other alloys in this system which also show a negative temperature coefficient of p. To gain some further insight into WL, and judge its suitability for explaining observations further above helium temperatures, we now con-

2 e

2.

Discussion

92 TABLE 1 Physicalparametersfor amorphousCa70Mg30_xAlxalloys,o, I, D from Sahnouneand Strom-Olsen[6]

15 22.5 30

cr (Sm-J(_+5%))

l (A)

L~(20) (A)

D (cm~s-1)

a(20 K)- a(4.2 K) atB=0T (Sm-' (+ 5%))

OWL(4 T)- OWL(0 T) atT=20K (Sm-' (+5%))

at T= 4.2 K

8.20 x 105 5.13x 105 2.92 x l0 s

6.7 4.0 2.3

202 190 130

3.0 2.0 1.25

240 596 540

154 136 78

276 268 204

sider our isotropic random flight model. The results (PN and S(N)) are summarized in Fig. 1 for flights up to 12 000 steps long. It is important to remember that eqn. (1) is only valid in the limit of long times, times for which the diffusion equation accurately describes the motion (i.e. the asymptotic region of PN)" Note that deviations from the asymptotic behaviour can be clearly seen for flights less than 10 steps long. For times shorter than this, quantum interference between all paths, not just between time reversed pairs of closed loops, becomes important and depends on the details of local atomic structure, etc. A n accurate description of the conductivity then requires a full quantum mechanical treatment and the WL theory is inadequate. We can make a quantitative comparison of our model with the magnetoresistance data using the results of ref. 6, eqn. (3) and Fig. 1 (and remembering that in three dimensions D= Vv2r/3). For Ca70Al30 we have the following physical parameters [6]: 1=2.3 A, Li(20 K ) = D r i ( 2 0 K) l/z= 130 A. Li(4.2 K) = 746 A. Since a field of 4 T corresponds to 64 A we have Ni(4.2 K) = 3.2 × 105= oo, Ni(20)=9600, and N8=2320. Using these numbers and Fig. 1 in (3) gives the following predictions for the WL contribution to changes in conductivity: OWL(20 K, 0 T ) OWL(4.2 K, 0 T ) = 1 6 0 Sm -1, and awL(4.2 K, 0 T ) - aWL(4.2 K, 4 T ) = 3 8 0 Sm -1. The agreement between these estimates and the experimental results is probably fortuitous given our neglect of spin orbit scattering and the uncertainty in S(N) and in the lengths l, and Li. However this shows that the random flight model provides estimates that are within 50% of the experimental results. Using this same line of reasoning we may estimate the maximum possible do due to ' ~ e a k localization" using (3) with [S(10)-S( ~o)= S(10)]. As noted above, a larger contribution would involve loops that are too short to satisfy the assumptions underlying the WL theory. For the

OWL(4T)-OWL (0T) (Sm-'(+5%))

case of Ca70A130 this corresponds to a change in conductivity of roughly 7000 Sm-1 (or approximately 2% of the room temperature value). This is not to say that quantum interference cannot be more important than this; simply that the simple WL theory for this interference is invalid for changes much larger than this. We note in closing that many of the discussions in the literature rely not on (1), but rather on an expression given by Kaveh and Mott in 1982 [7]. However, using the values of L i given above in the Kaveh-Mott expression gives 0(20 K, 0 T ) 0(4.2 K, 0 T ) = 1570 Sm -1, which is more than 10 times the value derived from the detailed analysis of the magnetoresistance at these two temperatures. We conclude that, while qualitatively correct, the Kaveh-Mott expression is not correct quantitatively, and should therefore not be used for quantitative analysis of resistivity data in metallic glasses. 3. Conclusion

In summary we conclude that the temperature dependence of the conductivity between 4.2 K and 20 K in these CaToMg30_xAlx alloys is not consistent with the combined effects of weak localization and electron interaction effects. Some other effect, also with a negative temperature coefficient, is needed to explain the data. Furthermore, we estimate that weak localization cannot be used to explain a change of substantially more than 2% to the conductivity in bulk CaToAl30 for any change in temperature or magnetic field. Acknowledgments

This work was supported by the National Science Foundation under grant number DMR8918139, the Natural Sciences and Engineering Research Council of Canada, and the Indiana University Foundation.

93

References 1 C.C. Tsuei, Phys. Rev. Lett., 57(1986) 1943. 2 M. A. Howson, B. J. Hickey and G. J. Morgan, Phys. Rev., B38 (1988) 5267. 3 K. D. Aylesworth, S. S. Jaswal, M . A . Engelhardt, A. R. Zhao and D. J. Sellmyer, Phys. Rev., B37(1988) 2425.

4 G. Bergmann, Phys. Rep., 107(1984) 1. 5 S. Chakravarty and A. Schmid, Phys. Rep., 140 (1986) 193. 6 A. Sahnoune and J. O. Strom-Olsen, Phys. Rev., B39 (1989) 7561. Note that the values for L, given in Table II of this reference are too large by a factor of (3)~/z. 7 M. Kaveh and N. F. Mott, J. Phys., C15 (1982) L707.