The Hall coefficient in disordered alloys of early and late transition metals

Journal of Non-Crystalhne Sohds 156-158 (1993) 307-310 North-Holland

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The Hall coefficient in disordered alloys of early and late transition metals J. Ivkov

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and E. Babi6 b

a lnstttute of Physws of the Untverstty, PO Box 304, Zagreb, Croatta b Phy~tcs Department, Faculty of Sctence, Unwerslty of Zagreb, PO Box 162, Zagreb, Croatta

The proposed explanahon of the positive Hall coefficient of amorphous Zr-TL alloys (TL = Co, N~ and Cu) m terms of spin-orbit interaction has been extended to other &sordered alloys containing early transit,on metals (TE) and rare earths The systems analyzed include hqmd Ce-Cu alloys, amorphous Ta-Ni and La-Ga alloys and ternary alloys based on T E - T L with normal metal or TL ad&Uons. All the observat,ons are consxstent w,th the earher proposition

1. Introduction

In the literature there is a considerable amount of experimental data on the Hall effect in disordered (i.e., amorphous and liquid) alloys based on early transition (TE) and late transition (TL) metals. These results enable us to construct a qualitative picture of the dependence of the Hall coefficient, RH, on composition, resistivity and temperature in these alloys. On the other hand, from a theoretical point of view, there is no unique explanation of the positive sign of R H in non-magnetic TE-rich alloys. There are two different (not mutually exclusive) theoretical approaches to the interpretation of the Hall efect results in T E TL-based disordered alloys. According to the first approach, the Hall effect in disordered T E - T L alloys is determined by subtle details of the electronic structure, i.e., by the S-shaped dispersion curve [1,2] or by the minimum in the electronic density of states [3,4] (for a review, see ref. [5]) that are both due to the strong electron scattering and the strong s-d hybrizidation that are characteristic for transition-metal-based alloys. Accord-

Correspondence to" Dr J Ivkov, Institute of Physics of the University, B0em~ka 46, PO Box 304, 41000 Zagreb, Croatia. Tel: + 38-41 271 211. Telefax: + 3 8 - 41 421 156. E-mad: ana. smontara@~rb.ac mail.yu.

ing to the second approach [6-8], there is, even in non-fernomagnetic transition-metal-based alloys, a strong magnetic contribution to R n that is due to the high resistivity of these alloys and to the influence of the strong spin-orbit interaction (characteristic for TE elements) on the scattering of the conducting electrons. In what follows we demonstrate that the second approach consistently explains the observed dependence of the Hall effect on composition, resistivity and temperature in non-magnetic transition-metal-based alloys. In this presentation, we have included also some R H data for the disordered alloys based on the light rare earths (RE = La, Ce).

2. Results

In our analysis, we have assumed as in ref. [9] that the Hall coefficient, R n, in non-fernomagnetic TE-TL-based alloys is (as in the paramagnetic phase of ferromagnets) equal to the sum of the normal Hall coefficient, R0, and RsX, where Rs is the anomalous Hall coefficient and X is the part of the magnetic susceptibility that corresponds to those electrons that yield the anomalous contribution. Further, we make a rough distinction between s- and d-band elec-

0022-3093/93/$06 00 © 1993 - Elsevier Science Pubhshers B V All rights reserved

308

J h~kov, E. Babd / The Hall coeffwtent m dtsordered alloys

trons [10] and approximate the unknown R 0 values with those that correspond to the 'free' selectron values estimated from the composition of the alloys. As to the anomalous (i.e., magnetic) contribution to RIj , we consider that it is mainly due to the d-band conduction electrons [11] contributed by TE elements. A more detailed discussion of the applicability of these approximations is given in ref. [9]. To prepare the forthcoming discussion, we emphasize two points. First, we note that the magnetic contribution to R H is always strongly dependent on the resistivity, p. This statement holds regardless of the exact interpretation of the influence of the spin-orbit interaction on the electron scattering (i.e., via skew-scattering or side-jump terms [12,13]). Second, we emphasize that in a proper analysis of the magnetic contribution to R n one actually ought to consider the difference (denoted below as AR H) between the measured R H and estimated R 0 values. 2.1. The dependence of R u on the alloy composition

In TE-based alloys, Rr~ is positive while in non-fernomagnetic TL-based alloys it is negative and extrapolates to the free electron values of the host element. Consequently, in TE-TL-based alloys, the sign and the magnitude of R n depends strongly on the alloy composition. This point is illustrated in fig. 1 where we show the dependence of the Hall coefficient, RH, on the alloy composition in amorphous Zr-(Ni, Cu) [5], Z r Co [5,7], T a - N i [16] and liquid C e - C u [14] alloys. The values of R H in liquid Ni, Co and Cu [14] are also included. We emphasize the almost identical dependence of RH on composition in C e - C u and Z r - C u alloys. When the resistivity and the strength of the spin-orbit interaction do not change too much with the alloy composition, it is possible to correlate the sign of R n and the electronic structure of the alloy. In amorphous Zr-Co,Ni,Cu alloys, for example, the Hall effect changes the sign from negative to positive values at Zr concentrations that are some 20 at.% higher than those at which the Fermi level moves from predominantly

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3-d TL band to the Zr d band [15]. A similar correlation has been found in amorphous Ni-Ta [16] and V-AI alloys [17] that are, from the chemical point of view, close to Z r - T L alloys. At the same time, in 'pure' TE T i - V amorphous alloys, R H is positive for all alloy compositions [18]. In the vicinity of the middle of the d-band, the magnetic contribution to R H decreases and changes sign as has been observed [19] in Mn-AI alloys (by contrast with V-A1 alloys). The interpretation of the dependence of the Hall effect on the alloy composition in Z r (Ni,Co,Cu) and Ta-Ni alloys shown in fig. 1 becomes rather simple if instead of R n we look at the AR n = (R n - R 0) values as a function of the number of d-electrons in the early transition metal d band, nd(VE) (fig. 2). The values of AR H for all four alloy systems coincide both at the highest and the lowest values of rid(WE). For nd(T~) > 1.5, AR~ amounts to (13 _.+ 1) × 10 -11 m 3 C -1 and extrapolates to zero a s nd(TE ) tends to zero. Further, the variations of AR n with nd(xE~ for Zr-Ni, Z r - C o and Ta-Ni alloys are the same within the dispersion of the experimental data and the uncertainity in the estimation of Ro. A more general analysis that will explain R H in other alloy systems must take into account the

J Ivkov, E. Babtd / The Hall coefficzent m dtsordered alloys

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influence of the resistivity [9] and the actual spin-orbit coupling on the magnetic part of R H.

2.2 The correlation between R H and electrical resistivity The magnetic contribution to the Hall coefficient is proportional to either the resististivity or to the square of the resistivity depending on the relative importance of skew-scattering and sidejump terms. The high value of the resistivity of the amorphous alloys and the high strength of the spin-orbit coupling in T E elements are the main causes why the magnetic contribution to R H is so important in amorphous T E - T L - b a s e d alloys. In particular, the difference in resistivity between the amorphous T i - C u and Z r - C u alloys almost completely accounts for the difference in their R H values [9]. The correlation between the electrical resistivity and the Hall effect is particulary pronounced in ternary T E - T L - b a s e d alloys. In amorphous T E - T L - M alloys (where T E = Zr or Ti, T L = Ni or Cu and M = AI or Ga), the Hall coefficient and the resistivity initially strongly increase on the addition of A1 or Ga to the base T E - T L [20,21]. On the M-rich side, RH will of course start to decrease and extrapolate to the values of pure amorphous A1 or Ga. Similarly, in ternary

309

Zr2Ni l_xTx alloys (T = Co, Cu; 0 < x < 1), R H either shows a maximum or is independent of variations of the resistivity and of the difference between R H for x = 0 and x = 1. In particular in Zr2Nil_xCo x alloys, where the resistivity increases somewhat with x, and where R H is considerably lower for x = 1 than that for x = 0, R n shows a shallow maximum around x-=-0.2. Since in ZrzNil_~Cu x alloys resistivity hardly changes with x and R H is higher for x = 1 than for x = 0, no such maximum is observed.

2.3. The temperature dependence of R , and ARI4 The Hall coefficient in non-magnetic amorphous alloys depends, as expected, very weakly on the temperature. Here we consider only the temperature dependence of R n at temperatures higher than those at which the quantum corrections to the Hall effect, in particular electronelectron interactions, are important. The data for the temperature dependence of R H at higher temperatures have been reported for about thirty amorphous Z r - ( C o , Ni, Cu), T i - C u and T a - N i alloys [2,7,16,22-24]. In all the cases except for the Zr36Ni64 alloy [22], the temperature coefficient of R H is either equal to zero (within the experimental error) or is negative regardless of the actual sign of R H. At the same time, in these alloys the temperature coefficient of resistivity, a, is small (of the order of 10 -4 ) and negative. This eliminates the possibility that the electron-electron interaction governs the temperature dependence of R H at higher temperatures. In our opinion, the temperature dependence of R H is caused by the temperature dependence of its magnetic part RsX. Trudeau et al. [7] have suggested that the temperature dependence of R H, particularly in Z r - C o alloys, is caused by the temperature dependence of the spin-orbit effects, i.e., by the valence susceptibility that decreases with the increase of the temperature. In our opinion, the more probable explanation for the observed temperature dependence of R H, i.e., AR H, lies in the correlation between the resistivity (that in alloys considered here decreases with temperature) and the magnetic part of the Hall effect.

310

J h,kov, E. Babt( / The Hall coefftctent m dtsordered alloys

However, because of the weak temperature dep e n d e n c e o f b o t h R H a n d t h e resistivity, t h e exact f o r m o f this c o r r e l a t i o n is h a r d to d e t e r mine experimentaly. N o w we e m p h a s i z e that, in Z r - C o [7] a n d T a - N i [16] alloy systems, t h e relative v a r i a t i o n s o f R H with t h e t e m p e r a t u r e a r e g r e a t e s t for t h o s e alloy c o m p o s i t i o n s for which t h e H a l l coefficient c h a n g e s sign a n d is t h e r e f o r e very small. This further indicates that the temperature depend e n c e o f R H actually d o e s n o t reflect t h e t e m p e r a t u r e d e p e n d e n c e o f t h e n o r m a l H a l l coefficient, R 0, b u t t h a t it is actually c a u s e d by t h e t e m p e r a ture d e p e n d e n c e o f R s X . Finally we mention the temperature depend e n c e o f R H a n d p in a m o r p h o u s L a - G a alloys with a G a c o n t e n t b e t w e e n 16 a n d 26 a t . % [25, 26]. In t h e alloys with t h e h i g h e r G a c o n t e n t in which t h e t e m p e r a t u r e coefficient o f resistivity is negative, R H significantly d e c r e a s e s with increasing t e m p e r a t u r e . A t t h e s a m e time, in t h e alloy with 16 a t . % G a , t h e c h a n g e o f R H with t h e t e m p e r a t u r e at h i g h e r t e m p e r a t u r e s is negligible a n d actually this is t h e alloy with t h e c o m p o s i t i o n close to t h a t for which t h e t e m p e r a t u r e coefficient o f resistivity c h a n g e s t h e sign. Evidently, t h e investigation o f t h e t e m p e r a t u r e d e p e n d e n c e o f R n a n d p in t h e alloy systems in which t h e r e is a c h a n g e o f t h e sign o f t h e t e m p e r a t u r e coefficient o f t h e resistivity w o u l d be very useful.

3. Conclusion T h e analysis p r e s e n t e d a b o v e shows t h a t t h e e x p l a n a t i o n o f t h e positive H a l l coefficient in a m o r p h o u s Z r - T L alloys [9] a p p l i e s e q u a l l y well to all d i s o r d e r e d alloys b a s e d on T E o r light r a r e e a r t h . In p a r t i c u l a r , t h e o b s e r v e d c o m p o s i t i o n a n d t e m p e r a t u r e d e p e n d e n c e s show t h a t the positive R H is a c o n s e q u e n c e o f the p a r t i c u l a r l y s t r o n g s p i n - o r b i t c o u p l i n g a n d t h e high resistivity o f t h e s e alloys. A c c u r a t e m e a s u r e m e n t s o f t h e t e m p e r a t u r e d e p e n d e n c e o f resistivity a n d R R on alloys with small positive a (such as LasnGa16) w o u l d p r o v i d e a definitive p r o o f o f t h e p r o p o s e d explanation.

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