SPIN SCATTERING EFFECTS IN RARE EARTH METALS

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SPIN SCATTERING EFFECTS IN RARE EARTH METALS

J. Rhyne, J. Cullen

To cite this version:

J. Rhyne, J. Cullen. SPIN SCATTERING EFFECTS IN RARE EARTH METALS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-246-C1-247. �10.1051/jphyscol:1971180�. �jpa-00214507�

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JOURNAL DE PHYSIQUE Colloque C I , supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C I

-

246

SPIN SCATTERING EFFECTS IN RARE EARTH METALS

J. J. RHYNE and J. R. CULLEN

U. S. Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland, 20910

RC.sum6. - On a effectue des mesures de rksistivitk Hall pH dans Tb, Dy, et un alliage Gd-Dy. Les rksultats ont 6tB analyses en ce qui concerne les coefficients normal RO et anorrnal Rs selon I'expression habituelle pH = Ro B

+

Rs 4 nM.

Dans Gd et Tb et dans l'alliage Gd-Dy, un accroissement anormal et un changement de signe du coefficient normal ont kt6 observb entre 0,4 Tc et Tc (Ro n'a pu &re obtenu dans Dy, le champ ktant trop limit&). Les valeurs de Ro obtenues B partir de la portion en champ fort de la courbe pH en fonctlon de H ont 6tk corr~gkes de l'effet de suscept~bllitk magnb tique au-dessus de la saturation. Les valeurs de Ro obtenues au-dessus de Tc sont en bas accord avec celles a basse temp6 rature. On a dkveloppe un modele dans lequel Ro est relie B la dependance en champ de la diffusion des Blectrons de conduction par les spins 4 f desordonnes au voisinage de To. Dans ce mod&le, Rs doit dkpendre lineairement du champ appliqub Utilisant la mkthode habituelle oh Rs est indkpendant du champ (au-dessus de Hsat) ce resultat se traduit par une augmentation de Ro qui est effectlvernent observb. Un effet analogue a ete reIev6 dans les mesures de magnktorksis- tance sur Tb qui prksente aussi une forte dependance en temperature ^t'i I'approche de Tc. Ces rksultats sont compares aux dependances en tempeature calculbs a partir du modkle de diffusion des soins dksordonnks.

Abstract. - Measurements have been made of the Hall resistivity p~ in the rare earth metals Tb, Dy, and in a Gd-Dy alloy. The data were analyzed for the normal (Ro) and extra-ordinary (Rs) Hall coefficients according to the conventional expression : p~ = Ro B

+

Rs 4 nM. In both the elements Gd and Tb and in the Gd-Dy alloy an anomalous increase in magnitude and a change of sign of the normal coefficient was observed between 0.4 Tc and Tc (Ro was not obtained in Dy due to field limitations). The Ro values obtained from the high field portion of the p~ vs H curves were corrected for the effect of the magnetic susceptiblllty above techn~cal saturation. Values of Ro obtained from data above Te were in close agreement with those at low T. A model has been developed which ascribes the Ro behavior to a field dependence of the scattering of conduction electrons by the disordered 4 f spins near Tc. In this picture Rs is expected to exhibit a linear dependence on applied field. Using the conventional treatment which considers RS field independent (above Hsrt) this result is manifested in an enhancement of R, as observed. A similar effect has been observed in magnetoresistancemeasu- rements made in Tb which also show a strong temperature dependence on approach to Tc. These results are compared to the temperature dependences calculated from the spin-disorder scattering model.

The Hall effect in ferromagnetic rare earth metals shows a number of features which can be related to the scattering of conduction electrons by 4 f spin fluctuations. We discuss some of these and present new Hall data on a Gd.,5Dy,,5 alloy and associated magnetoresistance work on Tb. The Hall results are conveniently described in terms of the conventional expression for the Hall resistivity :

. p H = Ro B

+

R , 4 n M (1) made up of normal R, and anomalous R, terms.The temperature dependence of the anomalous or magne- tization-dependent part is qualitatively very different between Gd [I] and Tb [2] (or Dy [2]) as shown in single crystal results plotted in figure 1 as a function of TITc, where Tc is the higher magnetic ordering temperature. In particular, the non-S state ions show a positive peak in R, near 0.5 Tc followed by a sign reversal a t higher temperatures. We expect [2] dis- tinctive behavior in Rs of the heavy rare earth ions with more than half-filled 4 f shells, reflecting an important 4 f orbit-conduction electron orbit coupling, absent in Gd.

The Gd.,,Dy.,, alloy shown in the figure is interesting in that its observed Hall resistivity is represented by

[ ~ H ( ~ / ~ c ) ] a l l o y = .75 [~H(TlTclJca

+

.25[~H(T/Tc)]Dy (2) at temperatures above 0.8 Tc, but disagrees signifi- cantly a t lower temperatures. In fact, the height of the positive peak near 0.5 T, is approximately 25 % of that in pure Dy, indicating a shorting out of the Gd contribution altogether. On the assumption of addi- tivity of p,, this alloy would show no positive values of R,.

a * - 1 ANOMALOUS HALL

\

1

COEFFICIENT

-140

r

^H^110001l

-160 -180

1

^, ^Cd:Th: T, ^:^T = ⁼^29:228" K ^K ^, ^:^:

1

DY,, ~ d , , : T, = 268" K \i

-200

0 .2 .4 .6 .8 1.0 T=-T,

T/Tc Fiqure I

FIG. I . - Temperature dependence of the basal plane anomalous Hall coefficient of Gd, Tb and a Gd-Dy alloy. The dashed curve is the result of a molecular field theory (see text).

In contrast t o the widely varying behavior of R,

across the series Gd-Dy, values of R, show a stri- kingly similar variation with temperature as shown in figure 2. Data on Gd, Tb, and the Gd-Dy alloy (Ro for Dy was not determined) all show essentially cons- tant negative values of Ro u p t o about 0.4 T,. Above this point Ro changes sign and rises sharply to a peak near T , (values of R, above 0.9 Tc cannot be separated

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971180

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SPlN SCATTERING EFFECTS IN RARE EARTH METALS C 1-247

ORDINARY HALL COEFFICIENT

H * [OOOll

0 1 2 3 4 5 6 7 8 9 T > T ,

T/T,

FIG. 2. - Temperature dependence of the ((normal ^PHall coefficient of Gd, Tb and the Gd-Dy alloy. The curve labeled ARs(T) is a calculated field dependent correction to R,.

reliably). On the over-simplified one band model in which R, = l/ne, the values shown in the figure would correspond to a change in carrier density from about 1 electron/atom to - 0.25 electron/atom. In terbium, data from the paramagnetic side when corrected for susceptibility effects [2] gave a negative value of R, = - 2 electrons/atom of comparable magnitude to the result found at 4 OK. Measurements above T, to determine R, were not made in Gd or in the Gd-Dy alloy.

We focus our attention on the temperature dependence of R,. The occurrence of the sign reversal and large change in magnitude of R, below T, are difficult to explain on the basis of conduction band polariza- tion effects particularly in view of the similar values of R, in Tb at 4.2 OK and for T $- T,. The values of R, over the anomalous range 0.5 T, to over 0.8 T, are found to be linearly related to the high field susceptibility indicating that the temperature dependence may be associated with spin ffuctuations. Several sources have been examined.

In the conventional analysis (eq. (1)) of the Hall constants, R, is evaluated from the Hall resistivity at Hi,, = 0 and is assumed field independent. R, is found from the slope above saturation with a correction made for the magnetic susceptibility above technical saturation. It is thus apparent that a linear field dependence of R, at high fields would be indistinguishable from an enhancement of R,. Such a linear H dependence in R, might appear as a result of the suppression of spin disorder scattering by the applied field as dis- cussed below. That such an effect can be significant was observed in magnetoresistance measurements made on the terbium sample which showed a value Ap/p of 0.6 % at 77 OK rising to 4.9 % at 240 OK both in an applied field of 26 kOe. Over the range from below '77 OK to over 200 OK, ap/aH is independent of field (above magnetic saturation) and varies at T2.

We have calculated the correction to R, linear in applied field by extenaing the standard [3] calculations

of R, based on spin disorder scattering to finite applied fields. A11 such calculations give

p & = R S 4 n M = ~ < 6 M 3 > (3) i.e., the anomalous Hall resistivity proportional to the mean cubed fluctuation in the total moment. The mean field calculation of

<

S M ~

>

reproduces the zero field pk (or R,) for Gd fairly well as shown in figure 1. Our calculation for the field-dependent correction to R,, including a temperature-independent contribution designed to account for R, in the low temperature limit, is plotted as AR,(T) in figure 2.

As mentioned previously this correction would be hidden in the experimentally determined R,, and we see that the calculated curve does reproduce the ini- tial upturn in R, correctly, but the overall size of the calculated effect (fixed by the size of the zero-field pk terms) is too small at higher temperatures. We should look for improvement of the theory in two different directions.

First, we remark that while the evaluation of

<

6M3

>

in the mean field approximation may lead to reasonable results for the zero field p i (or R,) temperature dependence, the requirement of repro- ducing the T dependence of the correction dp1(T)/8H is more demanding and may be beyond the scope of the mean field model. One may thus expect that a more rigorous calculation, while giving roughly the same temperature dependence as (eq. (2)) for pk, would led to significant improvement between theory and experiment for the field dependence of R, near the transition temperature.

There is an interesting relationship between pH and the linear magnetoresistance, based on the approxi- mate theory that leads to (eq. (3)). The same theory predicts a resistivity p proportional to

<

SM2

>.

It is exactly true that

Thus

Preliminary experimental results indicate very good agreement with (eq. (5)) up to .9T/T, in Gd.

Second, it is possible that, because of large aniso- tropy in the conduction electron lifetimes, the temperature dependence of the ordinary Hall mobility is different from that of the conductivity. This would give rise to a temperature dependent R, in the region of strong spin-disorder scattering. It is, however, extre- mely difficult to account for the sign change in R, on this Fermi surface model alone, and although the lifetime mechanism may assist to some extent, we believe that field dependence of the spin-disorder scattering is the important mechanism accounting for the temperature dependent R,.

References

[I] LEE (R.) and LEGVOLD (S.), Phys. Rev., 1967, 162,431. [2] RHYNE (J.), Phys. Rev., 1968,172,523 ; J. Appl. Phys., The values of RO given here have been modified 1969, 40, 1001.

to correctly include the effect of the susceptib~lity [3] MARANZANA (F.), Phys. Rev., 1967, 160,421 ; KONDO above technical saturation. ^(J.), Theoret. Phys. (Kyoto), 1962, 27, 772.