Critical scattering and low temperature deviations from Matthiessen's rule in the resistivity of antiferromagnetic ErZn12

Solid State Communications,

Vol. 12, PP. 455—459, 1973.

Pergamon Press.

Printed in Great Britain

CRITICAL SCATFERING AND LOW TEMPERATURE DEVIATIONS FROM MAUHIESSEN’S RULE IN THE RESISTIVITY OF ANTIFERROMAGNETIC ErZn12 A.M. Stewart* Istituto di Fisica, Universit~di Genova, Genova 16132, Italy and Department of Electrical Engineeringand Department of Physics, Imperial College, London S.W.7., England (Received July 1972; in revised form 7 December 1972 byP.G. de Gennes)

Critical effects in the spin disorder resistivity of the metallic antiferromagnet ErZn12 have been observed. The ordering temperature of 2.77 ±0.01 K is low enough for there to be no phonon contribution to the resistivity present. Below this temperature, the temperature derivative of the resistivity is described best by a divergence of a logarithmic form; above it, a divergence either of the type (T or of logarithmic form is consistent within experimental error. Possible deviations from Matthiessen’s Rule have been observed above the ordering temperature; they are proportional3 to the at higher estimated temperature. phonon resistivity at low temperature, and vary as T —

IN THIS paper experimental results are presented of critical scatteringeffects in the spin disorder resistivity’ of polycrystalline specimens of the metallic localised moment antiferromagnet ErZn 12, and also of possible deviations from Matthiessen’s Rule at low temperature caused by electron scattering off paramagnetic spins as well as off non-magneticimpurities,

any anomaly must be due to short-range fluctuations, and if so ap/aT should vary, above T~,like the magnetic specific heat. Suezaki and Mori6 have examined the resistance anomaly of an antiferromagnet and found that it resulted from long-range spin fluctuations around the magnetic wavevector Q at which the system was attempting to order. They predicted anomalies in ap/aT going as sign (t) t in the direction parallel to Q, and as sign7(t)haslnmade t I perpendicular to Q. More microscopically based recently Takada calculations of these effects and has come to the conclusion that for ferromagnets the Fisher—Langer— Mannari theory is to be preferred to the de Gennes— Friedel theory, and that in polycrystalline specimens of antiferromagnets apIaT should vary as sign (t) t In the above theories the coefficients of the dependences of aplar below the ordering temperature may be greatly different to those above, because of the presence of long-range order. —

There are three theories which attempt to describe the resistivity criticaland scattering magnetic ordering point.due Deto Gennes Friedel2near and aKim3 have examined the effect of long-range spin fluctuations in a localised moment ferromagnet within a molecular field approximation, and have found an anomaly in the temperature derivative of the resistivity, apfaT, which goes as sign (t)ln I t I where t = (T TC)/TC,TC being the ordering temperature. On the other hand Fisher and Langer4 and Mannari5 have argued that as long-range spin fluctuations will be suppressed, due to the finite mean free path of the conduction electrons, —

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—

~

Experiments on critical resistive effects in local moment ferromagnetic intermetallic compounds8 and dilute alloys9 tend to favour the de Gennes—Friedel theory, but this does not adequately describe the

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*

r~

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Present address: Center for Theoretical Studies, University of Miami, Coral Gables, Florida 33124, U.S.A. 455

456

THE RESISTIVITY OF ANTIFERROMAGNETIC ErZn12

Vol. 12, No.6

T (°K) 3.0

_________________________

3.5

4.0

.38

.

I 2.6

2.7

100

•

T

20~0-

2.8

T (°K)

FIG. 1. Measured polycrystalline resistivity of ErZn12 vs. temperature. (a) General form of the resistivity. (b) Resistivity in the critical region. (c) Resistivity below 1~on an expanded scale. 1, which seems more in accord behaviour of nickel’°~ with the Fisher—Langer—Mannari theory. Little work seems to have been done on antiferromagnets so far apart from the observation of large critical effects in the resistivity of the rare earth metal europium12 in which the antiferromagnetic transition is of first order13 and the examination of polycrystalline specimens of some rare earth compounds by Fote et al.14 Previous work’5 on the crystallographically ordered intermetallic compound ErZn12 has shown that it has a polycrystalline susceptibility which obeys a Curie— Weiss law, a paramagnetic Curie temperature of + 3°K and a paramagnetic moment close to the value of tripositive 4t’ erbium. Deviations from the Curie— Weiss law occur below 30°K,and the susceptibility has a peak at 2.8°Kindicative of some form of antiferromagnetic ordering. Because the ordering temperature is so low, the resistivity in the region of it is due to spin and impurity scattering only; the phonon contribution is negligible. The preparation and examination of the compound 15 An ingot, of average grain size is50~i described elsewhere. and good metallurgical quality, was spark-machined into a suitable size and its resistance was measured by the four terminal method while it was immersed in pumped

He4.The potential leads were spot-welded platinum wires and the voltage across them was measured with a Keithly Model 140 nanovoltmeter with a sensitivity of 5nV a resolution of 1 in iO~.Because of difficulty in measuring the dimensions of the irregularly shaped specimen, the absolute value of the resistivity could only be estimated to ±30per cent. —

The general form of the resistivity of polycrystalline ErZn 12 may be seen in Fig. 1(a) (see also the inset to Fig. 4). Above 4°Kthe resistivity behaves like that of a nonmagnetic metal with a quasi-linear slope of 0.054 ~2cm/°K at higher temperatures, and a resistivity of 15 .tflcm at 300°K.As the temperature goes below 4°Kthe resistivity rises up to a sharp peak at 2.77 ± 0.01°Kas shown in Fig. 1(b), and then below this ternperature decreases rapidly to a residual value of 0.282 i~2cm.The resistivity below 2.77°Kis shown on an expanded scale in Fig. 1(c). Whenever a resistance minimum is observed in a local moment system it is tempting to interpret it as afirst Kondo effect. However, we isreject thisagainst possibility, because if the resistivity plotted lnT a straight line is not obtained, and second because evidence has been put forward elsewhere15 that the s—f coupling constant in the RZn 12 compounds is positive in sign, at

Vol. 12, No. 6 I

—

THE RESISTIVII’Y OF ANTIFERROMAGNETIC ErZn12 I

1111111

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~

.01

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~

.006

1.0 (°K]

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FIG. 2 The logarithm of the modulus of the temperature derivative of the resistivity plotted against in IT T~I. The points above 7’, are referred to the resistivity derivative scale on the right, those below T~,to the scale on the left. The value of T~,is taken to be 2.75°K.The straight line has a slope of 1/3.

T)T0

~

p

—

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.

.004

.

.002

—

least for zero wavevector. This would lead to a decrease of the resistivity with decreasing temperature rather than the increase which is observed. We then interpret these results by assuming that the total resistivity t is the sum of four terms: P

=

Po

+P~+P~+~

(1)

Po is the residual resistivity due to impurities and defects, p8 the spin disorder resistivity, p~the phonon resistivity and ~ is a deviation from Matthiessen’s Rule. This latter is defined by the equation (1), and will be discussed at the end of the paper.

1

.o~

‘

.‘~

T— T~I

‘

(°Ic)

FIG. 3. The modulus of the temperature derivative of the resistivity vessus ln I T 7’~,I. The points above 7’~,are referred to the resistivity derivative scale on the right,thosebelow7,tothesca1eontheleft.7~,is taken as 2.77°K. —

through the experimental points of Fig. I against I T— T~I. T1, has been taken to be 2.75°K;with this value the points for T— T~ >0.1 °Kfall on a straight line with a slope of 1/3. The points below T~,,however, do nQt lie on a straight line. On the other hand, if I ap/aT I is plotted against ln I T 7’, I, as has been done in Fig. 3 with T~= 2.77°K,then the points below T,~,lie on a straight line and the points above it do also, apart from deviations near the critical point. Our conclusion is that the data below T~appear to indicate a logarithmic divergence, but that the data above 7, are consistent, within experimental error, with either a logarithmic divergence one having the form 3. An exponentorof 1/3, though, is preferred over other (T TJ” values. ,

—

Up to 2.77°Kthe variation of the resistivity is due to spin disorder scattering, which increases as the antiferromagnet loses its magnetic order. Just above the ordering temperature the resistivity is seen to be constant (apart from the critical scattering) thus giving a direct confirmation of Kasuya’s~theory. At higher temperatures the phonon contribution appears, and the compound behaves like a non-magnetic metal with a ‘residual resistivity’ of Po + P8~ The critical region is shown in Figs. 1(b) and 1(c). It is clear that ap/pT is negative above T 0 and positive below it, and that ap/aT has a discontinuity on a temperature scale of not greater than 0.02°Kor t 10-2. In order to estimate which type of critical diveigence best fits the experimental results, we have plotted in Fig. 2 the logarithm of I ap/aTI, which has been obtained by graphically differentiating the smooth curves drawn

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Measurements of greater sensitivity on single crystal specimens of higher purity will be needed to decide between these two alternatives, but it is already clear that the critical effects in the resistivity of the localised moment antiferromagnet ErZn12 are qualitatively different to those seen in the itinerant ferromagnet nickel;’°” in particular ap/aT is unambiguously

458

THE RESISTWITY OF ANTIFERROMAGNETIC ErZn12

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Vol. 12, No.6

scattering, for example pure LuZn12, is not known. However we have attempted to estimate deviations from Matthiessen’s of a similar compound Rule, at low lacking temperature spin disorder at least, and by impurity makingthe assumption that the ‘pure’ material possess a resistivity which obeys Grüneisen—Bloch relation 8).the This has been fitted to the (tabulatedvalues by Meaden’ measured of p and ap/aTof ErZn 12 at high temperatures in the linear regime, which may be done by observing that if the linear portion of the Grüneisen—

E I

0.145resistivity, zero 0D’ where 18itFor 0D intersects isErZn the Debye the temperature temperature 2 is extrapolated axis forat to the resistivity. Bloch relation for OD/5 > 12 T>°D°D/ = 168°Kis obtained; estimated this may be GrUneisen—Bloch compared to therelation 1 75°Kof is plotted zinc.18 The as curve

•

L

.015 I

10 T(°K)15

C of Fig. 4. The difference between the measured values amd is plottedfrom as curve B in Fig. 4, and is identified curve as the Cdeviation Matthiessen’s Rule.

10 1

20

I

This curve can be seen to vary up to 14°Kwith the FIG. 4. A log—log plot of the resistivity vs. temperature above 7’~.Curve A is the measured resistivity minus the value at 4.2°K.Curve C is the estimated phonon resistivity. Curve B is the difference between curves A and C. Lines having slopes corresponding to 7’3 and T4 power laws are shown for comparison. In the inset to the figure the resistivity behaviour is shown on a linear scale. negative above

T~.In our specimen the relatively large

residual resistivity may have smeared out the finer details of the resistivity divergence, as the nature of the physical situation will change when the spin correlation length becomes comparable in size to the impurity mean free path. Inhomogeneitesin the stoichiometry of the specimen, which have been shown to be less than 2 per cent by electron microprobe analysis, may also play a small part in softening the transition. Of course, the possibility that the critical behaviour may be dominated by changes17ofcannot the lattice parameter be ruled out. near Fisher—Langer4—Mannari5 the ordering temperature theory cannot be The judged until specific heat measurements are available, In the inset to Fig. 4 the resistivity of the compound is shown above its ordering temperature. The differences between the measured values of resistivity and its 4.2°K value (curve A of Fig. 4) can be seen to vary as UP to about 17°K.It is not possible to obtain the deviation from Matthiessen’sRule exactly, because the resistivity

same power of 7’ as does the estimated phonon resistivity, and above that it varies as ~ Caplin and Rizzuto19’~ have found a T3 dependence at low temperature of the deviation from Matthiessen’s Rule for non-magnetic impurities in aluminium, and this temperature dependence has been interpreted2’ as a consequence of momentum non-conservation in the scattering process, but Whall et aL ~ have noted that in gold based transition metal alloys the deviations are proportional to the phonon resistivity. We stress, though, that in ErZn 12 the deviations from Matthiessen’s Rule are due to scattering off paramagnetic spins as well as off static impurities. There may also be irregular contributions to the resistivity from the rare earth ions condensing into their lower crystal field states as the temperature decreases, thereby changing their elastic and inelastic scattering cross-sections. These effects have been observed in AuHo and ~Dy by Murani.~ It we must be made emphasized that the approximation which have in the above analysis will not be valid at the highest temperatures, where it gives a zero deviation from Matthiessen’s Rule, but it is hoped that it provides an indication of the low temperature behaviour which is characteristic of this compound. We also expect there to be deviations from ‘Matthiessen’s Rule’ in ErZn 12 below the magnetic ordering temperature. In other words, the total resistivity will not be the sum of the residual resistivity plus

Vol. 12, No. 6

THE RESIST1VITY OF ENTIFERROMAGNETIC ErZn12

the resistivity due to electron—magnon scattering. However, nothing is at present known about these effects.

459

Acknowledgements In addition to acknowledgements I should thank A.D. Caplin, J.A. and5 C. Rizzutolike fortosome very valuable madeMydosh, previously’ discussions and suggestions. —

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Dans l’antiferromagnétique métallique ErZn 12, on observe que le désordre de spin a un effet critique sur la résistivité. La temperature de Néel, 2.77 ±0,01 K, est suffisamment faible pour que la contribution des phonons a la résistivité soit négligeable. Des deviations possibles a la régle de Matthiessen ont été observées au-dessus de la temperature de Néel; ceiles-ci sont proportionnelles la résistivité due aux phonons basse temperature, et varient comme ~3 temperature plus élevée.

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