Hall effect and electrical resistivity of Tb75-Gd25 near the Curie point

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Hall effect and electrical resistivity of Tb75-Gd25 near the Curie point

J.M. Moreira, M. M. Amado, M. Ê. Braga, J.B. Sousa

To cite this version:

J.M. Moreira, M. M. Amado, M. Ê. Braga, J.B. Sousa. Hall effect and electrical resis- tivity of Tb75-Gd25 near the Curie point. Journal de Physique, 1984, 45 (4), pp.779-789.

�10.1051/jphys:01984004504077900�. �jpa-00209811�

Hall effect and electrical resistivity ^of Tb75-Gd25 ^near the Curie point

J. M. Moreira, ^{M. M.} Amado, ^M. Ê. Braga ^and J. B. Sousa Centro de Fisica da Universidade do Porto, ⁴⁰⁰⁰ Porto, Portugal (Reçu ^{le 11 mai} 1983, révisé le 14 novembre, accepte le 9 décembre 1983)

Résumé.

On présente des résultats de

de haute précision

la résistivite de Hall (03C1H),

dérivée par rapport ^{à la} température (d03C1H/dT), la résistivité électrique (p) ^et

^dérivée d03C1/dT pour

alliage polycristallin ferromagnétique Tb75-Gd25 ^et pour des températures 77-360 K. On

mesuré aussi la susceptibilite magnétique (~)

au

voisinage ^et au-dessus du point ^{de Curie.}

Les différentes contributions à la résistivité électrique ^sont séparées, ^{notamment les termes} de désordre de spin,

résiduel et de phonons;

^essaie

comparaison

la théorie. Pour la résistivité de Hall, ^{le termes normal}

(03C1oH) ^{et le terme} extraordinaire (psH) ^sont obtenus, ^{avec les} constantes de Hall correspondantes (R0, Rs).

Des anomalies bien marquees ^sont observées pour d03C1/dT ^et d03C1H/dT ^{dans la} région ^de transition, ^montrant

l’existence d’un point critique bien défini. L’analyse ^{montre que} (1/03C1c).(d03C1/dT)+ (A±/03B1). (1 ^{- |} 03B5|-03B1 ⁺ B±,

avec 03B5

= (T - Tc)/ Tc, (+) (-) signifiant ^T > Tc, ^T Tc, respectivement. ^{On obtient} Tc

244,79 K,

03B1+ ~ - 0,001, A+/A_ ~ ^{0,88, pour 1,2}

^10-4

^0,7

^{10-2 et 6,2}

^10-3 |03B5| 2,6

10-2.

Pour d03C1H/dT la situation est complexe, ^à

de l’effet du champ magnétique appliqué. ^Notre analyse ^montre

l’existence d’effets importants de fluctuation près ^de Tc,

le voit dans dRs/dT.

L’effet Hall extraordinaire de Tb75-Gd25 ^est analysé pour séparer les différentes contributions, c’est-à-dire, les termes de diffusion gauche ^{et de saut} latéral,

^tenant compte des effets de plusieurs hamiltoniens de diffusion

(Kondo, Smit). ^{Nous montrons} que 03C1sH ^dans Tb75-Gd25 peut ^être expliqué par des contributions Kondo (gauche ^et saut-latéral) ^{et Smit} (saut-latéral).

Finalement,

essaie d’obtenir la température de Curie du Tb75-Gd25 ^à partir ^des points de transition du Tb et Gd,

utilisant la théorie des alliages ^binaires magnétiques aléatoires de terres

Abstract

Fairly detailed and accurate data is presented for the Hall resistivity (03C1H), temperature ^derivative

(d03C1H/dT), electrical resistivity (p), and derivative d03C1/dT ⁱⁿ

ferromagnetic Tb75-Gd25 polycrystalline alloy,

a

wide _range of temperatures (77-360 K). Magnetic susceptibility ^data (~)

and above the Curie point ^{is also}

included.

The different contributions to the electrical resistivity

separated ^out, including ^the spin-disorder, ^residual

and phonon ^terms, the results being compared ^with theory. For the Hall resistivity, the normal (03C1oH) ^{and the extra-} ordinary (03C1sH) ^terms

^{obtained, with the} corresponding ^{Hall constants} (R0, Rs).

Very sharp anomalies

observed both in d03C1/dT ^and d03C1H/dT in the transition region, indicative of

well defin- ed critical point ⁱⁿ

concentrated random magnetic alloy. The critical analysis shows that (1/03C1c). (d03C1/dT)±=

(A±/03B1). (1 - |03B5|-03B1) ⁺ B±, with 03B5 = (T - Tc)/Tc, ^and (+) (-)

^T > Tc, ^T Tc, respectively. ^We obtain, Tc

^{244.79 K,}

^{03B1+ = - 0.001,} A+ /A_ ~ ^0.88,

^the range 1.2

10-4 03B5 0.7

10-2 and 6.2

10-3 |03B5| 2.6 10-2.

For d03C1H/dT the situation is complex, ^{due to} the effect of the applied magnetic field. Our analysis ^{shows the}

existence of important fluctuation effects

Tc,

given by ^the derivative dRs/dT.

The extraordinary ^Hall resistivity ⁱⁿ Tb75-Gd25 ^is extensively analysed ^{here, in order to} separate ^{out the diffe-}

rent contributions, i.e. the skew and side-jump ^terms, including the effects of several possible scattering ^Hamil-

tonians (Kondo, Smit). ^{We show that} 03C1sH ⁱⁿ Tb75-Gd25

be accounted for in terms of

Kondo (skew ^{and side-} jump) ^and

side-jump Smit contribution.

Finally, ^{the Curie} temperature ^of Tb75-Gd25 is related to the transition points ^{of Tb and} Gd, using ^the theory

of binary

earth random magnetic alloys.

Classification

Physics ^Abstracts

75.12E - 72.15G

1. Introduction.

The study of the critical behaviour of disordered _sys- tems has received considerable attention in the case of

a completely amorphous system [1] ] ^{and in the case}

of a crystalline system ^{with a} random distribution of

magnetic ^moments [2-5]. ^The alloy investigated ^here

falls in the second category, ^{with Tb and Gd} forming

solid solutions at all compositions (hcp structure).

Transport properties ^are particularly ^{sensitive to}

the _presence of atomic or magnetic order, ^{and so we}

studied the behaviour of the Hall resistivity (pH),

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004504077900

the temperature derivative (dpH/dT), the electrical

resistivity (p) ^{and the} temperature derivative (dp/dT)

over a wide _range of temperatures (77-360 K), ^{for a} highly concentrated Tb-Gd alloy (Tb7,-Gd25; ^fer- romagnetic). Particular detail was obtained in the immediate vicinity of the Curie point, Tc

^{244.8 K.}

Our results show _very sharp anomalies in the trans-

port coefficient derivatives near Tc, which indicates

a well defined critical point for this random alloy.

The Hall effect is particularly ^{useful to} investigate

the asymmetric mechanisms associated with the

scattering of electrons by magnetic ions. Such mecha- nisms lead to an extra contribution to pH [6], ^{the so}

called extraordinary ^Hall resistivity (pH) :

P’ ^being the normal resistivity ^{due to the Lorentz}

force on the electrons.

In rare earths, several mechanisms have been

proposed ^to explain ^the extraordinary ^{Hall effect.}

First, ^as it has been proposed by ^Kondo [7], ^{it can arise}

from asymmetric scattering by ^{orbital terms of the} exchange interactions between conduction and 4f electrons. These terms can be written as :

where Ai ^are appropriate interaction fields, gJ ^is the Lande factor for the ion i with total angular ^momen-

tum J ; ; ^{s and I are the} spin ^and angular ^momentum

of the conduction electron, the latter being ^referred

to an origin ^{centred in the ion. The} asymmetry ^of these terms (i.e. ,klH,,,Ik> klH I k >) gives

rise to angular deflections of the conduction electrons when the magnetic ^moments order below

Tc « Jz ) =F 0), ^{and so to the so} called skew scatter-

ing contributions to the Hall resistivity [7, 10, 11].

There is also a purely quantum mechanical effect : the electron wave packet effectively undergoes ^{a small}

transverse displacement ^{6r with} respect ^to its initial

direction, ^{the so called} side jump ^effect (S. J.) [12,13,15].

The second term of (2) (i.e. H2) gives ^the main contri- bution to this side jump ^effect [14].

The second possible origin ^{of the} extraordinary

Hall effect is the spin-orbit coupling of the conduction electrons (mainly associated with the d character of the conduction band). ^Smit [8] has shown that this

spin-orbit coupling gives ^{rise to « effective} asymmetric scattering ^{terms » for} any scattering potential ^and

thus to asymmetric scattering. ^{In rare} earths; ^when

the main scattering potential ^{is the} exchange ^inter-

action with the rare earth ions ( N ( g J - 1) s.Ji),

the skew scattering ^{and side} jump contributions

arising from this mechanism have been calculated

by ^Asomoza [36] : ^{the side} jump contribution has been shown to be much larger.

Besides the sign ^and magnitude of pH, ^we gave

particular ^{attention to the} investigation of the critical behaviour of the Hall resistivity. ^{This was achieved}

with the measurement of the temperature ^derivative

dPH/dT ^{in the} vicinity ^of Tc, ^which requires very high

resolution in the PH(T) measurements.

To the best of our knowledge, ^the measurement of

dPH/dT brings ^a novel contribution to the available

experimental ^{studies on} the Hall effect (see ^also [17]).

To complement ^the analysis of pH above Tc, fairly

detailed data ^on the magnetic susceptibility (x)

was obtained for Tb75-Gd25.

In section 2 of this _paper we describe briefly ^the experimental techniques and in section 3 we present the results for the electrical resistivity (p, dp/dT),

Hall effect (PH, dpH/dT) ^and magnetic susceptibility (x).

A thorough analysis of the data is presented ^{in sec-}

tion 4, including ^the separation of the different _p- contributions (4.1), the critical behaviour of dp/d T

near Tc ^(4.2), the discussion of the Hall effect in the para (4.3) ^and ferromagnetic phases (4.4), ^{and the}

critical behaviour of pH ^near Tc (4. 5) ; finally, ^the

variation of the Curie temperature ⁱⁿ Tbx-Gd1-x alloys is discussed in section 4.6, ^{in terms of the} theory ^{of random rare earth} alloys [2, 3].

2. Experimental techniques.

a) ^The sample ^{used in the} resistivity measurements was a parallelipiped ^with approximate ^dimensions

10 ^x 1 ^x 1 mm’, ^{and the} resistivity ^{was measured}

with a four-wire potentiometric ^method previously

described [ 18].

b) ^The high accuracy data ^on the Hall resistivity

were obtained with a lock-in technique, using ^ac

currents of about 0.5 A through ^the sample, ^and applied magnetic ^fields up ^to 7.96 ^x 105 A . m-1 (10 KOe). ^The sample ^{was a slab cut from the same} ingot ^as that used for the resistivity measurements, with dimensions 0.22 ^x 2.65 x 9.1 mm3. The _magne- tic field was set perpendicular ^to the slab and its inversion was made at each experimental point

to extract the Hall voltage :

The Hall resistivity ^{was then} calculated by ^{the usual}

formula PH

V Hall. (dj 1), ^{where d is the} sample

thickness and I is the electrical current. The tempera-

ture was measured with a copper constantan ther-

mocouple using ^a potentiometer ^{with a} sensitivity

of 0.01 pV. ^{Further details can be} found elsewhere [17].

The analysis of the Hall data was done using ^the

usual separation of pH into ^a normal and an extra-

ordinary contribution (see Eq. (1)). ^{In the} ferromagne-

tic phase, pH is expressed ^{as :}

where psH (7) ^{is here the} extraordinary ^Hall resistivity

of a monodomain sample, M,(T) ^the spontaneous magnetization ^{of a} monodomain at the temperature T, and M is the technical magnetization. ^{With our sam-} ple geometry ^the demagnetization ^{factor D is} nearly 1,

so that B ~ ₁₁₀ Ha ^and (1) is ^{written as :}

In the paramagnetic phase ^{and with our} sample geometry, (1) ^{can be} expressed ^{as :}

in the limit of small magnetization, ^when pH ^{can be}

written as :

When M is expressed ^{as a} function of the suscepti- bility X

Equation (6) is written as :

This relation enables a direct comparison ^between

the behaviour of PH(T) and that of x(T). ^{For such}

purpose, the susceptibility ^was measured in a Tb75- Gd25 sample (9.535 mg weight) obtained from the

same initial ingot used for the Hall effect. The _x- measurements were carried out at the Magnetism Group ^{of CFU} Porto, using ^{a standard} Faraday-

Curie method, ^{with a} highly sensitive Cahn balance to measure the small magnetic ^{forces on the} sample.

3. Experimental ^results.

3.1 ELECTRICAL RESISTIVITY (p, dp/dT).

Figure ¹

shows the temperature dependence of the electrical

resistivity ^{of our} polycrystalline Tb75-Gd25 sample,

with a typical ^{kink at the onset of the} paramagnetic

state (T,

^{244.8 ± 0.1 K, see} below). The residual

resistivity ^{of our} sample ^was po

6.2 pocn% ^which corresponds ^{to a} relatively high resistivity ^{ratio for}

this alloy, p (300 K)/p (4.2 K) ~ ^21. Besides po,

one can also estimate the phonon contribution

(a. T) ^{and the} spin-disorder electrical resistivity ^at

temperatures sufficiently above the Curie point (p’s ;

see section 4.1 for details). ^This separation ^{is shown}

in figure 1, ^{our data} giving ps

^87.2 pocm ^and

a. T

36.2 pqcm ^{at T}

^{350 K.}

In order to investigate ^more closely the details in the p(T) ^{curve we} measured the temperature ^deriva- tive dp/dT ^{over the same} range (77-360 ^K; Fig. 2).

Some interesting ^{features can be} pointed ^{out in the} dp/dT ^{curve of} figure ^{2 :} (i) ^{an almost constant value}

Fig. ^1.

Temperature dependence of the electrical resis-

tivity (p) ^of polycrystalline Tb7 s-Gd2s. Separation ^{of the} different p-contributions : ^aT (phonons), po (residual),

and p’ ⁼ p(T >> Tc) - aT - po.

Fig. ^2.

Temperature dependence ^{of the} temperature

derivative of the electrical resistivity (dp/dT) ⁱⁿ polycrystal-

line Tb75-Gd25.

of dp/dT ^{in the} ferromagnetic phase ^{over a wide}

range of temperatures (T 180 K, p practically

linear in T) ; (ii) ^a complex ^{structure in the curve}

near the Curie point, ^{below and} above; (iii) ^a sharp singularity ⁱⁿ dp/dT ^at Tc; (iv) ^{a constant value of} dpjdT ^at temperatures ^{in the} high temperature limit,

which can be ascribed to the phonon contribution to the electric resistivity ^{in the} paramagnetic phase (a. T). ^{Our data} gives

This value is of the same order of magnitude ^{as the} corresponding figures ^for single crystalline ^Tb (aa

0.13 JlQ.cm.K-1, Ctc = 0.085 JlQ.cm.K-1) ^and

single crystalline ^Gd (O(a

^0.095 gfl.cm.K-’, occ =

0.08 J.lQ. cm. K -1) [20].

If we calculate « for polycrystals of Tb and Gd using

the equation :

and then perform ^an weighted ^average of such values to take into account the atomic concentration of Tb and Gd in the alloy, ^we get ^a

^0.108 JlQ. cm. K -1

for Tb7s-Gd2s, ^{which is} fairly ^{close to the} experimen-

tal figure referred above.

3.2 HALL RESISTIVITY (PH, dpH/dT).

Figure ^{3 shows}

the temperature dependence ^{of the Hall} resistivity

from 77 K up ^to 360 K, ^{for two} values of the applied magnetic field, Ha = 3.88 ^{x 1 OS and 7.72 x 1 OS} A . m-1;

similar measurements were performed ^with Ha

0.80 x 105 and 1.59 x 105 A. m - 1.

The Hall resistivity ^{vanishes at a} temperature T*

175 K in call _cases, becoming positive ^below

this temperature, ^{with a broad maximum around}

135 K. The observation that PH(T*)

⁰ irrespec-

tive of the Ha ^{value can be} explained ^{as follows :}

under our experimental conditions Ha ^{is too small}

to produce magnetic saturation at T* (M ~ Ha/D,

where D is the demagnetization factor) ^{and then}

we have from (5) :

The temperature ^T* just corresponds ^{to the} parti-

cular value at which go R0(T*) = - pH(T*)/MS(T*).

Equation (9) also shows that _pH oc Ha ^at tempera-

tures below Tc ^{when field} penetration ^{does not occur.}

This is in agreement ^{with our} data for T % ^{225 K.}

At temperatures ^{close to} the Curie point, ^where significant ^field penetration ^occurs, pH is ^no longer

linear in Ha, ^{as shown in} figure ^4.

Figure ⁵ shows the derivative dPH/dT ^{as a func-}

tion of T, ^with dPH/dT obtained from local numerical differentiation of the PH(T) ^{data (3 points). For}

convenience the data have been normalized by ^the

factor PH(T,,.

The existence of critical features in the Hall effect is clearly ^evidenced by ^the very sharp dip ⁱⁿ dpH/d T

at T

244.5 + 0.5 ^K. This is consistent with the Curie temperature previously ^{obtained on} Tb7,- Gd25 single crystals, using magnetization ^data [21]

(Tc = 245 ^{± 2 K)} and electrical resistivity ^measure-

ments [19] (see ^section 4.2).

We also observe that the minimum in (1/pH(Tc)) x (dpHjdT) ^at Tc ^{gets less} sharp ^as Ha increases due to the corresponding ^{reduction in the} spin fluctuations in the system.

An interesting feature is that dpH/dT ^{vanishes at}

T**

135 K, irrespective of the value of Ha. ^Since

no field penetration ^{occurs at this} temperature, ^we

Fig. ^3.

^Hall resistivity of Tb,,-Gd2,

^{function of} temperature, ^{for two} different values of the applied magnetic ^field Ha.

Fig. ^4.

^{Plot of} PHI Ra

^T, for different values of H..

The coincidence of the experimental

outside the Curie

point region indicates that PH

Ha ^{in this} temperature range. However, considerable departures ^{from this} depen-

dence

near T c.

obtain from (9) :

Therefore the temperature ^T** corresponds ^{to the}

condition :

3. 3 MAGNETIC SUSCEPTIBILITY (X).

Figure ^{6 shows}

the magnetic susceptibility ^of Tb75-Gd25, ^{defined as}

where Hi is the internal field, H; = Ha - ^{D. M. Since}

M and H; ^are expressed ^{in A. m-1 this} susceptibility

is a dimensionless quantity. ^{The Curie} temperature obtained from the zero-field resistivity measurements

(see 3.1) ^{is marked} by ^{a vertical arrow in} figure 6,

at the region ^where x(T) exhibits its steepest ^variation.

At temperatures sufficiently ^above T c the suscep-

tibility obeys the Curie-Weiss law :

with C

7.17 K and 0

251 K in our polycrystal-

line sample (see Fig. 7). ^{In a} previous study, Bagguley

et al. [21] obtained for a Tb75-Gd25 single crystal

and basal plane susceptibility measurements, ^C

7.06 K and ob

^{257 K. The} slight discrepancy ^bet-

ween the two 0 values is attributable to the effect of the magnetic anisotropy, ^{which makes} Ðc ob ⁱⁿ Tb75-Gd25 (easy axis in the basal plane), ^{and thus} Ð polycrystal ^Ob*

4. Discussion.

4.1 ELECTRICAL RESISTIVITY (p).

^{In order to} sepa- rate the different contributions to the electrical resis-

tivity, ^{we start} with the known expressions ^for p(T)

in binary ^{rare earth} alloys [22] :

Pelect is the resistivity ^{due to the} difference in elec- trostatic potential between the magnetic components,

Fig. ^5.

Temperature dependence ^{of the} normalized Hall resistivity temperature derivative, (’/PH(Tc)).(dPH/dT) ^{for two}

different values of the applied magnetic ^field

Fig. ^6.

Temperature dependence ^{of the} magnetic

ceptibility (x)

^{and above the Curie} point ^of polycrystal- line Tb7 s-Gd2s.

Fig. ^7.

^Inverse susceptibility

^{T in} Tb 75-Gd25*

For T sufficiently ^above Tc, x obeys

Curie-Weiss law.

assumed to be temperature independent, pms is the saturated spin ^disorder resistivity ^{of the} alloy (in

terms of figure ^{1 we} have, _{pms = Po +Ps - Pelect}

PiO Pi is the extra resistivity ^{due to other} (unavoidable) impurities present ^{in the} alloy, ^and pm(O) is the residual

magnetic resistivity ^{due to the} difference in the spins

of the magnetic components.

The sum Pelect ⁺ Pm(o) ^can be estimated from available data on the resistivity of dilute Gd-Tb

alloys [23] ^{at helium} temperatures, which show that :

where A

9.7 pn. cm, B

^21.3 gf2. cm, ^{and AS}

is the difference between the spins of Tb and Gd.

This gives pelect = 1.8 gf2.cm, pm(O) ^{= 1.0} gf2.cm.

Using ^our experimental ^{value of} p(0) ^for Tb75-Gd25

and equation (14) ^{we obtain} p;

3.4 gf2.cm. Equa-

tion (15) ^now gives (with ^a

^0.105 JlQ. cm. K - 1 ;

T

350 K) Pms

88.2 gf2. ^cm.

Table I summarizes the electrical resistivity ^data

for Tb75-Gd25 (in gf2. cm; ^{a in} JlQ.cm.K-l) :

4.2 CRITICAL BEHAVIOUR OF dp jdT ^NEAR T,,.

^The dp jdT ^{data at} temperatures immediately ^above Tc

has been analysed ^{in terms} of the usual expression [25] :

with 8

(T - Tc)jTc, pr

p(Tc), A + ^and B, ^cons- tants ; q is the critical exponent characterizing ^the singularity ⁱⁿ dpjdT [26]. ^A computer ^fit using ^and adjustable Tr ^{value and} assuming ^{a 0} gave the best fit for Tc

^{244.79 K,} leading ^{to a} critical expo-- nent a = - 0.001 ± 0.002, ^{with a mean relative}

square deviation a

7.753 x 10-2. If one starts with a > 0, the best fit corresponds ^{to a}

⁺ 0.0007 + 0.002, giving ^a slightly higher ^{mean relative} square deviation -a

7.755 x 10- 2 ; ^{thus we take a}

-

0.001 ± 0.002 ^to represent ^our data above Tc, leading ^{to an} amplitude A,

^{6.918 x 10-4 K-1.}

For such small value of a the above expression ^{can be} approximated by,

As shown on figure ^{8 our data is in} good agreement with the logarithmic dependence ^{over a} considerable range of reduced temperatures, ^{1.2 x} 10-4 E 0.7 ^x 10-2, ^{with a} total of 47 experimental points

above T,,. On the inset of figure ^{8 we} show that the

mean deviation of the computer ^fit effectively goes

through ^a minimum for the particular ^choice Tc =

244.79 K.

So far we have analysed ^{the T} > Tc ^{data. In the} ferromagnetic phase (Fig. 2) ^a complex regime appears at temperatures ^{not too} far below Tc, ^{with a} pro- nounced minimum in dp/dT, ^followed by ^a steep rise of this derivative as T decreases further. Cer-

tainly ^{this trend is not a critical} feature, ^{but a mean}

field effect associated with the growth ^{of the} sponta-

neous magnetization ^{in the} sample. Also, close ins-

Fig. ^8.

The critical behaviour of (1/pc) (dpIdT)

^the

Curie point ^of Tb75-Gd25 obeys

logarithmic dependence

of the form A. Ln I 8 I + B, ^with

(T - Tc)jTc, A, B constants, ^and Tc = 244.79 K. Inset : the relative

deviation of the experimental ^{data goes through}

^minimum

for Tc

^{244.79 K.}

pection ^{of our data near} T, ^{reveals a} considerable

rounding ^{of the curve} just ^below Tc. ^The significant

enhancement of the rounding-off effects below Tc

is a well known fact for the experimentalists ^{in the}

field [27, 28]. These effects limit in practice ^{the tem-} perature range where one can confidently ^consider

our data for a meaningful analysis procedure. ^There-

fore we excluded from our T Tc analysis ^{the data}

below the minimum in dp jdT, ^{and the data in the} region ^where rounding ^{effects were} readily apparent, i.e. when the curvature d2 p/dT 2 changes sign. ^This

leaves a useful _range between 238.50 K and 243.30 K,

or between 2.6 ^x 10-2 and 6.2 ^x 10-3 in reduced temperatures.

Using ^{the same} Tr ^{and a} values obtained in the

paramagnetic phase (Tc

^{244.79 K; a} = - 0.001)

and using ^{the same} functional expression (Eq. (20)),

a computer ^fit gave ^a fairly good adjustment ^{of the} experimental data below T,, ^{with ?}

^{4.8204 x 10-3}

and A-

7.7899 ^x 10-4. We also found that by varying ^{a the value of -a} goes through ^a broad mini-

mum in the neighbourhood ^{of a}

^0.001.

In short, ^our analysis ^{leads to the} following ^{results :} (i) ^{a is} fairly ^{small for} Tb7 5 -Gd2 5; (ii) ^the scaling

relation a -

a + is consistent with the data, ^within

the experimental error; (iii) ^the amplitude ^{ratio has}

the value A+/A_

^0.88.

Let us analyse these results in the light of available theoretical renormalization _group (RG) calculations

[29-32]. One has first to find the appropriate ^number

of components (n) of the order parameter ^{in our} sys- tem (d

3). Since the results of Bagguley et ^al. [21]

have shown that it is a basal plane localized ferro- magnet ^{one takes n}

² (X Y model). RG calcula- tions [29, 32] ^{based on the} E-expansion ^{then lead}

to (E=4-d) :

which is in fact a fairly ^{small value for a.}

With respect ^{to the} amplitude ^ratio A+/A -, ^the

theoretical calculations are not always consistent.

For example, it is known that 1 st-order s-expansions apparently ^{lead to} better results than E2-expansions

[29]. ^Our experimental figure of 0.88 is in fact closer to the e-expansion prediction (0.99) ^{than to the s’}

expansion figure (1.08).

4. 3 HALL EFFECT IN THE PARAMAGNETIC PHASE. -

In the paramagnetic phase expression (6) ^{leads to the} following dependence of the Hall effect :

The staggered susceptibility x* includes the dema-

gnetization effects, i.e., x*

XI(I ⁺ D. x), ^{and in}

the last expression ^we used the fact that R. >> Ro

and D - 1 in our case.

At temperatures sufficiently ^above Tc, ^{when the} spin-fluctuation effects become negligible ^and Ro, Rs get constant, ^one expects ^{a linear} dependence

between PH(T) ^and x*(T). As shown in figure ^{9 this}

is the case for temperatures above 299 K

(s > ^{2.3 x} 10-1). ^{From the} corresponding slope ^and intercept ^we obtain the saturation values of Ro ^and Rs :

R. ^{has an} intermediate value between those of Tb and Gd [33, 34], ^{which seems} plausible ^{when we recall}

that Rg closely follows . the trend of the magnetic resistivity (see 4.4), and for this quantity ^{one has} pm(Tb) Pm(Tb7S-Gd2s) Pm(Gd).

Near T, ^one expects pronounced deviations from

the expression (20), ^since one can no longer ^{use the}

Fig. ^9.

^Plot of pH

x* shows that PH

X* ^for temperatures T Jit 299 K. This enables the determination of Ro and R. ^{in the} paramagnetic phase.

low-field x*(T) values, ^neither can RS ^{be taken as}

constant One _can, however, obtain information

on the behaviour of RS( T) by using calculated values of M(T) ^as shown in section 4.4.

4.4 SEPARATION OF THE DIFFERENT CONTRIBUTIONS TO

PH ^{IN THE} FERROMAGNETIC PHASE.

In order to compare ^our pH data obtained in low fields with available theories and with results obtained at magne- tic saturation, ^{it is} necessary ^to renormalize our

results according ^{to :}

Ms(T) ^{is the} spontaneous magnetization ^and M(T) ^is

the technical magnetization ^{at each T in our} experi-

ment. Using ^a Brillouin function to calculate M,(T)

and M(T) (including, ^{in this} case, the applied magne- tic field and demagnetization effects), ^we obtained the renormalized data for Tb75-Gd25 ^{shown in} figure ^10.

These values will be used in the remaining analysis

in this section.

As shown in section 1, several mechanisms can

contribute to the Hall effect, mainly À.t and À.2 ^aris-

ing from the orbital exchange ^terms Hl ^and H2

(with ^skew scattering ^{and side} jump contributions for both) ^and ps’ ^{arising from the} spin-orbit coupling

of the conduction electrons (neglecting ^{the much}

smaller p3ss) :

According ^to the calculation of Asomoza et al. [36],

where pm(T ) is ^the spin ^disorder resistivity, ( S,,. > T

is the spin polarization ^{at T and b is a constant}

related to the spin-orbit coupling within the conduc- tion band and independent ^{of the 4f moment.}

Fig. ^10.

^{Plot of} pH

^{T for} Tb7s-Gd2s.

In gadolinium, for which the contributions p, and _P2 do not exist (because ^L

0), ^{the whole} extraordinary Hall effect can be accounted by (23)

and the coefficient b can be easily ^extracted [36].

As Gd and Tb-Gd have nearly ^{the same band struc-}

ture it is reasonable to assume the same value of b

(i.e. b

^{0.613 x 10-3} gi2-l.cm-1) ^{for our Tb-Gd}

alloy. By using this value of b, experimental ^values

of Pm(T) ^{and a mean} field calculation of Sz )T

for Tb75-Gd25’ ^we have calculated psJ(T). ^{We have}

then subtracted p3sJ(T) ^{from the} experimental ^values of p’, ^{to obtain a curve} (Fig. 11) which should reflect

only the contribution of the Kondo terms to pH in

Tb75-Gd25’ ^i.e. P1 ⁺ P2

PH(Kondo).

On the other hand one can calculate pH(Kondo, Tb)

from available data of pH(Tb), ^{assuming the} validity

of the following expression :

For Pm(T) and S., .(Tb) >T ^{we have used} experimental

data published in the literature [37, 38].

Because no Kondo terms arise from Gd ions in

Tb7s-Gd2s, PH(Kondo) for ^this alloy ^{can now be} simply calculated :

As shown in figure ^{11 the curve} obtained in this _way

(dotted line) ^is reasonably ^{close to the} experimental points ^obtained by subtracting p3sJ (23) ^{from the} extraordinary Hall resistivity ^of Tb7s-Gd2s.

Having in mind the various separations ^involved

in the calculations, ^{the small} discrepancies ^observed

in figure ¹¹ bring considerable support ^{to the inter-}

nal consistency ^{of the} approach ^followed here; ⁱⁿ

particular ^{the use of} (24) ^and (25) ^as approximate

Fig. ^11.

Comparison ^between the Kondo Hall resistivity separated ^from

^{data in} Tb7s-Gd2s (0) (Eqs. (22)-(23)) ^and

the calculated curve (broken line; Eqs. (24)-(25)).

formulae to estimate the Kondo term in Tb and Tb7 5 Gd25-

In conclusion, ^our analysis ^{of the} PH(T) ^{data in} Tb7s-Gd2s is consistent with the existence of a

Smit-Fert type contribution [36] ^to PH analogous ^to

that existent in _pure Gd, plus ^{a Kondo like contribu-}

tion entirely attributable to the Tb ions and calculat- ed from the available Hall resistivity ^{data for} pure Tb [16].

4.5 BEHAVIOUR OF Rs(T) IN THE PARAMAGNETIC PHASE .oF Tb7s-Gd2s.

Using ^the high temperature ^{value of} Ro ^obtained in section 4.3 and assuming Ro ^constant

as a first approximation, ^{we can} separate ^{the extra-} ordinary contribution pH ⁱⁿ Tb75-Gd 25

Dividing pH by the technical magnetization ^{at each} temperature (see 4.4) ^we obtain the temperature dependence of Rs(T), ^{as shown in} figure ^12. Rs ^attains

a constant value Rs

^{121 x} 10-’o fl.m.T-1

at T >> Tc, ^{when the} spin-disorder scattering ^satu-

rates. This Rg value is consistent with the value deriv- ed previously from the relation between pH and x*, Rs

^{119.4 x} 10-10 n.m.T-1 (see ^section 4.3).

The broad maximum observed in the magnitude

of Rg slightly ^above Tc ^{can be} qualitatively ^under-

stood if one recalls that Rg ^is directly ^{related to the} magnetic electrical resistivity Pm (in ^{the skew scatter-} ing contribution, Rg ^oc pm ; in the side jumps ^oc pm)

and pm effectively goes through ^{a maximum} just ^above T, (see Fig. 1). ^{This is caused} by the persistence ^of short-range spin-spin fluctuations just ^above Tc.

Fig. ^12.

Temperature dependence ^of Rs in the parama-

gnetic phase ^of Tb7s-Gd2s.

One can obtain further insight ^{into the behaviour}

of Rs(T) ^near T, by taking log-temperature ^deriva-

tives of the two first terms in (2) (and rearranging terms),

Using ^{M and} dM/dT ^values calculated in the _para-

magnetic phase ^{with a} Brillouin function (J

6),

and using (Ilp’).(dp’IdT) values from experiment,

we derived values for. (ljRJ.(dRJd1) ^{in the para-}

magnetic phase, ^{as shown in the curve of} figure ^13.

Fig. 13. - Critical behaviour of (I/Rs).(dRs/dT)

^the

Curie point ^of Tb7s-Gd2s.

The critical features in R.(T) ^{are now} readily appa- rent as T approaches ^{the Curie} point However, ^one

cannot use directly ^{these results} to extract realistic critical indices associated with the behaviour of RS

near Tc, because the critical fluctuations are severely damped by ^the applied magnetic field used to obtain pH, and also because ^we have used ^{a mean} field _appro- ximation to describe M(T) ^{near the Curie} point

Further work is under _way to improve ^{the method}

of analysis ^of Rs( T) ^{data near the Curie} point-

4.6 CURIE TEMPERATURE OF Tb7s-Gd2s.

^{The cor-}

relation between the Curie point ^of Tb75-Gd2, (Tc

^244.8 K) ^{and those of Tb} (T,

^219.6 K)

and Gd (Tc

²⁹² K) ^{is related to the} problem ^of

random rare earth alloys. ^A simple ^molecular approach

has been worked out by Lindgard [2, 3], ^{in terms of}

the non-interacting magnetic susceptibilities ^{of the}

4f-moments of the two constituting ^elements (XO, xi).

The Curie temperature ^{of a} ferromagnetic alloy

is then given implicitly by [3] :

where Jii ^{is the q}

⁰ Fourier transformed exchange

interaction between the elements of type i ^and type j.

In rare earths, this interaction is indirect, just ^mediated by ^the spin-polarized conduction electrons. It can then be simply ^{related to} the average conduction electron

magnetic susceptibility (Xel) [35] :

where jt ^are the effective d f matrix elements and N(EF)

is the density ^of states at the Fermi energy for the

alloy, N(sF)

x. N1(EF) ⁺ (1

x). N2(sF). ^{If we ne-} glect crystal ^field effects, ^the noninteracting parama-

gnetic susceptibility ^is given by XP(T)

Ci/kT,

where Ci is constant, and then ^we obtain from (30),

If NI(BF)

N2(BF), this leads to a linear dependence

of T, ^{with the} alloy composition. Under such _assump- tion one would obtain T,

^{237.7 K for} Tb75-Gd251

a reasonable value when compared ^{with the} experi-

mental one.

5. Conclusions.

Our resistivity measurements enabled the separation

of the different contributions to p in Tb75-Gd2, (residual, phonon ^and spin-disorder resistivity).

The critical behavipur ^{near the Curie} point ^{has been}

studied with dp/d T measurements, giving ^critical

indices a-

cx+ ~ - 0.001 and an amplitude ratio.

A + /A _ N ^0.88. These results are satisfactorily ^inter- preted ^{in terms of the XY model} (d

3, n

2),

and of the available calculations from renormaliza- tion _group theory.

The Hall effect resistivity ⁱⁿ Tb7s-Gd2s ^{is inter-} preted ^{in terms of a} Smit-Fert contribution analogous

to that existent in Gd, plus ^a Kondo-like contribu- tion due to the Tb ions.

Our high resolution Hall effect data enabled the critical features to be experimentally displayed ^{in this} effect, through ^the dpH/dT derivative near the Curie

point.

In the paramagnetic phase, ^the extraordinary ^Hall

constant has been separated (Rs), using magnetic susceptibility measurements. Critical features were

also displayed ^{in the} derivative dRs/dT.

Acknowledgments.

The authors wish to express their gratitude ^{to Drs. G.}

Garton, ^{D. Hukin and D.} Bagguley ^{for the} supply

of several Tb-Gd samples, ^{and to Prof. J. M. Silva for}

the facilities given ^to perform ^the susceptibility

measurements. The financial support provided by

INIC (Portugal) ^{and NATO Res.} Grant 1481 is also

acknowledged.

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