Journal of Magnetism and Magnetic Materials 12.5 (1993) 103-109

North-Holland

Critical behaviour of the magnetoresistance of NdRu,Si, near the N6el point

M.A. Salgueiro a, B.G, Almeida a, M.M. Amado a, J.B. Sousa a, B. Chevalier b and J. Etourneau b ’ Centro de Fisica da Universidade do Port0 (INIC) and IFIMUP (MAT), Praga Comes Teixeira, 4000 Porto, Portugal ’ Laboratoire de Chimie du Solide du CNRS, 351 Cours de la Libkration, 33405 Talence, France

Received 10 August 1992; in revised form 24 November 1992

A detailed experimental study is presented on the temperature and applied magnetic field dependences of the

magnetoresistance (Ap/p) of the strongly anisotropic NdRu,Si, antiferromagnet, for temperatures near the NCel point

(5 x 1O-3 I I(T - TN)/ T, 1 5 lo-‘). The data are analyzed in terms of the available theoretical models describing the effect

of the critical spin fluctuations on Ap/p near TN. The field dependence T,(H) is also investigated.

1. Introduction

The Nd”+ ions in the intermetallic compound NdRutSi2 form a simple body-centered tetrago- nal lattice and their magnetic moments order initially below the Ntel temperature T,,, = 24K in a sinusoidally modulated phase with a wavevector k = 2~r/a (0.13, 0.13, 0) and the spins oriented along the c-axis [ll.

An extended account of the transport proper- ties of NdRu,Si, has been given recently 121, covering the cases of the electrical resistivity (p), thermoelectric power (S) and their temperature derivatives (dp/dT, dS/dT). These properties provide a clear picture of the temperature depen- dence of the magnetic order below TN, including the successive magnetic phases which occur in zero applied magnetic field with decreasing tem- perature: simple sinusoidal for 15.5 K < T < T,, square-modulated for T < 15.5 K and ferro (or ferri) for T < 8 K which coexist until 2.8 K with the square-modulated phase [1,21.

Correspondence to: Dr J.B. Sousa, Centro de Fisica da Univer-

sidade do Porto (INIC) and IFIMUP (IMAT), Praqa Gomes

Teixeira, 4000 Porto, Portugal. Tel: 31 02 90; fax: 31 92 67.

The effect of the magnetic field has also been studied with transport properties, leading to a detailed magnetic phase diagram of NdRu,Si, [3], which is in general agreement with the phase diagram reported in ref. [4] based on neutron diffraction and magnetization measurements per- formed on a single crystal.

We study here the behaviour of the spin fluc- tuations near the NCel temperature, under the presence of a magnetic field, using high-resolu- tion magnetoresistance (Ap/p) measurements. The magnetoresistance data are analyzed in terms of the available theoretical models describing the effect of the critical spin fluctuations in Ap/p near TN. The role played by the high anisotropy in NdRu,Si, is also considered.

2. Sample preparation and experimental tech- niques

An initial button of NdRu,Si, was prepared by arc melting of the constituent elements in a purified argon atmosphere and then annealed in a quartz tube under vacuum at 800°C for 15 days. The sample was identified by X-ray powder

0304.8853/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

104 MA. Salgueiro et al. / Magnetoresistance of NdRu,Si, near the N&e1 point

diffraction using a Guinier camera (Cu KAY radia- tion) and its composition and homogeneity was

checked by microprobe analysis. The study of this ternary silicide by neutron diffraction reveals that

10% of the Ru atoms occupy the crystallographic sites of the Si atoms and conversely [l]. A sim- ilar mixing has been observed in a TbRu,Si,

sample prepared under the same conditions

as NdRu,Si,, but it has not been observed in NdRh,Si, compounds. We notice that this mix- ing persists even after long annealing periods, e.g. 30 days (see section 4.3). Samples with adequate geometry t- 7 mm x 1 mm2) were subsequently

obtained using a low-power spark-cutting ma- chine.

The magnetoresistance was measured with a standard dc four-probe method in a 10 K closed-

cycle refrigerator, integrated in a fully automated system. The sample temperature was measured with a Au-Fe,,,,,,/ chrome1 thermocouple placed in good thermal contact with it. Both the voltage across the sample and the thermocouple emf were measured with high resolution (1 nV> using IEEE computer-controlled dc voltmeters [5].

The temperature was regulated, in a first stage,

by the digital PID temperature controller of the refrigerator, to within f 0.1 K, and in a second stage, by an analogic fine-control regulator with

the sensor and heater placed in the immediate vicinity of the sample. This latter apparatus uses an ac bridge and a lock-in amplifier detector. The temperature drift was typically of the order of 1 mK in periods of - 500 s. The computer program could automatically account for any long-term temperature drift, making the appropriate mag- netoresistance corrections using (previously avail- able) information on the resistivity thermal coeffi- cient (dp/dT) as a function of temperature.

Magnetic fields up to 16 kOe were generated by an electromagnet with a digital power supply (O-60 A) IEEE computer controlled. Very smooth field sweeps were possible (lowest field steps of 1 Oe>, and the sweep was stopped over a (pre- selected) short period of time before taking each magnetoresistance point, thus avoiding field-in- duced parasitic signals during each resistivity measurement.

3. Experimental results

Figure 1 shows, for the paramagnetic phase of NdRu?Si,, the magnetic field dependence of the

magnetoresistance (Ap/pl in the reduced tem- perature range from E = 6.5 x 1O-3 to E = 5.5 x

lo-‘, where E =(T- r,)/T, and T, = 24.3 K (see analysis in section 4.31. The magnetoresis- tance is always negative above T, and increases monotonically with the magnetic field, owing to the progressive reduction of the thermal spin

fluctuations. As expected, Ap/p increases as T

approaches T,.

At each temperature the magnetoresistance varies quadratically with the magnetic field.

Ap/p = -A( T)N”, (1)

as shown in fig. 2 for temperatures above T,. The coefficient A(T) increases steeply as T ap- proaches the NCel point. Below T,, the mag- netoresistance exhibits a more complex behaviour (fig. 31.

The isofield-magnetoresistance curves as a function of temperature, (Ap/p),, versus T, arc displayed in fig. 4. The results reveal a minimum in the magnetoresistance at the Ntel tempera- ture, T,(H), from the paramagnetic to the or-

Fig. 1. Magnetic field dependence of the magnetoresistance (Ap/p) of NdRu,Si, in the paramagnetic phase.

M.A. Salgueiro et al. / Magnetoresistance of NdRuJi, near the Nt!el point 105

0.02

-0.06

I , I L-a

50 100 150 200 250 Hz (kOe’)

-0.08 L 0

Fig. 2. Magnetic field dependence of the magnetoresistance of

NdRu,Si, plotted in the representation (Ap/p) versus Hz for T > r,. The full line represents the best fit with the

expression (Ap/p)= A(T)H’.

dered phase. TN decreases slightly with increas- ing H, according to a quadratic dependence [6],

TN(H) = TN(O) - aH*, (2)

where a = 9 X lo-” K 0eP2 and T,(O) = 24.3 K (see fig. 5).

4. Data analysis and discussion

4.1. Field dependence of TN

The field dependence of the NCel temperature is consistent, in order of magnitude, with the available mean-field calculations for antiferro- magnets [6,7]:

TN(h) - TN(O) TN(O)

= -f.h2, (3)

where h is the reduced field, h = mB/kT,(O), B = poH and m is the ionic magnetic moment of Nd3+; pu, is the magnetic permittivity. For isotropic antiferromagnets, the dimensionless constant f is given by the expression [61:

0.02

-0.02

-0.04

-0.06

_,,,,i, , , / , , , / , , , ,“‘~~;yyl

0 4 8 12 16 H (kOe)

Fig. 3. Magnetic field dependence of the magnetoresistance CAP/P) of NdRu,Sia in the magnetic phase.

(4)

where 13 is the paramagnetic Curie temperature. Introducing the experimental values for Nd- Ru,Si,, J = 9/2, T,(O) = 24.3 K and 8 = 29 K 181, we get f = 7.7 X 10e3, which is much smaller than the experimental value derived from the data in fig. 5, f = 0.036. This discrepancy may be related to the large magnetic anisotropy observed on NdRu,Si, [9]. A more stringent test demands

kOe

kOe

kOe

> 0 H=7kOe

. H=PkOe

-0.1 ,,,“,,‘,~~‘,,~l~,,“~, 20 22 24 T:PK) 28 30 32

Fig. 4. Temperature dependence of the isofield magnetoresis-

tance of NdRu,Si,.

106 M.A. Salgueiro et al. / Magnetoresistance of NdRu,Si, near the N&e1 point

I.. % I L 1

. . 11, I 50 100 150 200

HZ( kOe2)

Fig. 5. Quadratic magnetic field dependence of Nkel tempera-

ture. The full line represents the best fit with the expression

[T,(H)- 7’,(O)]= - aH’.

theoretical models that explicitly introduce strong anisotropic effects.

4.2. Critical behauiour of the magnetoresistance

Below T, we stress the initial positive sign of

Ap/p, changing into negative values at higher

fields (see fig. 3). The positive region widens as the temperature slightly decreases from T,.

The behaviour of the magnetoresistance near

the Ntel point of antiferromagnetic metals has

been theoretically studied with several levels of approximation. Yamada and Takada first devel-

oped a mean-field approach to the magnetoresis- tance, neglecting the role of the critical spin fluctuations [lo]. The same authors later devel-

oped a better model using the RPA approxima- tion for the spin system [ll] and taking into

account both the longitudinal and transverse cor- relations between spins at different sites. A more general treatment of the magnetoresistance, which includes renormalization group results to calculate the spin correlation functions [12] is due to Balberg and Helman [7]. The theoretical re- sults derived for the immediate vicinity of T, are summarized in table 1 and fig. 6 (for the case of ferromagnets see ref. [13]), based on the quasi- elastic approximation and the use of static corre- lation functions.

Table I

Reduced Regime Al’/{’ temperature

I c I < t(; ” (i)Itl i<h _ h 2, I <I )

(ii) I t / B II -h?lt&”

ltl >C(, (iii) / t / >-> 11 -/rlcl-’ z

J cc; is the Ginsburg reduced temperature, which separates

the truly critical regime (I E I <cc;) from the classical

(mean-field) regime of the fluctuations (1 E 1 > t(; ).

In the anisotropic compound NdRu,Si,, the

magnetic field leads to a complex magnetic phase

diagram below T,, with three distinct phases

within a narrow temperature range [3,4,9].

T

b)

Fig. 6. (a) Renormalization group results on the behaviour of

the magnetoresistance A&j/p in the immediate vicinity of T,

[6]. (b) Experimental behaviour of Ap/p for NdRu?Si, neat

r, and under small fields: (i) Ap/p a H’: (ii) for t > t(,.

AP/P a E -I”; (iii) for E < eCi, Ap/p a em’“: and (iv) for

E c cc;, Ap/p - constant.

M.A. Salgueiro et al. / Magnetorestitance of NdRu,Si, near the N6el point 107

I ’ I , I 1

FERRO

lo-

FERRO-LIKE

4- SIN MOD PHASE

2-

21 22 23 24

T, (K) Fig. 7. Magnetic phase diagram of NdRu,Si, near T,.

Our previous measurements of the resistivity derivatives versus temperature and magnetic field [31, (dp/dT),, (dp/dH),, show in low fields a succession of paramagnetic, ‘ferro-like’ (see be- low) and sinusoidally modulated phases, as illus- trated in fig. 7. For example, with H = 4 kOe, the sample goes from the paramagnetic to the ‘ferro- like’ phase at 24.27 K (N 0.3 K below T,(O)) and then into the sinusoidal phase at T E 23.8 K. Recent neutron diffraction studies performed on a single crystal of NdRu,Si, [4,9] below T, show the existence of an antiferromagnetic phase with wavevector k = 2~/a (0.13, 0.13, 0) in low fields and a ferrimagnetic phase in higher fields; this phase corresponds to the ‘ferro-like’ magnetic structure suggested by our transport property re- sults. In the temperature range 16 K < T < T,

the ferrimagnetic phase sets in at critical fields covering the range 5.5 kOe <H, < 8 kOe. Our transport property data (fig. 7) suggest that the ferrimagnetic phase persists up to T,(O) with H, decreasing to zero.

The onset of these phases may explain the reversal in the sign of the magnetoresistance in low fields, from negative (above TN) to positive just below T, (figs. 1 and 3). For example, under a field H = 4 kOe the reversal in the sign of Ap/p occurs at T = T,(O) - 0.8 K, in fair agree- ment with the crossing of the ferrimagnetic- sinusoidal line in the magnetic phase diagram. We notice that the sudden change in the sign of Ap/p just below TN is in disagreement with the

theoretical predictions for isotropic antiferromag- nets (see table 1 and fig. 6a).

Due to the mentioned complexities below T,,

here we restrict the magnetoresistance analysis to the paramagnetic region, where our data con- firms the main qualitative features predicted by the theory based on the quasielastic approxima- tion (see fig. 6b). First, the field dependence of the magnetoresistance is well described by a h2

dependence for regimes (ii) and (iii) (table 1). Under regime (i), i.e., for T very close to T,.,, the h2(lPa) dependence should occur, where (Y is the critical exponent of the specific heat. We do not have specific heat data for NdRu,Si, with suffi- cient accuracy and resolution to allow a direct estimation of (Y. However, (Y maybe indirectly obtained from the critical behaviour of dp/dT in zero field [2]. Such an analysis leads to an esti- mate 0.08 < (Y < 0.14, as shown in section 4.3.

The relative smallness of (Y prevents us from distinguishing, within our experimental resolution for Ap/p, between a h 2(‘-a) dependence and the simple h2 dependence shown in fig. 2.

The temperature dependence of Ap/p above T, exhibits the theoretically expected trends (fig. 6b): a finite negative minimum at T,, with Ap/p

independent of T over a small region around T,

(E -zz h regime; table 11, followed by a power-law dependence E-*, with A progressively changing from a critical to a classical value as E increases (see below). The numerical analysis of Ap/p

(equivalent, in our case, to the study of the linear plots in fig. 2) was performed in terms of the standard expression for critical phenomena:

(5)

where B and C are constants. A least-squares fit of the exponential data determines Z3, C and A for a given value of T,(O). As an initial parame- ter we adopt for T,(O) the temperature at which dp/dT (measured independently in zero field) has a maximum. Then T,(O) is refined within a small range around the initial value c&T, < 0.5

K), so as to minimize the variance of the fit. The stability of the parameters (B, C, A) was studied by successively removing experimental points above T,.

108 M.A. Salgueiro et al. / Magnetoresistarzce of NdRu,Si, near the N&e1 point

Our analysis shows that for T 2 26.25 K (E 2 0.9 X lo-‘) and up to the highest measured tem- perature (T = 40.84 K, E = 7 x 10-l) the magne- toresistance exhibits the expected classical be- haviour with A = l/Z. If we extend the analyzed data towards the vicinity of TN, we find a pro- gressive lowering of A, e.g. A = 0.3 if we keep all the data points above 25.03 K,and A = 0.23 when we use all the data points above 24.9 K. This result is not surprising since we are progressively penetrating into the critical region (E < co, where co is the Ginsburg reduced temperature) and the fit then gives an ‘average’ A-value, between the critical value A = LY << 1 (for NdRu,Si,l and the classical coefficient l/2. A more stringent check on the truly a-dominated regime is not possible with our present data, due to the experimental scatter, and also because as T -+ TN the E-~ regime is rapidly suppressed and Ap/p becomes temperature independent.

4.3. Critical behaviour of dp /dT

Provided we can use the quasi-elastic approxi- mation [141, the critical behaviour of the electri- cal resistivity derivative (dp/dT) should be the same as the critical behaviour of the magnetic specific heat and so we can estimate (Y from resistivity data.

The quasi-elastic approximation is usually as- sumed in the vicinity of a critical point, in view of the thermodynamic slowing down of the critical fluctuations. Correlation functions such as ( Si(O> . S,(t)), which enter the scattering cross section, may then be regarded as stationary and replaced by the static correlation (S, . (0) . Sj(O>). Geldart 1153 performed a detailed and critical assessment of the validity of this approximation near mag- netic critical points, making an evaluation of the lowest-order corrections to the resistivity due to inelastic scattering. Such corrections were found to be numerically significant only for very low spin, e.g., N 30% for S = l/2, and they vary as l/[S(S + 111.

In the particular case of NdRu,Si,, the local- ized 4f-spins in each Nd3+ ion, giving a total angular momentum J = 9/2, lead to a negligible value for l/[J(J + 111 (= 0.04). This justifies the

use of our dp/dT data near T, to estimate a, using the following dependences:

dp/dT=A(EI-“+B (T> TN)>

dp/dT=A’lEI~“‘+B’ (T<T,),

where E = (T- TN)/TN. If we take T, as an adjustable parameter and impose the physical condition a! = (Y’ [12]. the best least-squares fit- ting of the whole data (T > T, and T < T,) is obtained with T, = 24.31 K, leading to the fol- lowing values for the critical exponents:

(Y = (Y’ = 0.11 + 0.03.

The analysis covers the reduced temperature ranges 2.2 X lo-’ < ti < 1.2 X lop2 for T > T, and 1.8xlO-‘<I~I <2.2x10-’ for T<T,. Within the experimental uncertainty for N (a’), our results are consistent with the predicted value (Y = 0.11 for 3D Ising magnets [ 161.

The large uncertainty in the estimated N es- sentially results from the rounding effects near the peak of dp/dT (for E I 2 X lo-‘, which pre- cludes the use of a more extended E-range), and also from the scatter of the dp/dT experimental data. We cannot exclude a ‘smoothing’ of the transition due to chemical inhomogeneities and defects present in our ternary polycrystalline sample. In fact, some mixing between Ru and Si atoms has been inferred from the’analysis of the observed and calculated intensities of nuclear peaks in a previous neutron diffraction investiga- tion in a NdRu,Si, sample of the same batch [l]. In that study, the reliability factor R is better when we consider a statistical distribution be- tween Si and Ru atoms. Thus, for NdRu,Siz the improvement of the R-factor is from 12 to 7%, if 10% mixing is introduced in the calculations car- ried out with nuclear peak intensities recorded at 30 K. We also notice that the reliability factor goes from 10 to 5% with the same atomic Ru/Si disorder for a TbRu,Si, sample prepared under the same (nominal) conditions as NdRu,Si, [l].

4.4. Conclusions

From the A deviation from l/2, which starts around 25.5 K, we estimate a Ginsburg reduced

MA. Salgueiro et al. / Magnetoresistance of NdRu,Si2 near the Nkel point 109

temperature co = 4.9 X lo-* for NdRu,Si,, which has the right order of magnitude for com- mon magnetic materials.

Our magnetoresistance results above TN are consistent, within our experimental resolution, with the theoretical predictions based on the quasi-elastic approximation to describe the be- haviour of the magnetoresistivity near the transi- tion temperature [7]. Below TN we find a sudden reversal in the sign of Ap/p, which is attributed to the onset of new magnetic phases (field-in- duced) in the strongly anisotropic NdRu,Si, compound. We confirm the predicted H * depen- dence of T, in low fields.

Finally, an intermediate regime is observed in Ap/p above TN, with the exponent A gradually changing from its critical to the mean-field value as the temperature increases.

Acknowledgements

The authors acknowledge the financial support given by INIC and IFIMUP (IMAT Project, Pro- gramme Ciencia). The technical assistance of Francisco Carpinteiro is also gratefully acknowl- edged. This successful scientific cooperation was made possible by a combined effort of CNRS and JNICT, for which the authors are most grateful.

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