Effect of field and frequency on the temperature dependence of a.c. susceptibility of the (La, Gd)Ag spin-glass

Solid State Communications, Vol. 47, No. 2, pp. 147-15 I, 1983. Printed in Great Britain.

0038-1098/83 $3.00 + .00 Pergamon Press Ltd.

EFFECT OF FIELD AND FREQUENCY ON THE TEMPERATURE DEPENDENCE OF A.C. SUSCEPTIBILITY OF THE (I_a, Gd)Ag SPIN-GLASS S.N. Kaul and S. Methfessel Institut for Experimentalphysik IV, Ruhr Universitfft Bochum, 4630 Bochuttl, West Germany (Received 18 November 1982; in revised form 11 March 1983 by A. Blandin)

Response of the "sharp cusp" in the low-field a.c. susceptibility of the Lao.gGd0aAg alloy to the external magnetic field and frequency has been measured. The results have been discussed in the light of existing theories and compared with those previously reported for other spin-glass alloys. VARIOUS THEORETICAL MODELS [1-3], which either employ a phase transition approach [41 or invoke the spin-freezing mechanism [5 ], have been proposed to describe different properties of spin glasses but no single theory is complete in itself to account for the diversity of the experimental results obtained so far on such materials. Recently, the frequency, t,, dependence of the low-field a.c. susceptibility ×(T), near Tt, the freezing temperature, has received tremendous experimental attention [6] since such investigations are capable of ascertaining which of the different theoretical descriptions [71 of the spin-glass transitions (equilibrium phase transition, non-equilibrium metastable state, or themlally activated phase change) is more appropriate. The frequency or time dependence of Tt has been found to vary greatly [8-101 from one spin glass system to the other. This observation has led to various suggestions [ 11,121 regarding the nature of the nearest-neighbor magnetic interaction. Alternatively, varying strength of the frequency effect has been attributed [ 13] to the markedly different amplitude of Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in metallic spin-glasses (CuMn, AuFe, (La, Gd)Al2) or the differing concentration x in the insulating spinglasses (Eu~_xSr,,S, Eu~_xGdxS ). To test which of the above-mentioned interpretations of Tt(v) is correct, more experimental data on spin-glass systems that differ in the range and nature of the exchange interaction are needed. The present paper deals with the field and frequency dependence of ×(T) near Tt for the (La, Gd)Ag spin-glass. Lao.gGdo.tAg alloy was prepared from high purity La (99.99%), Gd (99.99%) and Ag (99.999%) by arcmelting in argon atmosphere. Different portions of the alloy button sealed under argon atmosphere in quartz tubes received homogenization annealing treatments at 900°C for time periods ranging from 4 to 24 hr and subsequent quenching in ice-water. X-ray analysis at room temperature revealed reflexes of only the cubic

CsCl-type structure with a lattice constant of 3.789 A. In conformity with the above observation, metallographic examination showed no traces of a second phase. The low-field (~ l Oe) a.c. susceptibility was measured on rod-shaped samples (2 m diameter and 8 mm length) for temperatures between 1.3 and 20 K in frequencies tip to 6 kHz using a conventional mutual inductance method whose details are given elsewhere [14]. The field dependence of X was determined by superposing a static magnetic field up to 100 Oe. Effects due to eddy current and skin depth, that become apparent for u > 100 l lz, were minimized by carrying out measurements for frequencies above 100 ilz on finely powdered samples with a grain size not exceeding 100tzm. We present here only the results obtained for tile in-phase component (X', hereafter referred to as ×) of the complex susceptibility on samples in which the high-temperature equilibrium state could be frozen-in (i.e., the ones quenched after annealing at 900°C for 8 hr or more). Tile out-of phase component of the complex susceptibility, ×", which attains its maximum value at Tt as observed by Mulder et al. [6, 15 ] in CuMn and AgMn spin-glasses and not below Tt as reported by Lundgren et al. [16] for the AuFe spin glass, did not exceed 1% of ×' in the temperature interval 1.3 K < T < 20 K. Figure 1 depicts results of the susceptibility measurements performed at v = 87 ~ in absence and presence of a static magnetic field, H, (up to 100 Oe) applied parallel to the a.c. driving field. The following points deserve proper attention: (1) the zero-field susceptibility, ×(0), as a function of temperature exhibits a sharp maximum which is taken to identify the freezing temperature, Tt (v = 87 Hz) = 3.35 -+ 0.01 K, (2) the height of this maximum, Xm(H),decreases while its position, Tt(H ), shifts to lower temperatures as H increases, and (3) in the temperature interval 4 K ~< T ~< 20 K, x(O) follows a Curie-Weiss law, ×(0) = C / ( T - 0), [Fig. l(b)] with C = (2.8 -+ 0.1) x lO-3emu Kg-t and 0 = -- 3.4 -+0.6 K. The Curie constant Cgives an effective

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148

EFFECT OF FIELD AND FREQUENCY ON (I.a, Gd)Ag S P I N - G L A S S T/z (K/at%) 0

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Fig. 1. (a) Field and temperature dependence of the susceptibility, X, (applied alternating field ~ 1 0 e , measuring frequency 87 Hz) for Lao.gGdo.t Ag near Tr. (b) X-t as a function of temperature for T > Tt and the molar susceptibility in the absence of a superposed static magnetic field for v = 87 Hz as a function of reduced temperature T/x with x = 10at.% Gd (dotted curve). The free-ion molar susceptibility of Gd s÷ (dashed curve) is also included in Fig. l(b) for comparison.

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Vol. 47, No. 2

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EFFECT OF FIELD AND FREQUENCY ON (La, Gd)Ag SPIN-GLASS 1.000

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Fig. 3. Log-log plots of the spin-glass order parameter q ( T ) and the reduced zero-field susceptibility X ( T ) vs the reduced temperature I T - Trl/TY. The least-squares straight line fits to the data yield q ( T ) = 1.5 [(Tt -- T)[ Ttl o with ~ = 1.02 and X ( T ) = 1.4[(T-- Tf)/TtI-'t with 7 = 1.5. transition. At this stage it is worth mentioning that much higher values of 6 ( 4 - 6 ) [20, 2ll and 7 ( 3 - 4 ) [21,27] have been found when an extended range ( 0 - 1 0 kOe) of the d.c. fields and the scaling arguments are used. Since even in such determinations the value of remains unaltered, i.e., B = 1, the scaling relation 5 = 1 + ('y//3) is not obeyed [21 ] in some spin-glass alloys. It remains to be verified if such values of 5 and are a consequence of the range of the d.c. fields and reduced temperatures used for the scaling fits or if they are a manifestation of the differing solute concentration or if both these factors are responsible. Though there are some indications of a cross-over to a higher value of "r for temperatures very close to Tt (see Fig. 3), we cannot ascertain the genuineness of this effect since in view of the fact that the value of ~, becomes increasingly sensitive to the choice of T r as T-+ T r, our data do not fix Tt to an accuracy needed to arrive at reliable values of "r for temperatures in the immediate vicinity of T r. Alternatively, in order to verify this point one needs a spin-glass system which exhibits a much sharper transition at Tr than that observed in this work. Frequency dependence of X is demonstrated in Fig. 4(a). Important observations are: (1) with increasing v, Tr shifts to higher temperatures and height of the susceptibility maximum decreases (by about 14%

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Fig. 4. (a) Temperature dependence of × for (La~_xGdx)Ag, (x = 0.1) measured at various fixed values of frequency. (b) The normalized susceptibility X(v)/× (v = 87 Hz) vs In v for different temperature values. The lines drawn through the data points are only a guide to the eye. from 0.6 Hz to 5.3 kHz), (2) for T~>4.2 K, X is independent of frequency, and (3) values of the critical exponents/3 and 3' determined, using equations (la) and (lb), in the temperature range 10 -2 GItl G 10 -I for different frequency values come out to be the same (within the error limits, -+0.04 for fl and -.+0.1 for 7) as those obtained for the data taken at u = 87 Hz when the observed values of C[Tt(v)] and 0 [Tt(v)] are used and t is defined as t = [ T - Tt(u)]/Tt(v). The latter finding lends a fire1 support to the phase transition interpretation of the susceptibility cusp. Within the framework of the superparamagnetic blocking model [51, Tt should be related to v through the Arrhenius law, v = Uoexp (--Eo/kaTt), where/~a is the activation energy. Such a law holds for our data but with unphysical values, (Ea/kB) = 680 +- 5 K and Uo ~ 109°Hz. These unphysical magnitudes can, however, be removed when in place o f an Arrhenius law, a Fulcher law, v = Voexp [--Ea/ kn(T r -- To)], is used to describe the frequency dependence of Tr. Following Tholence [13], we disregard the variation of Uo from one spin-glass system to the other

150

EFFECT OF FIELD AND FREQUENCY ON (La, Gd)Ag SPIN-GLASS

and use Vo = I013 Hz to deduce the values (Eo/kB) = 10.5 K and To = 2.94 K (~ 0.9 Tr) from the Tr vs [In Vo-- In v] -1 plot. These values compare favourably with those obtained for metallic spin-glasses [ 13 l. According to Tholence [ 131, the normalized frequency shift of Tr is given by

ATr[Tv(Alogtov ) e~ (1--To/Tv) ~x (l/Vo),

(2a)

where

Vo = 9nZ~J2S(S + 1)/16Erk~,

(2b)

is the amplitude of the RKKY interaction between impurity spin (S) pairs that are at a distance r,2 apart: HRKKY = 8k~VoF(2kFrt2),

(3a)

with F(x) = (x cos x -- sin x)/x 4.

(3b)

Here Er(kF) is the Fermi energy (Fermi wavevector), Z is the number of conduction electrons per host atom and J is the impurity atom-conduction electron exchange integral. For Lao.qGdo.IAg, we find A Tt/ Tt(A loglo v) = 0.0113 -+ 0.0002. This value is of the same order of magnitude as the ones previously reported for metallic spin-glasses (AgMn, CuMn and AuMn), (0.005) [6, t 5 ], Au Fe (0.01-0.018) [ 10, I 1 ], (La, Gd)B~, (0.005-0.026) [28], but substantially smaller than those for (La, Gd)AI2, (0.042-0.084) [281, (Y, Gd)AI2, (0.16-0.33) [28] and insulating spinglasses, (0.027-0.2)[9, 10]. In view of equation (2a), the above comparison suggests that the amplitude Vo should possess comparable values for the former set of alloys whereas for the latter set ilo should be at least one order of magnitude smaller. Contrary to this prediction, the Vo values for the pseudo-binary (La, Gd)B6, (La, Gd)Al,. and (Y, Gd)AI2 alloys are of similar magnitude [281. The value of Vo for (La, Gd)Ag estimated using Z = I, free-electron value of kF and the reported EF value for LaAg in equation (2b) comes out to be close to those found for the alloys just mentioned. However, it is not clear at this stage whether the Gd concentration in the present alloy is small enough for a meaningful comparison of the value obtained for Vo with those determined in the dilute limit of a RKKY spinglass. To clarify this point, measurements on the low Gd concentrations in this alloy system are in progress. According to an earlier suggestion of Zibold [11 ] and a more detailed model proposed by Hardiman [12], a large frequency shift in Tt is expected for ferromagnetic nearest-neighbour (NN) coupling, and a weak or even no frequency effect for antiferromagnetically coupled nearest neighbours. A weak frequency shift in Tt observed for the present alloy should, therefore, be a consequence of dominating antiferromagnetic

Vol. 47, No. 2

short-range interactions between the Gd ions. In conformity with this prediction, the statistically probable NN Gd ion-pairs are found to interact antiferromagneticaUy as inferred from: (a) the molar susceptibility, plotted against T/x (x is the Gd concentration) in Fig. 1(b), (dotted curve), of Lao.gGdo.lAg above Tf falls below the free-ion molar susceptibility of Gd 3÷ (dashed curve), and (b) a calculation of the oscillatory RKKY pair function F(2kFr12), equation (3b), using Z = 1 electron per LaAg molecule, free-electron value of kF and the presently determined lattice parameter value, reveals that the nearest neighbours couple antiferromagnetically while the next nearest neighbours couple ferromagnetically. A small but finite shift in Tt with frequency found in Lao.gGdo.iAg possibly results from the ferromagnetic next-nearest-neighbour Gd pairs. Finally, it shotdd be emphasized that the frequency dependence of x(T) around Tt. is of more interest than Tt(v ) in case a clear distinction between different spin glass systems that exhibit similar values for ATt/Tt( A Iog,o v) is sought. To illustrate this, the freqt, ency dependence of X (normalized to the value at 87 Hz) for several temperatures is shown in Fig. 4(b). It is noticed that the frequency dependence o f x [hereafter referred to as X(v)] decreases as the temperature is lowered below Tt. This behaviour is in accordance with X(v) observed in metallic spin-glasses (AgMn, CuMn and AuMn) [6, 151 but in contrast to the increase in X(v) as the temperature is decreased below Tt found in PdMn [291, NiMn [301 and AuFe [I0, 11 land insulating spin-glasses [9, I 0 ].

Acknowledgements -- Tile authors thank Professor J.A. Mydosh for his interest in this work and the referee for useful suggestions. One of us (SNK) thanks P. Stauche, G. Daub and tl. Bach for their help during sample preparation and characterization. This work was financially supported by the Alexander Von Humboldt foundation. REFERENCES I.

2. 3. 4.

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EFFECT OF FIELD AND FREQUENCY ON (La, Gd)Ag SPIN-GLASS

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