Anisotropy energy of spin glasses at T < Tg studied by transverse susceptibility

HAL Id: jpa-00209441

https://hal.archives-ouvertes.fr/jpa-00209441

Submitted on 1 Jan 1982

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Anisotropy energy of spin glasses at T < Tg studied by transverse susceptibility

F. Hippert, H. Alloul

To cite this version:

F. Hippert, H. Alloul. Anisotropy energy of spin glasses at T < Tg studied by transverse susceptibility.

Journal de Physique, 1982, 43 (4), pp.691-703. �10.1051/jphys:01982004304069100�. �jpa-00209441�

Anisotropy energy of spin glasses ^at

T Tg ^{studied by} transverse susceptibility

F. Hippert and H. Alloul

Laboratoire de Physique ^{des Solides} (*), Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France

(Reçu ^{le 18} septembre 1981, révisé le 16 dicembre, accepté ^{le 18 dgcembre} 1981)

Résumé.

Nous avons étudié les propriétés d’anisotropie ^{des verres de} spin CuMn, AgMn, CuMnAux, ^{AuFe, à}

T Tg ^{par des mesures de} susceptibilité transverse (~) ^en présence ^d’un champ statique H0 // ^z (Ho ~1 kG) et ^en

champ ^{nul. Nous} démontrons que, ^en présence d’une aimantation rémanente 03C3 créée initialement suivant _z, la

réponse ^du système ^de spins ^{à un} petit champ transverse ( z) ^{met en} jeu ^une rotation collective des spins qui ^est

entièrement caractérisée, ^{dans la} limite des faibles deviations, par la donnée d’un champ d’anisotropie macroscopi-

que HA. ^{A une} température ^donnée l’énergie d’anisotropie ^K

03C3HA ^ne dépend pas de 03C3. K décroît légèrement

avec la température ^{dans la zone étudiée} T ~ Tg/3. ^{Pour T ~} Tg ^{une valeur unique de K rend compte des} expé-

riences de ~, RMN, ^{et RPE} (en présence de rémanence comme dans l’état obtenu après refroidissement en champ nul). Lorsqu’un renversement de 03C3 peut ^être induit par application ^d’un champ H0 négatif, l’analyse ^{des valeurs de} ~ pour les deux orientations de 03C3, ainsi que les etudes de cycles d’hystérésis, suggèrent que l’énergie d’anisotropie

totale est la somme d’une contribution uniaxiale et d’une contribution unidirectionnelle. L’énergie d’anisotropie

par ^atome augmente ^comme la concentration c en Mn dans les verres de spin CuMn ^et AgMn ^{et croit} linéairement

avec la concentration x en Au dans les alliages ^ternaires CuMnAux, ^{en accord} qualitatif ^{avec les} prévisions ^théo- riques.

Abstract.

We have investigated ^the anisotropic properties ^of CuMn, AgMn, CuMnAux, ^AuFe spin glasses ^at

T Tg by performing transverse susceptibility (~) measurements in zero and small (~ ¹ kG) static field H0 //z. ^We

demonstrate that, in presence of ^{a remanent} magnetization ^03C3 initially developed along z, the response of the spin system ^{to a small} transverse ( z) ^{field involves a} collective rotation of the spins ^{which is} fully characterized by ^a macroscopic anisotropy ^field HA for small deviations. At a given temperature ^the anisotropy energy K

03C3HA

is found independent ^{of 03C3. A} slight decrease of K with temperature ^{is evidenced in the} investigated range T ~ Tg/3.

At T ~ Tg ^{the same} value for K is found to explain ~, NMR and ESR (both in presence of remanence and in the

zero field cooled state). ^{In cases where 03C3 can be reversed} by ^the application ^{of a} negative ^field Ho, analysis ^{of ~}

data in the states with 03C3 up and down, ^{in addition to} hysteresis cycles ^studies, suggests that the total anisotropy

energy might ^{be the sum of uniaxial} and unidirectional contributions. The anisotropy energy per ^atom is found to

scale as the Mn concentration c in CuMn ^and AgMn ^{and to increase linearly} with the Au concentration x in ternary alloys, ⁱⁿ qualitative agreement with theoretical predictions.

Classification

Physics ^Abstracts

75.40

1. Introduction.

In a magnetic alloy ^{such as}

CuMn the disorder of atomic positions ^{of Mn} magne-

tic impurities, ^{in addition to the} oscillatory depen-

dence of the exchange RKKY interaction [1] ^with

Mn-Mn distance, prevents any kind of long range

magnetic ^{order at T = 0.} However well below a

characteristic temperature Tg(- I Ja 1/kB), ^{the Mn} spins appear ^to freeze in the random directions of their local fields, ^{in the so called} spin glass ^{state, as evidenced} by impurity ^{Mossbauer spectra} [2] and host NMR [3,

4]. Whether this freezing ^{is a} progressive phenomenon

or a new kind of phase transition, ^as suggested by ^the

existence of a sharp cusp in the low frequency ^a.c.

susceptibility ^at Tg ^{[5, 6], is still an open} question [7, 8].

As long ^{as no} external field has been applied ^no macroscopic magnetization ^{can be detected in such a} spin glass. However, ^when cooling ^the sample through Tg ^{under a magnetic field H,, // z} which is then reduced to zero, ^a remanent magnetization ^a /’ He ^{is obtain-}

ed [9], ^which decays roughly logarithmically ^with

time [10, 11]. ^{Existence of remanence and} magnetic

after effects have been commonly interpretated ^within

a highly inhomogeneous picture [10, 12], by analogy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004304069100

with the Neel model for a collection of isolated ferro-

magnetic particles.

At T T i’ ^{6 is found to} remain in the direction of the cooling ^field Hc. ^The corresponding anisotropy

of the system which, ^{as the remanence} properties,

does not depend upon the direction of H, ^with respect

to the crystalline ^axis [13], ^{has been} probed by study- ing the response of the spins ^{to the r.f. field in a NMR} experiment. ^{The NMR data on the Cu nearest} peigh-

bours of a Mn impurity [4] ^{revealed that} the electronic

spins respond collectively ^{to a} small external field.

The anisotropy ^{forces are} characterized by ^a single macroscopic ^field HA // z ^which depends strongly

upon ^6. These results, ^{as well as the detection of} sharp hysteresis loops [14, 15], imply ^a rigid body descrip-

tion of the spin system for small applied fields, ^at variance with the inhomogeneous description ^needed

to describe the construction of a in large applied

fields [10]. ^The parameter ^{measured in a NMR} experi-

ment, namely the enhancement factor of the r.f. field,

is an indirect determination of the transverse (-L z) magnetization ^{due to} the collective rotation of the electronic spins. ^{One can as well} directly ^measure

the transverse magnetization [16] ^{or the low} frequency susceptibility (xl). ^We indeed checked the identity

between HA ^values deduced from NMR and _xl

experiments [17]. ^This anisotropy also induces a shift of the ESR line, either in presence of remanence [18],

or in the zero field cooled (ZFC) ^state [19, 20].

In this paper [21] ^{we intend to} present ^{a series of} extensive _xl measurements which allows :

-

the analysis ^{of the} relationship between ESR

and X, data,

-

the clarification of the properties ^{of the aniso-} tropy energy ^{K =} cHA ^and especially ^{its variations}

with remanence, temperature ^{and the nature of the} system,

-

an insight ^{into the} microscopic origin ^{of the} anisotropy ⁱⁿ spin glasses.

This paper is organized along ^the following ^lines.

First, in section 2, ^{we establish that a model based on}

the assumption ^of strongly coupled spins perfectly explains ^the response of the spin system ^{to a small}

transverse field. In section 3, the dependence ^{of K}

upon ^{Q at a} given temperature ^is investigated by analysing ^Q and Xj_ ^{data on the same CuMn} sample

for identical cooling processes. We then examine the relationship ^{between ESR} and X, in section 4 and the temperature dependence ^{of K in section 5.}

The link between the small angle ^{behaviour and the} hysteresis loops ^{is discussed} in section 6 while the

scaling ^{of K with Mn} concentration, and its variation with the addition of non magnetic ^Au impurities, ^are analysed in section 7. The microscopic coupling ^{at the} origin ^{of this} anisotropy, ^{as well as the} properties ^of

the macroscopic anisotropy energy, ^are discussed in section 8.

2. Transverse susceptibility ^{of a} spin glass.

^{Let us}

assume that, ^{for T} much smaller than T 91 ^{the electronic} spins ^{in a} spin glass ^{are frozen} along ^their randomly

distributed local fields and that, ⁱⁿ presence of a rema-

nence a initially developed along ^z, they respond collectively ^{to a small field. In this} picture ^a single parameter, ^the angle ^{0 between a and z, is} required

to fully characterize the magnetic ^{state. Let us describe}

the anisotropy ^forces by ^{an uniaxial} anisotropy

energy EA ⁼ (K/2) ^{sin2 0. In} presence of ^a transverse

(± z) ^field h, the total energy of the spin system ^is given by :

The equilibrium ^value of 0, ^{in the limit} of small devia-

tions, ^is given by 0,,q ⁼ h/HA ^where HA ⁼ Kla ^{is the} anisotropy field. Due to the rotation of the _{remanence_}

a transverse magnetization ^{uhjHA appears. Besides}

this collective motion slight distortions of the distribu- tion of local fields do exist, ^{as in an} antiferroma- gnet [22]. They ^are responsible ^{for the non zero iso-} tropic susceptibility (/;sJ of the ZFC spin glass ^at

T ⁼ 0. As long as we can assume that the response of the spins ⁱⁿ presence of remanence is the sum of the two previous contributions, ^i.e. x;so is not modified

by ^{the existence} of a, ^{the total} transverse susceptibility

will be given by :

This assumption ^is reasonable as the existence of 6, which never exceeds 5 % ^{of the saturated} high ^field magnetization ^{of the} alloy, ^{is not} expected ^{to drasti-} cally modify the distribution of local fields. When a

positive static field Ho ^is applied along ^z, equation (2)

is modified into :

We have performed xl measurements by ^{a mutual}

inductance technique. ^The cooling ^field H,, ^{as well as} Ho, ^were produced by ^a horizontal magnet (Hma. ⁼

13 kG, ^see figure 1). ^The magnitude of the vertical a.c.

field h ( v 5 ³⁰⁰ Hz) ^{does not exceed 1 G while} HA ^was

never found smaller than 300 G. Therefore the small

angle approximation ^is fully justified. ^{In order to use}

the same samples ⁱⁿ ESR, ^{NMR and} xl experiments

we have produced ^foil samples ^{about 40} pm thick.

They have been annealed, ^{in a} good ^{vacuum at}

500 °C for about ^one hour, ^{in order to eliminate}

stresses due to the rolling process. A well characterized

metallurgical ^{state is then} obtained, which indeed was

found to exhibit the same anisotropy properties ^{as the} original ^bulk sample.

In order to analyse ^the xl data it is _necessary to

check the validity ^{of the} previous decomposition ^{of xl}

Fig. ^1.

Geometry ^{of the} transverse susceptibility experi-

ment.

into two independent ^{terms. Let us} consider for the time being ^the susceptibility ^{Xiso involved in} equa- tion (3) ^{as an unknown} parameter Xo which can be

deduced, ^{as well as} HA, ^{from an} analysis ^of xl ^versus

Ho (see Fig. 2). ^When (XIXO) &#x3E;&#x3E; ^{1 the fit to} equa- tion (3) ^{is not} very sensitive to the exact value of _Xo.

However when Xl./ Xo ^{1 any error} in the determi- nation of _xo yields ^{a curvature in the} plot (xl - xo)-1

versus Ho. ^Then xo is determined with an accuracy

Fig. ^2.

Transverse susceptibility ^{data versus} Ho. ^The

range of negative Ho ^{has been limited in order to avoid a}

reversal of « (see ^section (6)). ^{On one} hand xl ^{data have been} plotted ^versus H 0 and the best fit to a hyperbola ^{has been}

indicated which corresponds ^to HA

^{870 G and a field} independent contribution _xo

9.7 x 10- 5 emu/g ^identical

to the isotropic susceptibility Xis. ^as directly ^{measured in the}

ZFC state. On the other hand we have plotted (xl - Xir 1,

i.e. the inverse of the transverse susceptibility ^{due to a} rotation, ^versus Ho.

of about 10 % ^{and is} always ^{found to} agree with the

isotropic susceptibility ^{measured in the zero field}

cooled state. This is indeed coherent with the facts that :

- x;so does not depend appreciably upon the static field Ho in ^our experimental conditions : T Tg and I Ho I I kG [17, 5],

-

the longitudinal susceptibility in presence of

remanence is found to coincide with _Xi,,. within a good

accuracy - 5 %; the initial slopes ^of magnetization

curves are indeed known to be identical whatever

Q [16,15].

We have therefore systematically ^measured Xiso in the ZFC state and plotted (åX)-1 ⁼ (xl - Xiso)-1

versus Ho. ^{In the case of} figure 2 for the CuMn 4.7 % (Tg ’" ^{30 K) at T =} 4.2 K in presence of a saturated

remanence, the straight ^line fitting ^{the data} yields ^from equation (3) ^{820 G} HA ^{950 G. We have taken}

care to start the _xl measurements at sufficiently long

time (one hour) ^after reducing H, ^{to zero in order to}

minimize the variations of r during the time of the

experiment. ^The validity ^of equation (3) ^{is confirmed}

as the inverse slope ^{of the} straight ^line fitting ^{the data}

in figure ^2, (2.95 ± 0.22).10-1 emu/g, coincides with

a ⁼ (2.7 ± 0.1).10-1 emu/g, directly ^{measured in the}

same conditions. Therefore equation (3) ^{with a well}

defined value for HA perfectly explains the response of the spins ^{to a small} transverse field at T Tg.

All the remanence is involved in the rotation pro-

cess. At this step ^we cannot assert from _xl data alone whether all the Mn spins ^{are involved} in the collective rotation. Some Mn spins, bearing negligible ^resultant magnetization, might indeed be excluded. However

analysis ^{of the} intensity ^{of NMR} signals [3, 4] ^revealed

that the fraction of spins ^which participate ^{to the}

collective rotation certainly ^{exceeds 50} % (’).

It is well known that HA strongly depends upon the

magnetothermal history ^{of the} sample [4]. ^{At a} given temperature, larger HA ^{values are} associated with smaller remanences. The anisotropy ^{energy K =} aHA might a priori vary with remanence, cooling process and temperature. ^{In a first} step ^{let us} consider the

dependence of K upon ^{Q at a} given temperature.

3. Dependence of K upon ^a at a given temperature T Tg.

As all the remanence is involved. in the rotation process the anisotropy energy ^can be equi-

(1) ^{Let us recall here the} arguments developed ^{in refe-}

rences [3, 41 ^to reach this conclusion. The fraction of spins

involved in the NMR signal ^{does not} vary by ^{more than a}

factor 2 whatever the Q value and the temperature ^{in the} range 0 T Tg/3. ^{As for T} Tg ^{nearly all the Mn spins}

are frozen and therefore contribute ^{to the ZFC} signal, ^the experimental accuracy allows then ^to conclude that more

than 50 % ^{of the} spins ^{are involved in the} collective rotation

detected for non zero remanence.

Fig. ^3.

Transverse susceptibility ^{due to a} rotation in zero

applied ^static field, A/(0), plotted ^versus the square of the remanent magnetization ^{in a} logarithmic ^scale. A x(0) ^{and Q}

are measured after identical magnetothermal history ^{at a}

time t after reduction of He ^{to zero. The} corresponding ^rema-

nence curves are shown in the insert. A straight ^{line with} slope unity ^{fits the} data, ^which demonstrates that

does not depend upon ^{at a} given temperature.

valently determined according ^to equation (2) ^as

K ⁼ A x(0) H1 ^{or K =} u2[AX(O)]-t, ^where

Ax(0) ⁼ xl - x;so in zero applied static field. The last formulation is the only available when HA ^is very large (practically HA larger ^{than a few} kG), ^{as a} direct deter- mination of HA from the variation of AX ^versus Ho

becomes inaccurate. In order to clarify ^the dependence

of the anisotropy energy upon the magnetic ^state we

have therefore performed systematically xl and a mea- surements on the same CuMn ^samples after identical

magneto-thermal histories. ^{In both} experiments, ^the

external fields have been applied during ^{the same time,}

decreased at the same speed, ^and Ax(0) ^{and a have}

been measured at the same moment of their time evo-

lution. All the magnetization measurements have been

performed by ^an extraction method in Grenoble, ^{in col-}

laboration with J. J. Pr6jean. ^{The data for} A x(0) ^{and u,} concerning ^the 4.7 % sample ^at 4.2 K for various

cooling processes, ^are summarized in figure ^{3. The}

variations of Ax(0) ^{versus U2 in a} logarithmic ^scale

fit a straight ^{line of} slope unity. This demonstrates that the anisotropy energy ^{K =} U2[LBX(0)]-1 ^does

not depend ^{upon a at a} given temperature. This result

Fig. ^4.

log J (filled symbols) ^and log [AX(O)] III (empty symbols) ⁱⁿ arbitrary ^{units versus} log ^t. Origin ^{of time is}

taken when the cooling ^{field has been decreased to zero.}

Parallel straight ^{lines can fit} the time variations of both

quantities.

is strictly demonstrated only for J values larger ^than

one tenth of the saturated remanence. Its validity ^for

lower Q values cannot be checked from _xl experi-

ments with our experimental ^set up. From the straight

line drawn in figure ^{3 we obtain K = 220} erg/g ^{for the} CuMn 4.7 % ^{at T = 4.2 K.}

Let us focus our attention on the two data points corresponding ^{to the} highest ^{values on} figure ^3.

They have been measured at t = 5 min. and t = 45 min.

after cooling ^the sample under 13 kG. The complete

time evolution of both Q and Ax(0) during ^{the first}

45 min. is detailed in figure ^{4 where} log J ^and log [AX(O)]’I’ ^are plotted ^versus log ^{t. The} important

feature is that parallel ^{curves fit the} time variations of both quantities. ^{This means that} ex(0) ^is proportional

to (J2(t), without any assumption ^{on the exact function} a(t) [23]. ^{In other terms} HA varies in time like (J-I.

Therefore time variations of _xl ^are by ^{no means} negli- gible, especially ^{when T is not} very low compared ^to Tg. This has lead us to take severe precautions, ^all along ^this work, ^{in order to ensure that J does not}

vary during ^the measurements of _xi versus Ho.

Therefore, ^{at a} given temperature ^T T g’ ^{a well-}

defined anisotropy energy ^K characterizes the aniso- tropy ^{of the} system. ^The transverse response of the

spins ^{is also} probed ^at higher frequencies (- ⁵⁵ MHz)

in NMR experiments [4]. We have checked in _many cases, ^as in reference [17] ^{that K} values deduced from xl and NMR experiments perfectly ^{coincide. At this} stage ^{it seems rather} important ^to clarify whether the

same values of K also explain ^{ESR data.}

4. Comparison ^{between ESR and Xl. data.}

^{Let us} briefly ^{summarize the} experimental ^{situation concern-} ing ^{ESR data.} We shall in a first step distinguish

between two opposite limits. First an ESR mode has been detected in presence of ^a high (saturated) ^rema-

nence, in nearly ^zero applied ^{field for a suitable choice}

of the frequency ^{of the} spectrometer [18]. ^{The reso-}

nance, is then entirely associated with the remanent

magnetization. ^{It is found to behave} quite similarly

to a ferrimagnetic ^resonance in presence of ^an uniaxial

anisotropy, fitting ^{in first} approximation

In this picture ^{the zero field resonance} frequency COA

provides ^a determination of the anisotropy ^{field. In an} opposite limit, ^{ESR has} been observed in a ZFC

sample in fields low enough ^to only ^induce negligible

remanence [19]. Data for various concentrations yield

then a non zero resonance frequency ^when Ho -1- ⁰

and fit OJ ⁼ ayHo ⁺ m(0) ^{for small} Ho ^{with a close to} 1/2.

While the resonance in _presence of remanence

is fairly easily understood by analogy ^{with a ferri-} magnet, interpretation of the ESR in the limit of

vanishing a requires ^{a detailed} analysis ^of equations

of motion in the spin glass ^state.

This has been done by Schultz et al. [19] (and by

Saslow [24, 25] using slightly ^different methods) ^{in the}

framework of a macroscopic ^{model based on the} following free energy :

where M is the total magnetization. ^{n is a unit vector}

which allows to characterize the rotation of the spin system and which is aligned along a in presence of

remanence. The two first terms of equation (4) repre- sent the magnetic ^free energy associated with the ^non remanent part ^{of the} magnetization ^when keeping

the possibility ^{of two} different values of the suscepti- bility along ^and perpendicular ^{to n. We shall here-}

after indeed assume, in agreement ^with experimental

results in CuMn (see ^section 2), ^that xo ⁼ XII ⁼ Xi,..

The third term of equation ⁽⁴⁾ is the Zeeman coupling

The last one modelises the anisotropy energy, N

being ^{a fixed unit vector directed} along ^the anisotropy

axis. In presence of remanence and in zero external field the minimum of the free energy is given by

n ⁼ N and M ⁼ Qn. In this case N is the direction z

of the cooling field and equation (4) is therefore

exactly ^{the free} energy implicitely ^used in section 2.

This can be easily ^{seen as,} writing ^for example

with

in presence of the static field H ⁼ Ho ^{N +} hu, ^where

u is a unit vector -1 N, ^{we indeed obtain the trivial} identity Mo ⁼ x;so H, ^and

which, in the limit of small 0 yields equation (3) [26],

that is :

Equations of motion for M and n, and then ^resonance

frequencies, have been derived from equation (4) by Schultz et al. [19] neglecting dissipation processes.

In the limit X" = xa = x;so their result for the two resonance modes is given by :

for arbitrary cooling conditions. Let us focus our

attention on the mode labelled + which is the one

usually observed up ^{to now.} In the limit of high ^rema-

nence, when the conditionv/k-/--xi7. a/xi.,. is ^satisfied, equation (6) yields

Table I.

Comparison between values of ^{K deduced} from ^{ESR and} xl data on the CuMn 4.7 % sample. ^{In the} first ^{columns, we have indicated the} parameters of ^{the ESR} experiments : cooling process, temperature, ^remanence, isotropic susceptibility. ^{From the resonance} frequency ^{in zero} external field, ^{we deduced a} value for ^{K, using equa-}

tion (6), ^{which is} compared ^{with K} deduced from ^{xl data at the same} temperature.

(a) ^{The error bars on a have been} slightly ^{increased in order to include time} variations, ^{as the time at which the}

resonance has been observed is not known accurately.

Assuming moreover a &#x3E;&#x3E; XHO equation (7) ^becomes m/y ⁼ Ho ⁺ K/Q, ^{i.e. a} ferrimagnetic like mode. On the other hand, ^for vanishing ^0’, equation (6) ^{is sim-} plified ^{into :}

which provides ^{an initial} slope ⁺ 1/2 ^{in the} plot w/ y

versus Ho ^while m(0)/y ⁼ (KIX)11’. ^This mode pre- sents a strong analogy ^{with the resonance in an anti-} ferromagnet ^{at T 0 0 as derived} by Keffer and Kittel [27].

As long ^{as (1 and} x;so ^can be measured independently

in the same conditions, ^{K is the} single parameter involved in the previous theory. Comparison ^of xl and ESR experiments ^therefore provides ^{a fundamen-}

.

tal test of its validity. This has been performed ^{on the}

4.7 % ^CuMn sample ^{as we could measure a} and X,

on the foil sample studied in ESR by ^{Monod and}

Berthier [18]. In presence of saturated remanence at T ⁼ 4.2 K the resonance is observed in zero

applied ^{field for v =} (2 ²⁵⁰ ± 50) ^Mz [28]. Using ^the

measured values _x;so ⁼ (9.7 ± 0.8). ^10-’ emu/g ^and

(1 = (2.45 ± 0.20). ^.10-1 emu/g, ^we deduce from _equa- tion (6) ^K

(260 ± 30) erg/g, ^{to be} compared ^with

K ⁼ (220 ± 20) erg/g ^obtained from X, ^{data of} figure ^{3. This} comparison, and similar ones, are’

summarized in table I. In the case of the saturated

remanence at T

1.5 K, w/y ^versus Ho ^{has been}

moreover determined [18]. ^{A careful} analysis ^reveals

that the slope ^is slightly reduced with respect ^to the paramagnetic gyromagnetic ^{factor y. The field} dependence ^of w/y ^{is in} perfect agreement ^with

equation (7). ESR in the same 4.7 % ^CuMn sample

has also been observed in the ZFC state at T ⁼ 2.2 K [28]. ^{The remanence induced} by ^{the field} applied ^to detect this resonance (H,. - ^{2 350} G) ^is

indeed negligible and the measured value of the shift of the ESR, ^{6H =} (w/y - Ho) = (960 ^± 40) G, yields

from equation (8) ^{K =} (280 ^± 35) erg/g. This result

perfectly ^coincides with the value of K deduced from ESR and _xl data in presence of remanence. So at

T Tg ^{the same value for K can} fit ESR and _xl data,

whatever the cooling conditions, in the framework of

equations (3) ^and (6). This establishes the validity ^of

the previous phenomenological macroscopic descrip-

tion of the spin glass. ^{Let us} point ^out that it is valid at T Tg ^{for a range of} frequencies extending ^{from a}

few Hz (a.c. susceptibility) ^{to a few GHz} (ESR). ^How-

ever it is clear that such an approach ^{will fail to describe}

the spin glass properties in the limit of extremely ^low frequencies (v ¹ Hz) ^when « internal » modifica- tions of the spin glass ^{state do occur, as} revealed for

example by ^{the existence of} magnetic after effects.

In consequence ^{one can} expect ^{that its} frequency

range of validity will be reduced at higher tempera-

tures.

5. Temperature dependence ^{of K.}

^{It is} possible

to observe the ESR through Tg and it has been already

known for a long time that the shift of the resonance 6H

only vanishes above Tg [30]. Assuming ^the validity ^of equation (6) ^{at all} temperatures, including ^T &#x3E; Tir

Schultz et al. deduced K(T)

K(0) [1

TII.5 Tg]

(where Tg ^{= 0.9} Tg ^{is the} temperature ^{of the} peak ^of

the susceptibility ^{in 3 kG} applied field), ^{from their}

data for various concentrations [20]. ^{Let us} point

out that K(T) ^{is found to be non zero above} Tg ^{in a} region ^{where no remanence effects are evidenced from}

static magnetization measurements [12, 31, 32]. ^It

would be particularly interesting ^{to do} comparisons ^of

xl and ESR experiments ^{at all} temperatures, ^{as the}

validity ^{of the} previous model checked at T Tg

becomes questionable ^at higher temperature.

However X, experiments, ^as performed here, require the existence of a finite remanence during the time of the experiment. They ^{are therefore not} appropriate

when T &#x3E; T g. Indeed when T increases towards T g

the time constants involved in the spontaneous decay

of a become shorter and shorter and finally ^{at T -} Tg

it is impossible ^to distinguish ^the paramagnetic ^res-

ponse from the remanence without analysing ^the

transients of the magnetization ^{once the} field has been switched off [33]. ^{In our} experimental ^set up ^we could

only perform experiments in the range 1.5-4.2 K.

It was therefore necessary ^to study ^{a more dilute} alloy than the 4.7 % sample (Tg - ^{30 K). We report}

here results concerning ^the 1.35 % ^CuMn sample (Tg - ^{12 K). In} figure ^5, AX(O) ^{versusu2 data are} plotted ^{at T = 4.2 K and T =} 1.5 K for various

cooling processes. The anisotropy energy K, ^determin-

ed as the inverse slope ^{of the} straight ^line fitting

the data, ^{is found to} vary with temperature :

K

(13.2 ± 1.4) erg/g ^{at T = 4.2 K and K =}

Fig. ^5.

Transverse susceptibility ^{due to a} rotation in

zero applied static field A x(0) plotted ^versus the square of the remanent magnetization for the CuMn 1.35 % ^at

T

4.2 K (filled symbols) ^{and T}

^1.5 K (empty symbols).

Straight lines fit the data at both temperatures demonstrat-

ing ^{that K}

a2[LBx(O)]-1 ^{does not} depend upon ^{a at a}

given temperature ^{but is a} decreasing function of tempera-

ture. The associated IRM curves are shown in the insert.

Fig. ^6.

Inverse of the transverse susceptibility ^{due to a}

rotation in zero applied ^{static field} plotted ^versus tempera-

ture in the range 1.5-4.2 K for ^a given ^{constant remanence.}

This is achieved by cooling ^the sample ^{in zero} applied

field from T

4.2 K to 1.5 K after having developed

the saturated remanence at T = 4.2 K. The variation of K oc [AX(O)] - ^{1 with} temperature ^is clearly ^evidenced.

(18.6 ^± 1.5) erg/g ^{at T}

1.5 K. A much more accurate method to study ^this temperature depen-

dence has been to establish a remanent magne- tization at T = 4.2 K and then to cool the sample

in zero applied ^{field. In such a} process the ^rema-

nence is constant as the magnetic after effects which occurred at 4.2 K are drastically slowed down when the temperature is decreased [34]. AX(O) ^{is found to}

decrease continuously ^{when the} temperature ^is lowered from 4.2 to 1.5 K (Fig. 6). ^After heating ^the sample ^{back to 4.2 K we found} again the initial value for AX(O), ^which corroborates the absence of time evo-

lution of Q during ^the experiment. The relative in-

crease of [A/(0)] ’ B and therefore of K = a’[AX(O)] - 1,

between 4.2 K and 1.5 K is about 50 %, ⁱⁿ agreement with the previous ^{results of} figure ^{5. The} temperature variation deduced from figure ^{6 is} consistent with

K(1) ⁼ K(0) [1

T/Ta] ^with To

(115 ^± 3) ^K.

This result is in rough agreement with ESR results [35].

So K(T) deduced from _xl experiments ^{seems to}

coincide with K(T) ^{deduced from} analysis ^{of ESR data}

at least in the range 0 T/Tg 0.3. Further experi-

ments are planned ^{in order to} investigate tempera-

tures closer to Tg.

6. Complete study ^{of AX versus} Ho in presence of reversals of Q.

In AgMn and in CuMn spin glasses

a more or less sudden reversal of «can be induced at T Tg/3 in presence of ^a high ^remanence by applica-

tion of negative ^fields [15]. ^When cycling ^{the static}

field Ho, nearly square and shifted hysteresis cycles ^are detected, ^as long ^as Ho ^{is small} enough ^to develop ^no

isothermal remanence, i.e. I (J I ^is kept ^{constant. For} example ^{in the case of} figure 7, ^a points ^{down when} Ho ^{is decreased} below - 550 G, and remains down if

Ho is then increased up ^{to -} 450 G. In these conditions xl measurements probe ^small angle deviations from the reversed equilibrium position. ^{As can be seen on}

Fig. ^7.

^{Inverse of the} transverse susceptibility ^{due to (J} rotation, (AX)-’, plotted ^versus Ho ^{for the} CuMn ^{4.7 %}

cooled at T

1.5 K under 3.2 kG. Ho ^is swept ^downwards from 700 G to - 1 200 G (squares) ^{and then} upwards (circles). ^The corresponding hysteresis cycle, ^as directly

measured [281 ^is indicated in ^{the upper part of the} figure.

As long ^{as a is} along ^{z, xl data fit} equation (3) ^with HA

(800 ± 50) ^{G .}

When ^Q has been reversed we find HA

(70 ± 50) ^G.

Data are found symmetrical ^with respect ^to Ho

^{370 G.}

The inverse slope ^{of the two} straight ^lines fitting ^the

data coincide with the value of Q directly ^measured (2.8 + 0.1 ) .10-1 emu/g.

figure ^{7 a} straight ^{line of} slope ^{a-’ fits} (LBX)-1 ^data

versus Ho :

but the anisotropy field deduced from this analysis HA ⁼ (70 ± 50) G, differs from the value HA ⁼ (800 ± 50) ^G measured when a points upwards, ^data being symmetrical ^with respect ^to Ho ^{= - 370 G and}

not to Ho ^{= 0.}

These features are common to all _xl or NMR

experiments ^as long ^{as a can be reversed. From these}

data alone one cannot infer anything ^{on the relation-} ship between the states with a up and down which

might merely ^have different and independent ^uni-

axial anisotropy energies. ^{But on} the other hand the existence of shifted hysteresis cycles ^{has been com-} monly interpreted by introducing uniaxial and unidi- rectional contributions to the anisotropy energy :

Such an equation ^indeed predicts sudden reversals of a

from 0 to n for Ho ^{= -} (Hax ⁺ Hd) and from n to 0

for Ho ⁼ HaX - Hd, ^where Hax ⁼ Kax/u ^and

Hd ⁼ Kd/U. ^{For small} angle ^{behaviour one obtains :}

where 8 ⁼ sign (a.z). ^This equation ^is equivalent ^to equations (3) ^and (9), ^with HA ⁼ Ha. ⁺ Hd ^and HA

Hax - Hd, ^{as in a small} angle approximation

one cannot distinguish between uniaxial and unidirec- tional contributions.

Experimentally ^{the centre of symmetry of} xl data

roughly coincides with the centre of the hysteresis cycle (which ^is directly ^{seen in the xl} experiments).

However the width of the hysteresis cycle ^is always

much less than 2 H ax ^{deduced from the fit of} xl data to equation (11). ^Therefore equation (10), ^where KaX ^and Kd ^are determined to fit the small deviations from 0=0 and 0 ⁼ _n, does not explain the reversals of a. The reversals of ^a are in fact complicated proces-

ses and not simple 7c rotations. Indeed an intermediate state, ^{where the} longitudinal ^{remanence is zero, can be}

stabilized. This state, for ^{which no} transverse magne- tization could be detected, ^{is rather divided in} magne- tic domains bearing opposite remanences [14, 36].

Therefore the single ^domain picture ^of equation (10),

with KaX ^and Kd determined from _xl measurements,

might ^still apply ^for large ⁰ although ^{it does not}

describe the reversals of a. Experiments, ^{where a}

true progressive rotation of a can be induced, ^would

check the angular dependence ^of EA (ESR, xl ^or torque measurements). ^At present torque experiments

Fig. ^8.

^{Inverse of the} transverse susceptibility ^{due to a}

rotation plotted ^versus Ho ^{for a} CuMn ^{4.7 % at T}

^{4.2 K}

(filled symbols) ^{and T}

^{1.5 K} (empty symbols) in presence of the same remanence : the sample ^{has been cooled under}

13 kG at T

4.2 K and then in zero applied field from 4.2 K to 1.5 K. The field was swept ^downwards from 1000 G to - 1 100 G (circles) ^{and then} upwards ^{in the same} range

(squares). ^Identical slopes ^{fit data at both} temperature ^which demonstrates the invariance of (1. The centre of symmetry of _xl data is multiplied by 1.7 when T is lowered from 4.2 to 1.5 K while HA only varies of - 15 %. ^{The width of the} hysteresis cycle ^is simultaneously ^reduced by ^a factor - 3.

were only performed ^{in a case} for which the unidirec-

tional contribution was dominant [13].

So, ^as long ^{as 6 can be} reversed, ^two parameters HaX ^and Hd (or HA ^and H’) ^are required ^to fully ^des-

cribe the small angle deviations of J from its two

equilibrium positions. ^{As the} independence ^of

K ⁼ KaX ⁺ Kd upon and its temperature ^variations

are well established, it is natural to wonder whether

Kax ^and Kd separately follow the same dependences,

and whether there is a general relationship ^between HaX and the width of the hysteresis cycle ^{AH. At a} given temperature alld ^{is found to be} roughly ^cons-

tant. This invariance could only ^{be tested from} xl data when reversals occur, i.e. for Q &#x3E; 6Sat/3, ^{and is}

far less accurate than the invariance of aHA. ^For example ^{for the} two states obtained by cooling ^the

4.7 % ^{CuMn under} 13 and 3.2 kG at T ⁼ 4.2 K the

respective ^{ratios of y,} HA, Hd ^{values are} (1.19 ± 0.03), (1.16 ± 0.12) ^and (1.48 ± 0.20). Similarly ^{at a} given temperature J AH ^{is found to be} roughly ^constant.

In order to clarify ^the temperature dependences- we

have performed ^{a detailed} study ^of Ax ^versus Ho at

T ⁼ 4.2 K and T ⁼ 1.5 K, for the 4.7 % CuMn, ⁱⁿ

presence of the ^same remanence, as ensured by ^the

fact that the data can be fitted at both temperatures by straight lines with identical slopes (Fig. 8). ^{We observe}

an important change ^of the unidirectional anisotropy

field : Hd

(220 ± 20) G ^{at T = 4.2 K and} Rd

(380 ± 20) ^{G at T = 1.5} K, ^while HA only ^{varies of}

15 %. The width of the hysteresis cycle ^{’is simulta-} neously ^reduced by ^a factor - 3. This tendency ^{to an}

increase of the displacement field of the hysteresis cycle

and to a reduction of its width for decreasing ^{T can also}

be detected in the magnetization ^{data of reference} [15].

It could be inferred from figure 8 that the observed variation of HA between 4.2 K and 1.5 K is entirely ^due

to a change ^{in the} undirectional anisotropy ^field Hd, ^the

uniaxial contribution Hax remaining unchanged. ^We

are not able at this stage ^to clarify ^{whether this is an}

universal behaviour.

In conclusion the hysteresis cycle characteristics do not reflect exactly ^the properties ^{of the} anisotropy

energy ^as deduced from X, measurements in the states with a up and down. Let ^us point ^{out moreover that}

the hysteresis cycles ^can only ^{be observed} in the limit where the negative ^field required ^{to inverse a is low} enough ^{to induce} only negligible negative ^isothermal

remanence. As soon as K is large, ^{as in AuFe} [17],

no cycles ^{are detected even} in presence of the saturated

remanence [34], ^{while K can be} determined in‘ _any system ^{from the zero field} measurement of _xi. The existence of a unidirectional contribution Kd ^{to the} anisotropy energy, which ^seems to be confirmed by

the unique torque experiment [13], ^{has received at} present ^{no decisive} interpretation. ^A given ^number

of spins, bearing ^no macroscopic magnetization ^and

not involved in the collective rotation, ^{could induce a}

constant field independent upon ^a direction, ^{as was}

first suggested by ^Kouvel [16]. ^As recalled in section 2

up ^{to now} NMR data only ^{allow to ensure that 50} %

at least of the spins ^are involved in the rotation induced

by ^{the r.f. field.}

7. Scaling laws of K. - 7.1 CONCENTRATION

DEPENDENCE OF K.

In order ^to clarify ^the depen-

dence of the anisotropy energy ^versus Mn concentra- tion in CuMn $pin glass, ^{we have extended our}

measurements to 0.6, 1 ^{and 9 at.} %§ ^Mn alloys. ^Results

are summarized in figure 9, where the anisotropy

energy per ^atom has been plotted ^{versus the Mn}

concentration. Comparison ^of alloys ^{of different}

concentrations in principle requires ^to extrapolate K(7’) ^{at zero} temperature. This was done for the three

highest concentrations using the variation of K(T)

inferred from _xl data in the _range 1.5-4.2 K. However for the two lowest concentrations, ^{as this} extrapola-

tion would not be accurate, ^we have indicated K as

measured at T ⁼ 1.5 K, ^which provides ^{a lower}

estimate of K(T ⁼ 0). ^{It is found to increase} linearly

with c at a rate of 12.8 ± ¹ (G. 9B/at.)/%, ^{at least for}

c &#x3E; 1.35 %. ^{For lower c a small} disagreement appears

(which ^{is indeed} stronger ^{if T} corrections are done).

Whether this is a spurious ^{effect linked} with residual

, metallurgical ^{defects or} impurities (see ^section 7.2) ^is

. at present unclear. The concentration dependence ^of K, 14.5 ± 0.6 (G./at.)/%, deduced from ESR data [19], ^{for c} &#x3E; 2 %, ^{is in} good agreement ^{with our} results which establishes once more the identity

between the values of K deduced from ESR and X, experiments.

We have also investigated the concentration depen-

dence of K in AgMn spin glass in the range 2.8-14.5 %.

We have deduced HA from the field variation of _xl and determined K as Ax(0) H 1. Temperature ^varia-

tions have been neglected. ^For the 5 and 10.4 % alloys,

for which Q has been moreover measured indepen- dently [37], ^{we checked} again ^the validity ^of equa- tion (3). ^The anisotropy energy per ^atom (see Fig. 9) ^is

found to increase linearly ^{with c at a rate of} (14 ± 1)

Fig. ^9.

Anisotropy energy per ^{atom versus} the Mn concentration in CuMn ^and AgMn spin glasses. ^The straight line indicates the best fit to a linear variation.

(G. pB/at.)/% very similar to the one measured in CuMn

spin glass. ^{Let us} point ^out that this linear relation holds even at high concentrations for which the

validity ^{of usual} scaling ^laws [9] ^{is not ensured} [37].

7.2 INFLUENCE OF ADDITIONAL IMPARITIES.

Anisotropy energy in spin glasses is enhanced by ^intro-

duction of metallurgical ^defects by ^cold working [38]

and by ^{addition of non} magnetic impurities. ^The study by Prejean et ^{al. of} hysteresis cycles ^on CuMn1% ^Au., samples (with ^x between 0.01 and 0.15 _at.% [39]) provide up ^{to now} the most systematic investigation

of the effect of non magnetic impurities. ^{The rema-}

nence properties, ^the isotropic susceptibility ^{and the} spin glass temperature ^{are not modified} by ^{the addi-}

tion of Au. But the width of the hysteresis cycles (determined ^{after the same} cooling process) ^{is shown}

to increase at a linear rate with gold concentration from its value in _pure CuMn 1 %. However the

study ^of hysteresis cycles presents ^a few intrinsic limitations. First they disappear ^{when x} &#x3E; 0.15 %.

Moreover, ^{as seen here} above, ^the anisotropy energy

Table II.

Transverse susceptibility ^data for CuMn1o/o ^{Aux alloys} cooled under 13 kG at T

1.27 K. For x &#x3E;, 0.1 %, ^{one cannot} detect the field dependence of xl with a sufficient accuracy and the hysteresis cycles disap-

pear. The only ^available quantity ^{is then the} transverse susceptibility Ox(0) ^{due to the} magnetization ^{rotation measur-}

ed in zero applied field. a calculated ^as AX(O) HA (last column), ^when HA ^{can be} directly determined, ^is found ^{to be}

constant and to be insensitive to the addition of Au, ^while HA, Hr ^and [A X(O)] - ’ ^are found ^{to increase} linearly ^{with x} from their values in pure CuMn. Our experimental conditions are similar to those of reference [39]. ^The slight diffe-

rence between a values is very likely ^{due to the} fact ^that He ⁱⁿ cycle exper.iments ^{has been switched} off ^{at a} slightly

higher temperature ^than in xl experiments (J. ^J. Préjean, private communication).