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The extra-dimension W = kT Ln (t/τ0) of phase space below the spin glass transition : an experimental study of the relaxation of the magnetization at constant field
in CuMn
R. Omari, J.J. Préjean, J. Souletie
To cite this version:
R. Omari, J.J. Préjean, J. Souletie. The extra-dimension W = kT Ln (t/τ0) of phase space below the spin glass transition : an experimental study of the relaxation of the magnetization at constant field in CuMn. Journal de Physique, 1984, 45 (11), pp.1809-1818. �10.1051/jphys:0198400450110180900�.
�jpa-00209926�
1809
The extra-dimension W = kT Ln (t/03C40) of phase space below the spin glass
transition : an experimental study of the relaxation of the magnetization
at constant field in CuMn
R. Omari (*), J. J. Préjean and J. Souletie
Centre de Recherches sur les Très Basses Températures (**), CNRS, BP 166X, 38042 Grenoble Cedex, France
(Reçu le 28 mars 1984, révisé le 16 juillet, accepté le 26 juillet 1984)
Résumé. 2014 Nos mesures de l’alimentation hors-équilibre d’un verre de spin CuMn à T Tc montrent que,
pour une histoire donnée de l’échantillon et dans un champ donné, l’intégralité de la dépendance en temps t et température T des relaxations peut être représentée par une courbe maîtresse unique dans un diagramme M(T Ln (t/03C40)). Nous discutons les implications d’un point de vue pratique et fondamental. Nous soulignons
la puissance de prédiction de cette loi ainsi que les servitudes que les expérimentateurs ne doivent pas ignorer.
Abstract. 2014 We present experimental evidence obtained in a CuMn spin glass below Tc showing that for a given history of the sample, the totality of the time t and temperature T dependence of the relaxation in a given field
can be condensed into a unique master curve of the variable Wc = T Ln (t/03C40). We discuss the implications from
a practical and a fundamental point of view. Particular emphasis is put upon the experimental aspects : the pre- dictive power of this scaling is stressed as well as the considerations that experimentalists should not ignore.
J. Physique 45 (1984) 1809-1818 NOVEMBRE 1984,
Classification
Physics Abstracts
75.60
1. Introduction.
The relaxation of the magnetization and of the energy in spin glasses has complex features. We have described several of these features in the framework of a pheno- menological model of two level systems (T.L.S.) [1- 4]. Many of our conclusions follow from very general reasoning and should survive in a more adequate description. This is in particular the case of the
T Ln (t/io) scaling which we anticipate and observe in the low temperature regime (t is the time of the
experiment). In the present paper, we concentrate upon the problem of the relaxation in a constant field
and stress the implications and consequences of the T Ln (t/io) scaling. We illustrate this point with
remanent magnetization data as well as with new
results in finite field
2. Phenomenological background
The relaxation of the magnetization (and of the energy)
is very slow and non exponential in a spin glass. The
Arrhenius law for thermally activated processes
is one of the few possibilities which permit us to link
times on an « atomic » scale (ro = 10-13 s) with the ordinary laboratory time scale (T - t with t in the range of seconds to hours). For this purpose we need activation energies W = kT ln t/io, larger by one
order of magnitude than the temperature of observa- tion (W 2-- 30 k T typically), and which are, a priori,
not very sensitive themselves to the variations of T.
We need a distribution of relaxation times t, hence of barrier heights W, in order to account for non expo- nential relaxations. This is enough to have many consequences. The relaxation of the magnetization M, in field H, at temperature T, can be formally described by an equation
PH(WE. ff) is the density of effective barriers which
hamper the relaxation of the magnetization in the
conditions of field and temperature of the experi-
ment. We can write the magnetization MWeff Bt /
associated with an effective barrier height W eff at time t
as follows :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450110180900
where the equilibrium contribution M(t -+ oo) is
assumed to be independent of Weff’
Now, equation (1) selects a particular barrier height We for which the relaxation time is equal to our measu-
rement time t. We have
With i as given by equation (1), exp -
t
is a veryfast function of W which abruptly varies from 0 to 1 when W is increased over We [1]. In practice, the barrier
height We separates the systems W We which
have relaxed in the time of experiment (i.e. giving the equilibrium contribution M(t oo)) from those asso-
ciated to W > We which preserve the memory of previous events (i.e. Mw(t = 0)). Equation (3) there-
fore simplifies to
Inasmuch as the function PH(W eff) is temperature
independent then M2, the frozen part of the magnetiza-
tion, depends only on the composite variable We
and on the previous history which imposed the initial conditions Mw(t = 0). We have previously presented
remanent magnetization [1] and associated energy relaxation data [3] which convincingly illustrated the validity of the argument for M2 in zero field when M(t - oo) = 0. In the following section we present
new data which have been obtained by applying a
finite field Ho on a previously zero field cooled sample.
In this case, there is no frozen magnetization prior to
the application of the field Mw(t = 0) = 0 and from equation (5) we get
3. The master curve MH(Tln TITO) for the magnetiza-
tion.
We have measured the magnetization of a CuMn 5 at % sample as a function of time in a constant field of 10 k0e
at different temperatures Ti below the temperature
Tc = 29 K of the susceptibility maximum. For each
run, the sample is zero field cooled from T > Tc
down to Ti T,,,. Once T i is stabilized, the field is increased isothermally up to its measurement value
of 10 k0e which is reached at time tic. The magnetiza-
tion in this constant field is then measured in the 100 minutes following ti which, from now on, will
be regarded as the origin of times at each temperature
(tri = 0).
Typical data of M vs. Ln (t - t) are shown in figure 1. The same data points lay on a unique master
(t - t; )
curve in the M vs. T Ln
t io t‘
B To / plot of figure 2 witha unique value of io which is io r>t 10-13 s but could have been chosen in the range 10-12 s > io > 10-14S with comparable success. For io outside of this range the data points, in terms of time for each temperature, do not fit a spline connecting the points relative to the
same time at different temperatures (dashed line of Fig. 2). This is how we have optimized zo ; we have
relied, for this purpose, mainly on those data between
Fig. 1. - The magnetization M vs. time of a CuMn 5 at %
obtained by applying (at different temperatures Ti) a field
H = 10 kOe to a zero field cooled sample. The origin of
time is taken at ti + e where ti is the moment when the
field is stabilized to its final value. A small correction 8 of less than 4 min (see appendix) was applied in order to align
the first data points with the subsequent log t trend. The
curve at 1.3 K presents some deviations due to field and temperature instabilities which are commented in figure 5.
1811
Fig. 2. - The same data as in figure 1 are plotted as M
vs. Ti Ln t in figure 2a. For r, = 10-13 s, the data points
io
lie on a unique master curve which tends to reach the field cooled response in the limit of large T Ln t values. This
superposition implies that the relaxations observed at two different temperatures would be given by the same curve
with two different logarithmic time scales as specified in
the upper part of the figure for T 1 = 3.16 K and T h = 4.01 K.
The insert shows the field dependence of the thermorema- nent magnetization (T.R.M.) and of the Isothermal Rema- nent Magnetization (I.R.M.). The field of 10 kOe which we
have chosen produces a sizeable remanence in our range of T Ln t values. The measured field cooled magnetization Mf., ,.(T) is found stable over our measuring window but does exhibit a slight temperature dependence. Figure 2b
is an enlarged version of a portion of figure 2a.
1 K and 7 K where the time dependence is strong
(Fig. 1) and-can be measured accurately (see section 5)
over long observation times. This permits us to
minimize with confidence the effect of an error on the determination of the origin of time (see section 4, Fig. I caption and appendix). At T = 200 mK, we did
not attempt a measurement of the magnetization vs.
time dependence because of the poor stability of the
temperature over one hour with the adiabatic dema-
gnetization cooling process which we use to reach this temperature. We have also represented, in the
same figure and for each temperature, the measured
values of the total field cooled magnetization Mf.c.
obtained by cooling the sample from T > T, down
to Ti in presence of a constant field of 10 kOe. In our case we find this field cooled magnetization to depend
very little on the temperature and to be very accura-
tely time independent, as the equilibrium should be.
There is however a slight but definite temperature
dependence of this response (from 1.519 e.m.u./g at 1.1 K to 1.512 e.m.u./g at 14 K) which represents only
0.4 % of the master curve variation and is scarcely appreciable on the scale of figure 2. The data, shown
in figure 2, indicate that the unfrozen response
M1(T Ln(t/To)) does tend towards the field cooled response for large T Ln (t/To) values i.e., presumably,
in the thermodynamic long time limit.
This, incidentally, seems to give credit to a common
belief that the field cooled magnetization provides
the equilibrium response. On the latter point, we do
not ignore objections [5] which have been raised.
We have ourselves [1, 4] extensively developed our position in previous papers. We have argued, in particular [4], that the correct way to reach equili-
brium is through the use of alternating field cycles of decreasing amplitude and that field cooling does not
in general lead to equilibrium unless the equilibrium susceptibility is strictly temperature independent.
The slight temperature dependence which is in general observed, in the field cooled response of the archetypal
RKKY spin glasses near Tc, can be taken as a measure
of the extent to which this procedure could be inade- quate.
Other features of the master curve are of interest.
For example, there is also a finite time and tempera-
ture independent limit when We = kT Ln (t/io) -+ 0.
This is the ergodic fraction of the response associated with those objects whose relaxation is not hampered by any energy barrier or is hampered by a barrier
that the external field can overcome at T = 0. Note that our procedure is symmetrical to the one which yields the thermoremanent magnetization (T.R.M.).
Here we cool in zero field and apply the field at a
constant temperature. In order to obtain the T.R.M.
we would cool in the field and withdraw the field at constant temperature. Both procedures are described by our equation (5). Here we have M(t -+ oo) = Meq(I)
and M(t = 0) = 0 and we measure the reversible part
MH in the field H. For the T.R.M., we would have M(t , oo) = 0 and M(t = 0) = M eq (H) and we
would measure the frozen part M2 in zero field :
If P( Weff) was field independent we would have
MH + T.R.M. = Meq(H). The T.R.M. would be the difference between the master curve and the equili-
brium curve i.e., a priori, Mf.,r - MH(T Ln (t/io)) on
figure 2. In fact, this difference, after one minute at 1.9 K, is of the order of 0.4 e.m.u./g i.e. considerably
smaller than the T.R.M. measured in the same condi- tions (~ 0.5 e.m.u./g, see insert of Fig. 2). This diffe-
rence signals the field dependence of the distribution
PH(Weff). However the general behaviour of
Me.c. - MH(T Ln(t/io)) is very similar to the general shape reported in T.R.M. studies [1] for comparable
field values. This raises the question of the detailed
shape of the MH(T Ln (t/io)) dependence. We observed
in reference [1] that the remanent magnetization, when
it is saturated, is well described by an exponential of We :
in a large but limited temperature range (T,,,13
T 2 Tr ,/3). In the unsaturated regime, the T.R.M.
is found linear in We [ 1 ). Here we see, from the insert of figure 2, that the T.R.M. is almost saturated in 10 kOe for T Ln (tlto) - 100 and our data of figure 2
would be consistent with a crossover from linear to
exponential behaviour occurring at about T Ln (t/To) -
100 K. Our ultimate goal is to have a better knowledge
of such questions through the determination of the field and magnetothermic history dependence of the shape of the master curve. It is clear, from figure 2, that the T Ln (tl7:o) scaling provides an essential simplification and permits exploration of an unmatch-
ed range of T Ln (t/io) values. Of primary interest
are the deviations from T Ln t/io scaling which must
occur, near the transition temperature, where the system is not able to maintain the large energy values
(W = kT Ln (tlto) - 30 kT ) essential to the validity
of the model. Our aim in the present paper is merely
to illustrate the advantages and the limits of this new
presentation of the experimental evidence. We will
for this purpose use the magnetization data on a 5 at %
CuMn sample already presented in figure 1 and some
remanent magnetization data on a 8 at % CuMn sam- ple. It is clear, that the qualitative point which we
are making here is not restricted to one concentration of a given system but should be generalisable to all spin glasses in the same range of TIT,,, values.
4. Grandeur and servitudes of the TLn (tlto) scaling.
The unity of the T Ln (t/io) scaling permits one to
condense a multiplicity of different solution which
are observed in terms of time at different temperatures
or in terms of temperature at different times. On the top of figure 2, we have indicated two actual time
scales tl and th corresponding respectively to a lower
and a higher temperature Ti and Th. We have chosen
Ti 1 = 3.16 K and T h = 4.01 K; then 1 second at T h corresponds to 52 min at TI. With these two scales
1813
we can illustrate three spectacular consequences of the T Ln (t/io) scaling.
(i) Point A, for example, is reached at Th one
minute after applying the field. Having measured the
curve MH(TH Ln (t/io)) at Th and given point A, we would correctly ascribe a time of 6.6 days to the initial field move which started the relaxation if we are
certain that the sample was always maintained at the lower temperature Tl.
(ii) Now, an extra 10 minutes at Th brings the magnetization from point A to point B producing the
same effect that 122 days would produce at T,. Thus,
the effect of a short annealing at a higher temperature is equivalent to shift the lower temperature time scale.
This point is illustrated with the data of figure 3. A heat pulse of a fraction of Kelvin during 10 to 100 s is suffi-
cient to shift by 100 min the time decay of the remanent magnetization observed at 4.2 K on a CuMn sample.
Fig. 3. - The decay of the saturated remanent magneti-
zation MRS of a CuMn 8 at % is observed at fixed tempera-
ture (4.3 K) vs. time t. A short annealing (- 50 s) at a slightly higher temperature (a few 10-1 K), produces an effect
similar to a long time (100 min) elapsed at constant T.
(iii) By contrast, by cooling the sample to a lower temperature, we stabilize a fast relaxation. This effect is illustrated with the data of figure 4. The remanent magnetization of a CuMn 8 at % is found to be stable during a long excursion from 25 K to 6 K and, in
addition the time and temperature dependence is
unaffected when we return to 25 K.
Incidentally, the facility which we have to modify
the time scale by changing the temperature permitted
us to explore the totality of the relaxation shown in
figure 2 despite the narrowness of our measuring
window. This window (typically 1 min to 100 min)
is marked at each temperature by the short segment where our data points are compressed in figure 2.
There are also quite a number of considerations which must be present in the mind of whoever wants to extract all the information that the relaxation data
Fig. 4. - The temperature dependence of the saturated
remanent magnetization MRS of a CuMn 8 at % sample
is shown around 25 K. The heavy dots below 25 K and the
light dots above 25 K were measured one min after the observation temperature was reached by increasing values.
They align smoothly on a single curve despite the fact
that a long time (75 min) has elapsed at 25 K producing an
observable relaxation and that a long excursion was then performed to much lower temperatures (6 K). The latter
treatment freezes the evolution of the T Ln t variable and the magnetization is unmodified when we come back to 25 K.
can provide. One such consideration is related to the definition of the origin of time. In practice, the variation
of a field and of a temperature involves some time interval which affects the definition of time zero.
However, in our experimental conditions, we have always found that we make a very small error by taking as time zero the precise moment ti where the field is stabilized at its final value. We justify this point in Appendix I.
Another consideration is the necessity to achieve
very good stability of both the temperature and the external field In particular, a low drift of the field affects the relaxation, while at the same time it drags
the huge reversible magnetization M1 (M,(H) o M,(H + AH(t))) : the latter effect can be important enough to completely hide the slowest relaxation discussed above. This point is well illustrated in
figure 5. In region II, the field and the temperature
are stable, and we observe the correct relaxation of the
magnetization. The observed relaxation tends to be inhibited by a very slow decrease of the magnetic
field (region I) while it is enhanced strongly by a slow
increase of the temperature (region III). Those effects,
if uncontrolled, hamper the determination of the relaxation M(H, T Ln (t/io) ) or of its logarithmic time
derivative S =
dM
specially in a range (T -+ T,,:)dLnt ’ p
where the greatest interest would be taken in their determination [7, 8] (see Fig. 6).
Fig. 5. - The time evolution of the field and of the temperature recorded during relaxation observed on our 5 at % sample
at 1.3 K (see also figures 1 and 2). Only in regime II are the field and the temperature strictly stable. A slight drift (4 x 10-4)
of the field in regime I or of the temperature (2 x 10- 3) in regime III produces measurable effects on the relaxation : hence the irregular behaviour of this relaxation as compared to other relaxations shown on figure 1 where the field and the tem-
perature were more accurately controlled.
Fig. 6. - The slope S =
dM
dLnt versus T at fixed time in field H = 10 kO: as deduced from the master curve of figure 2.A more relevant plot is that
of S _ dM
vs. T Ln t/io shown in figure 6b which condenses all the time and temperature dependences in constant field H. Notice the quasi-exponential regime following an initial plateau. We would hardly, fromthese data, dare to deduce a T(H) value where the cancellation of S would signal a transition to the paramagnetic regime (A.T. line) (see references [7, 8]).
