Rela,,~ation Phenomena of Random Spin System

in Itinerant Magnetic Fex TiS2

Y oshiaki Hara

Itelaxation Phenomena of Random Spin System
in Itinerant Magnetic FexTiS2

Y oshiaki Hara
Department of Materials Science, Faculty of Science, Hiroshima University ,Higashi-Hiroshima 739, Japan

Abstract
Relaxation phenomena of zero-field-cooled isothermal remanent magnetization(IRM) and field-cooled thermoremanent magnetization (TRM) in spin-glass (SG: x =0.20 and freezing temperature Tg = 41 K) and cluster-glass (CG: x = 114, Tg = 53 K)
phases of intercalation compound Fex TiS 2 have been studied using the anomalous
Hall effects over the time range 10- 1 - 104 s with waiting time tw=180 - 18,000 s at
low temperatures T below TIT g - 0.7. After an applied magnetic field is switched
off, the time decay of Hall resistivity (or remanent magnetization ) follows a powerlaw of the form PH = Arm within a limited time span, where A is a constant and m an
exponent. In both IRM and TRM, the exponent m depends appreciably on theapplied magnetic field intensity and temperature, for which we have obtained someuniversal relationship with a newly introduced quantity of "relative relaxedmagnetization" . Furthermore, the decay profiles over the wide time range, w here thedeviations from the power law occur, are analyzed using the existing "domain theory"
with some modifications of the theoretical expressions.
With the evaluatedparameters, discussions are given for the equilibrium relaxation spectra, overlap
length, and time-dependent maximum relaxation times that characterize the domain
growth and the dynamic properties in this material system.

1. Introduction
Intercalation compound Fex TiS 2 , formed by insertion of Fe atoms into van der Waals gaps ofthe host 1T-CdI2 type TiS 2 crystal, is an 'itinerant' magnetic material, whose salient feature ischaracterized by the presence of strong hybridization among the guest Fe atoms and the neighboring
host Ti and S atoms. 1) This material exhibits various magnetic phases depending on the guest Fe
concentration x, such as paramagnetic (x < 0.01), spin-glass (SG; 0.01 ~ x ~ 0.20), cluster-glass(CG; 0.20 < x ~ 0.40), and ferromagnetic phases (x > 0.40).2,3) In its SG and CG phases, it shows
a' cusp' or peak in the temperature dependence of static magnetization (J or ac magnetic susceptibility
X and a field cooling effect of the magnetization,4,5) similar to various SG and CG materials. For
example, Fig. l(a) shows the temperature dependence of the field cooled (FC) and zero-field cooled(ZFC) magnetizations (J for SG measured at H = 0.02 T, applied parallel and perpendicular to the caxIS.
For the case of HI/c, the ZFC curve shows a characteristic cusp, while the FC curve has aplateau of 0.2 emulg, followed by a paramegnetic decrease with raising temperature, indicating that theiron spins are frozen and align preferentially along to the c-axis by the FC process. On the other
hand, in the field perpendicular to the c-axis, both FC and ZFC (thogh not shown here) curves havevery small valuse of (J less than 0.02 emulg without showing any cusp. Thus Fex TiS 2 is a veryanisotropic spin-glass material with the easy axis parallel to the c-axis, quite similar to a typical Ising
spin glass of FeO.5Tio.503 .6) Figure l(b) illustrates the ZFC and FC curves for a cluster-glass
sample Fe 114TiS 2 at H=0.02 T for a parallel field direction (HI/c). The ZFC curve shows a peak,
while the FC magnetization decreases gradually from 9.6 emulg at 4.2 K as the temperature is raised.
As shown above, the field cooling effect for the SG and CG system is quite different. For these SG
0.3

10

a

\

Fe O.2TiS28

H = 0.02 T

...

Fe

0.2

b

Fe 1/4 TiS 2

H = 0.02 T

6

Cl

::J

E
C1>

4

b

0.1

2

J.
0.0

0

20

0
40

60

80

0

20

40

60

80

T(K)

T(K)

Fig. 1. (a) Fi.eld cooled (FC) and zero-field cooled (ZFC) magnetizations at H = 0.02 T, applied parallel andperpendicular to the c-axis, for a spin-glass sample Feo.20TiS2 and (b) those for a cluster-glass sample
Fel/4TiS2 at H = 0.02 T applied parallel to the c-axis.

2and CG phases of itinerant magnetic compound Fex TiS 2 , we have studied the dynamics of zero-fieldcooled isothermal remanent magnetization (IRM) and field-cooled thermoremanent magnetization(TRM) using a trans port method , in contras t to magnetic measurements done for various localized
systems, since this material show s an anomalous Hall effect, in which a Hall resi tivity PH can be
expressed by the well-known form,(1)
where RO and Rs are the normal and extraordinary Hall coeffic ients, respectively, and M(l{) themagnetization at magnetic field H. When an applied field H is turned off, the Hall resis tivity PH(t) is
proportional to the remanent magnetization Mr(t) at time t, as(2)

Now w ith regard to the dynamical treatments of various SG systems , there are two different
viewpoints. The one is the mean-field approach of Sherrington-Kirkpatrick (SK)7) and its replica
symmetry solution by Parisi 8,9), giving an infmite number of quas i-equilibrium states which are
hierarchically organized in a phase space - "hierarchical kinetic model". This model has been shown
to be valid for relaxation of TRM in insulating SG of Cd -Cr1.7Ino.3S410, 11) and dilute alloy Ag : Mn
(Mn 2.6 at. 0/0), 12) while from numerical and theoretical studies , Newman and Stein have pointed outthat the Parisi solution to the SK model cannot apply to short-range SG . 13) The other is aphenomenological approach b as ed on the existence of a distribution of droplets 14) or dynamical
domains, 15) which has been applied to the interpretation of the aging and time decay of TRM forvarious SG systems, such as CdCr1.7InO.3S41S) and Cu (10 at. % Mn) 16). The former model isconcerned with, in particular, the temperature cycle in the aging effect of TRM, while the latter deals
with its time dependence, which shows the existence of a clear crossover from dynamical processes
characterized by length scales smaller than the already achieved domain size (quasi-equilibrium
regime) to processes on larger time scales dominated by the continuation of domain growth throughthe movement of domain walls across the system (non-equilibrium regime). In the latter picture,aging is a manifestation of a slow domain growth below a glass temperature, where after certain
waiting time tw a characteristic domain size is reached. However, the time dependence of a domain
size s(t) is treated differently; Fisher and Huse (FH) 14) suggest a logarithmic dependence set) oc(logt) l/ljI from an acltivated dynamics scenario, while Koper and Hilhors t (KH) 15) assume a power lawset) oc !P.
In order to understand in some detail the dynamical nature of SG and CG phases of Fex TiS 2 , inthe present study we have carried through the dynamic me as urements of TRM and IRM through thetime decay of Hall resitivities and discussed our results within the framework of the realistic KH
domain theory .

2. Experimental
Single crystals of Fex TiS 2 were grown by a chemical vapor transport method ; 17) the glass
temperature for SG (x = 0.20) and CG (x = 114) is To = 41 and 53 K , respectively . Ohmic contacts
to the sample with a six-probe for Hall effect measurements were made by soldering an indium metal.
b3The procedures of dynamical measurements are T
shown schematicaly in Fig. 2. For a zero-field cooled Tg
IRM measurement (solid lines), a sample was cooled at
a constant rate (l Klmin) from a temperature 1.5Ta
down to a working temperature T «0.7Ta) without antexternal magnetic field (ZFC), followed by anHapplication of a pulsed magnetic field H p (~ 20 T)
applied parallel to the c-axis (magnetic easy axis) of theFC-------crystal after a waiting time for tw (=1800 s). Timedecays of the Hall voltages were measured using a
ZFC
digital storage oscilloscope (in time range 0-50 IDS) ort
by a nanovoltmeter (0.1-1000 s)~ both sample current
PHand magnetic field directions were reversed to exclude
any spurious contributions from misaligned contacts and
/' .
/'
:
/'
thermoelectromotive force generated at the I nIF ex TiS 2
/'
/'
t
..('"
interface agains t the Hall voltages. On the other hand,for a field-cooled TRM measurement (broken lines), thetsample was cooled similarly a temperature T under aFig. 2. Schematic measurement procedures
static magnetic fielld H FC ( =0.01-0. 14 T). Then after a·of ZFC-IRM and FC-TRM; (a) time
variation of temperature, (b) the external
wating time tw (=180, 1800, 18000 s), the magnetic
fields, and (c) the Hall resistivity correfield was switched off to zero and subsequently the timesponding to the magnetization of sample.
decay of TRM or the Hall voltage was recorded using
Solid lines for IRM measurements anddotted lines for TRM one.
the same measurement system as used for IRM
measurements over the time span up to 10,000 s. Afterthe measurements the sample was warmed up to 1. 5Tg in zero field and then the reversed magnetic
field was applied, followed by the above procedure to exclude any spurious signals.
A pulsed magnetic field was produced by a home-made condenser bank system, as shown in
Fig. 3, where a silicon thyristor is used for a switch, a series of electrolytic condensor banks are
1;)

1;)

~--

...----1 Control Box t - - - - - - - - - ,Ext.

.---+--, Trigger

20
DC Power
Supply
Pulse
Coil

Diode
L - -_ _- - - I~>----C-o-nd.....e-n-s-e-r_ _--'-_ _- - J

:I:

10

o

20

40

t (ms)

Fig. 3. (a) Schematic circuitof the system of application of pulsed magnetic fields. (b) Wave
fonn of pulsed magnetic field used in IRM measurements, wave duration are 30-75 ms.

charged through a high de power supply (charge capacity of 24 kJ at 1350 V), and a silicon diode isinserted parallel to a pulse coil in order to avoid the applied reversed voltage on the electrolytic
condensor. The pulse magnet itself was made of a solenoidal coil of copper wires (1.5 mm in4

diameter) wound around a fiber-glass reinforced plastics (FRP) bobbin; its inner and outer diameters,
and length are 25, 43, and 67 mrn, respectively. A typical pulsed magnetic field is depicted here,
having a half-wave pulse duration of 35-70 ms.

3. Res ults and Discussions
3.1.1

Isothermal Remanent Magnetization (IRM)
Figure 4 shows the recorder traces of the magnetic field dependence of (a) Hall resis tivity PH of

SG (x = 0.20) and (b) of CG (x = 1/4) at 4.2 K,
20~--~----~--~----~--~

res pectively. These PH - H curves show anappreciable hysteresis , corresponding to themagnetization curve:s;4) here only the 1st and 4th
quadrants of one complete cycles are depicted.
After the field sweep cycle, PH at H = 0 decays
with time t, indicating the exis tence of a relaxation
process in the transport quantity, which reflects aremanent magnetization M , as mentioned above.
We see that for SG, at higher magnetic fields H >8 T well above the saturation of magnetization,
the PH - H curve is almost linear with H having anegative slope, similar to the CG case s hown in
Fig. 3(b) .
Using Eq . (1) and the observed
magnetization curve" 4) we can estimate the valuesof RO = -7.4 x 10-9 and Rs = 1.3 x 10-6 for SG,
and Ro = - 1.2 x 10-9 m 3/C and R s = 1.0 x 10-6
m3/C for CG at 4.2 K, respectively. Theseexperimental results indicate that the dominantconduction carriers are electrons (RO < 0) and thedensity of state at dhe Fermi level for the downspin band is expected to be higher than that for the
up-spin band (Rs > 0) , according to Kondorsky
criterion for anomalous Hall effect. l8 , 19)

met;'.'.

a

15
10
5

E
C

O~~~------------------~

-5

SG

ex>I

-10
x 15

o

b

-- --

10 - .....

a.I_. -. ... ..-

-.

"":""

~.

~

5
0

.

-5
-10
-15

."
~~....~
0

2

CG
4

6

8

10

H (T)
Fi g. 4.

Recorrer traces o f the Hall res istiviti esPH( H) at 4.2 K for magnetic field sweeps up to

8.2 T, (a) for SG and (b) for CG samples .

After a pulsed magnetic field is switched off, the time decay of Hall resistivity is measured.
The experimental results for CG phase at 4.2 K and various peak fields Hp are illustrated in Fig. 5(a),where the Hall resistivities PH(t) at time t are shown over the time ranges 0.1-1000 s in logarithmic
scales. As can be seen, all the curves in the whole time range studied follow the power law of theform ,(3)

where A is a constant and m an exponent. In Fig. 5(b) are shown similar decay curves at various
temperatures over the time intervals up to 100 s for the field sweep up to Hp = 8.2 T . The decay510·1 ~ ...._

S.2T

9.5

7.01 .....-

E

c:

3.5

6.5

co

::,

1.5

X

1

T~

4.2 K

10

. .. -.

30.0 Ko~~oo~

..-~
....~

E

c:

. .. -..

1.0

b
-7

- 10

-8

....

..'~

a..'X.
a..'X.

1

0.5

0.05 ......

.....

--...~

I

10

0

1.4T~

:

I

0.01
-1
10

•

10

1

10

0--.

-9

10

10

-1

3

10

10

(s )

t

'

!
2

.....

42.5 K

0

10
t

1

10

2

(s )

Fig . 5 . (a) Time decays of the Hall resistivities PH(t) at 4.2 K for various sweep fields Hp for CG
sample and (b) those at different temperatures for the fixed field sweep up to Hp = 8.2 T.
20

25

c

at
20
~
0

15

:SG
:CG

15

X

~

0

X 10

10

E

E

5

5

0

0

12

12

b
E
C

9

E
C

co

9

co

6

6

0

X

X

a:x.

3

a..r. 3

0

0
0

2

4

6

8

0.2

10

Hp (T )

0.4

0.6

0.8

T ITg

Fig. 6. (a) The values of the exponent m of SG (solid circles) and CG (open squares), (b) theHall resistivity PH(t) at t = 1 s [PH ( 1) =A] in eq. (3) at 4 .2 K plotted against the peak field
Hp. (c) The values of m and (d) the Hall resistivity PH(\) for the fixed peak field Hp =8.2
T plotted against temperature; PH(l) for CG is equal to A at low temperatures T ~ 32.5 K,
which does not hold at higher temperatures T ~ 35 K.

6

curves follow well the power law of Eq. (3) at low temperatures T :s; 32. 5 K (as the temperature is
raised, the value of PH(t) is decreased and the exponent m is increased), while those at higher
temperatures T ~ 35 K cannot be expressed by the simple power law ; PH(t) at the highes t temperatureof 42.5 K is very small, but it is apparent that there still exists a very slow relaxation proces .
Though not shown here, systematic measurements of both H p- and T - dependencies of the decay
profiles were also performed for SG phase.
In Figs. 6(a) and 6(b) are plotted the Hp dependence of the exponent m and the Hall resistivity
PH(t) at t = 1 s [PH( 1) = A in Eq. (3)] at 4.2 K, respectively; solid circles for SG and open squares for
CG. One notes that with increasing Hp these quantities are varied smoothly up to the coercive force
He (= 3 T), and PH(1), in particular, shows a drastical change near H c' which corresponds to themagnetization curves. The temperature dependence of these values for the peak field Hp = 8.2 T isshown in Figs. 6(c) and 6(d). In the case of SG , the exponent m increases linearly with T, while for
CG it does not show a linear temperature dependence but appreciable increase with T up to 32.5 K;
correspondingly the values of PH(1) (= A for T :s; 32.5 K) show a drastic decrease near 32.5 K. We
should note that in the whole range of magnetic field and temperature measured, the exponent m for
SG is larger than that for CG, indicating that the relaxation of SG is faster than that of CG. To obtainany general behaviors involved, we have further analyzed the above experimental results in thefollowing way.

3.1.2

Relative Relaxed Magnetization for IRM

The dynamical properties of IRM in SG and CG phases are strongly depend on an appliedmagnetic field intensity, temperature, magnetic history, and a waiting time [the time spent at atemperature T after cooling a sample below Tao until the field is switched off (for the case of TRM) or
turned on (for IRM)]. 20) For discussions describedabelow, we show in Fig. 7 the schematic time profile ofHp(a) a pulsed field fl and (b) a magnetization M(t) attime t, normalized by the value at the peak field H p'
M(t)/M(H p)' where the origin of the time t = 0 and atime tz are defined when the pulsed field is switchedon and off, respectively. The magnetization decaysor relaxes with time from M(Hp) to zero at infinite timet -7 co. Now we introduce a new quantity ¢ (t) at

QL......--j-_ _ _ _ _....:::::::........_ _ _ _. . .

a.

J:time t after the pul';ed field is turned off, which werefer to as a "relative relaxed magnetization" (RRM).
Then from Eqs. (1) and (2), ¢(t) is expressed as,
Q~~------~--~-~

Q(4)

where Mr(t) is the remanent magnetization at time t;
for simplicity, we write the value at t = tz as ¢ (tz) = ¢.
7

Fig. 7. Schematic time profiles of (a) the pulsed
field H and (b) the magnetization M(t) at tim e
t nonnalized by the value at the peak field H p,
M(t) IMCHp), where the origin of the time isset when the pulsed field is switched on andtime t z is refined at pulse fi eld H is swithcedoff (see text).

Figure 8(a) shows the observed values of Hall resistivities PHCtJ at t = t z and PH(H p) at the peakfield Hp at 4.2 K plotted as a function of the peak field for CG phase. Using Eq. (4) with thesevalues and RO (= -l. 2 x 10-9 m 3/C), we have evaluated the values of RRM at t = tz' ¢, as depicted inFig. 8(b), where we see that the peak field dependence of ¢ is appreciable near the coercive force H c (=3 T) and it becomes constant at higher fields Hp > He- The temperature dependence of thecorresponding quantities for the fixed peak field Hp= 8.2 T are indicated in Figs. 8(c) and 8(d); thevalues of ¢ are evaluated by Eg. (4) assuming that the normal Hall coefficient RO is temperature
independent. With increasing temperature, PH(H p) is increased gradually to a constant at higher
temperatures, which is considered to be due to the temperature dependent term of Rs in Eg. (1), while

PHCtJ is decreased; correspondingly ¢ is increased appreciably from near zero at low temperatures,
approaching near unity at higher temperatures.
12

14

c

a

12

10

E

E

c:

8

c:

10

or0

6

or0

8

~

~

X

4

a...r.

X

6

a...r.

4

2

2
0

0

1.0

1.0

d

0.8

0.8

0.6

0.6

-e-

-e0.4

0.4

0.2

0.2

0.0

0

5

10
Hp

15

0.0

20

(T)

0

10

20

30

T

( K)

40

50

Fig. 8. (a) The values of PH(tz) at t =t z , PH(tz) (solid c ircles), and the Hall resisti vity a t the peak field PH (Hp )(open squares) for CG samples and (b) the values ofRRM at t =t z , 1> defined by Eq. (4), at 4.2 K plottedagainst the peak field Hp . (c) The values of PH(tz) and PH (H p), and (d) 1> for the peak field Hp = 8 .2 T
plotted against temperature.

Furthermore, in order to look for a more general relation between the dynamical quantities, in
Fig 9(a) we have plotted the values of m as a function of ¢ at the constant temperature 4.2 K (opencircles) and at the constant peak field Hp = 8.2 T (solid circles) in logarithmic scales. In both cases,
the experimental points lie well on two lines with different slopes, where the two lines cros s at thecommon characteris tic point ¢c = 0.34, as marked by arrows . At first sight, these two curves (solid8and open circles) are apparently different. Then in order to express the dynamical quantity m in theform of a universall function of ¢ , we have introduced a "scaled quantity" mT a , as(5)where a and f3 i are the exponents characterizing the dynamics of our CG system and C i is a constant(i = 1 for ¢ < ¢ c , i = 2 for ¢ > ¢ c), Figure 9(b) illus trates the values of m T a plotted against ¢with the best-fit value of a = -112 for the CG system. It is to noted that the experimental points (solidand open circles) lie well on a universal line with two different slopes f3 1 for ¢ < ¢ c and f3 2 for ¢ >Fig. 9.

Theexponentm in Eq.(3)atthe fixedtemperatureT=4.2 K (open circles) and atthe fixed peak fieldin logarithmic scales. Arrows mark thecharacteristic point <Pc = 0.34. The values of m T a plotted as a function of <P obtained at constanttemperature T = 4.2 K (open circles) and peak field Hp = 8.2 T (solid circles) for the CG system (x = 114)of the itinerant magnet ofFex TiS2; the inset shows the enlarged plots for the SG system (x = 0.20). The
best-fit values of a forCG and SG are -112 and 114, respectively. Arrows mark the characteristic point<Pc (=0.34 for CG and 0.90 for SG) (see text).

Hp = 8.2 T (solid circles) plotted as a function of <P

The best-fit values of f3 i and C i are listed in Table 1. For comparison, similar plots for the SG
system (Fe x TiS 2 ; x = 0 .20) with the best-fit value of a = 1/4 are also depicted in inset of the Fig. 9(b).
In this case the expression of Eg . (5) is also satisfied with a = 114 and the characteristic point of ¢ c =¢ c'Table I. Characteristic parameters in Eq. (5) characterizing the dynamical properties for the
CG and SG systems of Fe x TiS 2 (see text); Cj in units of Ka
System
CG
SG

a
-112
114

0.34
0.90

0.60
2.9

2.9
8.9

0.0087
0.19

0.10
0 .38

0.90, as indicated by arrow. Table I compiles the characteristic parameters for both CG and SG
systems. It should be noted that the exponent f3 2 for the CG system has the same value as that of f3 I
for the SG one in the restricted range of ¢ , 0.34 < ¢ < 0.90.
9In the following, we shall discuss the physical meaning of the characteristic value of ¢ c (= 0.34
for CG and 0.90 for SG) obtained above using the concepts of a simple Ising spin model and a sitepercolation picture.2 l ) For simplicity, we flfst consider a magnetic system consisting of N Ising
spins. At the initial zero-field cooled state (H = 0, T < Tg) , a magnetization M is zero, because thenumbers of the up- and down-spins are equal (= N /2). When an external field is applied up to a peakfield H p' the number of the up-spins is increased by an amount of n, while that of the down-spin isdecreased by n , and thus the magnetization at H p is proportional to 2n [M(H p) ex: 2n]. After theexternal field is switched off at t = tz' the spin system will relax with time t (Fig. 7). At t = tz ' apartial amount of up·-spins is decreased by an amount of n 1; the corresponding number of the downspins is increased by n 1. Then the remanent magnetization MrCtJ at t = tz is proportional to 2(n - n ,)[MrCtJ ex: 2(n - n 1)]' By definition, therefore, we have(6)

On the other hand, such a spin system can be described by a site-percolation picture, as follows.
The spin state at Hrp corresponds to a lattice with 2n sites, while the magnetic state at t = tz
corresponds to the situation, in which the n 1 sites of the lattice are occupied. Therefore, the volume
fraction v p is written by v p =n /2n = ¢ /2, where v is a filling factor of the lattice and p the fraction of
the filled sites . In our case, the experimental values of ¢ el2 for CG and SG systems are evaluated tobe 0.17 and 0.45, respectively , which are in good agreement with the well-known values of the"critical volume fractions" 0.16 ± 0.02 for a three-dimensional (3D) lattice and 0.45 for a twodimensional (2D) one. 21 ) In other words , we may say that the "percolation path" produced by theoccupied n 1 sites (or the inversion of the n 1 spins up to t = tz ) is of 3D nature in the regime 0.34 < ¢< 0. 90 and of 2D one in 0.90 < ¢ < 1 .
Therefore, the fact that the exponent f3 i for both CG and
SG systems (Table I) has the same value in the regime 0 .34 < ¢ < 0.90 suggests that f3 i is ameaningful parameter to characterize the dimensionality of the percolation path.

3.2.1 Relaxation of Thermoremanent Magnetization (TRM)
Figure 10 shows the typical time variations for SG at 16. 8 K of (a) the cooling-field H FC = 0.14
T and (b) Hall resistivity, PH - RoH, where the time is set zero when the cooling-field begins todecrease; the normal Hall coefficient has been determined to be RO = -7.4 x 10-9 m 3/C, which is
independent of temperature. The applied magnetic field H is decreased from the cooling-field H FC(= 0.14 T) at a constant rate 8H (= 10 Tis) and vanishes at time tz (= HFC/8H = 14 ms), as indicatedby arrows. We should note that t z is very short compared with a waiting time tw (= 180-18,000 s)
and the time span of relaxation measurements (0-10 4 s). As the magnetic field is decreased to zero,the Hall resistivity decays with time from PH FC - RoHFC = 1.4 x 10- 8 at t = 0 to PHCtJ = 1.15 x 10- 8
12m at time t z' and in zero field it continues to relax slowly. [Solid curve in Fig. 10(b) is calculatedusing Eqs. (24) and (25), which are obtained from the concept of domain theory, with the best-fit
parameters (see later)].
Similar measurements were carried out at various cooling-fields and temperatures. In Fig.
11(a) are plotted against H FC the values of PH FC - RoHFC and PH(tJ at TITa == 0.6 [T = 24.6 K for
SG (solid and open circles) and T = 32.4 K for CG (solid and open squares)]. As the cooling-field
b

10

0.2
~

a

0.1

:I:

0
b

E

c:
to

'0

><
:I:

0

ex:

1.0

a..r.

0.8

0

10

20

30

40

Fig. 10. Typical time variations at 16.8 K of (a) theexternal magnetic field from cooling-field HFC(i nitially at 0.14 T) and (b) the Hall resistivity (PH RofI) corresponding to the magnetization M(t).
Solid curve in Fig. lOeb) is calculated using Eqs. (24)and (25) with the best-fit parameters m = 0.027, to =7
x 10- 6 s, pz =0.9, t2pz1tl =0.0075 spz - I , and tt;H =40s (see text).

t (ms)

increased, t~e value of p~C - RoHFC is increased, indicating the enhancement of themagnetization, while the value of PHCtz) tends to saturate at higher fields. The difference betweenIS

1 ~----~----~~----~----~

10

a

b
0.8

8
a.

E

:r:

C

o

a: 0.6

6

C?

-

0

a.

-

~

:r:

X

4

a..:::I: 0.4--

a..:::I:2

0.2

o ~----~----~~----~----~
o
0.2
0.4
0.6
0.8
T IT gH Fe (T)Fig. 11. (a) Magnetic field dependence of the Hall resistivities PHFC - RoHFC and PH(tz ) at TIT g == 0.6(T =26.8 K for SG and those for CG =32.4 K) and (b) temperature dependence of PHK: - RoHFC andPH(tz) at a fixed magnetic fields HFC = 0.14 T .

PHFC - RoHFC and PHCtz) becomes appreciable with increasing H FC ; this difference is regarded as aquantity representing a 'relaxed' magnetization until the magnetic field vanishes (see later). On theother hand, in Fig. 11(b) we plot the temperature dependence of PH FC - RoHFC and PH(tz) ,normalized by the saturated values PH(H p) - RoH p using the data at 8.2 T (see Fig 8) , at fixedmagnetic field H FC = 0.14 T. The ratio [PH FC - RoHFCJ/[PH(H p) - RoHpJ shows a drastic decreaseat higher temperature for CG, in contrast to almost temperature-independent behavior of SG that11

reflects the temperature dependence of the magnetization (Fig. 1). In addition, we see that that ratio
for SG are extremely small compared to that for CG.
Figure 12 illustrates the typical decay curves of Hall resistivities or TRM for (a) SG at 16.8 K
and (b) CG at 32.4 K after the cooling-field is switched off with the waiting time tw=1,800 sunder
different field intensities H FC in log-log plots, where PH is normalized by the initial value (PH FC Rof!p). Here the Hall resistivities PH(t) at time t is expressed by Eg. (2) with H = 0, as PH(t)=4nRsMrCt), where M'rCt) is TRM at time t. As can be seen, in both SG and CG, all the experimental
points lie on straight lines up to a lapse time td marked by arrows, beyond which the deviations become
appreciable, which are in sharp contrast to the case of IRM. As done for a SG material of
EU0.4SrO.6S4,22) we define the time td, at which the deviations from Eq. (3) begin to increase morethan a quarter of the mean square errors, and we simply call it a 'deviation time' to specify the decay
profiles for TRM. For t < td, the decay curve can be written by the power law of time as Eg. (3),PH(t) = At -m with a constant A and an exponent m. Moreover, we have found no significant
0.9

a

b

0.8
0.7

0.8

0.6
U

LI.

:I:

0.5

o

0.7

II:

u

Ll.J:

0.4

a..:

0.3

SG

CG

T=16.8K

T =32.4 K
0.2

10

-1

10

1

10

3

10

5

10- 1

t (s)

10

0

10
t

1

10

2

10

3

(s)

Fig. 12.

(a) Time decay of the Hall resistivities PH for SG at 16.8 K with tw = 1800 s under various fields
HR:., norrnalized by (PHR:. - R oHR:.) at HR:. in log-log plots, (b) those for CG at T = 32.4 K. Arrows
mark the deviation time td, above which the experimental points deviate from the power law ofEq. (3).
Solid curves in Fig. l2(a) are calculated using Eq. (25) with the magnetic field-independent parameterspz = 0.9, l2pz/t, = 0.0075 sPz- 1, and field-dependent ones m, (d , and t6H (see text),

difference in the decay curves observed at different waiting times tw (180 - 18,000 s) in theexperimental conditions of H FC (= 0.01-0.14 T) and temperatures T (= 7.8 - 29.3 K) for SG; we havenot measured the waiting-time dependence of TRM for CG. [We shall discuss later these results interms of an "overlap length" introduced ,by Bray and Moore.2 3)] The exponent m for TRM is plottedagainst (a) the cooling-field at TIT g = 0.6 and (b) against temperature at H FC = 0.14 T in Fig. 13. It
is apparent that the exponent m for SG increases linearly with increasing 'the cooling-field and it also
increases drastically as the temperature is raised toward T,,; the behaviors for CG are almost the sameas those for SG.
0

12

16

16

•

b

a

H FC = 0.14 T
12

12

8

8

4

4

~

0

~

x

E

o

o

o

0.04

0.08

0.12

0.16

o

0.2

0.8

0.6

T/T g

H FC (T)Fig . 13 .

0.4

Variations of the exponent m with (a) the cooling-field HFC at various temperatures and (b)with temperature at different cooling-fields.

3.2.2 Relative Relaxed Magnetization for TRM
F or our discussion below, Figs. 14( a) and 14(b) illus trate the schematic time profiles of (a) an
applied field for field-cooling H FC down to a working temperature T «To-)and (b) a magnetization
o
a
H FC
:J:

0

t
b

u

1

u.

-::E

Fig. 14. Schematic time profile of (a) a field-coolingmagnetic field HFC and (b) a magnetization M(t) attime t and temperature T « Tg), normalized by themagnetization MFC at HFC, M(t)/MFC , where theorigin of the time is set when the applied field is
switched off to zero; CP' is refined by Eg. (7). (c)Schematic
tern perature
repenrenceofthemagnetizations M(tz) at time t = t z and M Fe at H =H Fe ,normalized by the value MFC at absolute zero [hererenoted by MFC(T -70)]; solid line indicates
Mr:c/MFC(T --,,0) and broken one M(tz) /MFC(T -70).
~ is a newly introduced quantity to characterize thedynamical properties for TRM, defined by Eg. (8).

M(tz)/M

~

::E

0

t

1

0

:\

.....to\U

M (t z )M FC ( T~O)

U.

::E

\

::E

c

~

\

0

,+
T

0

T

13

t

tz

0

Tg

M(t) at time t, no rn1alized by the mag netization M FC at H = H FC ' M(t) IM FC ' During the fieldcooling under H FC ' the magnetization M (t) is held almos t constant at M FC , and as the external field isdecreased at a cons tant rate to zero, it relaxes with time (Fig. 10). When the applied field is switched
off at t = tz' the magnetization attains to a value of M (tJ .

S imilar to the expression of ¢ in Eq. (S) for

IRM, w e here introdl uce a quantity ¢' for the present TRM case, w hich we refer to as ' relative relaxedmagne tization ', defined by(7)

U sing Eq. (7) with the experimental data of PH FC and PH(tJ (Fig. 11 ), we can evalu ate the values of¢' at different magnetic fields and temperatures.

As given in Eq. (S) for IRM, w e have attempted to

fmd any universal relation between the exponent m and the new ly introduced qu antity ¢' for TRM; but
in this case there is Dione.
Then in order to look for a general relation for TRM, we have taken into account the temperatu redependence of the nrragnetization in the Fe process, as shown schematically for SG in Fig. 14(c).
Here solid and dotted curves represent the magnetizations M(tJ at time t = tz and M FC at H =HFC
[Figs ~

l4(a) and 14(b )], normalized by the value M FC at absolute zero [here denoted by MFC (T -70)],

respectively.

The field-cooling under H FC proceeds along the dotted curve from a high temperature,say 1.ST 0"' down to a working temperature T, where as the-external magnetic field is decreased to zero,b

TRM is decreas ed along the dotted vertical line to the value M(tJ at t = tz (and further toward zero at
infmite time).

U sing this picture, we now introduce a new dynamical quantity ~ as a ' relative relaxedmagnetization' for TRM, as marked by arrow in Fig . 14(c), defined by~ = 1 - M(tJ I MFC (T -70) .

(8)

Since a saturation magnetization Msat(T) obtained experimentally by applying a pulsed magnetic fieldwith the peak field lfp=8.2 T is almost temperature independent below T g , ~ is rewritten to eliminate
the temperature-dependent extraordinary Hall coefficient Rs in Eqs. (1) and (2), as~= 1- [M(tJ I M sat( T) ] I [MFC(T-70)IMsat(T-70)]

(9)

with
(10)
(11 )
Using Eqs . (9)-(11), together with the experimental data of Fig. 11, we have evaluated thevalues of ~ at different temperatures and cooling fields.

In Fig. IS(a) is plotted m as a function of ~for SG in logarithrnic scales, where the experimental points lie on a single line with two different
slopes above and below the characteristic point of ~c = 0 . 21 marked by arrow , which is well describedby the following expression,14m = D~Y,(12)

where the best-fit parameters are determined to be D =0.30 and y = l.4 for ~ > ~c and D = 2.6 and Y=
2.8 for ~ < ~c' On the other hand, for CG, as shown in Fig. 15(b) in linear scales, we see that all theexperimental points lie on a single line with the best-fit parameter D =0.15 and y = l. O. Thus Eq.
(12) is regarded as a general expression to characterize the dynamics of TRM in the short time range
less than td for both SG and CG of our magnetic material and ~ is a good parameter to describe thefreezing state of SG and CG phases. We note that the y value for SG (y = l.4 and 2.8) is larger
than that for CG (y = l. 0), which indicates that the dependence of m on ~ is much stronger in theformer case than in the latter.

b

10

-1

6

N

0

E

T"""

x

4 -

E
2

SG
0.1

0.3

CG
0

1

0.0

~

0.1

0.2

0.3

0.4

0.5

~

Fig. 15 . (a) The exponents m plotted against ~ which can be expressed in the form ofEq. (l2)(see text),(a) for SG arrow marks the characteristic values of ~c = 0.21 and (b) for CG phase.

Here we discuss the quantity ~ using the concept of a single Ising spin model for TRM, where
for simplicity we consider a magnetic system consisting of N Ising spins, as done for IRM. At theinitial state above T g , a magnetization M is zero before applying H FC ; the number of the up- anddown-spins are equal (=N/2). When a cooling-field H FC is applied and the system is cooled down
to a temperature T below T g , the number of the up-spins is increased by an amount of nFC(T), whilethat of the down-spins is decreased by nFC(T), and thus the magnetization M FC is proportional to2nFC(T) . After the external field is switched off at t = tv the spin system will relax with time t. At
t = tz' a partial amount of up-spins is decreased by an amount of nH; thus TRM or M(t;) at t = tz is
proportional to 2[nFC(T) - nH]. By definition, we have

= { [MFC(T-70) - M FC ] + [M FC - MCt;)]}/ MFC(T -70).

15

(13)

We denote here the number of the increased up-spins nFC(T) at absolute zero by nFc(T~O)
corresponding to the magnetization MFC(T ~O).

The first term in Eg. (13), [MFC(T ~O) - M FC ] , isproportional to 2[nFC(T~O) - nFC(T)], which we simply write as 2nT' where nT is regarded as thenumber of thermally inverted spins out of the increased up-spins nFC(T-70). Here we may expectthat the value of 2nT of SG is much smaller than that of CG, from the temperature dependent
magnetization curves (Fig. 1).

In addition, the second term [M FC - M(t.J] , the difference in themagnetizations at temperature T before and after the external field is switched off [Fig. 14(c)], is
proportional to 2nH.

Then Eq. (13) is rewritten as,(14)

which in turn means that the quantity ~ is the ratio of the total number of the spins, inverted by thermal
excitation at H FC and by reducing the applied magnetic field, to the number of up-spins nFc(T-70)
that is increased during the field cooling down to absolute zero. Thus we believe that our newly
introduced quantity ~ is an important parameter to characterize the dynamical properties of TRM. It
should be noted that in this parameter, 'temperature fluctuation' effects are also taken into account in
the form of nT The universal relation of Eq. (12) for TRM is different from that for IRM of Eq. (5).
In both cases, however, the newly introduced' relative relaxed magnetization' , ~ and cp, are regarded
as good parameters to characterize the degree of relaxation in TRM and IRM, reflecting the magnetic
history during FC and ZFC, respectively.

3.2.3 Analysis OIf Time Decay of TRM Based on Domain Theory
N ow we shall consider the observed decay curves of TRM over the whole time range where thedeviations from the power law occur.

Since our results show the power law in a short time region,

we have analyzed them using the Koper-Hilhorst (KH) domain theory that has employed the powerlaw for the equilibrium relaxation function, 15) rather than FH one. 14) However, Koper and Hilhorst
have derived theoretical expressions with an idealized stepwise form for switching off of an externalcooling-field H FC ' whereas in our actual experiments it is decreased at a constant rate 8H to zero
within the time tz.

Thus we need some modifications of their expressions, as described below.

According to the KH model, the magnetization is expressed, by assuming that a linear response
theory is valid for relaxation of SG, as

M(t) = L1M(t) + NXeqH(t) = -NXeq

f~dt' R(t, t')H(t') +NXeq H(t),(15)

where l1MCt) is an excess magnetization, N the number of spins in a sample, and Xeq the equilibrium
dc susceptibility in zero field. Based on the experimental data24 ) and Monte Carlo simulations,25) the
relaxation function RCt, t) is written as,

R(t, t)= Rfrj(t - t)F(t, t) .

(16)

In thermal equilibrium Rfrj(t - t) is given by,16

Rm.(t - tj = [1 + (t - tjltO]-nz,(17)

where to specifies a minimum relaxation time in the equilibrium relaxation spectrum, as discussed in
section 3.2.4. A plausible choice for F(t, tj is a cutoff reflecting deviations from equilibrium, F(t,tj = exp[ -(t - tjl'rmax(s)] , where Tmax(S) is a maximum relaxation time in the relaxation spectrum of asize s domain, and a plausible generalization of F(t, tj using the time dependent domain size s((j (( <til < t) gives

F(t, t'; [s(t")]) = exp{-i~t" IT max [s(t")]}.

( 18)

t'

Due to the spin coherence within a domain, 'rmax(s) is assumed to be a function of domain size with atypical spin spacing a,(19)

where t 1 is a microscopic time constant and z a dynamical parameter.
Furthermore, Bray and Moore 23 ) have predicted that during the waiting time tw the domain sizes(t) cannot grow larger than an "overlap length" l6H for a given magnetic field jump i1H (in our case
H FC ), which is written as l6H - Ii1HI-2/(d - 2y) with the dimension of the system d and a constant y.
Taking account of the interplay between s(t) and l6H' the KH model considers the following twocases: For a small field jump, s(tw) < l6H' the domain size s(t) increases as a power of time,set) == a[(tw + t)lt2]P,(20)

where t2 is a microscopic time and p another dynamical parameter.

For a large field jump, where thelinear size of domains reaches its upper limit l6H during the waiting time tw' s(t.;) = l6H' and thus set)is written as, for t> tz'set) == a[(l6Hla) lip + (t - tJ1t2]P.

(20')

In this expression, the waiting time tw does not enter, indicating that it does not affect the domain size
after the switching-off of the external field. Since there is no tw dependence of the observed decay
curves, therefore, we use Eq. (20') for the time dependence of the domain size. In addition, a timetL\H is defined, during which the domain is growing up to a size of l6H' as(21)

With this quantity and using Eqs . (19) and (20'), the maximum relaxation time is given by ,
(22)

17

The time decays of TRM can be calculated using the above expressions. Taking account of thetime dependent magnetic field change at a constant rate 8H in our case, not stepwise as in the KH
model, we express the time variation of the magnetic field, with t z = H FC I8H, as
H(t) = H FC - t8H=0

for 0 ~ t ~ tz'

for t> tz.

(23)

In the time region t ~ t z we may take FCt, t ) = 1 since t - ( «'rillax(s ), and we get(24)
On the other hand, in the case for t> tz' F(t, t / ) 7:- 1, yielding(25)
The above integration cannot be carried out analytically and therefore we have made the numericalintegration to calculate TRM. For the numerical calculations of Eqs. (24) and (25), we have as sumed
the followin g conditions for fitting parameters: i) we take the observed exponent m , ii) pz is a universal
constant independent of temperature and magnetic field , iii) the ratio t2PZlti is a function of temperaturealone, and iv) both to and t6H depend on temperature and magnetic field.
As an example for SG, solid line in Fig. 10(b) shows the calculated curve of the Hall resistivity(orTRM) using Eqs. (24) and (25) with the best-fit values of m = 0.027 , to =7 x 10-6 s , pz = 0.9 ,t2PZlti = 0.0075 sIn -I, and t6H = 40 s in the time range t = 0-40 ms, where the cooling-field isdecreasing at the constant rate 10 Tis , in satisfactory agreement with the observation. Furthermore,
we have performed the numerical calculations of the overall time decays of TRM at various conditions(temperatures and cooling-field intensities) using Eq . (25) to obtain reasonable agreements with theexperiments, as shown by solid curves in Fig. 12(a) ; the parameters used for the calculations are thefield-independent ones (pz = 0 .9, t2pz/tl = 0.0075 sPZ - 1), and the field-dependent ones (m = 0.018 ,0.022,0.027 , to = 9 x 10-8 , 7 x 10-7 , 7 x 10-6 s, t6H = 350, 130, 40 s for H FC = 0.035 , 0.07 , 0.14
T, respectively). 1We shall discuss the obtained dynamical parameters below.

3.2.4 Equilibrium Relaxation -Power la w

N ow we shall pay attention to the dynamical parameters to and exponent m of the equilibrium
relaxation function Req(t - t) in Eq. (17). For a system with a distribution function of relaxation timer, per), the equilibrium relaxation function is given by ReqU - t) = fdrP(r)exp[U - t)lrl Using an
inverse Laplace transform, we can get P( r) corresponding to Rcq(t - t) with the power law form in Eq.
(17),per) = [tolnm)](rlto)-(nz + l)exp(-tolr) ,

(26)

18

where nm) is a Garnma function of m. In terms of the relaxation spectra Q(T), which is defined by
R (t- tj =fd (lnT)QCr)exp[(t- tjIT], we obtain
CX1

(26')

Q(T) = [lInm)]( T1tO)--mexp(-tolT) .

From the form of Eq. (26'), we see that to is a minimum relaxation time and m characterizes the
distribution intensity and width of the relaxation spectrum.
The best-fit parameters to obtained at different magnetic fields are plotted against temperature in
Fig. 16. With increasing temperature, the values of to increase appreciably from 10- 13 _10- 12 s ataround 8 K and tend to saturate above 18 K to the values as high as 10-6 -10-4 s. We also note that toincreases with the cooling-field H FC . The magnitude of to for our system is comparable with thosefor the insulating SG of CdCr1.7Ino.3S4 (to - 10-1S s)lS) and Feo.sMno.sTi03 (to = 10-6_10-s s
estimated from the reported decay curves )6).

10 10
10

-7

If)

0

-

--

10

7

N

0

-11
10
. : 0.14 T
0: 0.07 T
A: 0.035 T

10

10-15
5

25

15

4

10 1
35

10

T ( K)

-1

-2

10
m

Fig. 16. The best-fit values of minimum relaxation
time to of SG at different cooling-fields HR:. plottedagainst temperature .

Fig. 17 . The values of tzlto plotted against m;
symbols are the same as those in Fig. 16, arrowmarks me = 0 .033 .

N ow using the dynamical parameters m and to, the RRM parameter ~, as mentioned above, canbe expressed as given below. The magnetizations in ~ are written as MFC(T -70) = N Xeq(T-70)HFC~d M(t;) = NXeq(T)8H[-tol(-m + 1)][1 - (l + tito) -m + 1] - NXeq(T)HFC(titoyml(-m + 1), and thus ~IS rewritten as(27)

Since Xeq(T)IXeq(T-70) - 1 for general SG cases, including our results [see Fig. 11 (b)], we get a
simple form,
(27')

19

The values of t!to are plotted against m in Fig. 17. With increasing m , t!to decreases and becomes
nearly constant above mc = 0.033, marked by arrow , which corresponds to the characteristic value ~C(= 0.21) that represents the turning point where the minimum relaxation time to (or t!tO) depends on
the temperature or not.

3.2.5 Nonequilibrium Relaxation
In a nonequilibrium relaxation, the maxlIllum
relaxation time 1ffiax(t) [= (t6H + t - tJPZ(t2PZ1tl )-1] in
Eq. (22) is an important quantity that characterizes theupper cutoff in the relaxation spectra of timedependent domains. In Fig. 18 are plotted the bestfit values of t2PZ1tl against temperature in logarithmic
scales, which follow well a single line, as t2PZ1tl = 2.8
xlO-Sy2. Fig. 19(a) shows the values of t6 H, that

...

'r"

N

a.

...N

characterizes the overlap length 16H' at different
temperatures plotted against cooling-field H FC inlogarithmic scales, where one notes that theexperimental points lie single lines at fixed
temperatures, whose slopes are all equal to -1.5, ort6H - HFC-l.S .
In order to see the temperaturedependence of t6 H, in Fig. 19(b) we illustrate thevalues of t6HIHFC ·-l.S against temperature in log-log
plots.
As the temperature is increased, theexperimental points are decreased following a straight

10-3L-~~~~------~----~~
20
10
5
30
T (K)
Fig. IS. The best-fit values of t2pz1tl plottedagainst temperature in logarithmic scales,
which follow a single line, expressed as
t2pz1tl =2.8 x 10- 5 7'2.

a

10

(J)

:z::

...<1

1

In

~u

10

. u. 1020:I:...:z::
<1

10
10

-1

1
-2

10

-1

5

10

20

10

30

T ( K)

Fig. 19. (a) TIle best-fit values of t6H at different temperatures plotted against the cooling-field HFC inlogarithmic scales; symbols from open triangles to solid circles correspond to the temperatures 7.S, 9 .S,
II.S, 16.S, 21.S and 26.S K, respectively. The slope of each line is all equal to -I.S, which corresponds
to -21[(d - 2y)p] in Eq. (21). (b) The values of t 6H IHFC- 1.5 plotted against temperature.

20

line up to a temperature of about 20-25 K (:=Tg/2) , as t!1HIHFC -1.5 = l.2 x 10 4T-3, above which theyshow a steep decrease; such drastic temperature variations are also found in other quantities, such as

PHCtJ [Fig. 11(b)] and m [Fig. 13(b)]. We should note that the ratio t2PZ1tl exhibits no drastic
variation around that temperature (Fig. 18), while the value of t!1H or t2 [see Eq. (21)] changes
appreciably, which n1eans that the time constant tl is also reduced markedly above 20-25 K.

Theseresults indicate that thermal fluctuations in spin-glass system becomes appreciable above thistemperature.
From Figs. 18 and 19, we obtain '!inax(t)=3.6 x 104T-2 x (l.2 x 10 4H FC -1.5T-3 + t - tJO.9 forthe maximum relaxation time at low temperatures below 20-25 K.

The typical temperaturedependence of the maximum relaxation time '!inaxCt) for H FC = 0.14 T at different times (t = tz' 102 ,
103, 104 s) are shown in Fig. 20 in logarithmic scales; the curves at t = tz (the time when the external
field is reduced to zero) and 104 s follow a straight line, but those at 102 and 103 s do not lie on astraight line.

We see that '!inaxCt) is increased with time t, which indicates that the domain size in the

SG system grows as time laps.

Though not shown here, we have also found that at a fixed

temperature '!inaxCt) at t = tz depends strongly on the cooling-field as H FC -I .35, but it becomes
independent of H FC for t»

t z.
0.03

a

-

: 1: max (~)•• - •. : 1: max (1 0 4 5 )

10

0.02

7

£:
0

10

0.00

en

10

-

><
CIl

p

0.01

6

E 10

10

5

4

-10

-5

10

s

10

0.06

s

b

0.04

3

£:
0

0.02
10 2 '---_-'--_......_ _ _-'--_-"--_..1.....---1
6
8
101
15
20 25 30

0.00 ____--...._____....u._ _--1..._......LL...LJ....u"-'--_--lT (K)10. 15

10. 10

10. 5

100

105

10 10

1: (5)

Fig. 20. The calculated values of maximum
relaxation time Tmax(t) for HFC = 0. 14 T atdifferent times (t = t z , 10 2 , 10 3 , 1()4 s) plottedagainst temperature in logarithmic scales.

Fig . 21. Variations of the relaxation spectrum Q(T) andmax imum relaxation time Tmax(t) att =t z and 1()4 s with(a) the cooling-field HR: at the fixed temperature 16.8 K
and (b) with temperature at the fixed cooling-field HFC =0.14 T.

Moreover, Fig. 21 illustrates the relaxation spectra Q(t) for TRM cals:ulated using Eq. (26') withthe parameters to and m obtained, together with the maximum relaxation time '!inax(t) at t = tz (solidlines) and 104 s (broken lines).

As shown in Fig. 21(a), with increasing the cooling-field H FC ' thelower cutoff to shifts to a longer time side with the increased intensity and slope, while the higher
21

cutoff'rmaxCt) at t == t z becomes shorter but at t = 104 S it is independent of HFC . The relaxation
spectrum depends remarkably on temperature, as shown in Fig. 21 (b) for a typical case of H FC = O. 14
T. The lower cutoff shifts to a longer time side with raising temperature up to 16.8 K, above which it
becomes almost constant, and the relaxation spectrum is narrowed; the upper cutoff also shifts
gradually to a shorter time side. These results are in qualitative agreement with those obtained fromMonte Carlo simulations in the temperature range 0.6 < T1Too < 2 for two-dimensional Ising SG by
Nemoto and Takayama,26) where they have assumed that the longer relaxation time represents the
dynamical aspect associated with overturn of the spin cluster to which the spin belongs, while theshorter relaxation tilne represents the fast relaxation associated with the local excitation of each spin.
Finally, we have evaluated the dynamical parameters for TRM of our SG system. From t~H HFC-l. S, we have --2/[(d - 2y)p] = -l.5 in Eq. (21).
With this value, the parameters p and z are
evaluated using the theoretical value of y for two-diInensional (d = 2) and three-dimensional (d = 3)
Ising model by Bray and Moore. 23) Together with the theoretical values, our estimated values are listed
in Table II (p - 0.5, z - 2), which are in good agreement with those estimated by Koper and
Hilhorst 1S ) using the experiInental data for insulating SG of CdCrl. 7InO.3S4 by Alba et al. 24 ) It is ofinterest to note that these dynamical parameters p and z are independent of insulating Ising SG or ouritinerant magnetic S G.
Table II. Best-fit parameters obtained experimenJally of pz and -2/[(d - 2y)p],
together with the evaluated values p andz using the theoretical values ofy for two-dimensional (d = 2) and three-dimensional (d = 3) Ising SG by
Bray and Moore.2 3 )

d
3
2

y

pz

0.19
-0.29

0.9
0.9

-21[(d-2y)pJ
-1.S
-1.S

p

z

O.SI
0.S2

1.8
1.9

4. Conclusions
U sing the anomalous Hall effect, we have measured the tUne decay of zero-field cooled
isothermal remanent magnetization (IRM) and field-cooled thermoremanent magnetization (TRM) inthe spin-glass (SG: x = 0.20, To=41 K) and cluster-glass (CG: x = 1/4, To=53 K) phase of itinerantobmagnetic Fex TiS 2, using the pulsed magnetic fieldH p ::; 20Tor the cooling-field H FC = 0.01-0.14 T
over the time range 10-2- 104 s with the waiting time tw = 180-18,000 s below T1Too - 0.7. We havefound the following salient features of the dynamical properties in this material:(A) In the case of IRM, the temperature and peak field dependencies of PH(t) are analyzed toobtain a universal relationship, mT ex = C ¢ f3, for the parameters characterizing the dynamical
properties, where mt is an exponent for the power law PH(t) = At -m after a field is switched off, ¢ anewly introduced quantity, ¢ = [M(Hp) - MrCtJ ] 1 M(Hp) [M(Hp) is the magnetization at the peak field

Hp and Mr(tJ the remanent magnetization at time t = tJ. The exponent a = 114 and -112 for SG andCG, respectively. The exponent f3 for SG and CG phases, has different values for ¢ < ¢ c and ¢ >¢ c and ¢ c = 0.90 and 0.34, respectively, which corresponds to a two- and three-dimensional "criticalvolume fraction" in a site-percolation picture.
(B) In the short time regime t < td' the time decay of the Hall resistivity for TRM after theswitching off of the external field can be expressed in the form of power law PH(t) = At -m, whose the22

exponent m depends on both cooling-field H FC and temperature T, while for t > td deviations from thepower law become appreciable.

As found for both SG and CG phases, the magnetic field andtemperature dependent exponent m is written in the universal form, m = D~Y, w here ~ [= 1 -MUJ/ MFC(T-70)] is a parameter of "relative relaxed magnetization" (RRM) for TRM, with the bestfit values for SG of L) = 0.30, Y= 1.4 for ~ > ~C (= 0.21) and D = 2.6, Y= 2.8 for ~ < ~C, the Yvalues being larger them that for CG (D = 0.15, Y= 1. 0).

Moreover, using the dynamical parameters

appearing in the domajn theory, we can express the parameter of relative relaxed magnetization ~ thatcharacterizes the temperature and cooling-field dependence of the time decays of TRM for SG phase ina short time span, in the form ~ - 1- Ctito)-m/(-m + 1), which is a function of tito and the exponent m.
(C)

Using the "domilln theory" developed by Koper and Hilhorst, numerical calculations forthe observed decay curves of TRM of SG phase, over the whole time range have been performed withmodifications of their theoretical expressions.

As a result, we have found a satisfactory agreement

between the simulations and experiments, including the independence of the TRM decays on thewaltmg time.

The minimum relaxation time to in the relaxation spectrum is increased with increasing

temperature up to 18 :K , above which it becomes constant (of the order of 10- 5-10-7 s), while the upper
limit '!ffiaxCt) depends on time, temperature, and cooling-field H FC ' which is expressed empirically as'rmaxCt) = 3.6 x 104 T-2 x (1.2 x 104 H FC -1.5T-3 + t - tJO.9 that has been obtained from the observed
parameters t2pz/tl and t6H with the best-fit value of pz= 0.9.

Using these parameters, theequilibrium relaxation spectra are calculated; the spectra become narrowed with increasing temperatureand cooing-field, which are in qualitative agreement with the Monte Carlo simulations for twodimensional Ising S G .2 6)(D)With the evaluated value 2/[(d - 2y)p] = 1. 5 and the theoretical value of y for two-

dimensional (d = 2) and three-dimensional (d = 3) Ising model, we have obtained the dynamicalparameters p - 0.5 (a domain size increases as a power of time with the exponent of p) and z - 2 (theupper relaxation time depends on the domain size with the exponent of z), in agreement with those ofsome insulating Ising SG .

With regard to the nature of magnetism (localized spin or itinerant electron

picture), there are nlO differences in these dynamical parameters.

More studies of relaxation

phenomena, such as temperature cycle, will be desirable from another viewpoint of the hierarchicalkinetics for our itinerant magnetic material.

Acknow ledgmen ts
The author would like to express his sincere thanks to Professor M. Inoue for valuable advice
and continuing encouragement in the course of the present work.

He is also grateful to Professor H.

Takayama (Institute of Solid S tate Physics) and V. A. Kulbachinskii (Moscow State University) for
Valuable discussions.

The author also thanks Associate Professor M. Sasaki and Drs. H. Negishi,

M. Koyano, S. Ohara, G.X. Till, W.X. Gao , and K . Takase for fruitful discussions.

Part of thiswork was supported by the Grants from Asahi Glass, Ogasawara Science Foundations, and Grant-inAid from the Ministry of Education , Science and Culture of Japan.

Furthermore, he is indebted tocolleagues and students in Inoue Laboratory for useful discussions and their help in taking theexperimental data.

At last, he thanks Ms. Y. Tahara and M. Sakata and J. Koyama and Mrs. Y.

Nishigaki for their encouragements.

23

References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)

M. Inoue, H. P. Hughes and A. D. Yoffe: Adv. Phys. 38 (1989) 565.
H. Negishi, A. Shoube, H. Takahashi, Y. Ueda, M. Sasaki and M. Inoue: 1.
Magn. Magn Mater. 67 (1987) 179.
H. Negishi, H. Takahashi and M . Inoue: 1. Magn. Magn. Mater. 68 (1987) 27l.
H. Negishi, M. Koyano, M. Inoue, T. Sakakibara and T. Goto: 1. Magn. Magn.
Mater. 74 (1988) 27.
M. Koyano, M. Suezawa, H. Watanabe and M. Inoue, 1. Phys. Soc. lpn. 63(1994) 11 14.
A. Ito, H. Aruga, E. Torikai, M. Kikuchi, Y. Syono and H. Takei, Phys. Rev.
Lett. 57 (1986) 483.
D. Sherrington and S. Kirkpatrick: Phys. Rev. Bl 7 (1978) 4384.
G. Parisi: Phys. Rev. Lett. A73 (1979) 203.
G. Parisi: Phys. Rev. Lett. 50 (1983) 1946.
1. Hammann, M. Lederman, M. Ocio, R. Orbach, and E. Vincent: Physica A 185(1992) 278.
F . Lefloch , 1. Hammann, M. Ocio, and E. Vincent: Europhys . Lett. 18 (1992)
647.
M. Lederman, R. Orbach, 1.M. Harmnann, M. Ocio, and E. Vincent: Phys. Rev .
B44 (1991) 7403.
C. M. Newn1an and D . L. Stein: Phys. Rev. B46 (1992) 973 .
D.S. Fisher and D.A. Huse: Phys. Rev. Lett. 56 (1986) 160l.
G.l.M. Koper and H.l. Hilhorst: 1. Physique. 49 (1988) 429.
P. Granberg, L. Lundgren, and P. Nordblad: 1. Magn. Magn. Mater. 92 (1990)
228.
M.Inoue and H.Negishi: 1. Phys Soc. lpn. 53 (1984) 943 .
C. M. Hurd.: The Hall Effect in Metals and Alloys (Plenum Press, New York,1972) chap. 5.
E.1. Kondrskii: Soviet Phys. lETP, 28 (1969) 29l.
1. Ferre and N. Bontemps: Materials Science Forum 50 (1989) 2l.
R. Zallen: The Physics of Amorphous Solids (John Wiley & Sons , New York,
1983) chap. 4.
N. Bontemps and R. Orbach: Phys. Rev. B37 (1988) 4708.
A.1. Bray and M.A. Moore: Phys. Rev. Lett. 58 (1987) 57.
M. Alba, E. Vincent,l. Hammann, and M. Ocio: 1. Appl. Phys. 61 (1987) 4092.
A.T. Ogielski: Phys.Rev B32 (1985) 7384.
K. Nemoto and H. Takayama: 1. Phys. C 16 (1983) 6835.

24

CD Y. Hara, Vi. X. Gao, H. Negishi, and M. Inoue:
Relaxation Behaviors in Cluster-Glass Phase of Itinerant Magnetic Fe x TiS2 Crystals
Studied by Magnetotransport Measurements under Pulsed Magnetic Fields,
1. Magn. Nlagn. Mat.135 (1994) 311.

@ Y. Hara, H. Negishi, M. Sasaki, M. Inoue, and V. A. Kulbachinskii:
Time Decay of Thermoremanent Magnetization in Cluster-Glass Phase of Intercalation
Compound Fex TiS2 Studied by Use of Anomalous Hall Effect,
1. Magn. Nlagn. Mat. 145 (1995) 157.

G) Y. Hara, H. Negishi, M. Sasaki, and M. Inoue:
Relaxation of Thermoremanent Magnetization in Spin-Glass Phase of Itinerant Magnetic
Fex TiS 2,
1. Magn. Nlagn. Mat. (1996) (in press).

1) M. Koyano, S . Ohara, Y. Hara, M. Inoue, G. Kido, and Y. Nakagawa:High Magnetic Field Transport Properties of ll-Mo4011 Crystals,l. Phys. Soc. lpn. 62 (1993) 1676.
2) V. A. Kulbachinskii, P. D. Maryanchuk, 1. A. Churilov, M. Inoue, M. Sasaki, H.Negishi,
and Y. Hara:
Electronic and magnetic properties of diluted magnetic semiconductor

Hgl-xMnx Tel_ySey ,Semicond. Sci. and Technol. 10 (1995) 463.
3) H. Negishi, Y. Hara, M. Sasaki and M . Inoue:
Relaxation Phenomena of Remanent Magnetizations in Spin- and Cluster-Glasses ofItinerant Magnetic Fex TiS2 Crystals Studied by Hall Effect,l. Phys. Chern. Solids (1996) (in press) .