Reptation and hysteresis in disordered magnets

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Submitted on 1 Jan 1993

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Laurent Lévy

To cite this version:

Laurent Lévy. Reptation and hysteresis in disordered magnets. Journal de Physique I, EDP Sciences,

1993, 3 (2), pp.533-557. �10.1051/jp1:1993148�. �jpa-00246739�

J. Phys. I France 3

(1993)

533-557 _FEBRUARY 1993, PAGE 533

Classification Physics Abstracts 75.60 05.50

Reptation and hysteresis in disordered magnets

Laurent P.

Ldvy (*)

Universit4 Joseph Fourier, BPS3X, 38041 Grenoble Cedex, France and

Grenoble High ^Field Laboratory, BP166X, 38042 Grenoble Cedex, France

(Received

I st September1992, accepted in final form 21

September1992)

Abstract. In most disordered magnets, the hysteresis cycles obtained by the repetitive _ap-

plication of _a magnetic field _{are not} exactly reproducible. At low temperature, _a logarithmic growth of the remanent magnetization with the number of cycles

in)

is observed. This phe-

nomenon is known

as magnetic reptation. A review of the experimental studies of reptation in disordered ferromagnets and spin-glasses is presented. We emphasize the simple phenomenolog-

ical scaling laws governing the field and the temperature dependence of this phenomenon. We consider _two microscopic mechanisms which _may be responsible for this _process. The first _one relies _on the random dependence of the energy barriers

on applied magnetic field resulting from the balance between the fluctuations in the magnetic _pressure, the random

pinning

potential

and the surface tension of _a domain wall. In systems with long _range forces

(dipolar

interactions in ferromagnets, RKKY exchange in metallic

spin-glasses),

_a second reptation _process induced by the Onsager reaction field _appears to be particularly relevant _to experiments.

1. Introduction.

Hysteresis

is found in _many

substances, having

_a

large

number of low energy

minima,

which

cannot be

explored by thermodynamic

fluctuations. An external parameter, Such _{as a}

magnetic

field,

allows to Switch

irreversibly

from _one state to another. A first class of

hysteretic

materials

are nonlinear systems

(ferromagnets

_or other multi-stables

systems)

with several low _energy minima. For

example,

in _a

ferromagnet,

different

magnetic

domain structures

yield

different energy minima.

Thermodynamic fluctuations,

which decrease with the

sample volume,

do not allow the system to

explore

all the states,

leaving

it in _a metastable _energy minimum. For this class of

material,

it is

usually possible

to force the system in _a

unique ground

state with the external

magnetic

field

(I,e. by aligning

the domains with _a field

ha

much

larger

than the coercitive field

Hc).

In _a different class _are

glassy

materials which have _many low _energy

minima well

separated by large

_energy barriers below _a characteristic

freezing

temperature

Tg.

(* To the _memory of R. Rammal who inspired this work

2.0

II /

~/ /

~.~ Al ^~ ~

/ /

ts /

~ /

~/ / II

~l.O

^~ II'

o.5

O-O

O-O 0.2 O.4 O.6 0.8 1-O

h

/h~

Fig,I.

Evolution in _an

(m, h)

plot of

a minor hysteresis loop under the repetitive application of _a magnetic field ha. The magnetization in the field _{on state} at the n'~ _{cycle is} denoted _as Mn, while the

remanence in the field off state is _mn. When Mn and _mn grow at approximatively at the _same rate

as shown here, this is _a generic reptation process ^to be contrasted with the swing _process depicted in

figure 3.

The system ^is

effectively

frozen in _a metastable state. In contrast with

ferromagnets, only

a

small number of states _are

usually

accessible in

glassy

materials

by varying

an external field.

Aside from these

differences, metastability

and slow relaxation _{are common} features _among these systems. ^One _may therefore

question

to what extend their

hysteretic

behaviors share universal features. To

explore

this

issue,

_we consider in this work _a

particular

form of

magnetic hysteresis.

Let _{us suppose} that _a

magnetic

field

ha

is

applied

to an

initially demagnetized

system and then switched off. A _remanent

magnetization

_{mi can} be measured. If this _process is

repeated

_n times, it is

experimentally

found that the

magnetizations

Mn in the field _on

states

(h

₌

ha)

and _mn in the field off states

(h

=

0) keep changing

after the n~~

application

and removal of the _same

magnetic

field ha. This _process is illustrated in

figure

I in _an

(m, h) diagram:

the

magnetic cycles spiral

toward

higher magnetization.

In _very clean

ferromagnets,

_a

more

complex

behavior is observed: the

magnetization

in the field _on state

Mn decreases,

while the

magnetization

in the field off _{state mn} increases. At finite temperature, the

magnetizations Mn(t)

and

mn(t) slowly

relax

as a result of thermal activations _{over energy} barriers. At

sufficiently

low temperature, Mn and _{mn are} static and this

hysteretic

^behavior is _an intrinsic

zerc-temperature phenomenon.

It is known _as

"magnetic reptation"

after Ndel

original

work

ill.

Such

asymmetric cycles yield

the

asymptotic

laws for minor

loops.

In the

simple

_{cases, one}

can infer from such laws how the sytem rearrange itself. In turn, ^this

gives

_some constraints

on the

possible organization

of the states.

We wish to

explore

here the

physical

mechanisms involved in these _processes. In particular, what _are the

respective

role

played by

^disorder and interactions? To start with,

existing

ex-

perimental

studies

provide

_a rather

complete phenomenology

which _{may serve as a}

guideline.

A few classes of materials have been studied:

ferromagnets

_{[2 4]}

(Sect. 2)

and

spin-glasses

(Sect. 3)

[5]. ^{We will focus} on these studies, which

display

considerable differences in the observed behaviors. Studies _on other frustrated systems

(random-field

systems and antifer-

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNETS 535

romagnets

on frustrated

lattices)

_are

presently missing

and would also be of great ^interest.

Two models of

magnetic reptation

_are considered. In section 4, the

equilibrium

of domain walls in the random-bond _or random field

Ising

models is shown _to result from the balance between three forces:

(a)

the

magnetic

_pressure on the

wall, (b)

its surface

tension, (c)

the

pinning

^forces due to the disorder. A random

dependence

of the _energy barriers _on

magnetic

field results from the

fluctuating magnetic

moment of the interface. In other

words,

there is

always

_a fraction of the _energy barrier which vanish when _a

magnetic

field is

applied,

because the

magnetic

moment of interface _vary

randomly

with the interface

position.

This

gives

rise to

magnetic reptation.

In section 5, _we

study

the effect of rearrangments induced

by

the de- cay of metastable states: such processes are

particularly

relevant to systems ^with

long

_range

forces

(dipolar

interactions in

ferromagnets,

RKKY

exchange

in

spin-glasses).

A

spin

cluster

polarizes

the

surrounding spins through

these

long

_range forces. The induced

polarization

then

reacts on the cluster considered via the

dipolar

_or

exchange

field. This field is known _as the

Onsager

reaction field _or

cavity

field. We show how this reaction field introduces _a feedback process, which _as

anticipated by

Ndel

gives

rise to

reptation.

2.

Reptation

in

ferromagnets.

In most

ferromagnets,

it is

possible

to freeze out thermal fluctuations at

sufficiently

low tem-

perature ^{and the} zerc-temperature irreversible behavior is

readily

accessible.

Macroscopic hysteresis

_can be characterized

by

the

major hysteresis loop

obtained

by sweeping

the field from the saturated _states of the

magnetization

I-e- from

(ms,

h »

Hc)

to

(-ms, -h).

This _curve

crosses the H axis at the coercitive field

+Hc,

and the _m axis at the saturated _remanence

+ _mr. The _area within the

major loop

is _{a measure} of

hysteretic

losses. For each

magnetic field,

the system can be

prepared

in the two well-defined

magnetization

states _on the

major hysteresis loop.

In

addition,

another well defined _{state can} be obtained

by superposing

_an

ac field of

decreasing amplitude

to the dc field. Such

"anhysteretic"

states are

good starting points

for the minor

loops

^used in

reptation

studies. For fields ha <

Hc,

the initial

portion

of the

magnetization

curve may be

parametrized by

_a

ltayleigh

^law [6], m =

xh

+

Qh~/2

where _x is the reversible

susceptibility

and q, the

ltayleigh

coefficient, is _{a measure} of the irreversible

magnetization. Microscopically, hysteresis

arises from the irreversible motion of domain walls, known _as Barkhausen

jumps

_[7]. In

addition,

the irreversible rotation of the domain magne-

tization _can also

play

_a role for

granular

and

heterogenous

systems. Barkhausen

jumps

_occur

mainly

for

applied

fields close _to the coercitive field

Hc,

where

irreversibility

is

largest.

^The

reptation

studies described below _are made _on minor

loops (I,e,

within the

major loop),

_start-

ing

either from

(0, 0)

or from _any

point

_on the

anhysteretic

_curve.

Significant reptation

is

only

observed

by applying

_a field ha of the order of Hc

(see Fig. 2),

in the

region

where Barkhausen noise is

largest.

^Since

reptation

_{processes are}

qualitatively

^{different in clean and disordered}

ferromagnets,

_we start _our

description

with clean

ferromagnets having

^low

Hc, Prepared

in the

demagnetized

state.

In clean systems, the

magnetizations

Mn and _mn do not grow

monotonically. Instead,

Mn decreases while _mn grows. Both

quantities

saturate after _a few

(< 20) cycles

where _a limit

cycle

is reached. This

behavior,

shown in _an

(m, h) plot

in

figure

3, _was observed _{on a}

single crystal

of AIDo6Feo

94 with _a coercitive field of Hc ₌ 0.30 Oe [3]. ^Since Mn and _{mn vary} in

opposite directions,

the

corresponding hysteresis loop swings

_[8] toward smaller

slopes.

This

"swing"

of the minor

loops

is found in

single crystals

and clean systems ^with narrow

hysteresis

loops,

and low coercitive fields. The

relatively rapid

^{saturation of the minor}

loops

toward _a limit

cycle

is

exploited

in

magnetic recording, by using

a

high frequency biasing

field [9].

There,

each

magnetic particle experiences

in addition _to the

quasistatic

audio

frequency field,

_a

higher

20

1OO

I',

f ,

80 l

~ ,

8 ^'

~''("_

~° j_i~ ",)=,

J

)

^'".~7 IT~

40

20

~

0.5 1-O 1.5 2.O

h~/H~

Fig.2.

Evolution of the reptation rate r, defined equation

(1),

_{as a} function of the applied field ha for three steel-alloys

(after

Ref. [2]). Reptation is only

significant

for field strength above

Hc/2

and

decrease above Hc.

frequency biasing

field which oscillates ten to twenty ^{times while the} tape _passes in front of the head. In this _way, each

particle

reaches the well defined _remanence of the limit

cycle.

This

swing

of the minor

hysteresis loop

_may be attributed _to the

dipolar coupling

^between _a

small number of

neighboring

domains. This

interpretation

is

supported by optical

studies of the domain structure of YIG'S in

reptation cycles

_[10]. The

minority

domains

are observed _to be isolated from one-another and to shrink

homogenovsly through

the

sample.

^This will be

analyzed

further in section 4.

The introduction of dislocations and disorder

changes

the evolution of minor

loops

in _a fundamental _way. After lamination of the

single crystal

of Alo.o6Feo.94

Previously discussed,

the coercitive field is increased _to 1.75 Oe and the minor

loops

show _a monotonic increase of Mn and _{mn at}

approximatively

the _{same rate.} Neither Mn _{nor mn} show saturation and _no

clear limit

cycle

is reached after _an

arbitrary large

number of

cycles.

This is _a

typical reptation

behavior. The most

comprehensive

studies of this

phenomenon

_were made _on steel

alloys

_[2].

For values of

n

larger

than 20, the observed

growth

of the

magnetization

_can be fitted to

mn ml "

r[ In(n)]" ii)

for all the material studied. For values of _n below

20, significant

deviations from the

logarithmic

bebavior _are observed. Such deviations have been attributed _to the _same

physical

_processes

controlling

the

swing

of the minor

loops

in clean systems.

However,

this is not established _on firm

experimental grounds.

The

analysis

of

reptation

in the random

Ising

model

presented

in

section 4

suggests

that the deviation from

logarithmic

behavior reflects the

density

of _energy barriers _and is system

specific.

The

logarithmic

law

ii

is

only

observed in disordered

samples.

Previous studies used _an exponent ~Y of 0.5 to fit the

data,

in reference to _a model

proposed by

Ndel

ill.

We

reanalyzed

the

existing

data and found it _to be _more consistent with values of

o between 0.7 and 1.3

(cf. Fig. 3).

^The

dependence

of the

reptation

rate r with the

strength

of the

applied

field ha is shown in

figure

2. For values of ha < 0.5 _x

Hc,

the

reptation

rate is

quite

weak. It increases _very

rapidly just

below Hc reaches its maximum around h _Se Hc and

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNETS 537

2.O

_~ /

l.5 /$

$~

l'l'/

/~ _{/ ~}

Ws / / /

~ / /

%/ /

C /

~ ^l-O

/

o.5

~'~0.0

0.2 0.4 0.6 O-B I-O

h/h~

Fig.3.

Evolution in

an

(m, h)

plot of the minor hysteresis loop of Alo_osfeo,g, under the repetitive application of _a field ha _zS Hc ₌ 0.30 Oe

(after

ref. [3]). ^The magnetization Mn decreases while _mn

increase. The minor loop reaches _a limit cycle after 20 iterations. The minor loop swings toward

smaller slopes.

decreases above

Hc

for all the materials studied.

A variant of

reptation,

where the

applied magnetic

field ha is rotated between _two values 0 and R, has been studied [4]. ^The

magnetization perpendicular

to the inital direction of the

magnetic field,

shows _a

logarithmic growth,

similar _to what is observed in the standard _rep- tation

configuration.

This indicates that rotational

anisotropy plays

_a minor role in

reptation

process. This is also _an evidence

suggesting

that domain wall motion dominates the rotational

reconfigurations

of domains.

If the

cycles

_are started from other

points

_on the

anhysteretic

_curve,

reptation

can be _en- hanced _or

suppressed

to various

degree.

Since the observed

phenomena

_are more material

dependent,

the interested reader is referred _to the literature [3].

At finite temperature,

reptation

_can be

significantly

renormalized

by

thermal fluctuations.

It _can be difficult to separate

reptation

from relaxation effects _{as our}

spin-glass study

will illustrate. To this

end,

_we

actually give

_a well defined

procedure

which is also

applicable

to

ferromagnets.

3.

Reptation

in

spin-glasses.

Ferromagnetic

studies revealed that _a

genuine reptation

_process is

only

observable in

sufficiently

disordered

samples.

Since

reptation

_seems to be

intimately

tied to

disorder,

it could _prove to be _a useful

investigation

tool for other disordered

magnets,

such _as

spin-glasses.

To this

end,

this

experimental study

_[5] determines the

phenomenological

laws

governing

small

asymmetric hysteresis cycles

in dilute

magnetic alloys (Agmn

0.5 and 2.6

at.$l)

_{over a}

wide _range of temperature

(from

0.03 _x Tg ^to

Tg),

^time-scale

(10

to 2000

sec.)

^and fields

(from

0 to I

koe).

In contrast with

ferromagnets,

relaxation processes ^cannot be

completely

frozen out in

spin-glasses.

The effect associated with this

magnetic viscosity

have then to be distin-

guished

from

genuine reptation,

which

requires

_a

time-dependent study.

This

experimental

JOURNAL DE PHYSIQUE I T 3, N'2, FEBRUARY >993 19

6

/~'* ,

_~. _, ^,

,«,' ^~a'$,

^{, S}

~c ⁴

~

~

,t"$,,'

,~, ^{,a ~,}

,

"'

a/~

^'~'

2 ^{" a}

' ,°'

~, ^~°

,o'l'

A,

Io° _o~ o~ Io~ o~ _Io~

n

FigA.

_Dependence of the remanent magnetization after the n'~ cycle _{as a} function of

In(n) (data

from Ref. [2]). The dashed line, _a linear fit to

In(n),

gives _a ^{better fit} to the data, than the dotted line, _a fit to (In

n)~'~

work _stresses the

importance

of feedback _processes in the

spin-glass phase.

The

magnetization

measurements have been done with _a

SQUID magnetometer ill]

which enables to subtract _a constant term from the linear _response of the system. ^{This allows} to

keep

track of small

changes

in the irreversible

magnetization

between the

large swing

of the reversible _response to the

applied

field 0 _< ha < 60 Oe. Each of the _n

elementary magnetic cycles (Fig. 5)

^{consists of} a constant field

ha applied during

_a time to

(20s

< to <

1500s)

followed

by

_an

equal

time

period

to where _no field is

applied.

The data is taken after

cooling

the system in _zero field and

waiting

for _a time tw of the order of 2500s. The first

cycle gives

therefore the

dynamical

_response of the isothermal remanent

magnetisation (IRM).

The

magnetization

_response shown in

figure

5 at T

= 0.I _x Tg ^{and 0.6} x Tg ^has

always

_{a very} strong ^relaxation

throughout

the

spin-glass phase.

We first summarize the

qualitative

features:

(a)

After the first

cycle,

the relaxation of the

magnetization

is the _same from

cycle

to

cycle:

this

implies

_a re-initialization of the

dynamics by

field

cycling.

The relaxation is also

equal

and

opposite

in field-on and field-off states _as

expected

for linear _processes.

(b)

The

systematic

increase of the remanent

magnetization

_rn from

cycle

to

cycle

in the field off state is found to be

independent

of the times to and tw, ^I-e- the

hysteretic growth

of _rn

only depends

_on the number of

cycles. (c)

_rn

^obeys

a

simple scaling

law _{as a} function of

cycle

number

(n), field,

temperature ^and

exchange (proportional

to the

freezing

temperature

Tg).

^The

parametrization

of the

time-dependent magnetization

in the field-off state of the n~~

cycle

'~~n(I) "

°n(~)~apJ~(~/~0)1

+ _~n + _~l,

rn "

r'll(n), (2)

r =

Tgf(T/Tg)

summarizes these results. As

implied by

the re-initialization of the

dynamics,

the

origin

of time is

always

taken at the last transition between field states. The coefficients ~Yn(T), ^known as

magnetic

viscosities _are within

experimental

_errors

equal [an (T)

_{e ~Y(T)]} after the first

cycle in

_>

I).

The function f is _an

experimentally

determined

scaling

^law and

fl

is _a

dynamical

function

(close

to

linear)

which is

poorly

determined. In the field-on state, ^the

time-dependent magnetization

is the _sum of the reversible contribution in the n~~

cycle

_xn ha and the irreversible contribution

Mn~(t),

which

obeys

_a law similar to

equation (2).

Note from

figure

5, the

systematic

decrease

(resp. increase)

of the

magnetization

from

cycle

to

cycle

in the field-on

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNETS 539

/

_,/

~'

,

~ /

_/~

_~

! j 1 /

~

j ^{j i}

j

~ ~'~ ⁽

^{~ ~}

^~

( (a)T=O.lxT~

~°

/ / f

_-II

/ ^{f / f} _:~

i 20

~

. . ,

'~ '~ ~ ~

l

,

', ^~

^'

~ ^~

b T = O.6 x Tg

O 200 400 600 BOO

TIME (sec)

Fig-S. (a)

Magnetic _response of Agmn 0.5% at 0.3 Il _zz 0.1 _x

T,

to asymmetric field cycles schemat-

ically shown in the center. A large constant

(about

twenty times the full

scale)

is subtracted from the magnetization in the field-on state.

(b)

Same _response at T

= 1.8 KzS 0.6 _x

T,.

state at T

= 0.I _x Tg

(resp.

T

= 0.6 x

Tg).

^{This will be attributed to} a decrease of the reversible _{response xn} ha from

cycle

to

cycle.

Each term in

equation (2)

is _now

analyzed, starting

from the

dynamics

of the IRM

(first cycle,

n =

I).

At all temperatures ^below

Tg,

^{excellent fits} to the data

(see Fig. 6)

have been obtained

over

2)

decades of time

(300 sec.)

with _{a response} function linear in

In(t flu), although

other choices

(fl(z)

₌ I

lx, fl(z)

₌ I

/@)

also

give

reasonable fits. At

longer

times

(approaching tw)

and

higher

temperatures, the

decay

becomes

sub-logarithmic-

In contrast with the thermc-

remanent

magnet12ation (TRM)

in the

vicinity

ofTg

[12,

_13] the

dynamics

is

poorly

described

by

_{a poo>er} law t~~ at short times _as shown in

figure

6. The

magnetic viscosity,

_~Yi(T) is

approximatively

linear with temperature below 0. IS _x

Tg,

^reaches a maximum below 0.6 _x

Tg,

and decreases _as Tg ^is

approached

_as shown in the inset of

figure

6. The linear

dependence

in

TIn(t flu

leaves little doubt that the

dynamics

is activated at low temperature [14, 15].

After the first

cycle in

_>

I),

the

magnetic

viscosities _on

(T)

_{e ~Y(T) are} all

equal

_[16]. This shows that the field

cycling

re-mitiahze the system. ^If we are to describe the

dynamics

in term of the

decay

of low temperature excitations from metastable states, an additional _process

responsible

for the re-initial12ation must be invoked _so the relevant parts of the distribution of

metastable states which govern the

dynamics

_are unaffected

by

field

cycling.

The

dependence

of _ml, the

IRM,

with the

applied

^field ha is linear below 200e [17] ^and

a

quadratic

term becomes

significant

above 40 Oe [18]. Furthermore, _we find that

mi/ha

has below 0.IS _x Tg ^the same linear temperature

dependence

_as

x", suggesting

that the IRM

dynamics

reflects the linear _response of the system, ^{while the} TAM _may not. The

logarithmic

~ ..__

io~

O

o .

o

o o

'° 4 6 8 lo

TIK)

too ioi io2 io3

TIME (sec)

Fig.6. (a)

Comparison of the logarithmic (left

scale)

and _power law

(right scale)

fit to the IRM relaxation. Insert: Temperature dependence of the logaritmic vicosity of Agmn 2.6 at.% in the spin- glass phase.

dependence

of _rn with

cycle

number is tested

over 10~

cycles

in

figure

7a To determine the

dependence

of _rn with

applied field,

the

slope

_r of _{rn versus}

In(n)

is

plotted

in

figure

7b _as

a function of ha for various

cycle period

to- The

slope

is linear and within the

experimental

errors

independent of

the

period

to- ^Therefore the relevant

phenomenon depends

_upon how

many times the system ^has been

cycled in)

and not on how much time

into)

the system ^has spent in _a field _{nor on} how

long

the system has been

aged.

We _now

investigate

the

scaling properties

of _r with temperature ^and

exchange

J. Since J scales like

Tg,

^the

dependence

_on the

exchange

_can be studied

by looking

at

samples

of different

concentrations. For

example,

_Tg =2.9 K for 0.5 at. $l Mu and T~ = 10.5 K for 2.6 at.$l. The irreversible

susceptibility

_r,

plotted

in

figure

8b scales without _any

adjustable

parameters as

Tg

f(T/Tg).

Because of the Tg

prefactor,

_r does not scale _as the linear

susceptibility

which is _a function of

T/Tg

alone. To

interpret

the

physical significance

of this

prefactor,

_{we assume}

by analogy

with

ferromagnets

that

magnetic hysteresis

_can be

parametrized

at low fields with

a

Rayleigh

_{law [6] r o~}

[(<

h _>

+ha)~-

_< h

>~]

₌ ha~ +2 < h >

ha,

where the

phenomenological

field _< h _> is

interpreted

in

ferromagnets

_as the width of the distribution of local fields

ill.

While the

microscopic meaning

of _< h _> is not yet clear for

spin-glasses,

it should scale with the

exchange

J

(and Tg).

^Therefore a

Rayleigh

law

consistently explains why

_r scales

linearly

with ha and Tg ^{at low fields. When} ha >< h >, the

dependence

_on

applied

field ha should become

quadratic.

In the field _range studied _r decreases

linearly

with

decreasing

temperature:

reptation

is not _a zerc-temperature

phenomenon

in

spin-glasses,

in contrast with

ferromagnets.

Since _r and _o have below 0.10 _x Tg a linear temperature

dependence,

the

reptation

rate _r could be related to the

magnetic viscosity

_~Y. It is therefore instructive to _see how their ratio

depends

on temperature. As shown in

figure

8a, r/~Y ^scales as a function of

T/Tg

and is

approximately

constant above T

= 0A _xTg ^{but decreases}

smoothly

at lower temperatures. _{~Y(T) =} dmn

Id

Int and _rn

= mn

(to)

are not

proportional

to each other in the

spin-glass phase

below T

= 0A _x

Tg.

This suggests that

they

_are not sensitive to the _same part ^{of the} distribution of metastable

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNfITS ₅₄₁

(a I

io° ioi io2

5

~ (b)

3

~

a

a

O lO 20 30 40

ha (Oe)

Fig.7. (a)

Plot of _{rn versus}

In(n). (b)

Field dependence of _r for various cycle length: _open circles to ₌ 20 _s, solid circles to ₌ 40 _s, triangles to = 160 _s, rectangles to =1200 _s.

states.

In

spite

of the

dynamical

relaxation observed

throughout

the

spin-glass phase,

it does not appear

possible

to

explain

the observed behavior

solely

in _term of

dynamical

_processes, in

constrast with thermoremanent studies in the

vicinity

of Tg

jig, 20].

If _{we assume} that the

observed

logarithmic

relaxation _{m o~} ha

In(t/r)

is the linear _response to _a step

function,

the

time-dependent magnetization

in the field-off _state is obtained

by

linear

superposition

of the responses ^to each step as

mn(t)

₌

-«lT)ha f

'uilt

₊

2(I I)to)/Tl lull +12i -1)to)/Tl.

₁₃₎

Setting

_{n =} I

gives

the relaxation of the IRM,

mi(i)

=

«(T)h~in(i

+

iu/1) (4)

and the

reptation

_rn is obtained

by setting

t ₌ to

rn =

o(T) f

In

(I ))

zS

@

Inn + y.

is)

;=i ^I

where _y

= 0.577 is Euler's constant. The IRM has in this model _a

stronger sublogarithmic dependence

than observed in

experiments (particularly

at T «

Tg),

^{but the}

logarithmic growth

of _rn at

large

_n is

correctly reproduced.

Some other features _are not

properly

accounted for:

(a)

the ratio

r/o

₌

1/2

in this

model,

while the

experimental

ratio varies between 0.7 and

o , o o ~

l ^~

o,

~

' l ~

~

. «

°8 (°)

,o~

I-G

.

° ° °

«

~

* o

«

~

1.2

. O.5 at % Mn

~ O 8

. 2,6 at % Mn

04

16)

O

O O.2 O.4 O 6 O 8 O

T/Tg

Fig.8. (a)

Scaling plot of _{no versus}

TIT,.

Note that this ratio is constant at low temperature.

(b)

Scaling plot ^of

r/T,

_versus

TIT,.

This scaling function is linear at low temperature.

1.6 as a function of temperature. As noted

earlier,

_a temperature

independent

ratio of1.6 is observed at temperatures above 0A _x

Tg. (b)

The

reptation

rate found in the

experiment

scales with Tg

indicating

that the observed behavior has _a nonlinear

origin.

On the other

hand,

the linear _response is to first order

independent

of the

freezing

temperature below

Tg. (c)

The

change

of

sign

of the

reptation

rate _{as a} function of temperature in the field-on state shown in

figure

5 cannot be

explained

within linear response.

When a

spin-glass

is cooled in _a static field

H,

the

probability

distribution that _a

spin experiences

_a local field h is

asymmetric [PH(h) # PHI-h)]

_[21]. If _as

argued by

Ndel for

ferromagnets,

_rn reflects this

distribution,

_a

change

in the

growth

rate _r of the remanent

magnetization

should be observed if the system ^is

prepared

in _a static field H. The

dependence

of _r with H is

plotted

in

figure

9: _r increases first

quadratically,

reaches _a maximum at a field H ₌ 7000e _Se

J/15

and decreases at

higlier

fields [22]. ^{We found the maximum in} r to occur at

approximately

the _same field at T

= 0.03 _x Tg ^{and T} = 0.I _x

Tg,

^{I-e- Hmax is}

probably

not determined

by

thermal fluctuations. This suggests that this maximum arises when the field energy is

comparable

to _an internal energy. It should be noted that

hysteresis

in

spin-glasses

is

probably

not determined

by

local

properties

of

single spins:

Hmax is much smaller than the

exchange

field,

indicating

that the clusters

responding

to ha have _a

large

effective moment ~eR.

Assuming

the effective moment scales like the _square root of the number of

spins,

the reduction factor of IS from the

exchange

_energy would

correspond

to _a cluster of 225

spins

_very close to the

patch

size determined in numerical simulations

[23, 24].

When the

cooling

field H is

perpendicular

to the

reptation

field

ha,

the

reptation

_process

looks like _a

systematic

rotation of the

magnetization.

This "transverse

reptation"

_was found to

depend

_very

weakly

_on the

magnitude

of the transverse

cooling

field. To _a first

approxima-

tion,

the _tranverse

reptation

is _at low temperature ^{of the} same

magnitude

_as the

longitudinal

reptation.

As for

ferromagnets,

this suggests ^{that the rotational}

reconfiguration

of

spins

does

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNETS ₅₄₃

O

, -4

~

-8 ^. .

.

-t2~

O

o

H (Oe)

zation in the field-on state shows _a

striking "anti-reptation",

which is attributed to a

change

of the reversible

susceptibility

with field

cycling.

4.

Pinning

in disordered

Ising

models.

A convenient

framework

_to

study hysteresis

is the

Ising

model where disorder _can be introduced

quantitatively

_as random bonds _or random fields [25]. ^{The interface between two domains} may be

parametrized by

the

height

of the

interfjx)

^{with respect to a reference}

^plane.

^{A domain}

wall has _an intrinsic width

f

of order

f

_{= ~}

J/A,

where J and A _are

respectively exchange

and

anisotropy energies. Pinning

forces of this interface _{may come} from random

exchange

bonds which do not break the local

(Ising)

_symmetry and to first order do not

couple

to the order parameter. The interaction

potential

_per unit _area between the interface and the

impurities

may be written

as

V(x, z(x))

₌

~j _{v;fb(x x;)b(z(x) z;), (6)}

;

where _u; is the interaction energy with the i~~

impurity

_at

position

_r;

=

(x;, z;).

When the

impurity

I breaks the symmetry of the local order parameter, ^{it is}

equivalent

to _a random field H;. For

example,

in _a dilute

antiferomagnet,

its interaction _energy is _u;

=

2moH;,

where _mu the sublattice

magnetization.

This interface

potential

is

non-local,

z(X)

V(x, z(x))

₌ 2

/ mH(x,z(x))dz', (7)

o

where

H(x, z(x))

₌

£; H;b(x x;)b(z z;)

is the random field. Let 1I be the

typical

value of the random field _<

H(r)H(r')

_>=

H~b(r- _r'),

and _v the variance of the

potential.

In addition to

pinning, magnetic

_pressure and surface tension _{act on} the wall. At low temperature and _on

distances

large compared

to the interface

width,

the effective Hamiltonian

Ii

^z(x)

7i ₌

/ _{d~~~z -r(Vz)~}

2

/ _{m(r)hadz V(x, z(x)) (8)}

2 o

describes the interface at

long wavelength.

r is the interface stiffness which is

equal

to the surface tension _a for

isotropic

systems. For

ferromagnetic interactions,

the fluctuations in the local

magnet12ation

_{m are} not essential. On the other

hand,

for

randomly

diluted anti-

ferromagnets,

the _excess

magnetization m(r)

arises from the local fluctuation in the ion

density

and the Zeeman energy of the interface also fluctuates

randomly.

Let D be the _average distance between

impurities.

When the interface is broad

compared

to the

impurity separation (f

>

D),

the interface is

weakly pinned.

Provided the number of

impurities

in the interface is

sufficiently large,

the interface is

actually pinned by

the fluctua- tions in the

impurity density.

In this

limit,

it is convenient to measure the

pinning strength

in units relative to the elastic energy [25],

~ =

i~@~

_<g~

In _one

dimension,

the interface has _no surface tension. In this _{case, we} show that the

simple

balance between random

pinning

and

magnetic

_pressure is not sufficient to

give

_any

reptation

of the interface.

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNETS 545

1.o p~~

(

o.5

)

~

o-o

O-O O.5 1-O 1.5 2.O

i .o

o-B ~

O.6 E

O.4 O.2

~'~O.O _{O.2 O.4 O.6 O.8 1-O}

h

Fig.10.

^Schematic representation of the motion of _an interface, in _{a one} dimensional model. At point ^B the interface becomes unstable and jumps irreversibly to C.

4. I PINNING IN I-DIMENSION. We consider the motion of _a domain wall in _a one-dimensional random

potential V(z) previously

defined _as the wall

pinning

energy per unit _area. A motion of the interface

by

bz

changes

the

magnetic

_energy

by

bE ₌

-(m+ m-)habz

=

-pbz,

where p =

(m+ m-)ha

₌ 2mha is the

magnetic

_pressure. To remain in

equilibrium,

this energy

must be balanced

by

the

change

in

pinning

_energy

(dV/dz)bz,

I-e-

P(ha)

_"

~()~ ^(1°)

is the condition for

equilibrium,

as

long

_as the curvature

d~V/dz~

is

positive.

A

graphic

^de-

scription

of _a

reptation

_process is shown in

figure

10.

Initially,

the system is in

equilibrium

in A in absence of field. From A to B, the

equilibrium position

of the interface shifts

reversibly.

In B,

d~V/dz~ changes sign

^and the system

jumps irreversibly

from B to C. From C to D, the motion of the domain wall is

again

reversible, When the field is removed and

subsequently applied,

the interface _moves

reversibly

between D and E. The

magnet12ation

after each sub- sequent

cycle,

_mE, ^is

unchanged.

There is saturation after the first

cycle:

^{this model}

displays hysteresis

but _no

reptation.

4.2 HYSTERESIS IN HIGHER DIMENSIONS. On _a

length

scale

L,

the interface

gains

_{an en-}

ergy of the order of the fluctuations

bE;mp(L)

ⁱⁿ the

impurity potential, through

^the

impurities

within the interface width

f.

The balance between

bEjmp

^and the cost in the elastic _energy determines the

possible equilibrium shapes

of the interface. The

equation

of motion for the interface

height z(x)

measured with respect to _some reference

plane (x, 0)

follows from _equa- tion

(8)

~~~'~~ ~~x)

~~~~

where _y is the kinetic coefficient. Since _{we are} concerned with the local motions of the

wall,

for each

length

scale

L,

_{we use a}

wave-packet decomposition

for the

interface,

the random

potential,

and the local

magnet12ation [26,

27]

z(x)

₌

~jz>#>(x)

V(x,z)

=

~v>(z)#>(x)

(12) m(x,z)

=

~m>(z)#>(x)

where

#A(x)

denotes _a set of

orthogonal

functions of extend L, local12ed both in real and momentum spaces. The elastic _energy of _a mode I is of order

)rL~~~(zA /L)~.

It is therefore

possible

to

decompose

the Hamiltonian 7i, _on this basis _as 7i

=

£~

7iA where

i ^ax

7iA ₌

-rzA~L~~~

_VA

2L~~~zAmha 2Ll~~~~/~fliha qA(z)dz (13)

2

In

equation (13),

<

m(r)

_>% m and <

m(r)~

_{> <}

m(r)

>~e fli~ _are

respectively

the average and the variance of the local

magnet12ation

[28).

Similarly,

the normalized Gaussian variable

qA(z)

accounts for the residual part of the

fluctuating magnetization,

<

qA(z)q~i(z')

>=

) _exp(-(z z')~/i~)b~~i. (14) Following

Grinstein and Ma [27], ^the

dependence

of

a random bond

potential

_on the mode

amplitude

_zA is

readily

found to be

~d-ij

^1/2

VA " U

j

^6(ZA),

(IS)

while the interaction energy of the mode I with the random field is

ZA

VA =

2moHL(~~~~/~ / _{eA(z)dz, (16)}

o

where _E~ is the normal12ed

(to unity,

cf.

Eq. (14))

fluctuation of the random

potential

_or of the random field. Since the

potential

and

magnetization

fluctuations arise from the _same

impurities, spatial

correlations _may exist between them. Their correlation coefficient C _may be defined _as

~

QA(~)~A'(~')

>"

j _{~~p(~(~ ~')~/f~)}

bAA"

(17)

For random bonds and random

field,

the

equations

of motion for the

amplitudes

_zA become

respectively

~~~~~

"

~~~~~~~

₊

~

~j~~)

~~~

)~~~~

^{+ P~~ ~~~~}

~~zA(t)

=

-rzAL~~+2moHL(~~~~/~EA(zA)+ph (19)

Tat

^~

where

ph~ "

2L(~~~~/~ihhaqA(zA)

+ 21hha

(20)

N°2 REPTATION AND HYSTERESIS IN DISORDERED MAGNETS ₅₄₇

-O

O h~

Fig.ll. - Energy _{level diagram for} a local

is the

magnetic

_pressure. On short

length scales,

the surface tension term

always

dominates.

It is

straightforward

to check that

droplets

of radius R, ^smaller than Rc ₌

(d I)r/ph~

_are

unstable and

collapse.

Each local mode I has _a number of

possible

metastable _states. The

equilibrium position

zA~(x,

ha)

of the interface and _{an energy} EA~(ha) ^{of each} state is field

dependent. They satisfy

a non-linear

equation (fi7i/fiz

₌

0) which,

for the random field _case,

explicitly depends

_on the two random functions

(e, q)

^{associated with} random field and

magnetization

fluctuations

rzA =

2moHL(~~~~/~e(zA

+

2L(~~~~/~flihaqA(zA)

+

2mha. (21)

For each _new

equilibrium position

of the

interface,

the disorder and the Zeeman

energies

assume new random values. This leads to the

"spagetti diagram"

for the interfacial

energies

eA~(ha)

^{shown in}

figure

II. More relevant to the irreversible

jumps

of the interface _are the energy barriers between

equilibrium positions.

For _a mode of size L, metastable states have

an average

roughness w~(L)

₌ ^L~~~

f/[z(x) z(0)]~d~~~z,

w(L)

₌ ^{~~ ~}

((muH)~

+

(fliha)~

+

2CmuHiiiha)

^~~~

(22) f

~~~

Since

pinning

is

produced by

the fluctuations in the disorder

potential,

_energy barriers between metastable states _are of the _same order. From

equation (13),

these barriers scale with L _as

A(L)

₌

2(L~~~f)~/~ ((moH)~

+

(ihha)~

+

2CmoHfiiha)~~~ (23)

Such _energy barriers between

typical

metastable states _are not relevant to

reptation

_prc-

cesses, which

depend only

_on the _energy barriers which separate the low enelyy states. For such states, ^the _energy

gained through

the disorder

potential

is of

comparable magnitude

to the elastic energy. To estimate the surface

roughness

of such states, we minim12e the _average free _energy F for local modes of _area L~~~

2 r (i+r)/2 ^1/2

F = Fo

(L~-lf)~/~ j _(~)

₊

4Cihhaj (~)

⁺

^{(2ihha)~~ (24)}

f

D

f f