Phase diagram at a disordered interface between two ferromagnetic Ising systems

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Phase diagram at a disordered interface between two ferromagnetic Ising systems

G. Legal, A. Khater, T. Kaneyoshi

To cite this version:

G. Legal, A. Khater, T. Kaneyoshi. Phase diagram at a disordered interface between two fer- romagnetic Ising systems. Journal de Physique I, EDP Sciences, 1993, 3 (10), pp.2115-2122.

�10.1051/jp1:1993106�. �jpa-00246855�

J. Phys. I £Yance 3

(1993)

2115-2122 OCTOBER 1993, PAGE 2115

Classification Physics Abstracts

75.50R 75.10H

Phase diagram at

_a

disordered interface between two

ferromagnetic Ising _systems

G. LeGal

(~),

A. Khater

(~)

and T.

Kaneyoshi (~)

(~ Laboratoire de Physique des Mat6riaux

(*),

Universit6 du Maine, 72017 Le Mans Cedex, France

(~) Department of Physics, Nagoya University, 464-01 Nagoya, Japan

(Received

23 March 1993, ^revised 25 May 1993, accepted 7 June 1993

)

Abstract The phase diagrams at _a disordered interface between two ferromagnetic materials

are investigated by the _use of the effective-field theory. The conditions for the existence of _a magnetically ordered phase localized

on the interface _are examined and clarified.

1 Introduction.

The

study

^{of the}

magnetic properties

at the interfaces between two

crystalline magnetic

materi- als has received much attention in the past ^few _years. Most of the theoretical works _are directed to the examination of the

magnetic

excitations and the

phase

transition at the

crystalline

in-

terfaces between two

crystalline ferromagnets

_[1-3]. On the other hand, recent experiments show that in the interface

region

between the two

crystalline ferromagnets,

the two types ^of magnetic ^ions are

randomly

mixed [4, 5]. ^{The existence of the disordered interface often}

plays

an

important

role for the

magnetic

properties in the system. From the theoretical

point

of

view, however,

the effect of the disordered interface _on the

magnetic properties

in the system

as well _as

multilayers

has been little

investigated

[6-8].

Our _aim in this work _{is ~lia} the effective-field

theory

_{[9, 10]} to

study

the

magnetic phase diagrams

at the disordered interface between the two

spin -1/2 Ising ferromagnets

with differ- ent bulk

properties

and to look for the conditions

determining

the _appearance of the

magnetic

interface order when the bulks of the

sample

_are

paramagnetic.

The situation is similar to that of surface

magnetism

^where an enriched

phase diagram

^{with the}

possibility

^of _a surface critical temperature

T] higher

than the bulk temperature is obtained

ill].

In

particular,

the effect of

alloying

disorder at the interface _on the transition temperature ^for the interface

ordering

is

examined here. As far _{as we} know, there is _no

investigation

on such

problems

in the current

(*)

URA

no 807 CNRS

2116 JOURNAL DE PHYSIQUE I N°10

literature. We find that the critical condition above which the interface

magnetic

order is

possible

in the absence of bulk order is

strongly

influenced

by

the

alloying

disorder.

2. Model and formulation.

The system ^{considered here is}

composed

of _two semi-infinite

spin -1/2 Ising ferromagnets

A and B with different bulk

properties, matching

at the interface

plane.

The interface _possesses the _two kinds of atoms

randomly

mixed instead of _{one as} in _a

crystalline

interface. That is to say, the interface has the

alloying

^{disorder of} the type

ApBi-p

where _{p is} the concentration of A atoms. For

simplicity,

_we restrict _our attention to the _case of _a

simple

cubic

Ising-type

structure; the system is divided into the _two types ^of

planes parallel

to the

(100)

interface

plane

where _one refers _to A _atom

planes

and the other refers _to B atom

planes.

The Hamiltonian of the system is

given by

H =

~j _{J;j S]Sj(; (j,}

_I

(«J)

where the summation is _over all the

nearest-neighbor pairs only

once and

S]

is the

spin -1/2

operator

(S]

=

+I)

at _a site I

depending

_on whether the site is

occupied by

_an A _or B atom.

(;

is

a random

occupation

number

taking

values I _or o,

namely

_< (,&;A >~= p and < (;&;B >~= l p where

&m(a

₌ A _or

B)

is the Kronecker function and < >~ is the random

configurational

average.

J;j

is the

exchange

interaction which _can take three values JA>

JB

and JAB

(" JBA) corresponding

to the bonds

A-A,

B-B and A-B

(or B-A), respectively.

To evaluate the _mean values _<

S$

»~

(or

<

S$ >) (m

_E

A),

<

S(

»~

(or

<

S( >) (n

E

B)

and <

St

»~

(l

E

ApBi-p)

where < > expresses the canonical ensemble _average,

the effective-field

theory

_{[5, 8, 9] can} be

applied

to this

problem.

For the semi-infinite A

ferromagnet,

the

magnetization a$(m

_>

_I)

_per site in the _m th

plane parallel

to the

(100)

interface

plane

is

given by

at

_=<

_S[~i

_»~

=

(cosh JAR

+

at

sinh

JAR

_)]^~

(cosh JAR

+

at

sinh

JAR)) lp (cosh

JAR + mA sinh

(JAR ))

+

(I p) (cosh (JAB

R _{+ mB} sinh

(JAB R))]tanh (fix

_[~=o

(2)

and

at

_=<

_S$~~

_>

=

(cosh (JAR)

+

a)sinh _{(JAR)]~ (cosh (JAR)}

₊

a)_isinh _(JAR))

(cosh (JAR)

+

a)~i

_sinh

_(JAR)]

tanh

(fix)[~=o

for _u > 2,

(3)

where

fl

₌

I/kBT

and R ₌

0/0x

is the differential operator. The

magnetizations

_mA and _mB in

(2)

represent the

averaged magnetizations

in the

interface,

which _are defined

by

< (1 61a <

St

_»~

A

B)

_~~~

"~O "

< j~ &i~ >~ ^{~ ~~}

The

averaged magnetization

_per site in the interface is

consequently given by

m # p mA +

(1 P)"lB, (5)

N°10 PHASE DIAGRAM AT A DISORDERED INTERFACE ₂₁₁₇

where the

magnetization

_mA and _{mB can} be

expressed explicitly

_as

mA = b7

(cosh JAR)

+ _mA sinh

JAR)

₊

(I p) (cosh (JAB R)

+ _mB sinh JAB

R))]~

(cosh (JAR

+

at

_sinh

_{JAR)] (cosh}

_JABR ₊

at

sinh

(JAB R)]

tanh

(fix

[~=o

(6)

and

mB = b7

(cosh (JAB

i7 _{+ mA} sinh

(JAB i7) )

+

(1 p) (cosh (JB i7)

_{+ mB} sinh

(JB i7) )]~

(cosh

JAB

i7)

₊

at

_{sinh JAB}

i7)] [cosh (JBi7)

+

a)sinh _(JBi7)] tanh( fix)

_[~=o

(7)

Here, the

magnetizations at

_and

at (n

_E

u)

in the semi-infinite B

ferromagnet

_are also

given by

al

_"<

_S~]i

_>r

=

(cosh (JB

ⁱ⁷ +

at

sinh

(JB i7)]

^~

(cosh (JB

ⁱ⁷ +

afsinh _(JBi7

_)]

§J

(cosh

JABi7 _{+ mA} sinh JAB

i7)

+

(1 p) (cosh JBi7)

_{+ mB} sinh

(JBi7) )]tanh (fix

_[~=o

(8)

and

a)

_=<

_S~]~

_>

=

(cosh (JB i7)

⁺

a)

_sinh

_{(JB i7)]}

^~

(cosh (JB i7)

+ a~

$

sinh

(JB

i7)]

(cosh (JBi7)

+

a~/isinh _(JBi7)]

tanh

(fix)

_[~=o

(9)

3. Phase

diagram.

The transition temperature, _or

phase diagram,

for

determining

the _appearance of the

magnetic

interface order is obtained

by expanding

the

right

hand sides of

(2), (3), (6)-(9)

and

taking

only

the terms linear in

a), a)(u

>

I),

_mA and _mB.

Then,

_one should notice that for _{u - oc}

the

magnetizations a)

and

at approach

the bulk

magnetizations at

and

at,

which _can be obtained

by putting

_{am =}

a$_i

= am_~i =

af (o

₌ A _or

B)

into

(3)

_or

(9);

at

₌

[cosh (Jni7)

+

af

sinh

(Jai7)]~ tanh(fix)[~=o (10)

The bulk transition temperature

T)

for the bulk A

(or B) ferromagnet

is determined from

1 ₌ 6 sinh

(J~i7) cosh~ (J~i7)

tanh ^~

~

i, ₍₁₁₎

kBTc

~=o

which is

nothing

but the Zernike

equation

in the

simple

cubic

spin -1/2 Ising ferromagnet

[9,

12].

^{The bulk} transition temperature is

given by

~ ~b

~

= 5.0733

,

(a

₌ A _or

B) (12)

which is superior to the standard mean-field result

(kBT)/J~

=

6)

2118 JOURNAL DE PHYSIQUE I N°lo

As is discussed in the standard model of surface

magnetism [11, 13],

^{the linearized}

equations

can be solved

by introducing

the transfer functions _a and b in the semi-infinite A and B

ferromagnets

,A ,B

%

= a

,

~

= b for _u > 1

(13)

?v ?v

By

^the use of the linearized

equations

of

(3)

and

(10)

the parameters _a and b _are

given by

(1-4 KA) [(1-

4

KA)~

⁴

K(j~~~

~~

2KA

(1-4KB) ((1-4KB)~ -4K(j~~~

b=

~ ~

(14)

B

where the coefficients

K~ (a

= A _or

B)

_are determined from

K~=sinh

(J~i7) ^cosh~ (Jai7) tanh(fix)[~=o (15)

From these

procedures,

the linearized

equations

reduce to the

following

secular

equation

3i mA ~?

= 0

(16)

?j

ai

with

4 _p Ki 1

,

4(1 p)

K2

,

K3

,

K4

~ ~/~ ^~~l

~~~lz ₍₄

₊

~13

₁

~l

' ~~~~

p

Ri

,

(I p)Rz

,

0

,

(4

+

b)R3

1

where the coefficients

K;(I =1-8), Lj

and

Rj(j =1-3)

_can be obtained from

(6), (7), (2)

and

(8).

These _are

examplified by

such forms _as

(15)

and _can be calculated

explicitly by applying

the relation

exp(di7)#

₌

#(x

+

d). Thus,

the transition temperature of the disordered interface

can be determined from

det

3i

= 0

(18)

by taking

the

highest

solution.

4. Numerical results.

In this

section,

_some results _are

presented by solving (18) numerically. However,

_we have three parameters

(JA

JB

JAB

^{for the} numerical evaluations. Let _us here take JA > JB > 0 without

losing generality. Therefore,

the disordered

(or pure)

interface _may order _{at a} temperature

T(

which is

higher

than that of the bulk

T) _{given by} kBTf/JA

" 5.0733, _even when the semi-

infinite A and B

ferromagnets

are

paramagnetic.

^The situation is very similar to the surface

phase diagram

of _a semi-infinite spin

-1/2 Ising ferromagnet

which _can exhibit two successive

transitions, namely

the surface and bulk transitions, _as ^the temperature is lowered

Ill,

_13].

N°10 PHASE DIAGRAM AT A DISORDERED INTERFACE 2119

1.3

°.5 0.8

1.2 0.2

T[

^~.~

^~~~~

^°'~

i

i-o

°.9

Bulk Ferro

o-a

0 2 4 6 8 lo

~ab

~) ~a

1.3

1.2 ~~

T~ ^Para

q ^i-i

T~

i-o

Bulk Ferro

0.9

0 2 4 6 8

~ab

b) Ja

Fig. 1. a) Phase diagram

(transition

temperature versus JAB for the interface and bulk magnetic

orderings of the system with

JB/JA

" 0.5 consisting of the interface of the

ApBi-p

type ^and the two

semi-infinite A and B Ising ferromagnets, when the value of p is changed

b)

The _same phase diagram

as

a)

is plotted by taking _a larger scale.

In

figure

1, _we present the

phase diagram

of the system with

JB/JA

" 0.5 in the

(T)/Tf,

JAB

/JA)

_space,

changing

the value of _p. It characterizes the _state of the interface

magnetism, paramagnetism

and

ferromagnetism.

In

particular, figure

16 shows the details of

figure

la

by taking

a

larger

scale.

Furthermore,

for the system ^with

JB/JA

₌ o-I the _same

plots

_as those of

figure

I _are

given

ⁱⁿ

figure

2.

In the

figures,

the horizontal line

corresponds

to the bulk transition temperature

T)

of the semi-infinite A

ferromagnet.

For T _>

T),

the two

bulks,

_or the semi-infinite A and B

ferromagnets,

_are

paramagnetic

for all values of JAB ^{and the} same

happens

with the interface if JAB <

J[B,

where

J[B

is the critical value at which the interface and bulk

orderings

coexist

2120 JOURNAL DE PHYSIQUE I N°10

1.3

12 Para

0.6

~i

I-I C

~~

l-o

o.9

Bulk Ferro

o-a

0 2 4 6 8 lo

J~~ ~

a)

1.3

0.5

Para

1.2

~~

i_1

~~

i-o

Bulk

Ferro

0.9

0 2 4 6 8

~ab

b) _j~

Fig. 2. a) Phase diagram for the interface and bulk orderings _{as a} function of JAB in the system

with

JB/JA

" 0.I, when the value of _p is changed. b) The _same phase diagram _as

a)

is plotted by

taking _a larger scale.

for T

=

T). But,

if JAB _>

J[B,

an intermediate interface

ferromagnetic region (or

interface

magnetism)

_{appears, even} if bulk order is absent. The

region

of interface

magnetism

is widest for the disordered interface with p = 0.5.

Here,

_one should notice that in each

figure

the

T)

of

the _pure interface with _p

= 0.0 is

equivalent

to that of the pure interface with p = 1.o, since

only

the

exchange

interaction JAB exists between the semi-infinite A and B

ferromagnets.

As is seen from these

figures,

the critical value

J[B characterizing

the interface

magnetism dearly depends

on the values of _p and

JB /JA.

In

figure

3, ^{the critical} value

J[B

^{of the} system is

depicted

as a function of _p,

selecting

the two values of

JB/JA> namely JB/JA

" 0.5 and

N°10 PHASE DIAGRAM AT A DISORDERED INTERFACE 2121

o.I. Some

outstanding

features _{are seen} for the _curves: the critical value

J[B

^{for the} interface

magnetism

becomes smaller when the value of

JB /JA

increases and the minimum value of

J[B

is not found _at the most disordered interface with _p

= o.5.

5

J~/J

^=o.1

4

~~b

J~^I

j

= 0.5

~

^~

2

0 O.2 O.4 0.6 0.8

P

Fig. 3. The critical value

J(B

for the interface magnetism as a function _{of p,} when the two values of

JB/JA

_are selected.

5 Conclusions.

We have studied in this work the effects of disordered interface _on the transition temperature for interface

ordering

and the multicritical

point

within the framework of the effective-field

theory.

As shown in

figures

1-3, the existence of the disordered interface _may effect the

phase diagram severely. They clearly

indicate that the interface

magnetic ordering

_may be

possible

due to the

alloying

^disorder at the interface. In

fact,

in

figure

3, ^{the critical value}

J(B

takes _a minimum value at a certain value of _p. The _occurrence of

T)

>

T)

_in the disordered interface may be

observed,

like the _case of

T]

>

T)

_in surface

magnetism

which is

proved experimentally,

among

others,

in the

Gd(cool)

and

Tb(cool)

surfaces.

Now,

in order to compare the present ^{results with the related}

experimental

data, _one must note _some important facts. As _was discussed in [4,5], ^{the interface}

region

^between the two bulks is _more

complicated

than in the present ^model calculation. In this

work,

_we

simply

_assume

that the interface has _an

alloying

disorder.

Experimentally,

the interface of _some

multilayers (for example,

Co

/Cu)

consists of

only

_one mixed

layer,

while the interface of other

multilayers (for example, Co/Ni)

consists of two mixed

layers.

Even for the _one

layer

_case, the interface

which results from _numerous small atomic steps is

certainly

not _a flat but diffused interface with _a random distribution of the two types of atoms in the bulk. On the other

hand,

other

experimental

data indicate that the interface

region

^{looks like} _an

amorphous phase.

^Even if _so in the real

interface,

_we

hope

that the results obtained here will stimulate further

experimental

and theoretical work _on

magnetic properties

of disordered interfaces.

2122 JOURNAL DE PHYSIQUE I N°10

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