Bloch Walls with div I ≠ 0

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Submitted on 1 Jan 1959

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Bloch Walls with div I ̸ = 0

Jan Kaczér

To cite this version:

Jan Kaczér. Bloch Walls with div I ̸= 0. J. Phys. Radium, 1959, 20 (2-3), pp.120-123.

�10.1051/jphysrad:01959002002-3012000�. �jpa-00236000�

120

BLOCH WALLS WITH div I ~ ⁰

By

^JAN

KACZÉR,

Institute of Physics, Czechoslovak Academy ^of Sciences, Prague.

Résumé. 2014 On a calculé quelques types ^de parois ^de Bloch, pour lesquels la condition div I ~ 0

est satisfaite. On a montré que de telles parois peuvent ^{exister sous} certaines conditions bien qu’elles possèdent ^des énergies considérables. On a essayé de trouver une interprétation ^des parois

dites ^« invisibles ».

Abstract. ²⁰¹⁴ Some types ^{of Bloch} walls, for which div I ~ 0, ^are calculated. It is shown that such walls can occur under certain conditions even though they ^have higher energies ^associated

with them. An explanation of so-called invisible walls is attempted.

LE JOURNAL DE PHYSIQUE ^ET LE RADIUM TOME 20, FÉVRIER 1959,

1. The

present theory

^of Bloch walls

[1], [2]

between domains of

spontaneous magnetization

ⁱⁿ

unbounded ideal

ferromagnetics

starts out from the

assumption

^{that the vector of}

magnetization

^I

in the Bloch walls rotates in such a way that its

component in the direction of the normal to the Bloch wall is constant. In other

words,

^{the follow-}

ing equation

should hold

meaning

^{that the} Bloch wall is

usually

^{not accom-}

panied by

space

charges

which would

give

^{rise to}

energetically

^very unfavourable

demagnetizing

fields. This

assumption obviously

^{holds in the}

majority

^{of cases,}

particularly

^{if the}

crystal approaches

^{an ideal}

ferromagnet.

It

is, however,

clear that conditions in real

crystals

^{are often so}

complicated

^{that the}

simple

domain structure foretold

by

Landau and Lifshitz

only

holds in its main features but does not

explain

a number of

complex

^domain formations. Due to

the existence,

however,

of external and internal

surfaces,

inclusions and

inhomogeneities

^{of internal}

stresses

demagnetizing

^{fields are set} up which ^can

bring about

^{walls for which}

Eq. (1)

^{is not satisfied.}

In

previous

papers Néel

[3]

^{and the} present

author

[4]

^{arrived at the} conclusion that walls with

non-zero

divergence

^{I can exist in thin ferro-}

magnetic

^films.

In the

present

paper ^we shall show that under certain conditions walls can exist for which

Eq. (1)

is not satisfied. These walls have a greater energy

density

than classical Bloch walls but there is no reason to eliminate them from domain structure formation.

2. 1800 wall in an uniaxial

ferromagnet.

^{- lent us}

take an ideal unbounded uniaxial

ferromagnet

^with

the direction of _easy

magnetization

ⁱⁿ the direction of the z axis and

positive isotropic magneto-

striction x. Let us denote

by

^{0 the}

angle

^between

the direction of the

magnetization

^{vector and the z}

axis

( 6

^{is a function}

only

^of

z)

^{and 6 a}

homogeneous

stress

acting

ⁱⁿ the direction of the z axis. The free energy F is then a sum of four terms which are

functions of z : the

exchange

energy

FE,

^{the aniso-}

tropic

energy

FK,

^the magnetoelastic energy

Fo

and the

demagnetizing

energy FD. The different terms can be written as follows. : ^"

The

expression

^{for the}

demagnetizing

energy follows from the

invalidity

^of

(1)

^{and from the}

equation

on the

assumption

^{of a}

plane

^{unbounded wall.}

Here A denotes the

exchange

constant, ^{K the}

anisotropic

constant, ^{À the}

magnetostriction

^con-

stant and Is ^{the value} of saturation

magnetization.

The

angle

^{0 as a function of z is obtained}

by

variation of the ^total energy

given by

,-,

where we have introduced the abbreviations

3

₂ Àcr ⁼

L,

^{’ 2n}

12

^{= M. The}

boundary

^conditions

(see Fig. 1)

^are

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:01959002002-3012000

121

Solving problem (7)

^we obtain the

following expression

^{for the}

dependence

^{of the}

^angle

^{0 on z}

where

For the thickness of the Bloch wall 8 we obtain

A

difficulty

^{arises .if we want to} calculate the energy

density

y ;

integral (7) using

^solution

(9)

^is

divergent.

^{This is}

quite

understandable since ^{as a} result of the boundlessness of the wall the de-

magnetizing

^{field does not vanish at}

infinity but

becomes HD ^{=-- 4n} Is. In actual _cases,

however,

the walls ^are

always

^{bounded and} the field vanishes at

infinity

^{at least as}

1fr2,

^which guarantees ^{a finite}

energy

density

y.

3. 1800 and 90° walls in a cubic

ferromagnet.

^-

Let us now

investigate

^the

properties

^{of walls for}

which eq.

(1)

^{is not satisfied in a cubic ideal}

unbounded

ferromagnet.

3.1. 180o WALL IN A

(001)

^{PLANE. -we ch00se the}

coordinate system ^{as in}

Fig. 1,

^{the wall}

being perpendicular

^{to the z}

^axis,

^the direction of homo- geneous tension ^a

parallel

^{to the z axis. The}

angle

between the ^z axis and the direction of the magne- tization vector I will

only

^{be a} function of z.

In this case we can write for the individual

components

^{of the energy .}

The total energy is then

given by

^the

expression

By

variation of this

expression

^{with the boun-}

dary

^conditions

we obtain the solution in the form

where

In

Eqs. (19)

^and

(20)

^we introduced the _para- meter p for the sake of

simplicity.

For the thickness of the wall we obtain the

expression

3.2. 1800 WALL IN

(011)

^{PLANE. - The coordi-}

nate system ^{for this case} is shown in

Fig.

^{2. The}

axes of easy

magnetization

^are x, y, z ; the wall is

perpendicular

^{to the ç axis. The} stress acts in the direction of the Z ^axis.

The

angle

^of

magnetization

^{as a function}

of ç

^is

given by expression (18)

^with

3.3. 900 WALL IN

(011)

^PLANE. - The coordinate system ^{is chosen}

according

^to

Fis-.

^{3. The wall}

will be

perpendicular

^to the y

axis,

^{the stress in}

the direction of the _{y axis.} The axes of easy

magnetization

^{are in}

the x, y’,

^z’ directions. The individual

component

^{of the} energy

density

^will

be .

The

boundary

conditions are

FiG. 3.

By solving

the variational

problem

^{we obtain for}

the

dependence

^{of the}

angle

cp ^on y

On the basis of

(29)

we. obtain for the thickness of the wall

4. Discussion of results. ^- It follows

from

the above that walls for which

Eq. (1)

^{does not hold}

can

only

exist under

quite

^definite conditions.

Thus in the case of 1800 walls in uniaxial ferro- magnets

only

^{when p} > - 1. ^If the stress ^{6 ==}

0,

this means that K > M. This condition is not

satisfied for all materials. For

cobalt,

^for

example,

K ⁼ 5 .106

erg/cm3

^{and M =1. 3 X 101}

erg/cm 3,

thus K M and

Eq. (11)

^{for the thickness of the}

wall

gives imaginary

^{values. Neither is the pre-}

sence of stresses sufficient ^for

attaining

real values of the thickness ^of the wall.

For cubic

crystals

p ^must be

greater

^{than 0}

for

(100)

^walls

according to Eq. (22),

^{which cannot}

be satisfied in the absence of stress and on the

assumption

^{that .K} > 0. For

iron,

^for

example,

for which M =1. 8 X 107

erg/cm3

^{without stress}

p ^{= -} 43 and the condition p > 0 even with

large

stress cannot be satisfied.

Analogously

^{in other}

cases.

The situation for materials

having

^{a small con-}

stant of saturation

magnetization

^and

large

^aniso- tropy ^is différent. For

MnBi,

^for

example,

.NI ⁼ 2.4 X 106 and K

=10’,

^{thus M K and}

Eq. (11)

^{even without stresses}

gives

real values.

Walls with div I :A 0 ^{can therefore exist in MnBi.}

If, however,

^{the area} of the walls is ^not

infinite,

which means in essence that its linear dimensions

are

comparable

^{with its}

thickness,

^{then we can}

write

where

g is

^{a constant the} value of which

depends

^on

the geometry ^{of the}

demagnetizing

^{field and which}

lies between 0 g 1

going to

^zero for small wall surfaces. The conditions

for p

^{can in this case be}

satisfied even in materials where unbounded walls

are otherwise

impossible.

Such very small ^sur- faces of walls with ^non-zero

divergences

^{are found}

for

example

^{at the ends of Néel}

spikes.

^With L #

0,

it is all the more

likely

that in certain

parts ^{of a}

crystal,

^{where there are suitable} stresses, the necessary conditions will be satisfied and thus walls of the

proposed type

^{will be}

produced.

It can be assumed that the occurrence of 90°

walls of the

proposed type

^{is more}

probable

^than

the occurrence of 180°

walls, chiefly

in materials with

inhomogeneous

^internal stress, ^{where the}

direction of _easy

magnetization

^often

changes

^and

where there are therefore sometimes suitable condi- tions for the

production

^{of these} walls,: Similar 90°

walls were observed

recently

ⁱⁿ bent whiskers

[5],

where

inhomogeneous

^{stress was}

artificially

^pro-

duced.

5. The

problem

^{of invisible walls. -}

Experi.

mental evidence ^with

powder patterns

^{shows that}

some

walls,

^across which the normal component ^of

magnetization

^is

discontinuous,

^{are invisible}

(1).

We shall attempt ^to

explain

^this

invisibility

^{on the}

basis of our calculations.

As follows from our

results,

the thickness of the wall

depends

^{on the parameter p} and for certain values of this

parameter

^{this thickness} is very

large.

^{A result}

of this,

^of course, is that the

demag- netizing

^{field of such a wall and the}

gradient

^of

this field above the surface of the

sample

^are very weak and are no

longer

sufficient to attract col- loidal

particles.

Such walls would therefore not be made visible

by

the colloid

technique.

With the 90° walls calculated in para.

3.3,

^we

have in addition the fact that these walls have a

component

^{of the}

magnetization perpendicular

^to

the surface of the

sample,

^which

produces

^surface

charges

^of

density

In. ^These

charges give

^{rise to an}

additional

demagnetizing

^field above the surface of the

sample

^which decreases the value of the

demag- netizing

^field

produced by

^{the non-zero div I}

inside the ^wall. This could therefore be another

reason

why

^{these walls cannot} be made visible

by

the colloid

technique.

Quite

^another

explanation

of invisible walls is

given

ⁱⁿ the paper

by

^{Coleman and Scott}

[7],

^who

(’) ^Thé hypothesis ^on the existence of ^" invisible walls ^"

was expressed for the first time in the paper by ^Good- enough [6] ^{from 1956.}

123

found

experimentally

that the intersection ^of two classical Bloch

walls,

^{which lies in} the surface ^of the

sample,

^{does not attract} colloid and is there- fore invisible. On the basis of this observation the authors come to the conclusion that all invisible

walls,

for which the normal component ^{of the}

magnetization

^is discontinuous, ^{can be}

explained

ⁱⁿ

this way. It ^{seems ton} us,

however,

^{that this}

need not

always

^{be the case}

since,‘

^for

example,

with the maze-

pattern

^{the walls are visible.}

6. Conclusion. ^- The paper deals with the influente of the

demagnetizing

^{field on the} magne- tization vector inside the Bloch wall.

It is shown that such walls could be

produced

ⁱⁿ

real

crystals

under certain conditions even

though

their share in the

production

^{of domain} structure, due to the

high

energy densities

involved,

is pro-

bably

^small.

On the basis of this model an attempt ^{is made to}

explain

^{so-called invisible walls and the model}

proposed by

^{Coleman and Scott is discussed.}

REFERENCES

[1] ^BLOCH (F.), ^Z. Physik, 1932, 74, ^295.

[2] ^LANDAU (L. D.) and LIFSHITZ (E. M.), Phys. ^Z. Sowjet., 1935, 8,155.

[3] ^NÉEL (L.), ^{C. R. Acad.} Sc., 1955, 241, 533.

[4] ^KACZÉR (J.) : Czechosl. J. Phys., 1957, 7, ^557.

[5] ^{DE BLOIS} (R. W.) and GRAHAM (C. D.), ^J. Appl. Phys., 1958, 29, 528.

[6] GOODENOUGH (J. B.), Phys. Rev., 1956, 102, ^356.

[7] ^COLEMAN (R. V.) ^{and SCOTT} (G. G.), ^J. Appl. Phys., 1958, 29, 526.