Anisotropy of the magnetoresistance along and across domain walls in a ferromagnet

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Anisotropy of the magnetoresistance along and across domain walls in a ferromagnet

Yu. Zakharov, Yu. Mankov, L. Titov

To cite this version:

Yu. Zakharov, Yu. Mankov, L. Titov. Anisotropy of the magnetoresistance along and across domain walls in a ferromagnet. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.759-764.

�10.1051/jp1:1991167�. �jpa-00246368�

Classification

Physics

Abstracts 72.1 5G

Anisotropy of the magnetoresistance along and

_across

domain walls in

_a

ferromagnet

Yu. V.

Zakharov,

Yu. I. Mankov and L. S. Titov

Kirensky

Institute of

Physics,

Siberian

Academy

of Sciences,

Krasnoyarsk

660036, U.S.S:R.

(Received30 May

J989, revised2?

September

J990 and30 January J99J,

accepted

J

February J99J)

Abstract. We discuss _some

peculiarities

of the conduction electron motion in the

vicinity

of domain walls which lead _{to an}

anisotropy

of the

magnetoresistance.

We also discuss the _case of

single crystals

of

ErRh4B4

and

Homo~sg

where

magnetoresistance

vith the _same

qualitative

features has been observed.

1. Introdttcdon.

In recent years there

appeared

several papers

[1~3],

^{in which the} electroresistance of

single crystals

of reentrant

superconductors

in the

ferromagnetic phase

has been

investigated.

In the paper

by

Genicon et al.

[Ii

^the electroresistance _p of _a

single crystal ErRh~B4

_was ^observed in terms of the

magnetic

field H. In this paper a

large

effect of the

magnetoresistance anisotropy

was noticed. Koike _et al.

[2]

have been the first to

provide

electroresistance measurements on

single crystal Homo~sg.

The detailed

investigations

_on

single crystal Homo~sg

_were

performed by

Giroud _et al.

[3].

We have

already paid

attention

[4]

to the

possibility

that the p

(H) dependence

observed in reference

[Ii

could be due to the modification of the

trajectory

motion of the conduction electron in the domain wall

vicinity [5, 6].

Electrons which _cross the 180° domain wall _move

along

the infinite

trajectories (2~domain state) (Fig. la).

Under

magnetization

a

decreasing

domain width

D~

becomes

eventually

less than the

cyclotron

diameter 2 R. In this _case there _appear electrons

moving along trajectories, encompassing

three domains for _a

period (3~domain state) (Fig. lb),

and

simultaneously

the number of electrons in 2~domain states decreases. The

mobility

of the electrons in 3~domain states in the

y~axis

direction _appears to be much less than that of 2-domain _ones and this

causes an increase of the resistance

along

domain walls.

Here _on the basis of such _an

approach

the

magnetoresistance anisotropy

of _a

ferromagnet along

and _across the domain walls _was obtained and _some

peculiarities

of the

magnetization

process effect _on the

magnetoresistance

_was

analysed.

The results obtained _are

compared

with the

experimental

_ones

[I]

in _a

single crystal ErRh~B~

in the

ferromagnetic phase.

The

qualitative analysis

of the

magnetoresistance

in _a

single crystal Homo~sg

in the

ferromagnetic

phase

_was carried out too.

760 JOURNAL DE PHYSIQUE I M 5

8 3

~8

8 3 2

~

U~

u b _{o is} toMiM~

Fig.

I.

Fig.

2.

Fig.

I. Conduction electron

trajectories crossing

one domain wall

(a)

and

crossing

the

decreasing

domain at

D~

< 2 R

(b).

Fig.

2. The

dependence

of the electroresistance

anisotropy p~jp~~

on the

magnetization.

Curve I) at S

=

20 _;

2)

at S

=

10 _;

3)

at S

= 5. All _curves at 2

RID

=

0.8.

2.

Anisotropy

of the

magnetoresistance.

The resistance of _a

compensated ferromagnet

in the saturated state

p~=S~/«o,

_where

«o is the

conductivity

at induction B

=

0,

S

=

ilR, I

_{is the} free

path length,

R is the

cyclotron

radius. In the

demagnetized

state 2-domain electrons _move

along

the wall

(in

_y

direction) along

the infinite

trajectories.

The

conductivity

of the mentioned electrons is

«~~

«o. For the

sample

with the

plane-parallel

domain structure of domain width D when 2 R _< D

(that

is

satisfied

[4]

for the

single crystal ErRh~B4)

the

conductivity

is «~~~

(RID)

_«o at

SW

I,

then for the resistance in the

demagnetized

state

p)(I

_we obtained the ratio

Pj(~/Pn

"

D/Rs~ _j4j.

The resistance _across domain walls

pj$~ slightly

decreases because the localization size in the X direction for the 2~domain electrons increases. In the

vicinity

of _a wall in a

layer

of

width L 4 R

(when

4 R _<

D)

the resistance is _p~

=

~

p~

[7].

^The relative resistance of _a 4

sample

with the

plane-parallel

domain structure is

Pll~/Pn

"

(l PL/Pn)(4 RID), (I)

and _we obtain the estimate

pj(I/p~

= I

RID.

When 2R

< D it leads _to the

appreciable magnetoresistance anisotropy along

and _across the domain walls. This _was

really

observed in

a

single crystal ErRh~B~ [Ii.

We consider the

simplest magnetization

_process

by

the domain wall motion. lvhen the

decreasing

domain width

D~

becomes less than 2 R there appear 3~domain electrons. The localization size in the X direction for these electrons is

approximately

half _as much

again

_as

for the 2~domain _ones. It results in _some decrease in p~~ The

mobility

of electrons in the 3- domain states in the Y direction _appears to be much less than that of the 2-domain _ones.

Thus,

both p~~ ^{and the} ratio

p~jp~~

can

change

in _a wide range under

magnetization.

The

conductivity

of _a multidomain

ferromagnet

^is calculated

by

the Kubo method

[4, 6].

The

dependence

of the resistance _on

magnetization

was

computed

_on the

assumption

of _a

cylindric

Fermi surface

(the cylinder

axis is

along

the

z-axis).

In

figure

2 we

give plots

of the

dependence

of

p~jp~~

on the relative

magnetization M/fi~,

where

M~

is the

magnetization

value at which domains

disappear.

The kinks _on the

plots correspond

to the appearance of a

3-domain state. Then _a

gradual

transformation of 3-domain states into one~domain states

occurs under

magnetization,

and the

anisotropy vanishes,

the ratio

p~Jp~~-I

when

D~-0.

3. Influence of the domain structure

expansion

_on a resistance.

In reference

[Ii

the results of the resistance measurements _across domain walls

p~jp~

in terms

if

_the

magnetic

_field

are

given

in _zero field

p~jp~=0.9,

and in fields Har3kOe

p~jp~

₌ ^{I. In}

^figure

³ we

present

^the

dependence

of

p~Jp~

on

M/fi~

obtained

by

the transformation of

experimental [I]

_curves

p~(H) (curve I)

and the results of _our

computa~

tions at different values of 2

RID (curves 2, 3).

,' 3

'~

"

wR,

Fig.

3. The

dependence

of the resistance

p~Jp~

_on the

magnetization.

Curve

I) experimental

results

according

to

[Ii 2)

calculation at S

= 15 and 2

RID

_{= 0.2 ;} 3) ^the same at 2

RID

=

0.6 _;

4)

with the

account of the domain structure

expansion,

calculated

by

the formulae

(3)

at 2

RID

=

0.2.

For

improving

an agreement ^with the

experimental data,

_we consider the influence of the wall number

change

under

magnetization

_{on a} resistance. We have made

[8]

_an account of the effect of the domain structure

expansion [9,10].

For this purpose in the

expressions

for

p~~/p~

^the substitution D

-

Do #(q)

has been

made,

where

Do

is the domain width in the initial

demagnetized

_state, and the function

#(q)

is determined

by expressions [10]

:

#(q)

=

lf(I)/f(q)l~'~, f (q)

=

ij _(I

cos nflTq

) n~~, ₍₂₎

where _q

=

I _{m, m}

=

M/M~.

The numerical

analysis

shows

[8]

^that such _an account leads _to

only

_a small

displacement

of the theoretical _curves for

p~~/p~

^without

changing

their

character.

Really

for the

sample

with _a

plane~parallel

domain structure the

conductivity

_«~~ would consist of the domain

conductivity

_«~

=

«o/S~

« «o and the

conductivity

_«~

= «o of _a

layer

in the

vicinity

of the

domain

walls. The thickness of this

layer

L

changes

under

magnetization

_:

L

~

R when

D~

_> 2 R and L

~

D~

when

D~

_< 2 R. The relative resistance becomes

p~~/p~= (I+A)~',

_{where we use the} notation

A=S~R/D

when

D~>2R

and A

=

(8/3) (D~/D)(D~R)~'~ S~

when

D~

« 2 R. Here the factor

(D)R)"~

_{is a measure} of the Fermi surface

region occupied by

2~domain electrons localized in the

vicinity

of the

decreasing

domain. As

D~

₌ ^D

(I m)

then the value A

~

(l m)~'~

when

D~

« 2 R. With _account of

the domain structure

expansion

and the fact that when _m m0.7 the

approximation

D

= 2

Do/5(1 m)

holds

[10],

A

(l m)

when

D~

« 2 R. This makes the decline of the

curves p

)f~/p~ ^change

at _{m <}

I,

but

owing

to S~ _» I the

shape

of the _curves with and without

account of the domain structure

expansion

_may

substantially

differ

only

when

I _m

= 10~ ^~

= 10~ ^~

JOURNAL DE PHYSIQUE I T I, M5, MAil99l

762 JOURNAL DE

PHYSIQUE

I M 5

Consider the

dependence

_p

)f~/p~

_{on m.} After substitution in

(I)

D

-

Do ~ (q), expressing

all constants

through

the ratio

p)(~/p~

which may be taken from the

experimental data,

_we obtain

P)71Pn

"

(1 p]]~/Pn)/4'(q) ₍₃₎

Expression (3)

is valid until

D~

_> 2 R and 3-domain electrons _are absent. lvhen

taking

into account the domain structure

expansion,

3~domain _{states may appear} when

q# (q)

< 2

R/Do.

For

2R/Do<0.3

it is

possible only

at I

-m<10~~,

therefore the

dependence p,(°~/p~

(I

_=x,

y)

is determined

by

the _process of the domain _structure

expansion.

The value

p,(°~/p~,

^{I-e- the resistance}

change

in the

vicinity

of the domain

wall,

is connected with 2-

domain electrons. At 0.3 _w 2

RID

_w the contribution of the domain structure

expansion

to

pj~~/p~

is not essential. Such _an

approach

allows _us to obtain _a rather

good

_agreement with the

experimental

_curves for the

dependences p,(~~/p~.

The

graph pjf~/p~ considering

the

structure

expansion by

formulae

(3)

at

pj$~/p~

= 0.9

(I.e.

2

RID

=

0.2)

is

plotted

in

figure

3

(curve 4).

It should be noted here that the

computed

values

pj$~/p~

and

p)(~/p~

are determined

by

the values 2

RID

and S

respectively (when

SW

2),

and the measurements of ratios

p)(~/p~

^and

pj$~/p~

_can

give

information about the ratios

ilR

and

DIR.

4.

Qttafitative analysis

of the resistance of

Homo~sg.

Recently

Koike et al.

[2]

have demonstrated that the

resistivity

of _a

single crystal Homo~sg

^{in the}

ferromagnetic phase

at H=0 makes _up 989b of its value above 7~,.

Giroud _et al.

[3]

have

provided

simultaneous measurements of the resistance and

magnetization

of _a

single crystal Homo~sg

for several orientations of the _easy

magnetization

axis with respect to the

applied magnetic

field. It is of interest to note that the

dependence

p

/p~

_on the

applied magnetic

field _at T

= 95

mK,

which _was shown in

figure

6 of reference

[3]

(J.

Low

Temp. Phys.,

_see also the _same in

Fig.

I in

Physica),

is in _a

qualitative

agreement with that of

p/p~,

which is shown

by

_curves

2,

³ in

figure

3 of this _paper

(see

also

[8]).

Burlet et al.

[15]

have measured the resistance of _a

single crystal Homo~sg

under

heating

from T

= 0.085 K. At this

temperature

a

magnetic

field H

= 2 koe has been

applied

to reach

a

single

domain state, then the field _was switched off and the

heating

started. At T

= 0.12 K the resistance

dropped

and became _zero. Then it

began

to increase and reached the value 0.9 _p

~

at T

= 0.30 K

(see Fig.

5 of Ref.

[15]).

Such a resistance behaviour could be

explained by taking

into account the fact that after

switching

off the field the

sample

_was metastable and the

subsequent heating

_appears to be _a temperature

demagnetization.

Under

demagnetization

in _a

single

domain

sample

the nuclei of domains with another

magnetization

orientation appear and

begin

to increase. The relative volume of such nuclei is small and the width of the

magnetic peak

_may remain resolution limited. However _on the domain walls of these nuclei the conduction electrons in 2-domain and 3-domain states _are localized and this _{causes a} resistance

drop.

Under

subsequent

demagnetization

the number and relative volume of such nuclei continue

increasing.

A reconstruction of the domain structure may occur and _{as a} result _an

equilibrium

domain structure is reached. In

general

^the _process may be similar to that described in _our

previous

paper

[16],

where in

figures

4 and 5 the

computed

_curves 4 show the resistance

change

under

magnetization taking

the nucleation process into account. The behaviour of these _curves is in

qualitative

agreement ^with the observed _ones in reference

[15].

An

accomplishment

of _{a more}

thorough c6mparison

is

impeded by

_{some reasons.}

Firstly,

papers

[2,

3] had _no information about

measuring

current orientation with

respect

to the

crystallographic

_axes and hence to the domain walls. As the authors of reference

[3]

have

pointed

_out it is

explained by

an

imperfect shape

of the

crystals.

Secondly,

it is necessary ^to estimate the value of 2

RID

for

Homo~sg. According

to the data of

[I I]

the induction

B(0)

= 4.8

koe, V~

= 1.8 _x 10~

cm/s

and R

= 2 _x 10~^~ _cm. The

magnetic temperature

^is TM ₌ ^{0.7 K. The} energy parameter ^of

anisotropy

obtained from the data of reference

[12]

is

KD

=

0.4

K,

which

gives

_y

= 0. I

erglcm~.

For the

sample investigated

in reference

[2]

with

Zo

= 0.4 _mm the domain width estimated

by

the Kittel formulae

[13]

is

D=10~~cm.

_{We obtain the estimate 2}

RID

=4. From the

experiments

_on neutron diffraction

[14]

it _was found for the other

samples

that there exists _a domain structure with

D =1.5 _x

10~~

cm. In reference

[15]

_on

page103

it _was noted that at T~ 0.3 K _a

single crystal HoMo6S8

in

ferromagnetic

_state breaks into domains of width D

= 3

x10~~

cm.

Thus for

Homo~sg

it may be 2

RID

» I. In this _case the appearance

of, firstly,

the translational states of conduction electrons which _move

infinitely

_across domain walls

[17]

and, secondly, polydomain

states of electrons is

possible.

All this must lead to the conclusion that at 2

RID

» I the

anisotropy

of the

magnetoresistance along

and _across the domain walls in _a

single crystal Homo~sg

must not be _so

strongly pronounced

_as in

ErRh~B~.

But in

general

the resistance

change

under

magnetization

will be determined

by

_a redistribution of the conduction electrons _over

polydomain

states

(polydomain

states

always

substitute for the 2- domain states under

magnetization).

This has to lead to the resistance

change

effect of the

same

qualitative

character _as stated above.

We have to notice however that the

assumption

that

ilR

_» I is very

questionable

for these ternary

compounds.

This is the weak

point

which

probably

makes the alternative

explanation (by

nucleation of

superconductive layers

of width

fo along

domain

walls) plausible

in these systems ^{since it introduces} an additional channel which is not

present

^{in the}

paramagnetic

case. It _seems to _us that all this consideration _may be treated _{as one} of the

arguments

^for a

necessity

of further

investigations

^{of these}

systems.

Summary.

The result obtained _on the

anisotropy

of the

magnetoresistance along

and _across domain walls is based _{on a} model of transformation under

magnetization

of the conduction electron

trajectories

that leads to a

conductivity change

in the vicinities of the domain walls in

layers

of

a width

given by

the

cyclotron

radius. Such _an

approach

which takes the domain structure

expansion

into account allows _us to obtain _a rather

good agreement

with the

experimental

data

[I]

_on the

magnetoresistance

_across the domain walls of _a

single crystal ErRh~B~

in the

ferromagnetic phase.

We do _not consider _our

analysis

of the recent

experiments

_on the

magnetoresistance

of

single crystals Homo~sg

in the

ferromagnetic phase

_as finished because further

experiments

on more

perfect crystals

with simultaneous

investigations

of domain structures is necessary.

Acknowledgement.

The authors _are

grateful

to R. G.

Khlebopros

^{for useful} comments and discussions.

764 JOURNAL DE

PHYSIQUE

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