Bloch Domain Walls in (111)- Plate of Medium With Cubic Magneto Crystalline Anisotropy

8/7/2019 Bloch Domain Walls in (111)- Plate of Medium With Cubic Magneto Crystalline Anisotropy

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Conclusion. The nonlinear properties of STNO in

broad-band regime of phase-locking of forced oscillationsare analyzed. In this regime the phase-locking bandwidthof magnetization oscillations is directly proportional to theamplitude of an external signal at nonzero frequencymistuning. Transmission characteristics are linear and

amplitude of oscillations is greater than at zero frequencymistuning. All these properties allow to use the STNOworking at this regime as a stabilizer and/or tunablegenerator of spin-wave signals.

This work was supported by grant M/175-2007 ofMinistry of Education and Science of Ukraine.

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Perspectives of application of surface wave resonators for creation

of spin-transfer devices // Proceedings of the Seventh International

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phase-locking of two spin-torque oscillators: Influence of time

delay of a coupling signal // Proceedings of the EuropeInternational Magnetics Conference (Intermag 2008), Madrid,

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Submitted on 20 .09 .2008

UDC 537.622.4

B. Tanygin, stud.,O. Tychko, Ph. D.

BLOCH DOMAIN WALLS IN (111)- PLATE OF MEDIUM

WITH CUBIC MAGNETOCRYSTALLINE ANISOTROPY

(- ) (111)- -

Results of investigations of an energy and structure of the plane Bloch domain walls with opposite (right-hand and left-hand)directions and different paths of magnetization vector rotation in (111)-plate of cubic symmetry crystal with "negative" magneto-crystalline anisotropy are presented.

Introduction. The plane Bloch ( WMn =const, where

Wn is a unit vector along domain wall (DW) plane normal)

domain walls can realize in unrestricted crystal or in a

volumetric plate of the magnetically ordered medium in

cases of an absence or a smallness of demagnetizationfields [6, 7, 9, 10]. Structure ( M spatial distribution, DW

plane orientation, DW width ) of these DW is well investi-

gated in unrestricted crystals with "positive" ( 1K >0) [4, 5,

6, 8] and "negative" ( 1K


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~ 40 ~

[1, 10, 11]. Results of the research of the plane Bloch DW

structure in (111)-plate of the magnetically ordered medium

with "negative" cubic MA are presented in this work.

Type of the plane Bloch domain wall in spa-

tially restricted medium. Let m is a unit vector along

the magnetization vector M : / Mm M , where M is asaturation magnetization. At that 1m and 2m are unit vec-

tors along magnetization vectors 1M and 2M in neighbor-

ing domains: 1 1 / Mm M , 2 2 / Mm M . An angle 2

between 1m and 2m determines 2 -degree of DW:

1 22 arccos m m [5, 8]. Usually 2 -degree deter-mines DW type in spatially unrestricted medium [5]. In this

case the DW orientation is determined by an angle be-tween Wn and a plane of the vectors 1m and 2m [1].

Hereinafter different m turns between the vectors 1m and

2m at DW orientations in range / 2 / 2 [1, 3, 11]are considered.

The units vectors 1m and 2m are directed along differ-

ent easy magnetisation axes (EMA) those are coincidedwith crystallographic like directions in the cubic crys-

tals with 1K


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where

0 arccos cos sin sinF

and < 0 < . There are equalities

1 0RC and 2 0 RC

,

where

1sgn 1 sgn sgn sgnR WC Bm Bm m m A n B ,

RC

0 and RC

0 for right-handed (R-rotation) and

left-handed (L- rotation) rotations respectively [11].

Unique identification of m rotation kind in spatially re-

stricted medium gives parameter RC = RC F

, where

1F : RC 0 or RC 0 for R- and L- rotation

respectively. A degeneration of the DWs with opposite m

rotation directions takes place at equality cos 0 . At

that it is carried out always for 2 -DW or at a parity of the

function S for ,2

-DW. This case of them

rotationis marked as RL-rotation. If

1 2 0W n m m

then the vector m in the DW volume draws a part of a disk

surface (which coincides with the DW plane) with axis that

coincides with Wn and there is a possibility of a lapping of

the R- and L-rotation trajectories. It is a special case ofRL-

rotation. Further it will be marked as RL*-rotation.

A short (at 2 1 < ), middle (at 2 1 = ) and

long (at 2 1 > ) path of the vector m turn (further it

will be marked as S-, M- and L-path) in DW volume is dif-

fered [1,11]. A kind of path is determined by parameter

PC = 1 2sgn sgn cos R W RC C n m m :

0PC , PC 0 or PC 0

forM-, S- orL-paths respectively.

The function for opposite m rotation directions is

well approximated with use of the Legendre polynomials of

order 0,10q and coefficients qQ (table 1):

1 sinq q Rq

A K Q P C .

Table 1 . Approximation coefficients of the domain wall energy density function

qQ

q 2 2 2 0 1.32571 1.66885 1.93143

1 1.25209 0.938957 0.000000

2 0.487527 0.559086 0.0158357

3 0.339167 0.506716 0.000000

4 0.529678 0.558409 0.0829197

5 0.382304 0.003283 0.000000

6 0.0465596 0.227821 0.184272

7 0.353845 0.0951258 0.000000

8 0.378445 0.0141431 0.000000

9 0.218042 0.0325039 0.000000

An approximation standard error is 1.10 % , 0.91 %

and 0.41% for 71-, 109- and 180-DW respectively. The

above-mentioned expression can be used only for the

evaluative calculations. Orientation dependencies of den-

sity S for all types of DWs are shown in fig. 1. Here and

hereinafter it is considered a case 0 . For inverted ap-

propriate values R- and L- turn must be considered as

L- and R- turn respectively.

A degeneration of the DWs with opposite m rotation is

removed only for2

,2 -DW. At interfacial angle

=35.264 and 0 there is a coincidence of energy

densities of ,2 -DWs with arbitrary rotation direction

(fig. 1). For orientation 0 and =35.264 only one

equilibrium DW orientation takes place. It is equilibrium

orientation of the190,552 -DW. The

2,2 -DWs have very

large energy density at this orientation. At =90 the en-

ergy densities of DW with RL*-rotation are identical for

(111)-plate and unrestricted crystal. For such interfacial

angle the maximal energy densities of1

,2 -DWs ap-


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proximately correspond to the minimal energy densities of

2,2 -DWs at similar orientations. The 02 - and 712 -

DWs have similar equilibrium orientation, but the last have

less variety of such orientations.

S/0, arb.units

-2 -1 0 1 2a) , rad

3

3

10

8

6

4

2

0

S/0, arb.units

-2 -1 0 1 2b) , rad

1

2

1

2

3

10

8

6

4

2

0

S/0, arb.units

-2 -1 0 1 2

, rad

3

3

4

3

2

1

0

S/0, arb.units

-2 -1 0 1 2

, rad

3

3

4

3

2

1

0

c) d)

6

5

4

3

2

1

-2 -1 0 1 2

, rad

33

S/0, arb.units

e)

Fig. 1. Orientation dependencies for DW energy density for1

,2 - (a,c),2

,2 - (b,d) and 2 -DW (e) at R-(1), L-(2) orRL-rota-

tion (3): 0 1| |A K . All dependencies for 71-, 109- or 180-DW in unrestricted crystal are figured by dash lines. Thedependencies for 2 -DW (e) corresponds to 02 - (dash line) and 712 -DW (dot line). The interfacial angle between

(111)-plane and plane of the vectors 1m and 2m is equal to 35.264 (a,b), 90 (c,d,e)


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Magnetization distribution in plane Bloch

domain walls. A spatial distribution z of the turn angle

is determined as [5]:

z = 1/ 2

21sin , , , ,A A A e e d

,

where value 1 / 2RC depends on turn direction.Here subintegral function can be approximated by Taylor

series. Then at 0 the functional dependence z in

the volume of1

,2 - and2

,2 -DW looks like:

1 2 31 1 2

3 1 2 3 41 2 4

1 2 3 4

1 2 3 41 0 0 0 00

3 3 , , 4 3 2 sinp r r r p r p r r p

r p r r r r r r rpp i

p r r r r

zA r p a w c g f B r r r r

,

where

1

!

p

p

qA

p

;

q is the Euler gamma function, 1/ 2q p ,

1 2 3 4 1 2 3 4

!,

! ! ! ! !i

pr p

r r r r p r r r r

,

12 1 , ; ; cossgn 1

,2 2 1 / 2

tF X Y C Z tB t

t C

0 01 t t ; 2 1 , ; ; /F X Y C Z C is a regularized hy-pergeometric function,

1

2

tX

,

1

2Y ,

3

2

tC

, 2cosZ ; 0 1/A K

is a characteristic DW width [6]; p , 1r , 2r , 3r , 4r , t and q

are integers; ir marks the totality 1r , 2r , 3r , 4r .

The coefficients a , w , c , g and f define

4 3 21, , / cos cos cos cosAe K a w c g f and depend on DW orientation:

4sin 7 cos 4 8a , 3cos sin sin4 / 2w ,

3cos sin sin 4 / 2g ,

2 2sin 7cos 2 1 8 3 cos4 s in 2 16c ,

2 2cos 7cos2 2 cos cos4 9 16f

for1

,2 -DW;

4sin 3cos 4 28 cos 2 7 / 32a ,

3cos sin sin 2 7 3 cos 2 4w ,

2

sin 1 3 cos 2 5 cos2c

2

6 cos cos 2 / 16

,

2 2sin 2 sin 2 cos 1 3 cos2 2 3sin / 4g ,

2 2 42 1 3 cos cos2 sin cosf

25 3cos2 sin / 8 for2

,2 -DW.

At numerical calculation the function *z is ap-proximated. In this case it must be added:

* 01 z z Y , where 0<


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~ 44 ~

The monotonic m distribution is always symmetric i.e.

z z (curve 1 in fig. 2). The vector m

distribution is symmetric in volumes of ,2 -DWs at their

arbitrary orientation and also in 2 -DW volume at Wn

orientation along N direction. The possible 6 N vectors

is 1 3 2 3, , 2s s s s , 1 2 3 2, 2, s s s s and

2 1 3 12, ,s s s s .

2

1

0

-1

-2

-20 -10 0 10 20

0z/ arb.units

1

2

~ , rad

Fig. 2. Magnetization distribution in a volume of equilibrium1

,2 - (curve 1) and2

,2 - (curve 2) DW at S-(curve 1)

orL-(curve 2) paths: 0 1/ | |A K . The interfacial angle is: 1 90; 2 35.264

Plane Bloch domain wall width. At monotonic

(at 2 180 ) and non-monotonic (at 2 =180) vectorm distribution in the DW volume an expression for the DWwidth [8] can be obtained analytically:

02 6 5 3cos4L at 2 =71;

04 3 1 3cos 2L at 2 =109;

* 2 * 2 * * *0 2 2 2 2 33 2 / cos 1 8 tan 4 2 sin 3 tan z

*2z * 2 * 2 * *0 3 3 3 33 2 / cos 1 8 tan 4 2 sin 3 tan ,at 2 =180, where sinL is a length of turn

path [1],*2 and

*3 is the most distant inflection points of

z dependencies for 2 -DW. At non-monotonic magnetization distribution in the DW

volume the function z can has two or one inflection points

with 2 2, , / 0Ae for ,2 -DW or 2 -DWrespectively. An existence of the lasts leads to growth (limit-less in extreme cases) of the DW width. Unlimited DW width isrealized at passing of vector m turn trajectory through theadditional EMA those are in coordinate octants without the

vectors 1m and 2m . It takes place at critical orientations

( = CJ , where J is an integer) of DW plane:

1C CW R J W R J J C C n m n u , where Ju is a unitvector along EMA in coordinate octant without the vectors 1m

and 2m ( Ju 1m ; Ju 2m ). There are the next critical

orientations of the DW plane: 1 / 4C and 2 / 2

C

for 1 ,2 -DW ; 1 0C , 2C arccos 1/ 3 and3 / 2C for 2,2 -DW. In a case of 2 -DW the critical

orientations of the DW plane are the next: 1 / 2C ,

2 / 6C , 3 / 6

C and 4 / 2C .

At these orientations a DWs decay takes place: the"parent" DW is divided into a few "daughter" DWs. The

1,2 -DW divided into three

1,2 -DWs at 1

C or two2

,2 -DWs and one1

,2 -DW at 2C . The 2,2 -

DW is divided into two1

,2 -DWs at 1C

or two2

,2 -DWs at 2C or else two 1 ,2 -DWs and one

2,2 -DW at 3

C . In the case of 2 -DW it is divided

into1

,2 -DW and2

,2 -DW at the every critical orienta-

tion. All these "daughter" DWs separate domains with

magnetization vectors those are oriented along Ju .

All "daughter" DWs are formed by direct fission of them turn of the "parent" DW. At that it takes place a conser-

vation of the m turn path length kk

L L , energy density

kS S

k

and kk

where kis a number of the "daughter" DWs. Here kL , k

andkS are the proper parameters of the "daughter" DWs.


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. 12/2009 ~ 45 ~

All m turn paths of the "parent" DWs are the L- or M-paths and all m turn paths of the "daughter" DWs are theS-paths.

A stability and limited values of the DW width for theabove mentioned "unlimited" DWs is assured by introduc-tion of another items of energy of magnetically ordered

medium (e.g. magneto-elastic anisotropy energy term [8]).In this connection a research of DWs in crystal with a smallmagnetostriction is of interest [2].

Plane Bloch domain walls in (111)-plate. Pa-

rameters of all possible Bloch DW in (111)-plate of cubiccrystal with "negative" MA are presented in table 2. Upper

and lower signs of symbols " " or " " in these tablescorrespond to the R- and L-rotation respectively. Symbols" " and "2 " (table 1) correspond to unlimited DW widthand indicate a formation of one and two "domains " in DWvolume accordingly (metastable DW [6]).

In (111)-sample of a cubic crystal the M-path is an equi-

librium path of vector m turn only in the volume of 2 -

DW. The minimum energy density of

1

,2 -DW is

achieved =90, RL*-rotation and S- path. There is only

one minimum of the energy density of 2 -DW in (111)-

sample. The above-mentioned energy densities are identi-cal for proper DW unrestricted crystal. There is three DWs

(135,552 -,

255,352 -, 02 - and 712 -DW) in (110)- sample

with equilibrium parameters identical to a case of the unre-stricted crystal [5, 9].

Minimal values (at all ) of equilibrium energy density

S correspond to DW with RL-rotation and S-path of vec-tor m turn in their volume. With other things being equal

the2

,2 -DW have higher equilibrium energy density S

than1

,2 -DW. There is equilibrium1

,2 -,2

,2 - and

2 -DW with unlimited thickness.2

,2 -DW without de-

generacy of rotation direction (at =35.264) is character-ized by absence of the unlimited thickness possibility.

Table 2. Energy density and structure of the equilibrium plane Bloch DW with in (111)-plate

Inter.angle

DWtype

Trajectorytype

DW orientation, width and energy(equilibrium state)

,deg.

2 ,deg.

,deg.

,deg.

Rotation Path ,deg.

,deg.

0/ , arb.units.

0S ,arb.units.

S- 45.000 54.736 3.848 0.667RL- L- 37.881 59.911 14.663 1.74371 90.000 54.736

RL*- L- 90.000 35.264 2 5.538S- 3.236 51.500 12.357 1.309

R- L- -55.894 69.371 15.067 2.734

35.264

109 54.736 90.000

L- S- -16.188 70.924 8.453 1.060S- 90.000 90.000 4.264 0.461

RL*- L- 90.000 90.000 2 3.19771 35.264 54.736RL- L- 41.383 52.221 18.299 1.907

RL- S- 11.389 55.530 9.421 1.192S- 90.000 90.000 3.309 1.368109 54.736 35.264

RL*- L- 90.000 90.000 2 2.29030.000 90.000 1.829-30.000 90.000 1.82990.000 90.000 1.829180 90.000 0.000 RL*- M--90.000 90.000 1.829-90.000 90.000 1.829

90.000

180 19.471 70.529 RL*- M- 90.000 90.000 1.829

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Bloch Domain Walls in (111)- Plate of Medium With Cubic Magneto Crystalline Anisotropy