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67, №4, 2014 539.3

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Ы

Ч

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. .,

. .,

. .,

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ы : , ,

.

Key words: spin waves, layered ferromagnetic periodic structure, exchange interaction.

й й, Ա й й, Ս Սй й, й й

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Danoyan Z.N., Atoyan L.H., Sahakyan S.L., Danoyan N.Z.

Spin waves in periodic layered ferromagnetic structure when exchange effect is considered

The problems of existence and propagation of spin waves in a layered periodic ferromagnetic structure taking into account exchange effect are investigated. The dispersion relations, frequencies bands of existence and band gaps are derived.

,

, ё .

, ё ,

.

1. . ё

, ,

. (photonic,

phononic) (magnonic

cristals) ё (magnetic superlattices) [13, 18, 20- 22].

, ,

,

, . .

(band gaps),

. ,

[4, 7,-9, 18Ж,

.

[1, 2, 5,

10-16, 20-22Ж, , , . .

.

, ё ,

(2)

2. . ,

ё

h

1

ё

h

2.

Oxyz

, .1.

p , ё

Oz

.

,

0

H

ё

0 1

0

ρ

μ

M

(

μ

0

,

ρ

1 – )

0

H

ё , . .

Oz

. ,

z

1

( , , ),

x y t

2

( , , ), 0

x y t

1

x

,

y

,

t

, –

x

,

y

,

t

2

, ё

1

grad

1

H

 

,

H

2

 

grad

2, (2.1)

1

H

H

2 – ё

, .

: ,

[1, 2, 5],

- [13, 18, 21, 22]

.

. ,

Ф ймй .

1

h

0

(2)

(1)

 

0

M

z

0

H

2

h

1

h

1

y

h

2

y

 

h

x

(3)

, (band gaps).

2 1

  

h

y

h

.

,

,

.

3. , ы

-.

, ё ,

:

) , ы

1

0

 

y

h

( ,

) :

1 1

1 1

1

1 1

1 1 1

1

1 1

1 1 1

,

ˆ

,

ˆ

,

M

M

x

y

b

t

y

b

t

x





  





 

   



 



   

(3.1)

1 0

ˆ

b

 

b

 ,

0

M

0

H

0,

 

M

0

M

0

,

7 0

1.76 10

(  )-1 – ,

0

,

b

– ,

,

1 – .

b) ,

  

h

2

y

0

:

2

2

0

  

, (3.2)

2

– .

c) ы

y

0

   

x

:

1 2

1 2

,

1 1

,

1

0,

1

0

y

y





  

   

 

 

, (3.3)

, –

ё , –

(pinning) . ,

1

y

h

:

 

 

1

h

1

0,

1

h

1

0

. (3.4)

d) ы -

1

y

h

,

y

 

h

2:

1 1 2 2

1 1 2 2 1 1 1

( )

(

)

( )

h

(

h

),

h

( )

h

h

,

y

y



 

  

  

(4)

– :

iqa

e

,

a

 

h

1

h

2, (3.6)

a

– ( ),

q

– ё

, ё

, - Д29, 31Ж.

4. . (3.1)

:

 

 

1

,

1

,

1

, ,

i t px ry

e e

 

      

, (4.1)

, ,

  

,

r

– ,

p

.

(4.1) (3.1)

,

, ё :

2 2

2

ˆ

2 2

ˆ

2 2

1

0

r

p

 

b

 

r

p

  

b

r

p

, (4.2)

0

M

0

 

.

(4.2) :

1

r

 

p

,

r

2

  

p

1,

r

3

  

p

2 (4.3)

2 1 2

1 4

1

DE

p

   

, 2 2 2

1 4

1

DE

p

   

, (4.4)

1/ 2

DE

b

  

– - ,

,

, ё [1, 5].

, (3.1) :

 

1 2

1 1 2 3

i t px

p y p y

py

e

e

 

e

 

e

 

   

  

 

 

,

 

1 2

1 1 2 3

i t px

p y p y

py

e

e

 

e

 

e

 

   

  

 

 

, (4.5)

 

1 2

1 1 2 3

i t px

p y p y

py

e

e

 

e

 

e

 

   

  

 

 

.

, , 1

 py

e

1py

1 py

e

e

, ё

:

1 1 1

ˆ

ip

b

 

 

, 1 2 2 2 1

1

1 4 1 2

ip

 

 

 

 

, 2 3 3 2 1

1

1 4

1 2

ip

 

 

 

 

, 1

1

1

ˆ

p

b

 

 

, (4.6)

2 1 2

2

1

1 4 1 2

ip

 

  

 

, 3 2 3

2

1

1 4

1 2

ip

 

  

(5)

(3.1).

ё (3.2) .

( )

2

sy i px t

e e



  

,

s

– .

:

2 2

s

p

, (4.7)

. .

s

 

p

. ,

s

r

1

.

, (3.2) :

 

2

(

)

i t px

py py

e

e

e

 

  

  

 

. (4.8)

– .

(4.5), (4.8) (3.3), (3.4) (3.5) ё (4.4) (4.6),

1

,

2,

3, 

:

1 1 2 2 3 3

0

       

            

,

1 2 3

1 2 3

ˆ

1

ˆ

1 2

1 2

ˆ

1

0,

ˆ

1 2

1 2

b

b

b

b

       

  

 

 

   

 

 

 

  

 

 

   

 

 

 

1 2

1 2 3

1 2

1 2 3

1

1

1

ˆ

1 2

1 2

1

1

1

0,

ˆ

1 2

1 2

b

b

     

 

 

 

  

  

 

 

 

 

 

 

  

  

 

 

 

1 1 2 2 3

1 1 2 2 3

1

ˆ

1 2

1 2

1

0,

ˆ

1 2

1 2

b

b

     

 

   



   

 

 

 



 

   

   

 

 

 

(4.9)

1 1 1 2 1 2

1 1 1 2 1 2

1 2 3

1 2 3

0,

ph p h p h ph

ph p h p h ph

e

e

e

e

e

e

e

e

  

   

        

 

 

 

 

 

 

 

 

1 1 2 1

1 2

1 1 2 1

1 2

1 2 3

1 2 3

ˆ

1

ˆ

1 2

1 2

ˆ

1

0,

ˆ

1 2

1 2

p h p h

ph ph

p h p h

ph ph

b

e

e

e

e

b

b

e

e

(6)

1

1 1 2 1

1

1 1 2 1

1 2

1 2 3

1 2

1 2 3

1

1

ˆ

1 2

1 2

1

1

0,

ˆ

1 2

1 2

ph

p h p h

ph

p h p h

e

e

e

b

e

e

e

b

 

  

   

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

1 1 2 1

1

1 1 2 1

1 1 2 2 3

1 1 2 2 3

ˆ

1 2

1 2

0.

ˆ

1 2

1 2

ph

p h p h

ph

p h p h

e

e

e

b

e

e

e

b

 

  

   

  

 

 

 



 

 

 

 

 



 

 

 

 

 

 

 

 

2

1 4

   

.

(4.9), . . ё

, ,

:

2

2

f

1 0

 

. (4.10)

f

. ,

iqa

e

(4.10), - :

cos

qa

f

, (4.11)

q

,

p

, :

1

f

, (4.12)

- . ,

, ё , . ,

(4.10)

f

1

q

.

,

. ё ,

:

k

p

, 1

1 h d 

,

2 2

h d 

 .

. 2, 3 ё d1 d2,

,

k

- ё , . .

0.05

0.10

0.15

0.20

10.30

10.35

10.40

10.45

k

Ω

1

1

d

,

d

2

1

,

b

ˆ

0.025

(7)

, .

. 2 3

-, .

, ё 2, 3, ё ,

, , ё

, .

- .

N SCS 13-2C097.

1. . ., . ., . . . .: ,

1967. 368 .

2. . . .: , 1991. 560 .

3. . ., . . . : . ,

2006. 492 .

4. . ., . .

ё . // . . .

2010. .63. №2. .34-40.

5. . . . .:

,1973. 560 .

6. Danoyan Z.N., Piliposian G.T., Hasanyan D.J. Reflection of spin and spin-elastic waves at the ТnЭОrП МО oП К ПОrroЦКРnОЭТМ СКХП ЬpКМО. АКЯОЬ Тn RКndom and Complex Media.-Vol. 19. No. 4. November 2009, p. 567-584.

7. . ., . ., . .

.

// . . . 2011. .64. №3. .46-53.

8. . ., . ., . .

. // 7- . .: “

”. - , 2011,

.60-65.

9. . ., . ., . ., . .

( ) . // . VI

. . , - , 2008, . 115-125.

0.05

0.10

0.15

0.20

10.30

10.35

10.40

10.45

k

Ω

1

d 1

,

d2 3

,

b

ˆ

0.025

(8)

10. Damon R. W., Eshbach J. R., Magneto static modes of a ferromagnetic slab. //J. Phys. Chem. Solids, 19, p.308-320.

11. Maugin G. A., Hakmi A. Magnetoelastic surface waves in elastic ferromagnets. //J. Acoust. Soc. Amer., 77, p.1010-1026.

12. Hasanyan D.J., Bagdasaryan G.E., Danoyan Z.N., Sahakyan S.L. The vibration of the piecewise-homogeneous ferromagnetic space with a crack.– Proceedings of ISTC of International Seminar, Yerevan, 2000, pp. 89-92.

13. Gulyaev Yu.V., Nikitov S.A. Magnonic crystals and spin waves in periodic structures. Doklady Physics, vol.46, No. 10, 2001, pp. 469-471.

14. Danoyan Z. N., Chazaryan K. B., Piliposyan G. T. Surface gap wave propagation in layered electro-magneto-elastic structures.-Waves in Random and Complex Media., Vol.19, No. 3, August 2009, pp. 521-534.

15. Danoyan Z.N., Piliposian G.T. Surface Electro-Elastic Love Waves in a Layered structure with a piezoelectric substrate and a dielectric Layer. -International journal of solids and structures. 44 (2007), pp. 5829-5847.

16. Piliposyan G.T., Danoyan Z. N. Surface electro-elastic Love waves in a layered structure with a piezoelectric substrate and two isotropic layers. Int. Journal of solids and structures. 46 (2009), pp.1345-1353.

17. Danoyan Z.N., Piliposian G.T., Hasanyan D. J. Reflection of spin and spin-elastic waves at the interface of a ferromagnetic half-space. Waves in Random and Complex Media. -Vol. 19, No. 4, November 2009, pp. 567–584.

18. . ., . ., . ., . .

-

ё . // . . .

2014. .66. №4. .29-38.

19. . .

. // . . .

2009. №6. .405-416.

20. Danoyan Z.N, Atoyan L.H., Danoyan N.Z. Shear horizontal electro-magneto-elastic surface waves in a layered piezoelectric structure in the presence of an electric or

ЦКРnОЭТМ ЬМrООn. ,,TopТМКХ ProЛХОЦЬ oП ConЭТnuuЦ MОМСКnТМЬ’’. TСО proМООНТnРЬ oП

international Conference, 4-8 October 2010, Dilijan, Armenia, pp. 266-271.

21. S. A. Nikitov, Ph. Tailhades, C. S. Tsai. Spin waves in Periodic Magnetic Structures – Magnonic cristals. J.Magnet.Mater., v.23, 3, 2001, pp.320-331.

22. V.V.Kruglyk, A.N.Kuchko. Spectrum of spin waves propagation in a periodic magnetic structures. Physica B 339 (2003) pp. 130-133.

:

, .- . ,

: , 0019 , . , 24/2;

E-mail:zavendanoyan@gmail.com.

, . .- . , .

.

: , 0019 , . 24/2; E-mail: levous@mail.ru.

, . .- . , . . . .

- . .

: 0019, , . . 1, .: (+37477) 002-408,

E-mail: ssahakyan@ysu.am

. ., .

: , 0019 , . 24/2,

E-mail: zavendanoyan@gmail.com.

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