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Key words: spin waves, layered ferromagnetic periodic structure, exchange interaction.
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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.
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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
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0.20
10.30
10.35
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k
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d 1
,
d2 3,
b
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0.025
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.
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E-mail: ssahakyan@ysu.am
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E-mail: zavendanoyan@gmail.com.