Reflection of electromagnetic waves from magnetic having the ferromagnetic spiral

Transcription

1 Reflection of electromagnetic waves from magnetic aving te ferromagnetic spiral Igor V. Bycov 1a Dmitry A. Kuzmin 1b and Vladimir G. Savrov 3 1 Celyabins State University 51 Celyabins Br. Kasiriny Street 19 Russia Te Institute of Radioengineering and Electronics of RAS 159 Moscow Moovaya Street 11-7 Russia a bycov@csu.ru b uzminda89@gmail.com Keywords: electromagnetic waves spin waves acoustic waves ferromagnetic spiral reflection coefficient Faradey effect Abstract. Te spectrum of coupling electromagnetic-spin-acoustic waves for magnetic aving spiral magnet structure defined by eterogeneous excange and relativistic interactions ave been received. Te possibility of resonant interaction of spin and electromagnetic waves as been sown. Te electromagnetic waves reflectance for te layer of magnetic aving ferromagnetic spiral as been calculated for different angles of spiral. Te Faradey effect as been considered. Introduction Recently elicoidal (spiral) magnetic materials ave attracted researcers' attention for teir unusual pysical properties [1 ]. Te spiral magnetic structures contribute a number of features in te spectrum and dynamics of spin excitation in magnetic materials: band structure is observed te nonreciprocity effect is manifested i.e. difference between te velocity of wave transmission along and against te spiral axis. Previously te spin-wave spectrum was calculated witout taing into account of te effects of te electromagnetic retardation and te electromagnetic wave spectrum was calculated witout taing into account of te effects of te dynamic interaction of te electromagnetic field wit te oscillations of te spins in te ferromagnetic spiral structure [3 ]. Te coupled electromagnetic-spin electromagnetic-spin-acoustic waves in te magnetic structure simple spiral ad been investigated [5-7]. However te spectrum and dynamic properties of magnets in a pase ferromagnetic spiral are not studied enoug. In te present wor te spectrum of te coupled electromagnetic-spin-acoustic waves in spiral magnetic structure of type ferromagnetic spiral is investigated. Also te reflection of electromagnetic waves from a layer of magnetic material wit a ferromagnetic spiral depending on te angle of te ferromagnetic spiral θ determined by an external magnetic field and Faradey effect are considered. Researces of spectra of te coupled fluctuations in te modulated magnetic structures are spent in approac L >>a were L q- te spiral period a - te lattice constant. Te spectrum of coupled spin and electromagnetic waves. Te ground state of a crystal is described by a vector of magnetization wit components: M M sin cos qz M M sin sin qz M M cos (1) x y z were M is magnetization of saturation q - wave number of a spiral θ - an angle between a direction of magnetization and a spiral axis z. θ is defined by value of an external magnetic field. Wen θ = π/ te magnetic transforms from te pase of te ferromagnetic spiral into simple spiral wen θ = - in te ferromagnetic pase. Te free energy of te crystal pase of ferromagnetic spiral as te form:

2 1 M F F M M HM M M b M M u c u u xi H 1 z z z ijlm i j lm ijlm ij lm were M te magnetization of te crystal; bc te constants of inomogeneous excange anisotropy magnetostriction and elastic constant. Te term F H wic causes inomogeneous magnetization in te ground state for crystals wit excange spiral structure is: () F H 1 M xi and for magnetics wit a relativistic elicaloidal structure (3) FH () 1 MrotM were and 1 constants of inomogeneous excange interaction and inomogeneous relativistic interaction. In () it is taen into account tat te external magnetic field is directed along te axis of symmetry. From te minimum of free energy (1) we obtain expressions for determining te angle θ troug an external magnetic field H. M cos 1 me me M M cos q H (5) were: c c c c 1 1 b11 b1 M b33 b31 b11 b1 M c c 11 1 c33 b13 b1 b11 b1 M b M c c33c11 c13 c c c33 b c b11 b1 b33 b31 b33 b31 b11 b c c 11 1 c c b b b b b b b b c c c c For a spiral wit te excange interaction we ave γ > α < 1 q me b11 b1 M c11 c1 q. In te case of relativistic spiral α 1 α > q 1 b M q. me 1 Te tensor of te equilibrium deformations is: c33 1 uxx M b11 b1 sin cos c b b c b b c13 1 uyy uxx uzz uxx b33 b31 M cos c c b b u M sin cos qz u M sin sin qz u. xz yz xy c c Te linearized system of Maxwell's Landau-Lifsitz and motion of an elastic medium equations for Fourier components is: (6)

3 1 1 m cos me sin m me sin cosm q 1 q sin mz ( q) 1 3 i igbm sin u gm b sin qu q ig b b M sin q u q gm sin q gm cos z z 1 1 mz sin sin qm q qm q gm q q i gb M sin q u q q u q i st u bm sin mz q cos m (7) sl uz i b33 b31 M cos mz v m z mz. Here we introduce te following notation: v c/ - velocity of propagation of gm L electromagnetic waves in a magnet ε - dielectric constant sin s c s с 1 1 gm L cos me t l 33 L ( ) ( q ) ( q ) q L ( ) ( q ) ( q ) 1( q ) 1 3 me gm me gb M c 1 gm me sin 1 M cos me sin. Using te caracteristic values of te constants for magnetic wit excange spiral g = 1 7 s -1 erg -1 M ~ 1 3 Oe ~ 1-8 cm α~ -1-1 cm q ~ 1 7 1/cm and in te case of relativistic spiral α~ 1-1 cm 1 ~ 1 а ~ 1-8 cm q ~ 1 1/cm from equations (7) we obtain te spectrum of coupled electromagnetic-spin waves. Canging θ te range θ π/ we can calculate te spectrum for te ferromagnetic spiral. Figure 1 sows te dependence ω () for θ = π/ in te case of relativistic spiral. Fig. 1. Spectrum of coupled oscillations for θ = π/. Figure sows te te dependence ω () for different values of θ near =.

4 Fig.. Spectrum of coupled oscillations for different θ near = It is seen tat all spectra ave a band structure. At certain frequencies te gap (window opacity) is observed as for electromagnetic suc for acoustic waves. Tese band gaps appear due to te resonant interaction of spin acoustic and electromagnetic waves in a magnet. From Fig. we can see tat wit decreasing angle te electromagnetic band sifts toward lower frequencies and its widt decreases. Calculations sow tat in te case of excange spiral a band of opacity is muc narrower tan in te case of relativistic one. Note also tat te magnitude of te interaction of spin acoustic and electromagnetic waves depends on te angle θ. Te reflection of electromagnetic waves from a layer of magnetic in pase of ferromagnetic spiral. Let us now investigate te reflection of electromagnetic waves from a alf-infinity layer of magnetic material in pase of ferromagnetic spiral. Consider te normal incidence of electromagnetic wave q z. Restrict ourselves to small wave numbers << q. Te system of boundary conditions include te continuity of normal components of te magnetic and electric fields te tangential component of te electric and magnetic fields te vanising of te derivative ( e) ( i of te magnetization and te absence of stress at te boundaries of te magnet: H H ) E ( e) ( i ) E B ( e) ( i ) n Bn D m n x ( e) ( i ) n Dn n. Indexes (i) and (e) denotes quantities inside and outside te magnet respectively n normal to te surface. Taing into account te number of roots of dispersion equation in tis approximation te system of boundary conditions in te cyclical components of te magnetic field becomes: j R j; R j; jz ; j1 j1 j1 3 3 j v j j jjz j1 j1 ; ; () i j v s cos sin ; ; j l j j jz jz j1 st j j1 st j j1 sl j (8)

5 Fields R define wave reflected from te surface of te magnetic dispersion equation. i solutions of te Fig. 3. Frequency dependances of reflectance for different θ Solving (8) wit (7) we find te reflection coefficient of electromagnetic waves R R. Figure 3 sows te frequency dependence of te reflection coefficient of electromagnetic waves from a layer of magnetic material wit a ferromagnetic spiral at different angles. From te frequency dependence for different angles θ can be seen tat wit increasing angle θ of ferromagnetic spiral (and tus increase te external magnetic field) increases te band gap (window opacity) and sift it to iger frequencies. Also tere is a pea at frequencies of resonant spinacoustic interaction wic sifts to iger frequencies wit increasing θ. Te Faradey effect. Let us now consider te Faraday effect i.e. dependence of te angle of rotation of te polarization plane from te external magnetic field. Let te magnet in a pase of "ferromagnetic spiral" falls linearly polarized wave. It can be represented as a superposition of two waves of Fig.. Te dependence of te angle of rotation of te polarization plane of te external magnetic field

6 different circular polarization. From (7) for waves of different polarizations obtain different wave vectors. Since te wave vectors of te waves are different so te refractive indices also will be different terefore will be observed te rotation of te polarization plane by an amount were Δ = + l. Based on te solutions (7) depends on te frequency of te incident wave and reaces maximum values near te gap. Te dependence of te angle of rotation of te polarization plane of te external magnetic field for magnetic layer ticness of 1 cm is sown in Figure. Summary Studies on te electromagnetic-spin-acoustic waves in magnetic materials wit a elical magnetic structure defined by te inomogeneous excange and relativistic interactions in te pase of ferromagnetic spiral sowed tat te spectrum of te coupled waves as band structure. Te band gap depends on te angle of te ferromagnetic spiral and ence on te external magnetic field. Increase in te angle (magnetic field) leads to an increase in band gap te maximum gap (window opacity) is observed at θ = π/ i.e. at te pase transition ferromagnetic spiral simple spiral. Te possibility of resonant interaction of spin acoustic and electromagnetic waves as been sown. Te value of te interaction of te waves depends on θ. Te frequency dependence of te reflection coefficient of electromagnetic waves from te plate of te magnetic wit a ferromagnetic spiral at different angles of te elix as been calculated. As te angle increases te opacity region broadens and sifts to iger frequencies. Te angle of rotation of te polarization increases near te band gap. Tere is a pea of rotation of te polarization plane wic sifts to lower fields if te frequency increases. References [1] S. Kobayasi Pys. Rev. Lett (11). [] T. Micael S. Trimper Pys. Rev. B (11). [3] D. I. Sementsov A. M. Morozov Fiz. Tverd. Tela 591 (1978) (rus). [] D. I. Sementsov Optia i spetrosopiya 5 37 (1981) (rus). [5] I.V. Manzos I.E. Cupis Pys. Low Temp. 1 6 (1988) (rus). [6] V.D. Bucelniov I.V. Bycov V.G. Savrov JMMM (199). [7] I. V. Bycov D. A. Kuz min V. V. Sadrin and V. G. Savrov Bull. RAS: Pysics (1).