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SUMMARY:Dipole-exchange oscillations in transversely magnetized ferromagne
 tic nanowires of elliptical cross-section
DTSTART;VALUE=DATE-TIME:20190524T111500Z
DTEND;VALUE=DATE-TIME:20190524T113000Z
DTSTAMP;VALUE=DATE-TIME:20260311T071539Z
UID:indico-contribution-614@indico.knu.ua
DESCRIPTION:Speakers: Maksym Popov (Faculty of Radio Physics\, Electronics
  and Computer Systems)\nThe investigations of the nanosize magnetic partic
 les and arrays continuously attract academic and technological interest. T
 heir magnetic properties can be varied over a wide range by modifying the 
 nanoparticle shape\, size\, curvature and separation [1]. While static mag
 netic properties of a single particle are mostly well understood\, their h
 igh-frequency spin-wave dynamics still lacks comprehensive theoretical exp
 lanation for some important cases [2]. In the given paper the effect of pa
 rticle curvature on the dipole-exchange oscillations frequency is investig
 ated \nWe have considered an infinitely long ferromagnetic curvilinear cyl
 inder with elliptical cross-section given by semiaxes *a* and *b*\, made f
 rom ferrite with magnetization $4\\pi M$ and biased with external magnetic
  field *H*\, applied along cylinder semiaxis *a*. We introduced the modifi
 ed elliptical coordinate system\, according to [3]\n\\begin{equation}\nz =
  \\left(\\rho  + \\frac{{c^2 }}{{4\\rho }}\\right)\\cos \\phi \, \\\, y = 
 \\left(\\rho  - \\frac{{c^2 }}{{4\\rho }}\\right)\\sin \\phi\, \\\, \\phi 
  \\in [ - \\pi \,\\pi ]\, \\\,\n\\rho  \\in \\left[\\frac{c}{2}\,\\infty \
 \right)\,c=\\sqrt{a^2-b^2}\n\\end{equation}\nIn these coordinates the magn
 etostatic potential inside the ferrite is given by  $\\Psi _1 (\\rho \,\\p
 hi ) = \\sum\\limits_{k = 1}^\\infty  {\\left( {M_k U^k (\\rho \,\\phi ) +
  N_k V^k (\\rho \,\\phi )} \\right)} $\, where  $U(\\rho \,\\phi ) = \\lef
 t( {\\rho  + \\frac{{c^2 }}{{4\\rho }}} \\right)\\cos \\phi  + \\frac{1}{{
 \\sqrt { - \\mu } }}  \\left( {\\rho  - \\frac{{c^2 }}{{4\\rho }}} \\right
 )\\sin \\phi $\, $V(\\rho \,\\phi ) = \\left( {\\rho  + \\frac{{c^2 }}{{4\
 \rho }}} \\right)\\cos \\phi  - \\frac{1}{{\\sqrt { - \\mu } }}\\left( {\\
 rho  - \\frac{{c^2 }}{{4\\rho }}} \\right)\\sin \\phi$\, and $\\mu  = \\fr
 ac{{\\omega ^2  - \\gamma ^2 H_i (H_i  + 4\\pi M)}}{{\\omega ^2  - \\gamma
  ^2 H_i ^2 }}$ - is the diagonal part of tensor magnetic permeability\, $H
 _{i}  = H - 4\\pi M \\cdot b/(a + b)$\, and $\\gamma$ is the gyromagnetic 
 ratio.\nAfter applying standard boundary conditions at the lateral surface
  of the cylinder we get a pair of independent secular equations \n\\begin{
 equation}\n{(1 - i\\sqrt { - \\mu } )\\left( {a + \\frac{{ib}}{{\\sqrt { -
  \\mu } }}} \\right)^k  + (1 + i\\sqrt { - \\mu } )\\left( {a - \\frac{{ib
 }}{{\\sqrt { - \\mu } }}} \\right)^k } = 0\,\n\\end{equation}\n\\begin{equ
 ation}\n{ - (i + \\sqrt { - \\mu } )\\left( {a + \\frac{{ib}}{{\\sqrt { - 
 \\mu } }}} \\right)^k  + (i - \\sqrt { - \\mu } )\\left( {a - \\frac{{ib}}
 {{\\sqrt { - \\mu } }}} \\right)^k }  = 0  \\\,\\\,(1)\n\\end{equation}\nf
 or the modes with symmetric and antisymmetric spatial distribution with re
 spect to the *Z* axis. \nEqs. (1) are to be solved for $\\mu ^{(nk)}$\, an
 d the spin modes eigenfrequencies $\\omega ^{(nk)}$ can then be retrieved 
 from the abovementioned expression for $\\mu $. Figure 1 demonstrates the 
 effect of cylinder cross-section aspect ratio on the normalized eigenfrequ
 encies. All calculations were made for the YIG biased with external field 
 *H*=3 kOe. The aspect ratio *b/a* was taken as a variable parameter. A str
 ong influence of sample’s cross-section curvature (from prolate cylinder
  to oblate) is clearly visible. \nThe exchange interaction\, which plays c
 rucial role for nanosized particle\, was accounted for by the substitution
  $H_i \\to H_i  + Dk^2 $\, where *D* is the exchange stiffness and *k* is 
 the spin-wave transversal wavenumber. The latter was extracted form the pe
 ak value of the spatial Fourier transformation of the mode's dynamic magne
 tization.\nFinally\, the presented theory was applied to explain the exper
 imental results\, published in [4] for the dependence of the spin-mode fre
 quencies on the intensity of the transversal magnetic field measured for t
 he array of nickel nanowires with length *L*=175 nm and radius *R*=35 nm b
 y Brillouin light scattering (BLS). The experimental values were found to 
 be in a good accordance with theoretical calculations. Also\, our theory c
 onvincingly reproduces the characteristic fine structure of the BLS magnet
 o-optical response. \n\n![FIG1][1a]\nFIG. 1\n \n[1] P. Toneguzzo\, G. Viau
 \, F Fiévet\, Handbook of Advanced Magnetic Materials\, vol.3: Fabricatio
 n and Processing\, Springer\, New York\, 2006\, p 217-266.\n[2] R. Arias\,
  D.L. Mills\, Phys. Rev. B\, 63 (2001) 134439. \n[3] I.V. Zavislyak\, G.P.
  Golovach\, M.A. Popov\, V.F.Romanyuk\, J. Comm. Tech. and Electronics. 51
 \, (2006) 203-211.\n[4] A.A. Stashkevich\, Y. Roussigne\, P. Djemia et al.
 \, Phys. Rev. B\, 80 (2009) 144406.\n\n\n  [1a]: https://indico.knu.ua/eve
 nt/4/images/117-Popov_Fig1_sm.png "Fig. 1"\n\nhttps://indico.knu.ua/event/
 4/contributions/614/
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URL:https://indico.knu.ua/event/4/contributions/614/
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