Walker’s modes in ferromagnetic finite hollow cylinder

Journal Pre-proof Walker’s modes in ferromagnetic finite hollow cylinder Patrizio Ansalone, Vittorio Basso

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S0921-4526(19)30755-0 https://doi.org/10.1016/j.physb.2019.411872 PHYSB 411872

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Physica B: Physics of Condensed Matter

Received date : 20 June 2019 Revised date : 25 September 2019 Accepted date : 6 November 2019 Please cite this article as: P. Ansalone and V. Basso, Walker’s modes in ferromagnetic finite hollow cylinder, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.411872. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

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Walker’s modes in ferromagnetic finite hollow cylinder Patrizio Ansalone1 , Vittorio Basso

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Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce, 91, 10135 Torino, Italy

Abstract

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We study the magnetostatic Walker’s modes in a ferromagnetic finite hollow cylinder magnetized along the cylinder axis. We derive the analytic expression for the magnetic scalar potential and the magnetization components in the case of the surface non reciprocal azimuthal modes which are uniform along the cylinder axis. Keywords: magnetostatic modes, ferromagnetic hollow cylinder, surface waves 1. Introduction

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magnetization field were missing. These analytic expression are useful when the coupling with external radiofrequency antennae has to be designed[17]. In the present contribution we solve the Walker’s equation in cylindrical coordinates and we extend the results of [16] by deriving the analytical expression for the magnetic scalar potential and the magnetization components.

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The properties of spin waves in non-simply connected domains are of great interest for the possibility to exploit interference effects or to induce additional phases by an electric field, like in the Ahronov-Casher effect [1, 2, 3, 4]. These effects are seen as the possible working principles of future magnonic devices working as logic ports in which the phase of the spin waves are expected to carry the information to be processed [5]. From the point of view of a theoretical description the geometry of an hollow cylinder represents a simple but meaningful example also because nanorings, nanotubes and other kinds non simply connected geometries are being currently being realized and studied [6, 7, 8, 9, 10]. By neglecting the effect of ferromagnetic exchange spin waves and modes are entirely due to the magnetostatic field and the Walker’s equation holds [11]. The problem is a classical one that has been largely investigated in the literature going from the spheroids [11] to ferromagnetic thin films [12, 13, 14] and to cylinders [15, 16]. Hollow cylinders have been considered by Das and Cottam who derived the dispersion relation and shown that the uniform modes along the cylinder axis are surface non reciprocal azimuthal modes concentrated at the inner or outer surfaces of the cylinder [16], however the analytic expression for the magnetic scalar potential and for the

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∗ Corresponding

author Email address: [email protected] (Patrizio Ansalone) Preprint submitted to Journal of LATEX Templates

2. Walker’s equation

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In this paper the magnetostatic modes are described by the Walker’s partial differential equation, whose solution describes the physical behavior of the magnetic scalar potential Φ. The Walker’s equation (the detailed derivation can be found in the literature [13, 14]) is based on the linearization of the magnetization dynamics and on the magnetostatic approximation of Maxwell’s equations. By neglecting the damping the magnetization dynamics is given by the precession equation for the magnetization M ∂M = −µ0 γL M × H (1) ∂t where γL is the gyromagnetic ratio for the electron spin (γL ' 1.76 · 1011 T−1 s−1 ) and H is the effective field. The magnetization vector is decomposed as M = Mk + M⊥ (r, t), where the time dependent component M⊥ (r, t), which is assumed to have a small amplitude, can be assumed to be perpendicular to the time independent component Mk . By assuming the same decomposition for the magnetic September 23, 2019

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where χ = − ω ), κ = 2 ωωM /(ωH − ω 2 ) with ωM = γL µ0 Ms and ωH = γL µ0 Hk . If one is interested to spin waves and modes at long wavelengths the exchange interaction can be neglected and the time dependent component of the effective field will only contains the magnetostatic field which can be expressed in terms of the magnetic scalar potential as H⊥ = −∇Φ. This leads to the Walker’s equation

Figure 1: Geometry of the ferromagnetic hollow cylinder with internal radius r− and external radius r+ . With axial static magnetization Mk and constant magnetic field Hk along z and time dependent components M⊥ (r, t) and H⊥ (r, t) in the (x, y) plane.

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The solution of the Walker’s equation for a ferromagnetic body of finite shape is obtained by adding the boundary conditions for the magnetic field and the magnetic induction B at the interfaces between the body and the outside media. Tangential components of the magnetic field H⊥ × n must be continuous at interfaces (n is the normal at the surface) while normal components of the magnetic induction B⊥ · n must be continuous for free boundary conditions or zero for metallic surface layers.

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3. Solutions for the finite hollow cylinder

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In the following we apply the Walker’s theory to the geometry of an hollow cylinder with inner and outer radii r− and r+ , respectively, and length Lz , as shown in Fig.1. The appropriate coordinate system to describe the problem is the cylindrical one (r, ϕ, z) defined by the versors (er , eϕ , ez ). By appropriately solving the time independent magnetostatic problem along each of the cylindrical axis one may formulate a set of equations for the magnetostatic modes for axial, azimuthal or radial magnetization ([18]). In the present paper we will limit ourselves to axial magnetization, i.e. the parallel direction Mk coincides with the z-axis and the dynamics M⊥ (t) develops in the (er , eϕ ) plane as in Ref.[16]. To obtain uniform magnetization along z

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in a finite cylinder we consider two additional ideally soft ferromagnetic blocks placed at z = ±Lz /2 to provide the closure of the magnetic flux. We also limit ourselves to study the modes instead of the waves propagating along z, therefore we select metallic boundary conditions at z = ±Lz /2. The modes are obtained by solving the Walker’s equation

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∇2 Φ = −∇ · χ∇Φ ¯

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field H = Hk + H⊥ (r, t) one can linearize the precession equation. Once the time independent part of the problem is appropriately solved, one can take an harmonic behavior for the time dependent components (i.e. M⊥ (t) = M⊥ exp(−iωt)) then obtaining a linear relation between the amplitudes M ⊥ = χH ¯ ⊥ where χ ¯ is the susceptibility tensor. One obtains   χ −iκ χ ¯= (2) iκ χ

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(1 + χ)



1 ∂2Φ ∂ 2 Φ 1 ∂Φ + + ∂r2 r ∂r r2 ∂ϕ2



∂2Φ = 0 (4) ∂z 2

for both the ferromagnetic material and the outside air regions (χ = 0) and by imposing the appropriate boundary conditions. By using the method of the separation of variables we write the solution as the product Φ(r, ϕ, z) = Φr (r)Φϕ (ϕ)Φz (z) and substitute into Eq.7. We then can separately solve the equation for Φz (z) and for Φϕ (ϕ). Along z the equation is 1 d 2 Φz = −qz2 (5) Φz dz 2 from which we obtain an harmonic behavior exp(iqz z) with real qz . Metallic boundary conditions at z = ±Lz /2 impose the quantization of the wavenumber qz = πm/Lz with m integer. The z dependence of the type sin(qz z) for m odd and cos(qz z) for m even. Along ϕ the equation is 1 d2 Φϕ = −n2 Φϕ dϕ2

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which is the Bessel equation. In air (χ = 0) the solutions are given by the modified Bessel functions In (qz r) and Kn (qz r). In the ferromagnetic material we can either have the modified Bessel functions In [qz r/(1 + χ)] and Kn [qz r/(1 + χ)] if (1 + χ) > 0 or the Bessel functions Jn [qz r/(1 + χ)] and Yn [qz r/(1 + χ)] if (1 + χ) < 0. The type of solution is determined by the boundary conditions. One writes the magnetic scalar potential for the three regions: (a-) inside the cylinder hole, (m) inside the ferromagnetic materials, (a+) outside the cylinder. At the interfaces, r = r± , the boundary conditions require the continuity of magnetic scalar potential. This condition will be satisfied only if n, ν and qz are the same for all three regions. In air the radial part must be consistent with the required asymptotic behavior Φ → 0 as r → 0 for the region (a-) and Φ → 0 as r → ∞ for region (a+), respectively. The correct choice is then:

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4. Surface azimuthal modes

Φr,a− (r) = Φr,a− In (qz r)

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Φr,a+ (r) = Φr,a+ Kn (qz r)

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In the ferromagnetic materials we can have either surface modes (with (1 + χ) > 0)

Φr,m (r) = Φr,m− In



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+ Φr,m+ Kn



 qz r 1+χ (10)

or bulk modes (with (1 + χ) < 0)

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 qz r Φr,m (r) = Φr,m− Jn + Φr,m+ Yn 1+χ (11) The four independent coefficients Φr,a− , Φr,a+ , Φr,m− , Φr,m+ are reduced to two by imposing the continuity of Φ at both interfaces while the remaining two condition requiring the continuity of the radial component of the magnetic induction, Br , at both interfaces impose a dispersion relation between the ω and the numbers n, ν and qz characterizing the specific mode and the value of another qz r 1+χ





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With qz = 0 the solutions are simplified as Φr,a± (r) = Φr,a± r∓n and Φr,m (r) = Φr,m− rn + Φr,m+ r−n so that one easily obtains the following dispersion relation for the surface azimuthal modes (see also [16])

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∂Φr ∂ 2 Φr r2 qz2 + r − Φr − n 2 Φr = 0 ∂r2 ∂r 1+χ

coefficient, then leaving only one free coefficient associated with the arbitrary amplitude of the specific mode considered. In the case of large Lz or of long wavelengths one has qz → 0. In this limit when expanding the Bessel functions for very small argument (x  1) one finds that both modified and normal Bessel functions of the first kind are dominated by the xn power function, In (x) ∝ xn and Jn (x) ∝ xn while both modified and normal Bessel functions of the second kind are dominated by the x−n power function, Kn (x) ∝ x−n and Yn (x) ∝ x−n . This leads us to consider the specific case qz = 0 as the simplest case displaying the formation of the surface azimuthal modes and allowing a simple analytical solution which still contains the main physical ingredients of the problem.

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giving the harmonic behavior exp(iνnϕ) with n integer and positive and ν = ±1. We then obtain the following the equation for Φr (r)

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v " u  −2n # 2 u ωM r+ t 1− ω = ωH (ωH + ωM ) + 4 r−

(12) shown in the plot of Fig.2 top. It should be noticed that the mode with n = 0 exists with qz 6= 0 but is suppressed in the limit qz → 0. The non reciprocal field displacement is clearly shown by deriving the analytic expression for the magnetic scalar potential in the ferromagnetic material

 cn (r/r0 ) sn (r/r0 ) φm (r) = φm,0 + pν cn/2 (r+ /r− ) sn/2 (r+ /r− ) (13) √ where r0 = r+ r− and we have defined the functions cn (x) = (xn + x−n )/2 and sn (x) = (xn − x−n )/2, in which the coefficient 

pν = ν

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2ωH + ωM [1 − (r+ /r− )−n ] 2ωH + ωM [1 + (r+ /r− )−n ]

(14)

changes sign with ν. The radial dependence of the potential is also shown in the plot Fig.2 bottom. The non reciprocal field displacement depends on n and (r+ /r− ) as given by Eq.(14) and

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Figure 3: Magnetization amplitudes Mϕ (full line) and Mr (dashed line) obtained by the susceptibility of Eq.(2) and using Eqs.(12) and (13). The plot shows normalized amplitudes in the case r+ /r− = 1.3 and ωM = 3ωH . Top panel: ν = −1 Hallbach-type internal modes; bottom: ν = 1 external modes. The different curves are labeled with the value of n.

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is also well seen when plotting the components Mr and Mϕ of the precessing magnetization as a function of r/r0 for different values of n as shown in Fig.3. The magnetostatic problem in presence of more general boundary conditions, as for example the metallic boundaries placed on outer and inner radii (R− < r− and R+ > r+ ) and confining the magnetic field to finite air gaps, presents several similarities to the analogous case of thin films [13]. Also this general case can be easily solved by the same method used here [18] and the solution provides the behavior of the dispersion relation in cases closer to the measurement conditions.

Figure 2: Top: dispersion relation for surface azimuthal modes in a hollow cylinder with ωM = 3ωH from Eq.(12). Lines era a guide for the eye given by the analytical continuation of Eq.(12). Bottom: corresponding normalized magnetic scalar potential with r+ /r− = 1.3 at ν = +1 (full line) and ν = −1 (dashed line) for different values of n from Eq.(13).

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5. Discussion and conclusions While it was already clear from the work of Das and Cottam that the uniform modes along the cylinder axis are surface non reciprocal azimuthal modes [16], the explicit analytic expression for the magnetic scalar potential and for the magnetization field was missing in the literature. These analytic expression have been derived and shown in the present paper. The surface modes developing at the inner or outer surfaces of the cylinder. These modes have several similarities with the Damon-Eshbach surface modes of thin films. Fig.3 also shows the sketches of the magnetization configurations corresponding to n = 1. With ν = −1 we obtain a configuration which resemble the magnetization

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[1] Y. Aharonov, A. Casher, Topological quantum effects for neutral particles, Phys. Rev. Lett. 53 (1984) 319–321. doi:10.1103/PhysRevLett.53.319. URL https://link.aps.org/doi/10.1103/ PhysRevLett.53.319 [2] Z. Cao, X. Yu, R. Han, Quantum phase and persistent magnetic moment current and aharonov-casher effect in a s = 12 mesoscopic ferromagnetic ring, Phys. Rev. B 56 (1997) 5077–5079. doi:10.1103/PhysRevB.56.5077. URL https://link.aps.org/doi/10.1103/PhysRevB. 56.5077 [3] T. Liu, G. Vignale, Electric control of spin currents and spin-wave logic, Physical Review Letters 106 (24). doi:10.1103/physrevlett.106.247203. URL http://dx.doi.org/10.1103/physrevlett.106. 247203 [4] K. Nakata, K. A. van Hoogdalem, P. Simon, D. Loss, Josephson and persistent spin currents in bose-einstein condensates of magnons, Phys. Rev. B 90 (2014) 144419. doi:10.1103/PhysRevB.90.144419. URL https://link.aps.org/doi/10.1103/PhysRevB. 90.144419 [5] A. A. Serga, A. V. Chumak, B. Hillebrands, Yig magnonics, Journal of Physics D: Applied Physics 43 (26) (2010) 264002. URL http://stacks.iop.org/0022-3727/43/i=26/a= 264002 [6] R. Hertel, Curvature-induced magnetochirality, in: Spin, Vol. 3, World Scientific, 2013, p. 1340009. [7] A. Goussev, J. M. Robbins, V. Slastikov, Domain wall motion in thin ferromagnetic nanotubes: Analytic results, EPL (Europhysics Letters) 105 (6) (2014) 67006. doi:10.1209/0295-5075/105/67006. URL https://doi.org/10.1209%2F0295-5075%2F105% 2F67006 [8] R. Streubel, P. Fischer, F. Kronast, V. P. Kravchuk, D. D. Sheka, Y. Gaididei, O. G. Schmidt, D. Makarov, Magnetism in curved geometries, Journal of Physics D: Applied Physics 49 (36) (2016) 363001. [9] J. A. Ot´ alora, M. Yan, H. Schultheiss, R. Hertel, A. K´ akay, Curvature-induced asymmetric spin-wave dispersion, Physical review letters 117 (22) (2016) 227203. [10] J. A. Ot´ alora, M. Yan, H. Schultheiss, R. Hertel, A. K´ akay, Asymmetric spin-wave dispersion in fer-

romagnetic nanotubes induced by surface curvature, Physical Review B 95 (18) (2017) 184415. L. R. Walker, Resonant modes of ferromagnetic spheroids, Journal of Applied Physics 29 (3) (1958) 318– 323. doi:10.1063/1.1723117. URL http://dx.doi.org/10.1063/1.1723117 J. R. Eshbach, R. W. Damon, Surface magnetostatic modes and surface spin waves, Phys. Rev. 118 (1960) 1208–1210. doi:10.1103/PhysRev.118.1208. URL https://link.aps.org/doi/10.1103/PhysRev. 118.1208 A. Gurevich, G. Melkov, Magnetization oscillations and waves, CRC Press, Boca Raton, 1996. D. D. Stancil, A. Prabhakar, Spin Waves. Theory and Applications, Springer, New York, 2009. R. I. Joseph, E. Schl¨ omann, Theory of magnetostatic modes in long, axially magnetized cylinders, Journal of Applied Physics 32 (6) (1961) 1001–1005. doi:10.1063/ 1.1736149. URL http://dx.doi.org/10.1063/1.1736149 T. K. Das, M. G. Cottam, Magnetostatic modes in nanometer sized ferromagnetic and antiferromagnetic tubes, Surface Review and Letters 14 (03) (2007) 471– 480. doi:10.1142/s0218625x07009505. URL http://dx.doi.org/10.1142/s0218625x07009505 J. Peuzin, J. Gay, Magnetostatic hole modes, 9th European Microwave Conference, 1979doi:10.1109/euma. 1979.332731. URL http://dx.doi.org/10.1109/euma.1979.332731 V. Basso, P. Ansalone, In preparation.

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of the Hallbach cylinder, this is the internal mode. With ν = 1 we simply obtain a uniform magnetization configuration. It must be noticed that both configurations are characterized by a uniform rotating field in the inner bore of the cylinder. The explicit analytic expression for the magnetic scalar potential and for the magnetization field can be useful when the coupling with external radiofrequency antennae has to be designed [17]. In future work it will be of interest to derive the effect of a static radial electric field on the surface modes in view of the phase manipulation of spin waves in non trivial geometries.

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Journal Pre-proof *Conflict of Interest form

Conflict of Interest and Authorship All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

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This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

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The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

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Istituto Nazionale di Ricerca Metrologica Istituto Nazionale di Ricerca Metrologica

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Patrizio Ansalone Vittorio Basso

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Author’s name