V S Tkachenko - Spectrum and reflection of spin waves in magnonic crystals with different interface profiles - страница 1

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PHYSICAL REVIEW В 81, 024425 (2010)

Spectrum and reflection of spin waves in magnonic crystals with different interface profiles

V. S. Tkachenko,1 V. V. Kruglyak,2 and A. N. Kuchko1,2 1Donetsk National University, 24 Universitetskaya Street, Donetsk 83055, Ukraine 2School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom (Received 22 June 2009; revised manuscript received 22 December 2009; published 27 January 2010)

Using the transfer-matrix method, we have developed a theory of exchange spin waves in a thin cylindrical magnonic crystal (periodically layered all-ferromagnetic nanowire) with diffuse interfaces. The magnonic spectrum and the frequency dependence of the reflection coefficient of spin waves from a junction between a homogeneous magnetic nanowire and a magnonic crystal have been calculated and compared. Diffuse inter­faces with linear and sinusoidal profiles of variation in the uniaxial anisotropy value have been considered, also allowing for asymmetry in the relative thicknesses of either main layers, or interfaces, or both. We have found that, although the thickness and profile of interfaces have a significant effect on the size and position of the magnonic band gaps, the smoothing of interfaces does not lead to disappearance of the band gaps. At the same time, the profiles and relative thicknesses of interfaces might provide additional means by which to design magnonic crystals with a desired magnonic spectrum.

DOI: 10.1103/PhysRevB.81.024425 PACS number(s): 75.30.Ds, 75.70.Cn

I. INTRODUCTION

Magnonic crystals are materials with periodically modu­lated magnetic parameters and represent spin-wave (magnon) counterparts of photonic,1 phononic,2 and plasmonic3 crys­tals. The study of spin waves and magnonic crystals (so-called "magnonics") has been intensively growing recently as a new research field at the interface between nanomag-netism and electromagnetic metamaterials. So, many excit­ing results have recently been obtained in magnonics by experimental,4-9 theoretical,10-17 and computational18-20 means.

Generally, the theory of waves in layered media is well developed. In particular, a number of exactly solvable mod­els have been proposed and thoroughly investigated (see, e.g., Refs. 21 and 22 for review). In part, the advance in this and some other research directions has been aided by the remarkable analogies with the quantum-mechanical theory of an electron in a one-dimensional (1D) potential. The analogy can be drawn in some cases of 1D problems of spin-wave propagation in nonuniform magnetic fields23-25 and also in the discussion of nucleation fields in composite magnets.26-30 In the latter case, due to the dominating role of the aniso-tropy field, the analogy holds also for two-dimensional (2D) and three-dimensional (3D) problems. Nonetheless, in a gen­eral case, the results obtained for other excitations and qua-siparticles (e.g., phonons or electrons) are not readily trans­ferred to the case of magnonic crystals. This is mainly due to the complicating role of the long rage magnetodipole inter­action, combined with the increased number of material pa­rameters that affect the dispersion of magnons in such media. The situation is even more complicated in the case of 2D and 3D magnonic crystals, for which only few theoretical results have been published so far,10-13 with strong approximations often made. Alternatively, the spectra of spin waves in mag-nonic crystals can be obtained from micromagnetic simula­tions (with only 1D case having been considered to date18,19), or using the dynamical-matrix method,20 which can be thought of as micromagnetics in the reciprocal space. On the

1098-0121/2010/81 (2)/024425 (7) experimental side, there is a record of studies of periodic magnetic multilayers31-33 while more recent investigations concentrated on planar arrays of magnetic elements34-37 and magnonic crystals.4-9

The studies confirmed that the spectrum of exchange, magnetostatic, and dipole-exchange spin waves in magnonic crystals forms band structure, as opposed to the discrete spectrum observed in systems of noninteracting magnetic thin films or elements. The next issue to consider is how the magnonic band structure is affected by various kinds of im­perfections, which are inevitably present in realistic samples and devices of magnonics, and how to suppress any destruc­tive effects of such imperfections. So, in Refs. 38 and 39, Ignatchenko et al. studied the modification in the magnonic band spectrum due to random variations in the period of a magnonic crystal. Localization of spin waves by isolated de­fects was investigated in Refs. 40 and 41. Here, we again note the useful analogy between the problem of magnonic spectrum and the problem of nucleation.29,42 In Refs. 39 and 43, the effects of the presence of damping and of its modu­lation in magnonic crystals were studied.

In addition, realistic multilayered and planar magnonic crystals are likely to have diffuse interfaces.44 In other words, transitions between constituent layers of a magnonic crystal or between elements made of different materials9 are likely to have finite thickness. Magnetic properties of such transition regions are generally different from those of the two adjacent materials and can continuously vary across the thickness of the interface. Alternatively, such interfaces could be created artificially, e.g., aiming to create a particular magnonic band spectrum. In Refs. 44 and 45, Ignatchenko et al. investigated propagation of waves in periodic multilayer materials with diffuse interfaces. The magnonic crystal was considered as a model magnonic crystal with a periodically modulated value of the uniaxial anisotropy constant. Both the spectrum and the scattering parameters (i.e., transmission and reflection coefficients) of spin waves were found to de­pend strongly upon the thickness of interfaces. In the model, the spatial modulation was described by a Jacobian elliptical sine function. However, the model is not solvable exactly,

©2010 The American Physical Society

024425-1

PHYSICAL REVIEW В 81, 024425 (2010)

and so the perturbation theory had to be used instead, assum­ing a small modulation of the anisotropy value. In contrast, exactly solvable models were used in Refs. 46-48, to study the effect of diffuse interfaces upon the spectrum of spin waves propagating in magnonic crystals with an arbitrary large amplitude of modulation of the anisotropy value. The difference between results obtained using the two approaches can be drastic. For example, the first-order perturbation theory predicts vanishing of all n > 1 band gaps for a sinu­soidal magnonic crystal,48 which is not the case in the exact solution in terms of Mathieu functions.46,49

Here, we address the question of the effect of the particu­lar functional form of the variation of interface's properties across its thickness upon the magnonic band structure of a thin cylindrical magnonic crystal (periodically layered mag­netic nanowire with a circular cross section). In particular, we compare models with linear and sinusoidal variations of the anisotropy value in the transition regions. The aim is to study the effect of thicknesses of the constituent layers and interfaces as well as the structure of the interfaces, upon the spectrum and reflection coefficient of spin waves from a semi-infinite magnonic crystal. The here presented study of the reflection of spin waves from a semi-infinite magnonic crystal will be particularly important for measurements and practical implementation of propagating spin waves in mag-nonic devices.

In Refs. 50 and 51, imperfect interfaces were modeled as regions with increased damping. It was found that the overall damping of spin-wave modes in magnonic crystals described by such models can vary from band to band much stronger than in the case of damping uniformly distributed within layers, as considered in Ref. 43. However, the damping of spin waves is beyond the scope of this particular study, where we focus on the functional profiles of interfaces.

II. MODEL

Due to space constraints, future networks of magnonic waveguides are likely to consist of waveguides with a nar­row cross section.15 Hence, we model the magnonic crystal as an infinitely long wire consisting of periodically alternated adjacent uniform layers of two types. The latter layer types differ only by the strength of the uniaxial anisotropy while the easy axis is parallel to the axis of the wire with the associated unit vector denoted as n. This assumption is a reasonable approximation, backed up by the recent progress in fabrication of multilayered magnetic nanowires.52 So, in Ref. 52, the authors successfully grew multilayered Co nano-wires in which only the strength of the uniaxial anisotropy constant was modulated. The exchange parameter a, the gy-romagnetic ratio g, and the spontaneous magnetization M0 are assumed to be constant throughout the sample.

Furthermore, we assume that the so called "main" layers of the magnonic crystal are separated by "transition" layers in which the strength of the anisotropy varies as

FIG. 1. (a) The coordinate dependence of the anisotropy value in the magnonic crystal is schematically shown for linear (thick solid line) and sinusoidal (thin solid line) profiles of the interfaces. (b) The geometry of the problem is schematically shown.

A,3 = (P) +

др

др

P2,4 = (P) ±-T&z) z1,3 + lL < z < z2A + lL,

(1)

where (p) = ^/+ and Др= \P+-p- are the average value and the amplitude of modulation of the anisotropy, respectively, l is the number of the period, and z1,2,3,4 are the coordinates of the layer boundaries within the period, as shown in Fig. 1. The upper and lower signs in the expression correspond to the first and second indices, respectively. L is the period of the magnonic crystal, and d13 and ^ 4 are thicknesses of the main and transition layers, respectively. One can see that the model allows one to study not only asymmetry in the thick nesses of the main layers but also asymmetry in the profiles of the interfaces, e.g., magnonic crystals in which sharp and smooth interfaces alternate (<52(4) = 0, <54(2) Ф 0). The Z axis is chosen to be parallel to n.

Equation (1) can describe both interface profiles studied here. For linear profile, one has

z - z2,4

2,4

For sinusoidal profile, one has

%(z) = cos

z - z2,4 ir- .

<%,4 _

(2)

(3)

III. SPECTRUM

To describe dynamics of magnetization M(r, t) in the magnonic crystal, we use Landau-Lifshits equation53

2

PHYSICAL REVIEW В 81, 024425 (2010)

dt

j m x |[

dM

= - g\ M X | [H + P(Mn)]n + hm + (a^

111

(4)

where H is the bias magnetic field parallel to n, hm is the demagnetizing field, g is the gyromagnetic ratio, and a is the exchange constant. Assuming sufficiently strong bias and an­isotropy fields, the magnetization in the ground state is uni­form and parallel to the axis of the wire. To consider small amplitude spin waves, we can represent magnetization as

M/r, t) = nM0 + m/r, t),

(5)

where irij are small deviations of magnetization (\nij\<M0) in layer j=1 , . . . ,4 from the ground state, assuming that the length of the magnetization vector is constant \Mj(r,t)\ = M0. We assume that the diameter of the wire is smaller than the exchange length, and so, magnetization dynamics are uniform in the circular cross section of the wire.54,55 Then, it is appropriate to describe the nonvanishing components of the dynamic demagnetizing field, which are perpendicular to the axis of the wire, in terms of ballistic demagnetizing fac­tors. The static demagnetizing field in an infinite cylinder magnetized along its axis is already zero. Hence, one has

hm=-DM, where D is the tensor of ballistic demagnetizing coefficients that has only two nonvanishing components

Dxx = Dyy = 2l.

We linearize Eq. (4) using Eq. (5), introduce Fourier com­ponents mj(z, t) = mwj(z)exp{iwt} where t and w are time and frequency, respectively, and do a standard substitution of variables v=mx + imy. Then, we obtain the following equa­tion that describes propagation of spin waves in each layer of the magnonic crystal

kj (z)Vj(z) = 0, kj(z):

C - h -2i- fl-(z)

(6)

where C = w/gM0 and h=H/M0.

Solutions of Eq. (6) and their derivatives must be continu­ous at the boundaries of the main and transition layers as well as everywhere else. This ensures that, first, in the re­gime of dominating exchange interaction, the magnetizations on the opposite sides of the boundary are parallel, and sec­ond, the normal component of the vector of the density of energy flux is continuous.53 Besides, the solution of Eq. (6) must satisfy the condition of periodicity, i.e., solutions Vj at the period boundaries z=0 и z=z4=L must differ only by a phase factor56

V(0) = eiKLv(L),

(7)

where K is the quasiwave number (Bloch wave number). To find the spectrum of spin waves, we use the method of

transfer matrices Mj,56 which link values of the dynamic magnetization in the beginning and the end of each layer

(   V(z) 1 \dv(z)/dz! .

j-1

V(z) dv(z)/dz

)|

(8)

FIG. 2. (Color online) The two lowest bands of the spectrum of spin waves described by Eq. (9) are shown for magnonic crystals with the linear (black dotted line) and sinusoidal (red dotted line) interface profiles and for ideal magnonic crystals with sharp inter­faces (solid black line), all with p+ /P- = 2 and S2=S4=S. Here and in the following we assumed the value of the exchange constant of 3 X 10-12 cm2, which is characteristic for Co.

Using Floquet-Bloch theorem Eq. (7), it is then easy to write the spectrum of spin waves in the magnonic crystal (Fig. 2) in the following form:

cos(KL) = -2M, (9) where MM =Sp(M)[or MM=Tr(M)] is given in the Appendix.

IV. REFLECTION COEFFICIENT

From the point of view of application of magnonic crys­tals in magnonic devices, e.g., as spin wave filters, it is im­portant to know how spin waves are reflected from the boundary between a semi-infinite magnonic crystal and the adjacent semi-infinite uniform ferromagnetic medium. To calculate the coefficient of reflection of spin waves incident upon the boundary, we use the recurrent method proposed in Ref. 57 for investigation of scattering of neutrons from a semi-infinite periodic potential. The semi-infinite magnonic crystal and the recurrent method are schematically illustrated in Fig. 3. The magnetic properties of one period of the semi-infinite magnonic crystal, including the profiles of the tran­sition layers, are the same as in the case of the infinite mag-nonic crystal considered in the previous sections. The uniform ferromagnetic medium has anisotropy value of P-while the rest of its parameters are equal to those of the magnonic crystal.

The amplitudes v0 and vr of the incident and reflected spin waves, respectively, are connected as

a

FIG. 3. The recurrent method used for calculation of the reflec­tion coefficient is illustrated for the case of the semi-infinite mag­nonic crystal with a linear profile of the interface. The incident and reflected spin waves are schematically shown by forward and re­verse arrows.

0

where R is the coefficient of reflection of spin waves from the semi-infinite magnonic crystal. The same relation will hold for spin waves fin and fin incident on and reflected from period n

fin = R fin.

Using recurrent relation

fin = Tfin-1 + pfin

we obtain

fin T

(10)

(11)

fin-1      1- pR'

where p and t are the reflection and transmission coefficients through a single isolated symmetric period of the magnonic crystal.

In a similar way, we also obtain

fin-1 = R fin-1 = pfin-1 + Tfin

and hence

R=p+

t2R

(1-pR)

According to Ref. 57, the solution of this equation is

V(P +1)2-t2 - V(p-1)2-T V(p +1)2- t2 + V(p -1)2- T.

R

(12)

Here, p and t are the reflection and transmission coefficients of spherical waves for a single isolated period of the mag-nonic crystal Using the method of transfer matrices, one can obtain explicit expressions for p and t as

p= V1-4M2

■2 M.

(13)

(14)

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V S Tkachenko - Spectrum and reflection of spin waves in magnonic crystals with different interface profiles