V S Tkachenkoa' - Spin wave reflection from semi-infinite magnonic crystals withdiffuse interfaces - страница 1

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Metamaterials 3 (2009) 28-32

Spin wave reflection from semi-infinite magnonic crystals with

diffuse interfaces

V.S. Tkachenkoa'[1], V.V. Kruglyakb, A.N. Kuchkoa

a Donetsk National University, 24 Universitetskaya Street, Donetsk 83055, Ukraine b School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK

Received 24 April 2008; received in revised form 20 October 2008; accepted 30 October 2008 Available online 7 November 2008


The frequency dependence of the coefficient of spin wave reflection from a semi-infinite magnonic crystal with a periodically modulated value of the uniaxial anisotropy and a finite thickness of interfaces has been investigated, assuming a linear distribution of the anisotropy value in the interfaces. The analysis shows that the performance of magnonic devices employing magnonic crystals as a filtering element can degrade as the thickness of interfaces increases, e.g. due to the process of diffusion between constituent layers of the magnonic crystals. © 2008 Elsevier B.V. All rights reserved.

PACS: 75.30.Ds; 75.70.-i

Keywords: Magnonic crystal; Spin wave; Periodic structure; Magnonics

1. Introduction

Superlattices are artificial structures with a peri­odic modulation of one or several material parameters. Superlattices with periodically modulated magnetic parameters show magnonic band structure and are called magnonic crystals (MCs) when the spin wave (SW) wavelength of interest is comparable to their period of the MCs [1]. For SWs with wavelength much greater than the period of the MC, magnetic superlattices behave as magnetic meta-materials with properties different to those of the constituent layers [2]. Thus, the study of SWs in MCs (so-called magnonics [3]) is an intensively devel­oping direction in the physics of magnetic phenomena and meta-materials.

The primary source of interest to magnonics is the opportunity to use SWs propagating in MCs as data car­riers within elements of SW logic devices. In particular, it was suggested that the use of MC instead of continu­ous SW wave guides could help to decrease dimensions of such SW logic devices while maintaining their con­trollability by applied magnetic field [4]. In the context of electromagnetic meta-materials, MCs could offer a way of designing magnonic resonances so that regions of negative magnetic permeability and possibly of neg­ative refractive index [5,6] could be created near the resonances, which could be tailored to reach frequencies of several THz [4].

As materials having spatial modulation of magnetic parameters, one can use yttrium-iron garnets grown

1873-1988/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.metmat.2008.10.001


V.S. Tkachenko et al. / Metamaterials 3 (2009) 28-32

using different chemical and physical methods, e.g. using liquid phase epitaxy with variable temperature regime [7], alloys of Fe100-x-Nix and CoW0-x-Px [8], and many other magnetic alloys with spatially varied concentration of the magnetic ions in layers, thus allowing for differ­ent values of magnetic parameters in layers with different x.

A great number of works has been dedicated to inves­tigations of wave multilayered materials with infinitely thin interfaces [9-15]. However, in many cases, the assumption of infinitely thin interfaces is a severe idealization, since diffusion normally occurs between homogeneous layers of a multilayered structure, which can manifest itself, e.g. in formation of magnetically "dead layers" at the interface between magnetic and non-magnetic layers [16]. The diffusion is proportional to the concentration gradient at the interface and hence is fastest in multilayers prepared with sharp interfaces. On the other hand, multilayers could be prepared with smeared interfaces from the beginning, thus reducing the speed of subsequent inter-diffusion, provided of course that such multilayers retain their useful functionality, e.g. the magnonic band spectrum in the case of MCs. That is why models of MCs with finite thickness of inter­faces have attracted an increasing interest of researchers recently.

For example, in Refs. [17-19] the influence of the interface thickness on the spectrum of spin waves (SW) in MC has been investigated, with a conclusion that the spectrum of waves depends significantly on the thickness of the interface. In Ref. [18], a possibility of recover­ing the magnetic structure of a multilayer material from its SW spectral characteristics has been demonstrated, assuming standing waves. However, measurements of scattering of propagating SWs from such samples might provide additional information regarding the MCs. So, in Ref. [20], the reflection coefficient of SWs from a MC with ideal interfaces was calculated. In Ref. [21], the authors investigated MCs with an imperfect exchange interaction between its adjacent layers. The main aim of the present work is to investigate the effect of the inter­face thickness on the coefficient of SW reflection from


2. Model of the material

Let us consider a semi-infinite MC represented by a system of two types of alternating homogeneous ferro­magnetic layers of equal thicknesses. Each of the layer types is described by a different value of the uniax-ial anisotropy. The direction of the easy axis of the anisotropy is assumed to be perpendicular to the plane

Fig. 1. The coordinate dependence of the anisotropy value в is schematically shown for a semi-infinite MC with interfaces of finite thickness and with a linear variation of the anisotropy value within the interfaces. The inset shows the same for a single period of the magnonic crystal.

of the layers. Also, it is assumed that the homogeneous "basic" layers of the MC are separated by inhomoge-neous "transition" layers of finite thickness in which the value of the uniaxial anisotropy varies as

e(z) = <

в1,5 = (в) 3 = (в) +

' 2 2

Z0,4 + nL<Z<Z1,5 + nL

Z2 + nL<z<Z3 + nL

a         io\ _i_ Ав z - z2,4 .j .j

в2,4 = (в) ± "2--1- Z1,3 + nL<Z<Z2,3 + nL


where L is the period of the MC. Fig. 1 schematically shows the coordinate dependence of the anisotropy in

the MC.

For the purpose of this calculation, the dynamics of the magnetization M (r, t) can be described by the Landau-Lifshitz equation [22]:

= -g[M x Heff],

where Heff is the effective magnetic field.

#eff = {H + в {Mn)) П + hm + —[ a—

r \     dr I



where a and g are the values of the parameter of the exchange interaction and the gyromagnetic ratio, respec­tively. Н is the value of the external magnetic field, which is applied parallel to the easy axis direction, and nHis the unit vector in the direction of the external mag­netic field, hm is demagnetizing field determined from

the magnetostatic Maxwell equations rot(h m) = 0,    div(h m) = -4n div(M).


In a general case, hHm is coordinate-dependent and has a constant value only in a limited class of samples, namely in samples of ellipsoidal shape [23]. Here, we consider the case when the lateral dimensions of the sample are much greater than its total thickness in z direc­tion and when the magnetization dynamics are uniform in the plane of the layers, i.e. the in-plane component of the SW wave vector is equal to zero. Furthermore, we assume that the saturation magnetization is constant throughout the sample Mj 00 = M0 where j is the number of a particular layer. In this case, the demagnetizing field is constant and leads to the so-called shape anisotropy equal to -47tM0 and with axis perpendicular to the layers of the MC and hence parallel to the easy axis. Hence, for the sake of brevity, we assume that the shape anisotropy is accounted for in function в^), and therefore in equa­tion (3) hm = 0.

If the absolute value of the magnetization was differ­ent in the two basic layers and varied continuously within the transition layers, hm,z would vary so that the normal component of the magnetic flux density hmz - 4nMz was constant. Again, this could be accounted for by assuming that function в^) also includes this contribution, making the same approximation that the variation of hm,z in the transition layers could be described by a linear function.

Let us consider small deviations mmj (j = 1, 5) of the magnetization from the ground state, which is a uniform magnetization parallel to the easy axis,

Mj(z, t)=nM0 +mhj(z, t), \Mj(r, t)|=M0, \mhj\ « M0.


Linearizing Eq. (2) with respect to Eq. (5), intro­ducing the temporal Fourier components mi j (z,t) = m jtlo (z) exp [icot] and then introducing variable = mx,co + imycC, we obtain the following equation that describes propagation of SWs in each layer of the MC

+ k2(z)/xj(z) = 0,j = 1,      5,kj(z)

Q - h - ві (z)


where we introduced dimensionless frequency Q = co/gM0 and magnetic field h = H/M0. This allows us to avoid assuming any particular value of M0, which for a wide range of magnetic materials of interest has values M0 ~ (10 to 103)Gs. In the following calculations, for the sake of brevity, we assumed that the period

of the MC is L =5mkm, the exchange parameter is a =10-8cm2, and the value of the external magnetic field is h = 0. Yet, the formulae derived are more general and can be applied to a much wider range of samples.

At the interfaces, the solution of Eq. (6) must sat­isfy the boundary conditions of the continuity of the magnetization and its derivative [15].

In Ref. [24], Ignatovich developed an original recur­rence method by which to derive the dispersion in and scattering of particles from a semi-infinite periodic potential. Here, we apply this method to the problem of SW scattering from a semi-infinite MC shown in Fig. 1.

Let us consider a spin wave incident on the boundary of the MC. The amplitudes of the incident wave /л0 and of the reflected wave /лг are connected as

where R is the coefficient of SW reflection from the MC.

Using the method proposed in Ref. [24], one can obtain


л/(р + 1)2 - T2 -y/(p - 1)2 - T2



here p and t are the reflection and transmission coeffi­cients of SWs for a single period of the MC. Using the same method, it is also possible to obtain the Bloch wave vector in terms of the latter scattering coefficients

exp (iKL)=—= --====

V(t + 1)2 - p2 W(t - 1)2 - P2


Using the method of transfer matrices [25], one can obtain explicit expressions for scattering coefficients p and t. According to this method, transfer matrix Mj is put in correspondence to each layer of the MC. This matrix connects the values of the magnetization and its derivative at the beginning and the end of that layer:


Л (z) d (z)/dz



Л (z) d (z)/dz


For the basic layers, these matrices are [26]:


1,5 =



sin -

V 2;

- k1 sin


cos (k3d)

k1 d

co4^J /

k3 sin (k3d)


k3-1 sin (k3d)   cos (k3d)


For the transition layers the transfer matrices are

(     P2,4 Q2,4 Л

M2,4 = (      P, \-1      Q      j , (11)

P2,4 G2,4 Л

2,4 /

-P2,4£-1 Q2

where for the linear profile of the interface considered here [26]:

Pj = Г Q)3-1/3 [Ai«j)31/2 - Bi )] , Qj = Г (3)3-1/2 [Ai(Zj)31/2 - Bi (Zj)] ,






dAi(Zj) dBi(Zj)

dz dz 31/3 dAi(Zj)    dBi(Zj)

Z2,4 (z)

dz 8


=--(Q - ± ,

a £

Ai(Z) and Bi(Z) are the Airy functions, and £ = )1/3.

The transfer matrix for one period of the MC and its trace can be written as

m = П Mj,   MM = Sp m,



Then the explicit expressions for the reflection and trans­mission coefficients will be

p = V1 - 4M2

t = -2M, where

cos (k1d) P2 + (£k1)-1 sin (k1d) P2

(13) (14)

cos (k3d)

+ cos (k1d) Q2 - (£k1)-1 sin (k1d) Q2

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