A N Kuchko - Spectrum of spin waves in a magnonic crystal with a structure defect - страница 1

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ISSN 0031-918X, The Physics of Metals and Metallography, 2006, Vol. 101, No. 6, pp. 513-518. © Pleiades Publishing, Inc., 2006.

Original Russian Text © A.N. Kuchko, M.L. Sokolovskii, V.V. Kruglyak, 2006, published in Fizika Metallov i Metallovedenie, 2006, Vol. 101, No. 6, pp. 565-569.

^^^^^^^^^^^^^^^ THEORY


Spectrum of Spin Waves in a Magnonic Crystal with a Structure Defect

A. N. Kuchko", M. L. Sokolovskii", and V. V. Kruglyak*

"Donetsk National University, ul. Universitetskaya 24, Donetsk, 83055 Ukraine bSchool of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom

Received December 12, 2005

AbstractProblem of the propagation of spin waves in a one-dimensional magnonic crystal with a structure defect has been analyzed theoretically. As a magnonic crystal, we consider a multilayer magnet representing an infinite system of plane-parallel alternate adjacent exchange-coupled slabs of magnets of two types. As a single structure defect that breaks the translational symmetry of the crystal, we consider a layer that is different from the layers composing the magnonic crystal. A dispersion law for spin-wave modes localized at the defect has been obtained. The appearance of several defect modes in the band gap has been analyzed depending on the parameters of the defect layer.

PACS numbers: 75.30.Ds

DOI: 10.1134/S0031918X06060019


Physics of multilayer magnetic systems is one of the most intensely developing fields of the physics of magnetic phenomena. Technical progress in the field of obtaining films with thicknesses as small as a few nanometers and the possibility of obtaining struc­tures with a predetermined distribution of compo­nents depending on the film thickness resulted in the creation of a new class of materialsartificial multi­layer magnets, or magnonic crystals (MCs)which are spin-wave analogs of photonic and phononic crystals [1-4] and represent materials with periodi­cally modulated magnetic parameters. To date, numerous works devoted to the investigation of spin waves (SWs) and magnetostatic waves (MSWs) in MCs have been published. These works have shown, in particular, that in a system consisting of a very large number of periodic alternate layers and pos­sessing a property of translational invariance there arises a band structure for SWs and MSWs instead of a discrete set of frequencies characteristic of the case of separate layers.

In real MCs, various structure defects (SDs) can exist, which lead to a local change in the magnitude of anisotropy and/or exchange interaction and in other parameters of the magnet and thereby violate the translational symmetry of the MC. Thus, it was shown in [5, 6] that a local change in the thickness of one of the layers composing an MC results in the appearance of a discrete level in the band gap. In more detail, the properties of localized modes in the photonic and phononic crystals in the presence of a single SD have been analyzed in [1, 3, 7-11]. The aim of this work was the derivation of a dispersion relation for a magnonic crystal with a structure defect and the investigation of the effect of its param­eters on the number and properties of spin-wave modes localized at this defect.


We consider the propagation of an SW in a one-dimensional MC with a single SD. As an MC, we take a multilayer magnet representing an infinite system of plane-parallel alternate adjacent exchange-coupled uni­form and uniformly magnetized (to saturation) layers of two types with thicknesses a and b, constants of exchange coupling aa and ab, constants of uniaxial anisotropy Pa and pb, gyromagnetic ratios ga and gb, and saturation magnetizations Ma and Mb. As a single SD breaking the translational symmetry of the MC, we take a layer, embedded into the multilayer material, of thick­ness d with magnetic parameters ad, pd, gd, and Md that are chosen in an arbitrary way, irrespective of the cor­responding parameters of the ideal structure (Fig. 1). We assume that the easy axis (EA) of each layer lies perpendicular to the layers of the MC. A dc uniform applied magnetic field H is assumed to be oriented along the EA. The Cartesian coordinate system is cho­sen so that the axis OZ be perpendicular to the layer planes.

n = -2; n = -1;   n = 0 \ n = +1; n = +2

ab d p(z)A


z-2 z-1 z0























z1 z2 z3 Z

Fig. 1. Geometry of a one-dimensional magnonic crystal (MC) with a single structure defect (SD) and the coordinate dependence of the parameters of the material p = a, P, g, M.



To describe the dynamics of the magnetization M(r, t) of an MC, we use the Landau-Lifshitz equation [12]

<dM = -g [ M x He],


where g is the gyromagnetic ratio (g > 0), and HE is the effective magnetic field. At the chosen orientation of the magnetization in the direction perpendicular to the plane of layers, the surface magnetodipole fields reduce to the demagnetizing field. The corresponding magne-tostatic components in this case can be included into the energy of magnetic anisotropy, which makes it possible to write the expression for the effective magnetic field in the form

HE = [ H + p( Mn)] n + aAM, (2)

where n is the unit vector along the OZ axis.

Consider small deviations m(r, t) of the magneti­zation from the ground state—uniform magnetiza­tion along the EA. To this end, we write the magne­tization as

M(r, t) = Mvn + mv(r, t), где |mv| < Mv, (3)

where the subscript v = a, b, d denotes different types of layers.

For a monochromatic plane SW with a frequency со and wave number kv which propagates in a direc­tion perpendicular to the layer boundaries, we can write

mv(r, t) = mvexp{imt}, mv = A+v exp {ikvz } + Av exp {-ikvz },


where Av are the amplitudes of SWs in the vth layer. In the approximation that is linear in mv we obtain from Eq. (1) the following expression for the disper­sion law of SWs in the homogeneous material of the vth layer:




With allowance for (5), Eq. (1) takes on the form

dz2     a-\gvMv  Mv PV

In the approximation of strong interlayer exchange coupling without spin pinning at interlayer boundaries, the solutions to Eq. (6) should satisfy boundary condi­tions of the following form [14, 15]:


-M------ a


-M------ b

Ja д m

Jb dmb

Mb d z


where Jv = av M v /2.



To find the spectrum of SWs in an MC, we use the method suggested in [8]. We introduce a two-compo­nent vector column W(z) = (m(z), X(z))t. Using the transfer-matrix method [16], we can write the following expression that relates the magnitudes of the vector W(z) at the beginning and at the end of each of the homogeneous layers (a, b, or d):

W (zv) = Sv W (z+v),

cos (kvv) Jvkvsin(kvv)


Jv kv


cos (kvv)

where Sv is the matrix of transformation of the homo­geneous layer of type v, and zv is the beginning, and = zv + v is the end coordinate of this layer.

By introducing a designation Wn = X(zn), where zn is the coordinate of the beginning of the nth period (Fig. 1), the change undergone by W upon the propaga­tion of the wave through one period of the MC can be written as

Wn + 1 = TnWn, (8)

where Tn is the unimodular matrix of the transformation of the nth pariod (n Ф 0) or of the defect layer (n = 0) of

the MC.

It is obvious that the transformation matrix for the defect layer is T0 = Sd, and the transformation matrix for





the nth period of the MC is Tn * 0 = SbSa. Thus, for the

components of the matrix Tn

* 0

we can write

Following (8) and introducing a vector Vn = (mn, mn _ 1)t, the last relation can be reduced to the form

V = Y C

y n        ± n - 1 ^ n '



cos (kdd),

n * 0

= X = cos(kaa)cos(kbb)

^J-k. sin (kaa) sin (kbb);


Ц0 = cos (kdd), ц * 0 = Ц = cos (kaa) cos (kbb)

-J---b--k---- b

-J---a--k---- a





°n *0 = a = Jk



+ ---------- cos(kaa)sin(kbb);




Zn * 0 = Z = - Jaka sin (kaa) cos (kbb) - Jbkbcos(kaa)sin(kbb).

With allowance for (4), the magnitude of the vector W at the boundary z = zn can be written as

W = PC ,



where Cn = (A(n+), A(;) )t, A™ = AV exp{±ikvzj,


P = M




iJvkv -iJvkv

Because of the continuity of W at the boundary z = zn, the index v can be chosen arbitrarily. Equation (9) per­mits us to write Eq. (8) in the following form:

W = R C




where Rn = Tn P.

Combining Eqs. (9) and (10), we obtain the magni­tudes of the magnetization at the boundary of the nth period (mn = m(zn)) in the form

mn  = A<n+ + A<n^ =  (Rn ) 11      + 1 + (Rn )12 An++ 1.

where Yn



Mv Mv

(Rn)11 (Rn)12

Eliminating Cn from Eq. (12), we obtain, using

Eqs. (9) and (10), that Tn1PYn+1 = PYn-1 Vn. Pass­ing in the last equation to the magnetizations in the lay­ers, we obtain a discrete equation that relates the ampli­tude of the magnetization at the boundaries of adjacent layers in the form


-mn+ 1 +





Xn  ,  Mn -1 ,

— +-\mn

55 n    55 n -U


In the case of a defect-free MC, the transformation matrices Tn are the same for any period n. Therefore, the last equation is simplified and reduces to the form

mn + 1 + mn-1 =

where F = Tr(Tn * 0) = X + ц.


Using the Bloch theorem and substituting mn ~ exp{inKl} (here, к is the quasi-wave number, and l = a + b is the period of MC) into Eq. (14), we obtain a well-known dispersion relation for the spectrum of SWs in a defect-free MC (see, e.g., [5, 16]:

2cos(Kl)= F.


It is obvious that this equation will have real roots if the condition |F | < 2 is fulfilled and that it is this condi­tion that determines the magnitudes of the allowed fre­quencies of SWs and the band character of the spec­trum. The boundaries of the allowed bands can be found from the equation |F| = 2.


Following [8], let us rewrite Eq. (13) in the form (1 + 5 Kn) mn + 1 + (1 + 5 Kn-1) mn-1 = (F + 5Qn + 5 Nn-1) mn,

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