V S Tkachenkoa' - Spin waves in a magnonic crystal with sine-like interfaces - страница 1

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Journal of Magnetism and Magnetic Materials 307 (2006) 48-52

www.elsevier.com/locate/jmmm

Spin waves in a magnonic crystal with sine-like interfaces

V.S. Tkachenkoa'*, V.V. Kruglyakb, A.N. Kuchkoa

^Donetsk National University, 24 Universitetskaya Street, Donetsk, 83055, Ukraine hSchool of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

Received 27 January 2006; received in revised form 13 March 2006

Available online 17 April 2006

Abstract

The spectrum of exchange spin waves in a magnonic crystal (periodic magnetic multilayer) with diffuse interfaces is derived for a model with a cosine-like profile of the uniaxial anisotropy value at the interfaces. The dependence of the band gap size upon the interface thickness and the depth of modulation of the anisotropy value is analyzed. In particular, it is shown that diffuse interfaces may lead to a magnonic spectrum in which band gaps have size equal to or even larger than those in the model with sharp (infinitely thin) interfaces (Kronnig-Penney model). ^^^^b © 2006 Elsevier B.V. All rights reserved.

Keywords: Magnonics; Magnon; Superlattice; Nanomagnetism

Due to the wealth results of investigations of photonic crystals [1] and other similar objects with artificially created periodicity [2-6], periodic magnetic structures have re­ceived a renewed attention. Such structures could be used to control the propagation of spin waves (SW) [7,8].By analogy to photonic crystals, periodic magnetic media are referred to as magnonic crystals (MCs). The knowledge of the SW spectrum of magneto-photonic crystals (dielectric MCs that support magnetic field controlled photonic band gaps [9]), is also required if they are to be used within devices operated at a GHz clock rate.

While most of the theoretical efforts have been devoted to investigation of MCs with infinitely thin interfaces [10-14], interfaces of realistic samples always have a finite thickness. For example, resonant X-ray magnetic scattering analyses show that the magnetic interface can be thicker than the corresponding chemical interface [15]. On the basis of the full Ginzburg-Landau free energy functional, Schwenk et al. [16] considered the effect of finite temperatures upon both statics and dynamics of magneti­zation in an all-ferromagnetic periodic multilayer. In particular, they showed that, even when multilayer inter-

^Corresponding author. Tel.: +441392264163; fax: +441392264111. E-mail addresses: kuchko@dongu.donetsk.ua (V.S. Tkachenko), V.V.Kruglyak@exeter.ac.uk (V.V. Kruglyak).

faces are chemically sharp, the static magnetization magnitude profile is always smooth so that magnetic "transition" layers are formed between the basic constitu­ent layers. In addition, the interfaces can contain some roughness, which was predicted to significantly modify SW spectrum in the case of thin magnetic films [17,18]. Therefore, a question arises whether the realistic multi­layers can efficiently act as MCs.

Gorobets et al. [19] used a harmonic function to approximate the coordinate dependence of magnetic parameters of a MC with diffuse interfaces. A similar model was used for the investigation of a planar metallic MC by Kolodin and Hillebrands [20]. Ignatchenko et al. [21,22] used a model based upon the Jacobian elliptic sine function, which can describe interfaces of arbitrary thickness, to demonstrate a strong dependence of the magnonic spectrum and coefficients of reflection and transmission of SWs upon the thickness of interfaces. Although being more realistic than that of Gorobets et al., the model of Ignatchenko et al does not allow one to obtain an exact solution, and so is based upon the perturbation theory in the limit of small modulation of the parameters of the MC. At the same time, the theory of waves in inhomogeneous media is well developed [1,2,4], and can be used to build an exactly solvable model for investigation of SWs in a MC in which both the thickness

0304-8853/$-see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.03.039

of interfaces and the amplitude of modulation of the MC parameters have finite (maybe quite large) values [23].In the present work, such a model is developed for a one-dimensional MC in which the distribution of the value of the uniaxial anisotropy constant at the interfaces is approximated with a cosine function, while the anisotropy value is assumed to be constant within the basic constituent layers of the MC.

Let us consider an infinite one dimensional MC represented by a system of alternating uniform magnetic layers of two different types but of the same thickness d. The layers have different values b— and b+ of the uniaxial anisotropy constant b, while the values of the exchange interaction parameter a, gyromagnetic ratio g, and the saturation magnetization M0 are assumed to be constant throughout the sample. We assume that the "basic" layers of the MC are separated by "transition" regions of thickness d in which the value of the uniaxial anisotropy constant changes continuously between values b— and b+. The OZ axis is chosen to be perpendicular to the plane of the layers. The easy axis (EA) and the internal magnetic field H both lie parallel to the OZ axis. n is the unit vector in the direction of the internal magnetic field H. Here, we neglect magnetic damping. The effects associated with its presence were considered in Refs. [19,20,24-26].In particular, the case of an interface dominated SW damping was discussed in Refs. [25,26].

To describe the dynamics of the magnetization M(r,t), we use the Landau-Lifshitz equation

dM

Hi

— —g

M x \(H + b(Mn))n -

d

dM

dr\ dr

(1)

For the present geometry, the dynamic magneto-dipole field is absent [7]. Magnonic crystals properties of which are dominated by the magneto-dipole interaction are studied, for example, in Ref. [8].

Let us consider small deviations my (j — 1... 4) of the magnetization of each layer from the ground state given by a uniform magnetization parallel to the EA

My(r, t) nM0 + my(r, t), I my I « M0.

(2)

Linearizing Eq. (1) with respect to my, and introducing Fourier components my(r, t) — my(r) exp(ifflt) and then variable m mx + imy, we obtain the following equation

d2m(z) dz2

+ k2(z)m(z) — 0,

(3)

where k(z) y/(Q h b(z))/a, O ю/gM0, and hH/M 0.

We have to find such a function b(z) that function k(z) is periodic with period L, is continuous, and has continuous derivatives. In addition, function b(z) (and hence k(z) ) must be constant within the main layers of the MC, and it must be possible to find an analytical solution of equation (3) in the transition layers. These conditions are satisfied by

Ab 2

= <

z0 + nL<z<z1 + nL, z1 + nL<z<z2 + nL, z2 + nL<z<z3 + nL,

the following function

b(z)

'    b b— <b>

b2 (b)+ f cos[f (Z Zi)\

b3 b+ (b)+ Db

b4  (b)  Db cos [f (z — Z4)]    Z3 + nL<Z<Z4 + nL,

(4)

where (b) (b+ + b—) /2, Ab b+ b—. zy(y1... 4) are the coordinates of the boundaries between the basic and transition layers (z0 — 0, z1 d, z2 d + d, z3 — 2d + d, z4 — 2d + 2d — L), n0, ±1, ±2,and L is the period of the MC. The graph of function b(z) is shown in Fig. 1.

At the interfaces zy, the solution of Eq. (3) must satisfy the boundary conditions that in the exchange limit [12] are

my L — my+11

, dz

dm

y+1

dz

,y1 ...4.

(5)

The physical meaning of the first of them is that, in the exchange-dominated regime, the directions of the magnetization on both sides of the interfaces are parallel. The second condition ensures that the normal to the interface component of the energy flux density vector is continuous [12].

Furthermore, the solution of Eq. (3) must satisfy the periodicity condition, i.e., the dynamic magnetization at the boundaries of the period z z00 and z z4 L can differ only by a phase factor [2]

m(0) exp{iKL}m(L), (6)

where K is the Bloch wave number.

Let us find solution my(z) — m(zy—1 <z<zy) for each layer. In the basic layers (for z0<z<z1 and for z2<z<z3), the solution of Eq. (3) has the form of the plane waves

my (z) m+ exp{+ikyz} + m— exp{ ikyz} ,y1,3,

(7)

where m+ and m— are the wave amplitudes. In the transi­tion layers (for z1< z< z2 and for z3< z< z4), Eq. (3)

в (z)

<p>

Fig. 1. The coordinate dependence of the anisotropy constant is shown for one period MC, consisting of an infinite number of such periods, i.e., ... —1—2—3—4—1—2—3—4—...

zy

в

+

в

transforms into

d2my(z)

dz[1]

■ +

O <b> — h + (— V/2 Ab cos(f (z z,)

0 y— 2 4.

my( )

(8)

The substitution of the independent variable transforms Eq. (8) into the Mathieu equation [27,28]

d[2]my (t)

dt[3]

+ (a 2qy cos(2t)^ my (t) — 0,

(9)

where a 4d[4](O (b) h)/ая[5], qy — (— 1У/2_1 d[6]Ab/af[7]. Eq. (9) has solution

my(z) — /^(^ qy) + m—c{t, qy),y — 2, 4 (10)

where c(t qy) and s(t qy) are the general Mathieu functions.

In order to find the spectrum of SWs in the sample, we use the method of the transfer matrix [2,29]. If applied to individual atomic layers, as for example in Refs. [12,17,18,] this method can be used to calculate numerically SW spectrum of a MC with arbitrary interface profile, while here an analytical solution is obtained.

Let us introduce the following two-component column-vector

m*   \m(z)\ m* dm(z)

U (z)—[ s(z)   , °(z)^.

(11)

The values of the magnetization and its derivative in the beginning of each layer are expressed via their values in the end of the layer using the transfer matrix My

Uzy—1 MyUz

(12)

For the basic layers (y— 1 3), the transfer matrices are

cos k +d       —k— + sin k— +d

M 1,3

k—1 sin k— +d        cos k— +d

(13)

where k— + —     O — h — b— + /a.

For the interface layers (y— 2 4), the transfer matrices are /   Py —(k)Qy\

M2 4 —

where

(k)—1P'y Qy

py c'[°0, qy^js(ty, qy^j s' (0, qy^j c(ty, qy

Qy c (0, qy) s{ty, qy) s(0, qy) c      , qy)

P' c' 0 q s' t q s' 0 q c' t q

Q' c 0 q s' t q s 0 q c' t q

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