Localized magnetic excitations for a line of magnetic impurities in a transverse Ising thin film ferromagnet

Physics Letters A 373 (2009) 3678–3683

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Physics Letters A www.elsevier.com/locate/pla

Localized magnetic excitations for a line of magnetic impurities in a transverse Ising thin film ferromagnet R.V. Leite a,∗ , L.O. de Oliveira Filho a , J. Milton Pereira Jr. b,d , M.G. Cottam c , R.N. Costa Filho c,d a

Universidade Estadual Vale do Acaraú, Centro de Ciências Exatas e Tecnologia, Av. Dr. Guarany, 317, Campus do Cidão, 62040-730 Sobral, Ceará, Brazil Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió, Alagoas, Brazil c Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7 d Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil b

a r t i c l e

i n f o

Article history: Received 30 July 2009 Accepted 3 August 2009 Available online 7 August 2009 Communicated by V.M. Agranovich PACS: 75.30.Hx 73.30.Ds 75.30.Pd

a b s t r a c t A Green’s function method is used to obtain the spectrum of spin excitations associated with a linear array of magnetic impurities implanted in a ferromagnetic thin film. The equations of motion for the Green’s functions of the anisotropic film are written in the framework of the Ising model in a transverse field. The frequencies of localized modes are calculated as a function of the interaction parameters for the exchange coupling between impurity-spin pairs, host-spin pairs, and impurity-host neighbors, as well as the effective field parameter at the impurity sites. © 2009 Elsevier B.V. All rights reserved.

Keywords: Ising model-transverse Green’s function Spin wave Impurities modes Ultra-thin films

1. Introduction

In recent years there have been numerous experimental and theoretical studies of spin waves in antiferromagnetic and ferromagnetic thin films (see Refs. [1–3] for reviews). In some bulk magnetic materials, light scattering experiments have clearly demonstrated that the introduction of an isolated magnetic impurity or a low concentration of random impurities into crystals modifies the spectrum of the elementary excitations. This occurs because the introduction of defects or impurities will break the translational symmetry of the lattice and thus modify the microscopic scheme of interactions in the material. Green’s function theories for impurities in bulk materials have been developed by several authors in the context of the Heisenberg model [4–9], which allows the identification of defect modes outside the bulk band of spin waves (SW) and of resonance modes within the SW band. For the Ising model the defect modes have

*

Corresponding author. E-mail address: [email protected] (R.V. Leite).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.08.003

also been predicted to arise for energies below the bulk band of the ferromagnetic host material (for a review, see e.g. [10–14]). Localized magnetic impurities can also be employed to model the dynamics of regular arrays of sub-micrometer-scale magnetic dots and wires created on a host medium. Such structures have attracted a great deal of interest, due to their fundamental properties and potential technological applications in magnetic sensors and memory devices [15,16]. Experimentally, laterally confined magnetic structures on the sub-micrometer and nanometer scale can be fabricated by various methods. For example, arrays of magnetic stripes with different geometrical forms can be created by using lithographic patterning procedures. In recent years the static and dynamical properties of such systems have been studied intensively [17,18]. One way to model the dynamic properties of heterogeneous magnetic structures is by treating them as localized impurities in a magnetic host medium. The goal of this Letter is to investigate the SW spectrum of excitations associated with a magnetic impurity line (or wire) implanted in a ultra-thin ferromagnetic film, in the framework of an Ising model in a transverse (or in-plane) magnetic field. The line of impurities is assumed here to be oriented perpendicularly to the layers of the film. We show that this ordered array of impurity

R.V. Leite et al. / Physics Letters A 373 (2009) 3678–3683

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Fig. 1. Representation of the interaction scheme for a line of impurities at x = y = 0 in a Ising ferromagnetic film in a transverse field. The impurities (black circles) are exchange coupled with each other and with their nearest neighbors in the host medium.

sites leads to modification of the SW properties that are distinct from those associated with an isolated impurity. 2. Model and Green’s function formalism We consider a finite-thickness ferromagnetic slab or film with a pair of parallel (001) surfaces, with lattice parameter a and a simple cubic structure. The position of any site of the lattice is indicated by the vector

r = (x, y , z) = a(l, m, n),

1 2

z J l,m S lz S m −



l,m

hl S lx ,

(2)

H I = Ho,d + Ho,o .

(3)

Here Ho,d is the term for coupling between an impurity at a given site labelled o and its neighboring host sites:



( Jo − J )

o

 d



S dz S oz − (ho − h) S ox ,

(4)

( J I − J ) S oz S oz  ,

(5)

o,o

where J I is the interaction between two neighboring impurity. The Zeeman contribution in Eq. (4) describes the effect of the transverse magnetic field ho at each impurity, which can assume the value hS at the surface of the medium and h otherwise. To evaluate the SW properties of the system we construct the β Green’s functions  S lα ; S m ω , where α and β are the components of spin operators, and ω is a frequency label. The Fourier amz plitudes  S lz ; S m ω at frequency ω satisfy the usual equation of motion [19]:



l

where S lx,m and S lz,m are the x and z components of the spin operator S, with S = 12 for all sites. The first term in the right-hand side of Eq. (2) contains the contribution due to the exchange interaction. Throughout this Letter we assume that the summation runs over nearest neighbor sites. The second term on the right-hand side of Eq. (2) refers to the effect of the transverse magnetic field in a given site l (where this field is h for host sites in the bulk of the film and h S for the host sites at the surface). The Hamiltonian can be written as H = H0 + H I , where H0 is the Hamiltonian for the “pure” ferromagnetic host medium, whereas H I corresponds to the perturbation due to the impurities:

Ho,d = −



Ho,o = −

(1)

where l, m and n are integers, with −∞ < l, m < +∞ and 1  n  N, respectively. Thus n = 1 and n = N denote the two surface layers, where N is a positive integer. A perpendicular line of localized impurities spins is embedded in the medium at sites with l = m = 0 (see Fig. 1). The localized spins are described by the Ising model in a transverse external field and the Hamiltonian of the system can be written as

H=−

where the d index labels the nearest neighbor host sites. The exchange constant assumes the values J o = J  for the interaction between an impurity and a host site inside the medium, and J o = J S when both sites are at the surface. For the pure material the exchange is J S between neighbors in the surface and J otherwise. Likewise, the term Ho,o in Eq. (3) comes from the coupling between two impurities. It is given by

ω S lz ; S mz



1 

ω = 2π

z S lz , S m



+







z S lz , H ; S m ω.

(6)

In an infinite pure system described by Eq. (2), previous calculations have shown that a second-order phase transition should occur at a critical temperature T C , which is given in mean-field theory [21] by tanh(h/2k B T C ) = h/3 J , where k B denotes Boltzmann’s constant. For T < T C the average spin orientation at each site can have nonzero components in both the x and z directions, whereas for T > T C the orientation is along the x direction. In thin films the behavior can be modified due to the surface spins [2]. Likewise the presence of impurities in the system is expected to change the critical temperature to T Ci , which in some cases may be greater that T C . In order to simplify the calculations, in this work we focus on the high-temperature regime (T > T C ). The corresponding unperturbed Green’s function can first be obtained by solving the above equation with H replaced by H0 and then decoupling the higher-order Green’s functions on the right-hand side using the random phase approximation. This solution in the absence of impurities is well known and is given

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G0 =

by [20]:



 z

S lz ; S m ω 1 

=

M



 G n0,n (q , ω) exp iq · (rl − rm ) ,

(7)



ω − h + 4hR J γ (q ) 2

2

x



hR x J

G n0,n (q , ω) =

−1 2π J

(8)

,

where from mean-field theory, we obtain the spin averages at the bulk and surface sites as R x = 12 tanh(h/2k B T ) and R xS = 1 2

2π J

⎛

To obtain the explicit solutions of Eq. (8), it is convenient to express it in a matrix form

where A( N ) and 1( N ) are N × N matrices and

A

.. .

.. .

=

··· ··· ···

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . .. ⎟ .. .. ⎟ . ⎟ . . ⎟ · · · d −1 0 ⎟ · · · −1 d −1 ⎠ · · · 0 −1 d + 

[ω2 − h2 + 4hR x J γ (q )] hR x J hh S R x R xS

  JS − 4γ (q ) −1 .

(N )

x

(1 − x2p )(x − x−1 )

(16)



x p (xn − x2p −n ) 1 − x2p

(17)

,

δ1,n M 1,1 − δ N ,n M 1, N det M

M

 −1

N ,n

;

(18)

δ1,n + δ N ,n det M

− Mn,n M p −n , p −n + M p −n ,n Mn, p −n ; −δ1,n M N ,1 + δn , N M N , N = ,

G n0,n (ω) =

2p −|n−n |

(13)

where p = N + 1, and x is a complex parameter defined by

x + x−1 = d and |x|  1. The formal solution of Eq. (9) can be written as

(20)

det M

−1



M 1,1 B 1,n − M 1, N B N ,n

2π J



(21)

,

det M

−1 2π J

−



B n,n +

− Mn,1 M N , N + M N ,1 Mn, N det M

M n, N M 1,1 − M 1, N M n,1 det M

(14)

−1



M N ,1 B 1,n + M N , N B N ,n

2π J

B 1,n



B N ,n ,

for 2  n, n  N − 1, and

−x , (1 − x2p )(x − x−1 )

(19)

for 1  n  N − 1;

(12)

x

1 − x2p



G 0N ,n (ω) =

2p −(n+n )

xn − x2p −n

M −1 n,n = δn,n +

G 01,n (ω) =

J

+



M −1 1,n =

(10)

These parameters d and  contain the information regarding the bulk and surface, respectively. Following previous calculations for Heisenberg ferromagnets in semi-infinite [20,22] and film [23] geometries, we write the matrix (N ) A( N ) as a sum of two matrices A0 and ( N ) . This decomposition is useful because the inverse of A( N ) can be deduced using standard mathematical techniques. Denoting B( N ) = [A( N ) ]−1 , we obtain

B n,n =



(11)

,

J

 − x|n−n |

0

M N −1 , N M N ,N

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

where det M = ( M 1,1 )2 − ( M 1, N )2 and 2  n  N − 1 and 1  n  N. Finally, with the aid of Eqs. (18)–(20) and (13), we obtain

ω2 (h S R xS − hR x ) − hh S (hR xS − h S R x )

n+n

+ δ N ,n



In the above we have introduced the definitions:

d=−

1

.. .

M n,n = δn,n + δ1,n

(9)

2π J

.. .

⎞

M 1, N M 2, N

Using Eq. (13) the elements of M can be written using the recurrence relation



 ( N ) 1( N ) [A]( N ) G0 =− ,

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(15)

with 1  n, n  N. The elements of the matrix M−1 can now be expressed as

3. Matrix equation for the Green’s function

(N )

0

M N ,1

tanh(h S /2k B T ), respectively. Also γ (q ) = is a structure factor appropriate to our choice of lattice.

d +  −1 0 ⎜ −1 d −1 ⎜ ⎜ 0 −1 d

M 1,1 M 2,1

⎜ ⎜ ⎜ .. M=⎜ . ⎜ ⎝ M N −1,1

1 [cos(q x a) + cos(q y a)] 2

⎛

(1 + B)−1 B,

where we omit the N indices. The next step is to invert the matrix M = (1 + B), which can be conveniently written in partitioned form as

q

where q = (q x , q y ) is a two-dimensional wavevector and M is the number of sites in any layer parallel to the surface. The vectors rl and rm indicate the positions of two given sites l and m. The Fourier amplitudes G n0,n can be found by solving

−1

(22)



det M

,

(23)

for 2  n  N. Therefore we have obtained explicit expressions for the unperturbed Green’s functions at high temperature (T > T C ) in terms of the exchange parameters, the variable x (which is a function of the frequency w) and the quantity  (which describe the surface properties). 4. Impurity modes The presence of localized impurities breaks the translational invariance of the system in the xy-plane. Consequently, the calculations must be performed in real space. The Green’s functions for the ferromagnetic film with impurities can be constructed from Eq. (6) using the full form of the Hamiltonian H. The new Green function G l,m (ω) obeys the equation

A l, j G l,m (ω) = δl, j −

2π  R lx



P l, j − U l, j − U l, j G l,m (ω).

(24)

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Here

 (ho − h)δl,o ]2 o δl, j , h + [h + o (ho − h)δl,o ]  J l, p R lx δl, j , P l, j = A l, j =

ω2 − [h +

(25) (26)

p

U l, j =

 ( J o − J ) R lx (δl,o δd,m + δl,d δ j ,o ),

(27)

o,d

express the interaction between the impurity site located in o and the neighbors on the host d, while

U l, j =

 ( J I − J ) R lx (δl,o δ j ,o + δl,o δ j ,o )

(28)

o,o

gives the interaction between two impurities sites located in o and o . By rewriting Eq. (24) in matrix form, we obtain the Dyson equation



 −1

˜ 0 (ω) G

 − V(ω) G˜ (ω) = I,

(29)

˜ 0 (ω) and G˜ (ω) denote matrices whose elements are where G (2π / R lx )G l0,m (ω) and (2π / R lx )G l,m (ω), respectively, with G l0,m (ω) given by Eqs. (22) and (23). Here V(ω) is an effective potential depending on impurity term H I , with elements V l, j (ω) =

ω2 − h2 h

δl, j − Al, j − P l, j − U l, j − U l, j .

Fig. 2. Localized SW frequencies as a function of the exchange parameter J S between the impurity at film surface and its neighboring host sites to a ferromagnetic film with: (a) 5 layers, and (b) 10 layers. The exchange parameters between the other impurities and their neighbors on the host as well as between two neighboring impurities were kept constant. Parameters are in the text.

(30)

The Dyson equation relates the matrix Green’s function G(ω) for the transverse Ising model system containing impurities to the corresponding G0 (ω) for the pure system. The spectrum of localized modes is found numerically by calculating the frequencies that satisfy the condition





˜ 0 (ω)V(ω) = 0, det I − G

(31)

which represents the poles of G(ω) for the system containing impurities. Since each impurity spin couples to only four host spins as well as to one or two other impurity spins (depending on its position relative to the surface), the determinant of Eq. (31) reduces to a determinant of dimension 5N. 5. Numerical results In the high-temperature phase  S rz  = 0 at all sites, and the average orientation of the spins is along the x direction. The average value of S x is obtained from mean-field theory where, for systems with spin 1/2,  S rx  = R rx ≡ tanh(hr /2k B T C ), where hr = h S at the sites at the surface, hr = h, for sites in the bulk and hr = h for host sites with localized impurities. For a pure ferromagnet the critical temperature is given by tanh(h/2k B T C ) = h/3 J . Specifically, we consider J /h = 1.0 and a temperature corresponding to k B T /h = 2.5 (since k B T C /h ≈ 0.65). The impurity modes can be classified as resonant modes when the frequencies are found inside the bulk band of the host, and they are called defect modes when they occur outside the bulk band. In all cases, we considered only the defect modes with frequencies below the lower limit of the bulk band obtained for the host, ω/ J ≈ 0.63, which is indicated in the graphs by the dotdashed horizontal line. To analyze the theoretical results we may regard the impurity line as a “wire” introduced into the ferromagnetic material. Thus we can study the behavior of different “wires” inserted in the material by changing the parameters related to the impurity line like J S , J  , J I , and h . For convenience, in all figures, we take h S /h = 1 for host sites at the surface.

Fig. 3. Localized SW frequencies as function of the exchange parameter J  : (a) 5 layers, and (b) 10 layers. The other parameters are described in the text.

First we analyze the dependence of the localized SW frequencies on the impurity surface exchange parameter J S . The other impurities parameters are kept constant with J  = 2.0 J , J S = J , and the impurities are coupled together through a constant exchange interaction J I = 0.6 J , and the field parameters are hS = 0.5h = 0.3h. In particular, we are interested to see how the defect modes associated with the surface behave. For a film with five layers the results are plotted in Fig. 2a, which shows two surface defect modes (effectively one at each surface of the film) going to zero as the impurity surface exchange increases. The other defect modes are effectively constant. In Fig. 2b we consider a ferromagnetic film with ten layers, and we conclude that despite the increase of the number of modes, the overall behavior is analogous to Fig. 2a. The modes are denser, however, characterizing the formation of a band. We now keep a constant ratio J  / J S = 2 and show in Fig. 3a the localized SW frequencies as a function of the exchange parameter J  between an impurity and its host neighbors in a ferromagnetic film with five layers. The other parameters are the same as in Fig. 2. The figure shows the behavior of the modes for different

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R.V. Leite et al. / Physics Letters A 373 (2009) 3678–3683

Fig. 4. Localized SW frequencies as function of the exchange parameter J  for a 10 layer film: (a) J S = 0.5 J , and (b) J S = 1.5 J . The other parameters are described in the text.

Fig. 5. Localized SW frequencies as function of the effective external field h on the impurities in a ferromagnetic film with: (a) 5 layers, and (b) 10 layers.

“wire” materials. As can be seen in the figure, all modes go to zero for large enough J  . However, the bulk-like and surface-like defect modes behave very distinctly. This is even more evident when we consider a ten layer film as shown in Fig. 3b. It is clear that the surface defect modes have a “crossover” (mode mixing and repulsion) with the defect bulk modes. They behave like regular surface and bulk modes in films where the surface modes start below the bulk band and exit above it. This band of defect modes is also very much dependent on the surface parameters of the host film. Thus Fig. 4 shows, for a film with ten layers, how the “band” of defect modes is affected by the parameter J S . In Fig. 4a we used J S = 0.5 J , showing that the effect of the surface defect modes on the bulk defect modes is not very pronounced. However, for J S = 1.5 J , Fig. 4b shows the layer effect due to an impurity line with the surface host parameter large compared with the bulk exchange. In this case, the surface defect modes of the system “cross” the bulk defect band and eventually go to zero at higher value of J / J . Next, we analyze the influence of the effective external applied magnetic field on the SW frequencies for impurity modes. Fig. 5a shows the behavior of the defect modes as a function of h /h for a 5-layer film. We consider variations with respect to the effective field in the surface, keeping the fixed ratio h /hS = 2, and

Fig. 6. Localized SW frequencies as function of the exchange parameter J I between the neighboring impurities in a ferromagnetic film with 20 layers: (a) J S = 0.5 J , and (b) J S = 1.5 J . The exchange parameters between the impurities and their neighbors on the host were kept constant. Parameters in the text.

J S = 1.25 J S and J  = 2.5 J . These parameters are different from those in the previous figure to show more clearly the effect of the field on the defect modes. The graph shows a set of five localized modes with frequencies below the lower limit of the bulk band. The frequency of the modes increase with the field. Again, the surface defect modes have a behavior that is distinct from those related to the bulk. Fig. 5b shows a ferromagnetic film with ten layers. As before, the modes get denser for the thicker film. Finally Fig. 6 shows the behavior of the localized modes as a function of the exchange parameter between the neighboring impurities in a ferromagnetic film with twenty layers. the other parameters used were J s = 0.5 J s and J  = 2.25 J . The figure shows two distinct defect modes related to the surface modes of the host film. It is interesting to note that the defect bulk modes do not all behave similarly. Some of the modes decrease with J I while others increase in a manner that seems to be related to their localization characteristics. When comparing the behavior for two distinct values of J S (as in Figs. 6a and 6b), the results show that only the surface defect modes are significantly affected by this parameter. 6. Conclusions In this Letter we presented calculations for the localized SW spectrum of a linear array of coupled magnetic impurities implanted in a otherwise ideal ferromagnetic film. The results were obtained in the context of the Ising model in a transverse field and show the localized modes as function of the different exchange and field parameters of the system. We employed a Green’s function method and a generalized random-phase approximation (RPA) for the Ising ferromagnet. Numerical results were shown for non-resonant modes that lie outside the bulk SW energy bands. As the number of impurities is equal to the number of layers of the film, the number of defect modes is equal to the number of SW modes of the film. The defect modes even form a band of defect modes with surface-like and bulk-like defect modes. The results show the effects of the different parameters on the localized defect modes. Depending on the value of the surface parameter, there is a pronounced “crossover” between the defect surface modes and the defect bulk modes. That is the main difference from previous results for one and two impurities in ferromagnetic films.

R.V. Leite et al. / Physics Letters A 373 (2009) 3678–3683

Acknowledgements The authors would like to acknowledge the financial support of the Brazilian agencies FUNCAP and CNPq. References [1] A.C. Bland, B. Heinrich, Ultrathin Magnetic Structures I, Springer-Verlag, Berlin, 1994. [2] M.G. Cottam (Ed.), Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices, World Scientific, Singapore, 1994. [3] P. Grunberg, in: M. Cardona, G. Guntherodt (Eds.), Light Scattering in Solids V, Springer, Heidelberg, 1989. [4] T. Wolfram, J. Callaway, Phys. Rev. 130 (1963) 2207. [5] H. Ishii, J. Kanamori, T. Nakamura, Prog. Theor. Phys. 33 (1965) 795. [6] Yu. Izyumov, Proc. Phys. Soc. 87 (1966) 505. [7] N.-N. Chen, M.G. Cottam, Solid State Commun. 76 (1990) 437. [8] N.-N. Chen, M.G. Cottam, Phys. Rev. B 44 (1991) 7466. [9] N.-N. Chen, M.G. Cottam, Phys. Rev. B 45 (1992) 266.

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