Localized states in thin films with internal disturbed sublayer

164

Surface Science 200 (1988) 164-171 North-Holland, Amsterdam

LOCALIZED STATES IN THIN FILMS WITH INTERNAL DISTURBED SUBLAYEP W. MACIEJEWSKI Institute Of Physics, A. Mickiewicz University, Matejki 48/49, 60-769 Pozna/~, Poland Received 10 June 1987; accepted for publication 30 October 1987

A model of two layers sandwiching a thin disturbed sublayer is considered. The behaviour of the localized states and the conditions of their generation are discussed in a geometric approach to the characteristic equation. It is shown that the disturbed sublayer can generate two types of localized states.

1. Introduction

In standard theories of excitations in layered systems, the transient regions near the surfaces and interfaces have been approximated by single atomic planes [1-6]. The assumption of short-range perturbation is, however, not valid in many finite systems of physical interest, because both spontaneous and artificially stimulated interdiffusion may produce inhomogeneous zones which form a significant part of the whole system. Such systems are, e.g., short-wavelength modulated multilayers [7-10], metal-insulator-metal junctions [11-13] or two layers sandwiching a thin sublayer. The latter systems were used very recently to test the coupling between magnetic layers across a nonmagnetic sublayer [14-16] and these investigations stimulated our interest in similar systems. A typical feature of the excitation spectrum of films with layer inhomogeneities is the occurrence of a number of localized states (LS) with e1~ergies outside the bands of bulk excitations. The properties ~ LS have been discussed in the literature mainly for semi-infinite systems with surface-induced inhomogeneity [17-21]. As far as we know, there have been no papers devoted to the theoretical analysis of LS generation by an internal disturbed sublayer. Our aim is to propose an analytical method for studying the conditions under which such states are generated. 2. Model and method

To formulate explicitly the problem of elementary excitations in disturbedlayer systems, we consider standing spin waves in a finite Heisenberg ferro0039-6028/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

IV. Maciejewski / Localized states in thin films with internal disturbed sublayer

165

magnet with simple cubic structure and (001) orientation of the film surface• The two homogeneous layers and the disturbed sublayer contain L1, L 2 and d atomic planes respectively (fig. 1). The atomic planes are labelled by the index I and each plane contains N lattice sites labelled by the vector r. The sublayer is a disordered binary alloy composed of host (h) atoms of the homogeneous layers and impurity (i) atoms• The atom concentrations Ch(I ) and ci(l ) (Ch(/) 4" Ci(/) "- 1) change continuously along the normal to the film• The magnetic properties of our model are described by the following anisotropic Hamiltonian: .~=-2

E

J t , . r , ' ~ , ' ~ ' , ' __ 2

{ l,,l'r" }

)'.

_ ueff6% Kt,.t,,,~Sf, ,, __ /xs ~,__. st, ,-,` at,,

{ lr.l'r" }

(1)

h"

where Y'-t,,,,s~ denotes the summation over nearest neighbours. H~ff consists of the external magnetic field, the inhomogeneous demagnetization field and the surface anisotropy fields. The host and impurity atoms are characterized by the spins S h and Si and by the Land6 factors gh and gi- The exchange interactions are described by the parameters Jhh, Jhi, and Jii for isotropic interactions and Khh, Khi, and K a for anisotropic interactions. To derive the formulae convenient for the analysis of the LS, we use the methods reported in refs. [22-24]. Namely, at first, we apply the zeroth-order Murray approximation within each atomic plane of the disturbed sublayer. As a result, one formally obtains two-dimensional translational symmetry within these planes. Next, we resort to the in-plane Fourier transform Grr of the Green function <<,~t+ I~-;,,)) and use the random phase approximation. Finally, we obtain the equations of motion with the following dynamical matrix: a, 1, 0 1,0,1

1,0,1 1, D1, B1 B 1 , /)2, B2

(2)

Bd_1, Dd, 1

1,0,1

1,0,1 O,l,b

L]+L2+d

166

W. Maciejewski / Localized states in thin films with internal disturbed sublayer

Fig. 1. The structure of a layered system composed of two homogeneous layers (LI and L2 atomic planes) sandwiching a thin disturbed sublayer (d atomic planes), ~ denotes the lattice constant.

The matrix .,~ contains d disturbed rows, which are due to the presence of the disturbed sublayer, and the two surface parameters a and b, which describe effectively the perturbations induced by the surface anisotropy fields. The form of matrix ~ ' is also characteristic of other problems of elementary excitations (e.g. electronic, vibrational) in finite layered systems with internal inhomogeneities. The full evaluation of the excitation spectrum requires the diagonalization of ~ which is possible only by numerical computations. However, the problem of localized states can be discussed analytically with the help of the method of dynamical parameters proposed in ref. [23]. This method consists in the transformation of the eigenvalue problem of the matrix

.~Uk(l ) =ekUk(I),

ek = 2 cos k,

(3)

to the eigenvalue problem of the reduced matrix ¢¢~,r=~, whose dimension is decreased and is given by: a, 1, 0 1,0,1

1,0,1 . . ~ red

(4)

~l(k), ~3(k), 1 1,0,1

1,0,1 0.1, b

LI+L,

IV. Maciejewski / Localized states in thin films with internal disturbed sublayer

167

The reduced matrix contains only two internally disturbed rows with three dynamical, i.e., energy-dependent parameters: -~l(k), ~2(k), ~3(k) defined as follows: d-I

l-ls .@1(k) ..

=

(Sa)

j=1

det(%Sn, - . ~ ' ) '

det(,,Sn, - J ~ " ) "@2(k) = d e t ( % S j / - . ~ " ) '

(5b)

det(%Sn,-.,d' " ' ) "@3(k)= det(%Sn,_,¢~, ) '

(s~)

where the submatrices ~ " , .,d'", ~ ' '" are defined as: D1, B1 BI, D,, BE

Bd_~, D~_,, Bd-~ /92, B2 @

•

O

¢PP

__

*)

Bd-2, D~_~, Bd_~ B d _ l , Dd

I DI,

B1

B 1 D2, B2

Bd-2, Dd-1 --'lhe structure of the matrix ~ ' " ~ hnplies that the parameter ~1~,~)"~ " '- -1----'--" the effective coupling between the homogeneous layers of the system, whereas ~2(k) and ~3(k) describe the effective pinning of the interfacial atomic planes• Thus, we can obtain the following characteristic equation

F,(~,(k), L1, L 2, k)-aF2(~,(k), L,, L 2, k) -bF3(~,(k), L1, L 2, k)+ abF4(-@,(k), L1, L2, k ) = 0 , i = 1 , 2 , 3,

(6)

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It'. Maciejewski / Localized states in thin films with internal disturbed sublayer

where F1--[~2(k)~3(k)-

~ 2 ( k ) ] sin(L 1 - 1)k sin(L 2 - 1)k

--@2 ( k ) sin( L 1 - 1) k sin( L 2 ) k

-.,@3(k) s i n ( L l ) k sin(L 2 - 1)k + sin(L 1) k sin( L 2 ) k,

(7a)

F2 = [~2(k).,@a(k)--@2(k)] s i n ( L 1 - 2)k s i n ( L 2 - 1)k -..q92(k ) sin(L 1 - 2)k sin(L2)k --@3 ( k ) sin( L 1 - 1) k sin( L 2 - 1 ) k + sin( L 1 - 1) k sin( L 2) k,

(7b)

F3= [ ~ 2 ( k ) ~ 3 ( k ) - ~ 2 ( k ) ] s i n ( L 1 - 1)k sin(L 2 - 2)k - ~ 2 ( k ) sin(L 1 - 1)k sin(L 2 - 1)k - ~ 3 ( k ) s i n ( L l ) k sin(L 2 - 2)k + sin( L 1) k sin( L 2 - 1) k,

(7c)

/74 = [~2(k) ~ 3 ( k ) - - @ 2 ( k ) ] s i n ( L 1 - 2)k s i n ( L 2 - 2)k - -,@2( k ) sin( L 1 - 2) k sin( L 2 - 1 ) k

- ~ 3 ( k ) s i n ( L ~ - 1)k sin(L 2 - 2)k + sin( L 1 - 1) k sin( L 2 - 1) k.

(7d)

3. Results The characteristic eq. (6) always has (L 1 + L 2 + d) distinct roots k,,, which correspond to the wave numbers specifying the energies of the excitations. In our model, three types of wave numbers can be distinguished: (i) real k,, (0, w) with energy lying in the band of bulk excitations; (ii) complex k n = itn, t ~ (0, oo) with energy lying below this band; and (iii) complex k,, = ~r- itn, t ~ (0, oo) with energy lying above this band. The complex wave numbers correspond to states with spatially localized amplitudes. It i~ obvious that the dynamical parameters strongly influence the types of roots, k~, of eq. (6) and, in particular, they determine the possibility of the appearance of LS in the spectrum. In the following, we shall deal only with low-energy LS; however, the results obtained are also applicable to high-energy LS. Formulae (5) imply that the dynamical parameters always have d singularities, which correspond to the eigenstates of the isolated disturbed sublayer. We have found [23] that the singularities occurring for the wave numbers k = it can be formally related to LS in the spectrum~ Namely, if the dynamical

W. Maciejewski / lmcalized states in thin films with internal disturbed sublayer

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parameters exhibit S singularities for t = tp (p = 1, 2,..., S), then there are S localized states in the spectrum. The number of the above LS depends only on the thickness and microscopic properties of the disturbed sublayer. Moreover, the energies of these states are well separated from the band of bulk states and their amplitudes are localized within the disturbed sublayer, i.e., they are rapidly damped inside the homogeneous layers. Therefore, it cml be assumed that the influence of these sublayer-localized states on the spectral and thermodynamic properties of the whole layered system will be negligible. Moreover, the characteristic eq. (6) implies that the spectrum can also contain some other LS. The energies of these states are greater than the energies of sublayer-localized states. Their amplitudes are localized at the interfaces between the disturbed sublayer and the homogeneous layers and they are slowly damped inside the layers. We call these states the interfacelocalized states (ILS). It can be expected that the presence of ILS in the spectrum significantly influences many properties of the layered system. Therefore, we shall now consider the condition under which ILS are generated, We propose a geometric approach to the characteristic equation, which is based on the following statements: (i) For k = it the characteristic equation determines isoenergetical surfaces O(t) in the five-dimensional space F5 spanned by the surface parameters a, b and the dynamical parameters -~1, ~2, -~3 treated formally as independent variables. (ii) For the values of the wave number corresponding to ILS, the expressions (5) are parametric equations of some continuous curve A (t) in the space F5. Therefore, a given value of a wave number (t = t') will be a root of the characteristic equation if, and only if, the isoenergetical surface O ( t ' ) intersects the curve A ( t ) strictly at the point t = t'. To specify the conditions under which this situation will take place, we analyse the intersections of the surfaces O(t) with the plane of surface parameters (a, b). As follows from (6), these intersections are rectangular hyperbolae (#(t)) with asymptotes parallel to the a, b axes and given by b = aF aF 4 -

-

(8) F3 "

The above expression indicates that for given values of ~1, -~2, and ~3, the branches of the hyperbola ~(t = 0) divide the ab-plane into three regions of well-defined numbers of branches ~(t) passing through each point on the ab-piane. Moreover, for some critical values of the dynru-nic~ parameters (~1 = ~ r , ~2 = ~ r , -~3 = ~ r ) the shape and position of the hyperbola 0(0) show a peculiar behaviour. Namely, if the dynamical parameters tend to the critical values, then the asymptotes and curvature radii of the hyperbola ~(0) tend to infinity. Simultaneously, the number of branches t~(t) passing through the points of the ab-plane increases or decreases by one. This behaviour indicates that the space F5 i~cludes the i_soene:getical surfaces O(t), which can

170

W. Maciejewski / Localized states in thin films with internal disturbed sublayer

Fig. 2. The regions of existence of interface-localized states (ILS). The numbers in brackets are the numbers of ILS in the spectrum (the Dl-axis is perpendicular to the D2D 3 plane).

intersect the curve A(t). Such intersections correspond to the generation of ILS in the spectrum. Therefore, the key point of our analysis is the determination of the critical values of the dynamical parameters. As follows from (8), the critical values .~{:', D~r and -9~r are determined by the condition F4(k = 0 ) = 0 .

(9)

The above condition is the second-order surface equation, which describes the surface of a double cone in the rectangular coordinate system -91, -92, -93- The dotted conical surfaces in fig. 2 divide the D1~2-93 space into three regions~ which correspond to a well-defined number of ILS in the spectrurn. The presented figure enables us to formulate some conclusions concerning the. influence of the dynamical parameters on the generation of ILS. Namely, for L 1- 1

(lOa)

L 2 -- 1

(10b)

~2(0) > L1_ 2 ' or

-93(0) > L 2 - 2 '

the disturbed sublayer generates one ILS. The localization of this state (i.e. the value of t) increases with increasing ]-91(0)[. Moreover, if the conditions (10) are fulfilled simultaneously, the disturbed sublayer can generate a second ILS, when the value of 1-91(0) 1 is sufficiently small. This implies that the weak value of the effective coupling between the homogeneous layers of the film favours generation of two ILS in the spectrum. In conclusion, one can state that the number of ILS in the spectrum can give us some insight into the details of the interactions in layered systems.

IV. Maciejewski / Localized states in thin films with internal disturbed sublayer

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Acknowledgements The author wishes to thank Dr. A. Duda for critically reading the manuscript and for her useful remarks. This work was sponsored by the project 01.08 of the University of L6d~.

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