Interface magnetic and collective electronic modes in randomly layered metallic structures

Vacuum/volume 41/numbers 4-6/pages 1414 to 1415/1990

0042-207X/9053.00 + .00

Printed in Great Britain

© 1990 Pergamon Press plc

Interface magnetic and collective electronic m o d e s in randomly layered metallic s t r u c t u r e s J Barna~* a n d M Z i r n p e l , Institute of Physics, Technical University, Piotrowo 3, 60-965 Poznan, Poland and P. G r i i n b e r g , Kernforschungsanlage, IFF, 5170 J(Jlich, FRG

Interface magnetic modes and collective electronic excitations (plasmons) in layered structures consisting of alternating magnetic metal and nonmagnetic insulator have been analysed theoretically within the classical approximation. The film thicknesses have been assumed to be distributed randomly. All collective states, magnetic and electronic, are then localized on a finite number of layers, contrary to the corresponding periodic case (superlattice) where they are of extended nature. The localization length and density of states of the magnetostatic modes and plasmons have been calculated numerically.

Excitations in layered systems are a subject of increasing interest. In most of the relevant papers the authors dealt with periodic structures, finite or infinite superlattices. In such a case the excitations are of extended nature. If, however, the structure is random, for instance with random film thicknesses, the excitations localize on a finite number of layers. This localization phenomenon is similar to the quantum-mechanical localization of single-particle electronic states in disordered potential I or to the localization of classical electromagnetic and acoustic waves 2. Here, we analyse plasmons and magnetostatic waves in the Voigt geometry, which may propagate in magnetic m e t a l - n o n magnetic insulator layered structures with randomly distributed film thicknesses. As is known from the corresponding periodic limits3-6 such waves consist of surface modes in each layer. The appropriate wave functions have maxima at the interfaces and decay exponentially inside the films. The considerations are restricted to the classical continuum approximation. Consider a semi-infinite structure ABABAB occupying the half-space x > 0, where the films of the materials A and B in the nth (n = 1, 2, 3. . . . ) elementary bilayer {AB} are of the thicknesses d~") and d(2"), respectively. The dielectric properties of the metallic material (A) are described by the dielectric function

e, (~o) = E~ ( 1 - ~o~/o9 ~)

(1)

where eI is a constant and cop is the hulk plasmon frequency. The dielectric properties of the material B are described by the constant e2. The film thicknesses d~n) and d~") are assumed to be independent random variables distributed uniformly in the intervals (d I - Al, d l + At) and (d2 - A2, d2 + A2), respectively. Here, dl and d2 are the appropriate average film thicknesses. The magnetic films (A) are assumed to be magnetized in the film plane and along the z axis of the co-ordinate system. Their magnetic properties are then described by the following non-

vanishing components of the permeability tensor ~/lxx(O)) = ]./ l yy ((.O) ~ ~1.1.((/)) = 1 + ~ 0 ~ ' ) m / ( ~

-- 09 2)

~lxy(O)) = --~lyx((D) ~- i/~lx(CO) = -iQmco/(Q 2 - o~2)

(2a) (2b)

and #l= = l • Go and Qm are defined here as: Go = y/aoHo and Qm = y ~ M o with H o, M o, /~o and 7 being the external static magnetic field (applied in the film plane and along the z axis), the spontaneous magnetization, magnetic permeability of vacuum and gyromagnetic factor, respectively. For the material B we assume /'/2ij = ~ij' The excitations considered here are described by the magnetostatic and electrostatic equations. In the approximation considered here there is no mixing between plasmons and magnetostatic modes so one can consider them independently. To solve the magnetostatic and electrostatic equations we introduce the magnetic, @re(r), and electric, @e(r), scalar potentials. Inside the metallic layer of the nth bilayer one can write (I)~e)(r) - t'*m(e) r~ (-) + exp[k Ix ("') --

.a_ -* ,i m(e) (,) i

x exp( --Iklx('))]exp(ikll . rll )

(3)

where krl(kll =key) and rll are the two-dimensional in-plane wave and position vectors, respectively, ~(") m(e)+ and A~IeT are constants, whereas x (") = x - E i : ~ ' - l ( d t i ) + d~)) for n > l and x ( ' ) = x for n = I. A similar expression can also be written for the nonmetallic layers (B). Applying now the magnetostatic and electrostatic boundary conditions at the interface inside the nth bilayer and at the interface between the nth and (n + l)th bilayers one can write ["- ~1(n -I- 1) ~ I | ' " m(e) i , ~ (n + I) Lea re(e)

:

T(n., + 1)

~m(e)

, -A](n) eal' r n -t(e) i A (n) Lea re(e)

(4)

where the elements of the 'transfer matrices' --mte)"F(n' n + 1) are

T~i~ T,I~=j = exp( _+ Ikld~~>) (cosh(lkld~">) _ Fr~(o) sinh(Ikld~ °)) (5a) *Also at: Kernforschungsanlage, IFF, 5170 Jfilich, FRG.

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T("" ') = -T- a ~(~) rn(e)" +12121]

exp(-T-]k[d~"))sinh(]kld~"))

(5b)

J Barnag et al: M o d e s of r a n d o m l y layered metallic structures

where the lower sign refers to the indices in square parentheses and

10 3

Fe = el (°9)/2/;2 q" 821281 (o~)

10 2

I

I

i

i

a) J

G ~ = #±(09)/2 - (1 - #2(o9))/2#± (~o) + #x(Og)sgn(k)

0

with s g n ( k ) = +1 for k ><0. Equations (4) and (5) give a recurrence solution to the problem provided the boundary conditions at the free surface are specified. Since we are interested in the localization behaviour of the wave functions, which does not depend on the boundary conditions, we assume here A(X)+ . ~ d ) - ~-- 1. m(e) ~ _ Z*m(e) In the disordered case the solution for A m+[~-) grows exponentially with n for n--* oc. The corresponding localization l e n g t h )-m and 2~ for the magnetostatic and plasmon excitations, measured in the superlattice units d = d, + d2, may be defined as

"Jl-

l/(n

I

~

I

I

""

II

\

b)

r

"

L-, \1i

,'

II Ii II d

2O 30 400

0.25

0.50 to/wp

075

1.v0

Figure 2. Localization length 2 e (a), and the density o f states (b), for

the plasmon excitations at Ikld = I, A I/d I = d,/d2=0.5,

A2/d 2

=

0.1, el/'~2 = 1.3 and

1) ln[(IA~A I

--

,A(n)--x m(e)

I

10 0 rm

!

i

I

10o 10-1

G + = G~- = el (~o)/2e: - ee/2e I (co)

f

I

101

F m =/~ (o9)/2 + (l - #x(CO))/2/l± (09)

1/2re
I

-0) I)f(IA,,-,(oT I+

(l)IAm~)

I)]

(6) I0 s

Numerical calculations have been performed for structures consisting of 105-10 6 elementary units. The localization lengths 2~ and 2o are shown in Figure l(a) and Figure 2(a) vs frequency of the appropriate excitation. In the periodic limits 3 6 there are two bands of plasmons and one band of the magnetostatic waves, which is also visible in Figure 1 and Figure 2. The corresponding density of states of the magnetostatic modes and plasmons has been calculated by the node counting method and is also shown in Figure l(b) and Figure 2(b). All states are localized and the density of states exhibits features typical to the one-dimensional tight-binding model. However, the bands are not symmetric now. The divergences in the D O S at the band-edges, which occur in the periodic structures, vanish now

10 ~

I

i

,

,

,

,

10" 103

102

101 100

I

0

02

I

0'.. x 0.6

o'8

lo

Figure 3. The localization length of plasmons at w/cop = 0.5 ( - - - ) and magnetostatic modes at o / Q o = 2 ( ) vs the strength of disordei" x = A l / d j = A2/d 2. The other parameters are same as in Figures 1 and 2.

I

10 3 10 2 E .<

and the bands become broader due to some tail states in the appropriate gaps. There is an anomalous behaviour of the localization length at the upper band-edge of the magnetostatic modes. This results from the peculiar behaviour of the magnetostatic modes in the corresponding periodic case. The dependence of the localization lengths on the disorder strength is shown in Figure 3 for a given frequency of the magnetostatic and plasmon modes. F o r sufficiently strong disorder the excitations are localized on a few layers.

101

10 o

!0 -1 0

10 o

~

20 ~0

References I

1.7

1.8

~

I

1.9 2.0 ~IQo

I

2.1

2.2

Figure 1. Localization length 2m (a), and density of states DOS (b), of the magnetostatic waves for [kid = 1, f~o = 1.0 c m - ' , f~m = 2.15 cm - l , Ai/dj =A2/d 2 =0.1 and dl/d2 = 1.

i T Ishii, Prog Theor Phys, Suppl, 53, 77 (1973). 2 p Sheng, B White, Z Q Zhang and G Papanicolau, Phys Rev, B34, 4557 (1986). 3 F Bechstedt and R Enderlein, Phys Status Solidi, (b)137, 109 (1986). 4B L Johnson, J T Weiler and R E Camley, Phys Rev, B32, 6544 (1985). 5 R E Camley, T Rahman and D L Mills, Phys Rev, B27, 261 (1983). 6 p Gr/inberg and K Mika, Phys Rev, B27, 2955 (1983).

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