bulk modes in complex layer structures

Solid State Communications, Vol. 51, No. 10, pp. 793-796 , 1984. Printed in Great Britain.

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

BULK MODES IN COMPLEX LAYER STRUCTURES

R. Hotz and R. Siems

Theoretische Physik, Universit~it des Saarlandes, Federal Republic of Germany

(Received 14May 1984 by P.H. Dederichs)

Normal modes, dispersions and spectral densities were calculated for macroscopic layer structures with an arbitrary number of layers per periodicity unit. The influence o f the arrangement of layers on the band structure is discussed.

1. INTRODUCTION

THERE IS a growing interest in the elastic vibrations of systems consisting of consecutive layers of different materials with layer widths of macroscopic extent [3, 5]. Such structures are found in "natural" materials (e.g. ferroelectrics with parallel-sided domains, martensitic structures, twinned materials, etc.), as well as in artificial ones (composite materials). In actual systems there is no perfect periodicity in the sequence of layers. To get insight into the consequences of deviations from simple a, b periodicity, we consider in the present paper a periodic arrangement of blocks, each of which contains a larger number of different layers (Fig. 1).

2. BOUNDARY CONDITIONS AND TRANSFER MATRICES

We study a system of layers perpendicular to n = [100] and of infinite extent in t h e y and z directions (Fig. 1). The first object is to determine the normal modes of vibration for frequencies so low that the con- tinuum theory o f elasticity may be applied in every layer.

The equations o f motion for an isotropic homogeneous subvolume (layer) are

uAu + (u + x)v(V.u) = oti.

k and ~ are Lam~'s constants, p is the density and u the displacement field. Solutions with a harmonic time dependence (frequency ~ ) may be expanded in terms of longitudinal (l) and transverse (t) plane waves.

The normal modes o f a system of layers are still characterized by a simple exp ( iK'r) space dependence parallel to the layer (K = [0 k2 k3]), while the dependence on the perpendicular coordinate x l is more compli- cated. We first consider plane waves with a given wave vector component K parallel to the layers. We may then choose a coordinate system such that K = K[0 1 0].

The problem defined by the equations of motion and the boundary conditions between adjacent layers separates into two parts, one for the displacements

"l m

~ 2 ,, n I "° X n

I

Fig. 1. Periodic solid: N layers per period.

[Ul u2 0] in the plane spanned by K and n and the other for the displacements [0 0 u3] normal to this plane. In this paper we shall be concerned only with the more interesting first case. With the present choice of the coordinate system the third components of all vectors are zero and we shall omit them for the time being.

The displacement fields u(r, t) of the ~2, K normal modes are, then, in every layer composed of four plane wave displacements corresponding to l, t waves with

2 2 2 2 wave vectors [+ •oK 0] and with Ko = f2 /co--K , c~ = (2# + k)lp, c~ = ta/p.

For explicit calculations new linear combinations of the four basis functions are introduced. These are com- bined to form a 2 x 4 matrix U. For the nth layer one has

u(f2, K ; r , t ) = e-i~2t+iKyu(g2,K;n,~)B(n), (1)

with

U ( Q , K ; n , ~) =

iC[ iKtS[/K --iKS~/K t iC; 1

--KS[/K t C] C; KtS;/K ]

and t

x = X + ~ , Co = cos(t%~), Sa = sin(Kay),

where X, K o and ~ correspond to the nth layer (cf. Fig. 1). The displacements of the ~2, K normal modes in the

793

794 BULK MODES IN COMPLEX LAYER STRUCTURES

nth layer are thus specified by the 4-component vectors B(n).

The four conditions U i ( t / + 1 ) = ui(n ) and Oli(n + 1) = oli(n) at the boundary between adjacent layers determine B(n + 1 ) as linear function of B(n):

B ( n + 1) = H - I f n + 1)H(n)L(n)B(n) . (2)

L(n) depends on the material constants and the width, H(n) only on the material constants of the nth layer:

0 .

S tKt /K ,! "

loO 1 1

b - - c

0 0

I c! s',~/l~ o - S l /~z Cl 0 L =

0 Ct

0 - - S t K / ~ t

C O = cos(nod), Sa = s in(nod) ,

b = ~22p/K 2 - 2 p , c = 2p.

3. NORMAL MODES OF AN UNBOUNDED SYSTEM WITH N LAYERS PER PERIODICITY INTERVAL

Let there be N layers n = 1 , . . . , N in the periodicity interval of length D. According to Bloch's theorem one has for the displacements o f the normal modes (with a real constant k l ) :

u (k; r + O [100], t) = eik'Du(k; r, t). (3)

In the language of lattice dynamics, equation (3) describes an optical phonon with wave vector k = (kb K), in a system periodic with a (macroscopic) lattice constant D in the x direction and continuous (i.e. with a lattice constant small as compared with the reciprocal of the wavevector) in the y (and z) direction. By iterating equation (2) one sees that equation (3) corresponds to an eigenvalue equation

MB(1) = eik 'DB(1) (4)

with

M = H- ' (1 )M(N)M(N-- 1 ) . . . M(1)H(1),

and

M(n) = H(n)L(n )H- ' (n )

from which four eigenvalues X i are determined. The following statements hold for M: (1) det M = 1,

i.e. XlX2X3X4 = l , (2) X i are real or they occur in complex conjugate pairs, (3) with Xi, also 1/X i is an eigenvalue.

Vol. 51, No. 10

Statement (1 / follows from det L = 1 and H(N + l) = H(1).

Statement (2) is true since L and H are real. Statement (3), finally, is proved by observing that

for every layer L -v = S I S -1, H -1' = VHS-IK2/p~2 2, H' = S H - I V - ~ p ~ 2 / K 2 where a prime indicates the trans- posed matrix and V and S are n-independent orthogonal matrices whose only nonzero elements are: V~3 = V24 =

- V31 = V42 = $12 = --$21 = $34 = -$43 = I. Since thus,

M-l ' ( t / ) = VM(n)V -1,

i.e.

M -~' = {H'(1)VH(1)}M{H'(1)VH(1)}- '

the matrices M -~' and M are similar and, therefore, have the same set of eigenvalues. This proves the third statement.

The set of four eigenvalues Xi of M has, therefore, one of the following forms (with real constants r, s, ¢, ¢):

r e i~, re -i~, ei~/r, e-i~/r; (5a)

r, 1/r, s, I/s; (5b)

r, 1/r, e i~, e-i'¢; (5c)

e i~, e -i~, e i~ , e -iV . (5d)

For an unbounded material only eigenvalues with ] hil = 1, i.e. only e -i~ type solutions are permitted [cf. equation (3)]. Counting only solutions with positive or zero kl one has, thus, no permissible solutions for r 4:1 in equation (5a) and for r 4: 1, s 4:1 in equation (5b), one for r :f 1 in equation (5c), and two for ~0 4: ~ in equation (5d).

We determined the allowed values o f ~ = k i d for given values o f ~2 and K. This corresponds to the cal- culation o f ~s for a given vector k = (kl, K) in the standard lattice theoretical version of this problem.

From these consistent triplets ~2 ~, k b K one may construct dispersion surfaces ~21(kl, K), curves represent- ing the minima ~min(K) and maxima ~max(K) (with respect to k l ) o f the frequency as functions of K, or, finally, densities of state defined, e.g. by the Bowers - Rosenstock relation. Examples are presented and discussed below. Dispersion surfaces meet at those points in g2, k l , K space where two different eigenvectors exist, i.e. where two eigenvalues coalesce.

For every eigenvalue there is at least one eigenvector B(1) and to every B(1) there correspond N - 1 vectors

B (n) = H-l(n)M(n -- 1)M(n -- 2 ) . . . M(1 )H(1)B(1 )

with n = 2 , . . . , N, which determine the displacement field in the whole material.

Next the transition to a fixed coordinate system is

Vol. 51, No. 10 BULK MODES IN COMPLEX LAYER STRUCTURES

performed (in which the wave vector components k 3 are, in general, not zero; the displacements u are, however, still in the planes spanned by n and the respective K). This transformation is necessary if quantities as, e.g., Brillouin intensities are to be evaluated [5, 10].

In the equations formulated above, K is now inter- preted as [0 k2 k3] . The normal mode displacements are then

u J ( k ; r , t) = e-i~2JtTUJ(k; r) = e i(kr-s21t)TUj(k;x)

with

tIJ(k;x) = e -ikl(xn + OU(~2J, K; n, ~)B(n).

The variable x may be expressed by the local coordinate in each of the layers (cf. Fig. 1). The only nonzero ele-

ments of T are Tu = 1, Tz2 = T33 = k2/K and Ta2 = - - T 2 3 = ka/K with K = (k2 + k3) 1/2.

The length of the eigenvector B(1) [and thereby the size of all other B(n)] is chosen in such a way as to nor- malize the displacements in the following way D

dx p(x)UJ(k;x)lfl*(k;x) = D. 0

This guarantees the orthonormality of the normal modes:

(2~ ' ) -3 f dr p(r)uJ(k; r, J'* ' t )u (k ; r , t ) = 6 ( k - - k ' ) ~ jj',

where the integration extends over all space.

4. DISCUSSION OF RESULTS

Due to the elastic isotropy within the layers the constant surfaces in k-space have cylindrical symmetry about the kl-axis, i.e. for any band g2 depends, besides on kl, only on K.

For small differences of the material constants of

795

3

i i / i i i / g>-'/'/I/~/'I///" / 0 1 2 K*

Fig. 2. Dispersion surfaces: u: = f2d/2ct, K* = Kd/2, d = 2, d ' = 2.6,/a'//a = 4, P'/P = 1.5, X' =/~', X =/~.

• Dispersion curves, at K = 0, of transv. and long. modes respectively.

the layers the geometry of the dispersion surfaces can be obtained approximately by projecting the dispersion surfaces of the longitudinal and transverse modes of the average (homogeneous) material onto the first Brillouin zone, i.e. onto the region with Ikll < n/O and K < ~. This construction elucidates the fact that in the actual analysis - where the eigenvalues exp (iklD) of M were determined for given values of ft and K - there are at most 4 different values of kl (two positive and two negative). For given values of f2 and kl, on the other hand, the number of K values increases with ~ .

For a simple two layer periodicity unit, the dispersion surfaces of low frequency modes can be well visualized. As an example the acoustic and the two lowest optical modes are shown in Fig. 2 for markedly different material constants. For very small wave vectors the dispersion of the acoustic modes is that of a homogeneous hexagonal material with averaged elastic

,,,,,,,,,,,,,, 1

0 . 2 . 4 . 8 . 8 K ~ B . 2 . 4 . 6 . 8 K ~ B . 2 . 4 . 6 . 8 ~

Fig. 3. Regions wi th propagating modes in the ~ -K* -p lane for two-materials-systems: Parameters as in Fig. 2; d = 2. (a) N = 2, d' = 2.6; (b) N = 4, d' = 2 and 3.2; (c) N = 20; d' = 1 . 8 . . . 1.8 and 9.8.

796 BULK MODES IN COMPLEX LAYER STRUCTURES Vol. 51, No. 10

constants [ 1 ]. Consecutive surfaces are connected at points marked by full circles.

For K = 0 the modes are either transverse or longitudinal. For the lowest bands modes can be classified according to the connection of their dispersion surfaces with either t- or/-modes at K = 0. For higher modes this is no longer possible since one coherent dispersion surface may end at a combination of transverse and longitudinal dispersion curves on the K = 0 plane. This can be seen in Fig. 2, where the uppermost of the displayed dispersion surfaces ends, for K = 0, at a dispersion line consisting of an l- and a t-part.

There is a marked influence of a variation of the number N of layers per periodicity interval. Figure 3 show the allowed frequency ranges (shaded) as functions of the wave vector component K parallel to the layers. A dark shade indicates regions in which two propagating modes exist. The diagrams are projections of dispersion surfaces onto the K, f2 plane. Displayed are results for systems consisting of different arrangements of two types of material. The width of the layers of one material and the average width of the others is in all three cases the same. The coarse structure of the diagram is determined by the dominant layer width d. The fine structure, however, depends on the number N of layers per period. With increasing N the number of forbidden regions increases and their extensions change.

An example of the explicit x dependence of the normal mode displacement u2 is shown in Fig. 4. For ~2 < ctK, i.e. co < K* (cf. figure caption), modes exist only for special combinations of the parameters (Stoneley waves). For K* < co < K* c't/ct the modes are bound to soft layers; the displacements in the hard layers decrease exponentially with the distance from the layer surface.

The calculations presented here are the basis of a theoretical determination of Brillouin intensities from layered materials. In Fig. 4 only the mode at co = 4.91 is strongly Brillouin active. It turned out, that Brillouin intensities are rather sensitive to the distribution of layer dimensions [10]. Brillouin scattering from N = 2 structures was presented in a previous paper [5].

Acknowledgement - This work was performed within the frame of the Sonderforschungsbereich "Ferro-

W

5 . 3 6

5 . 3 4

4 . 8 1

4 . 7 7

- " - " V v

~ t V v

• V /N

3 . 3 5

3 . 8 B

2 . 5 8 )X

A

Fig. 4. Displacements u~ (x) for normal modes with k l =

0, K* = 2.4 at t = y = z = 0: Model parameters as in Fig. 3(b).

elektrika", which is supported by the Deutsche Forschungsgemeinschaft.

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5. R. Hotz, J.K. Krtiger & R. Siems, Solid State Commun. 46, 2 ,155 (1983).

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bulk modes in complex layer structures

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