Simultaneous surface Green's-function matching for discrete systems with N interfaces

N

surface science

ELSEVIER

Surface Science369 (1996) 367-378

Simultaneous surface Green' s-function matching for discrete systems with N interfaces L. Fern~mdez-Alvarez ", G. Monsivais b, S. Vlaev °, V.R. Velasco a,. a Instituto de Ciencia de Materiales, CSIC, Cantoblanco, 28049 Madrid, Spain b Instituto de Fisica, UNAM, Apdo. Postal 20-364, 01000 Mdxico, D.F., Mexico o Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria

Received 14 April 1996;acceptedfor publication 27 June 1996

Abstract

The surface Green's-function matching (SGFM) method for discrete systems is extended to the case of arbitrary multilayer structures, so that matching is simultaneously effected at an arbitrary number N of interfaces. In this way all the SGFM formulas are obtained in a compact 2N x 2N supermatrix form, ready for practical calculations. As an illustration, the vibrational properties of multilayer one-dimensional systems are considered. Keywords: Green's function methods; Heterojunctions

1. Introduction

The most c o m m o n types of heterostructures often studied are single heterojunctions, quantum wells and superlattices. In these systems one must solve a matching problem at one or two physically distinct interfaces. The control now achieved in the growth of heterostructures, by means of different techniques, has made possible the existence of more complicated heterostructures of interest, for which it is necessary to match at a larger number N of non-equivalent interfaces. Some examples of those heterostructures are an arbitrary sequence of wells and barriers, a digital quantum well, a polytype superlattice or a multilayer system of T h u e - M o r s e or Fibonacci type. * Corresponding author: Fax: + 34 1 3720623; e-mail: [email protected]

Green's function methods, like surface Green'sfunction matching ( S G F M ) [ 1 - 3 ] and the interface response theory [ 4 - 1 1 ] , are very useful for the study of this kind of system. In surface Green's-function matching ( S G F M ) the extension from one to two interfaces is performed by defining the simultaneous bi-projection on the two-interface domain [ 1 - 3 ] , which is the entire interface domain where matching is effected. An extension of this formalism to structures with N interfaces has been recently presented for continuous systems, where the problem is formulated in terms of differential equations [12]. Discrete systems, on the other hand, are described in terms of matrix algebra and this presents specific features in the way matching is effected, making it mathematically a different case. The purpose of this paper is to carry out the extension of the S G F M method to the N-interface problem in the discrete

0039-6028/96/$15.00 Copyright© 1996Elsevier ScienceB.V. All rights reserved PII S0039-6028 (96) 00896-5

368

L. Ferndndez-Alvarezet aL/Surface Science 369 (1996) 367-378

case. This will be embodied in a compact algebra in terms of 2N x 2N supermatrices which result from the simultaneous projection at all interfaces involved. Section 2 presents the formulation of the problem, giving the general form of the Green's function of the entire heterostructure in terms of the Green's functions of the constituent media. We also show how the simultaneous matching is done, and obtain the matching formula and those giving the local density of states in the different layers of the heterostructure. In Section 3 we demonstrate how the method works in practice with an application to the longitudinal phonons in simple one-dimensional systems. Final comments are presented in Section 4.

2. F o r m u l a t i o n

of the problem

for discrete

bulk Green's functions at the N interfaces in order to obtain the Green's function G~ of the full system in terms of the given Gj; The definition of interface and boundary domains requires precise specification in the case of discrete systems. The projector of the interface domain separating the constituent media ( j - l ) and (j) will be denoted as ij; it consists of r j_l, the right end of medium ( j - 1 ) , and lj, the left end of medium (j) (Fig. 1). The boundary of medium j consists of lj and r j, its right end. The projector of the domain consisting of lj and rj will be denoted J j . The projectors naturally coming into the analysis are the ij for continuous media, where r j_ 1 and lj coalesce into ij, while they are the J 1 for discrete media, as will be seen presently. A technical point must be noted. On the left (Fig. 1) we have a medium Lwith bulk Green's function Gr. and on the right a medium R with bulk Green's function GR, so we have N interfaces and N + 1 physically different media. As in the case of the quantum well I-3,12] it is convenient to define domain 1 formally so that medium 1 is, by definition, medium L / R on the left/right and {71 is GL/GR in L/R. We then have

systems

We display in Fig. 1 the system under study and the relevant notation. Each constituent medium has a bulk Green's function Gj. T h e purpose of our analysis is to match in compact form all these L • 4im •

•

1I

,i

44~uu

i

•

• m)

i

• nn]

: 1i i

.

.

.

.

44~11m.

•

oq

po ~

m,

1

4 41nn•

GL

N

J

2

.11111

Io • i

[ s

bo

od

i

R • Ou4n, J~

!

n

i

Io •

• ob.u~

u

i1

iu

Ioo : :

oo I

Gj

:

|

i

1

1

?o.

•

o •

i

000

i

bo

i

o~

l

•

•

-O0~Ji,

•

•

•

•

i t

i

i

• ell*i,

i

le •

• eli* J*

Io •

• oII,,.ID.

i

1 1i 441" "

• uI

• 4,88 •

•

bo

oc~

l~

r~

g

i

i

~% 12

IlUl

r~ IR

Fig. 1. Sketch of a general heterostructure with N interfaces and N domains. Note that domain 1 is disjoint and consists of Lon the left and R on the right. The layer projectors ... lj,rj ... can be combined either as interface projectors ij or as boundary projectors ~¢j, as explainedin the text.

L. Ferndndez-Alvarez et aL/Surface Science 369 (1996) 367-378

formally N interfaces and N media. We shall indicate by vj + 1 the number of principal layers [3] of each material ( j = 2 , 3. . . . . N) sandwiched in between the medium L/R ( j = 1). A principal layer is defined so that it interacts only with its nearest-neighbor principal layers, and this accounts for all the interactions in the crystal. This may contain one or more atomic layers depending on the range of interactions of the physical model employed. After Fourier transform parallel to the interfaces, which introduces a two-dimensional crystal momentum x, one atomic layer is described with as many basis states as are needed to describe the state of one atom. This determines the size of the diagonal term, a matrix in general, representing the local layer projection of, say, a Hamiltouian or Green's function matrix. If the principal layer contains, for example, two atomic layers, then the size of the matrix representing the principal layer projection is doubled. Henceforth the term layer will be simply used to denote principal layer. We then define the interface i and boundary J domain projectors il=rL+12;

iN=rN+IR;

Jl=rL+IR;

Jj =Ij +r 2

ii = r j _l+ls,

(1)

(j=2,3 ..... N),

and, out of Gj, the boundary projection

=Jjajj.

(2)

Now, the total interface projector can be alternatively viewed as

J = Z b = Y isl

(3)

J

In the continuous case, only the first alternative is relevant and appears naturally in the analysis [ 12]. In the discrete case it is more meaningful to think of the second alternative. Then, in the space of J j, (~3 is formally the 2 x 2 matrix

~ ~ Gj= k A .

(4)

In fact this consists in general of four blocks, as each element can be in general a matrix, so Gj is formally a 2 x 2 supermatrix. With this proviso we shall denote this, for brevity, as a 2 x 2 matrix.

369

Likewise Gs, the ultimate goal of the analysis, is formally a 2N x 2N matrix in the space of J and objects like Gj and its inverse (~j-1, always defined in the small space of ~¢j 1-3] by

jj

;ljj. j j

= jj

=jj,

(s)

or, in an obvious simplified notation, by G i 1 • (~j = J j ,

(6)

must ultimately appear in the final formulas as partial blocks within the large 2N x 2N matrix in the large space of J , at the corresponding locations labelled by j. However, most of the incidental algebra, including matrix inversions, is carried out in the small spaces of the different J2" Again, due to the different topology of domain 1, the form of t~l is ¢

= [

0

] "

(7)

Note that the large matrix containing all the Gj, or their inverses, is block diagonal with the J j as labels, corresponding to the second form of Eq. (3). The large matrix representing, say, an ETB Hamiltonian or a dynamical matrix in discrete lattice dynamics, is also block diagonal, but with the ij as labels, corresponding to the first form of Eq. (3) (dotted lines in Fig. 2, top). Thus the final form of (~, is tridiagonal by blocks, as in the continuous case (Fig. 2, bottom) and the first part of the S G F M analysis is agaih isomorphic for both continuous and discrete media [3]. The differences appear when it comes to effecting the actual matching in order to obtain the formula for t~-1. Now, the dependence on the 2D wavevector x and on the eigenvalue, e.g. E or coz, will be understood throughout. We shall only display explicitly the dependence on the layer position variable n. A symbol like would indicate the value of when n is at the position corresponding to the j - l t h interface (li) and n' at that of the jth interface (rj). We shall indicate by nj that n is in the domain j, so that only elements intervene in the analysis, but of course all elements are ultimately obtained. Again, requires special consideration, depending on

370

L. Ferndndez-Alvarez et aL /Surface Science 369 (1996) 367-378

£,

£2

I

I

r

rL

n t|. . . . .

i,

1

n

i

I

n

itz

~'z

g8

* • • i

0

0

~:

~

r8

'r4

le

0 -°o

~"

£,

£s

i

•

•

t

0

•

0

•

i

"

[

i D O B

- :

-

-

e

•

•

o

,ra D J leo

e

I

Q 0 0 0

~4

(a) i'~t i,

ir

~f

Cz

'/,s

~4

0

0 0

0 0

0

~6

(b) Fig. 2. Top: discrete media. The successive layer projectors rL,12, ..., r: _ t,lj ,rj, ..., lR can be combined as either interface (ij) or boundary (J~) projectors, as explained in the text. The squares represent the nonvanishing elements of blocks of ~i and Gi ~ (--), los (...) and t~ -~ (...--...). Bottom: continuous media. In this case the only meaningful projectors are the interface projectors ij and full line blocks represent non-vanishing elements of all th e t~j, (~)[1 and of G~-~.Thus in both cases G~-I is block-tridiagonal.

whether 1 means L or R. Thus

We then define r o w 2N-supervectors as follows:

(riLl Glln'L) =
( , n j l G j l J 2 ) = [0,0,...,0,(njlGjllj),(njlGjlrj),0 ..... 0].

(9)

= 0 = ,

< ngl Glln'R> = < nRIGRIn'R>.

(8)

This

is, when put in the large format

371

L Ferndndez-Alvarez et aL/Surface Science 369 (1996) 367-378

of 2N-supervectors, but in the space of~¢) it would be only a 2-supervector. Each component is an n x n matrix. Likewise, we define the column 2Nsupervector, ( J j [Gj In)> with the two non-vanishing components (l] [(~j In) > and . As particular cases we have again

This equation is a part of the 2N x 2N supermatrix Gs, but Eq. (12) can also be viewed as an equality between 2 x 2 supermatrices, and in their own space we can carry out the desired matrix algebra, including matrix inversion as discussed above. In this reduced space we have

(13)

= = [,O..... 0,0], (10) = = [0,0 ..... O,], and the corresponding ones for the column 2Nsupervectors. Then the form of the general elements of G~ is given by

Although G~ appears here, the fact of having (~f 1 on the left and on the right automatically carries the corresponding projector ,¢j on both sides, so the Gs appearing in a formula like Eq. (13) is, by definition, the LHS of Eq. (12). Therefore

(njlG, ln))=+(njlGj]Jj>'(Tf ~

= + <3C~lGln)>,

. (11)

Projecting the second of Eq. (11) we obtain

= <~CklGkln'k>,

=

as in the usual S G F M analysis [ 12]. The meaning of the reflection, It(j), and transmission T(j,k), matrices are the usual ones in the S G F M method and we must stress that by definition, these have the nature of T-matrices in the sense of scattering theory. This is why the propagators after interaction with the interfaces are the known "unperturbed" bulk propagators, so the problem is reduced to finding the reflection and transmission matrices. It is clear by inspection of Eq. (10) that the four non-vanishing elements of It(j) in the large 2 N x 2 N supermatrix format are those labelled (/j,/j), (lj,rj), (rj,lj) and (rj,rj), while those of T(j,k) are (lj,l~), (l~,r~), (rj,l~) and (rj,r~). The case j or k = 1 again has a special form. Firstly, consists only of the two non-vanishing disjoint elements (rL,r~) and (l~,lR). Secondly, ( J l l 7"(1,k)[J~> consists of only two non-vanishing disjoint pairs of elements, one formed by those with labels (rL,/~), (rr,r~) in row 1 and another one formed by those with labels (IR,l~), (l~,r~). Likewise T(j,1) consists of two non-vanishing disjoint pairs of elements, one constituted by elements (l~,rL), (rj,rL) and the other by elements (/j,/~), (rj,l~). We can project the first of Eq. (11) and obtain

,

16"jIJj > + <,-,¢jIGj [Jj > <~jl~j)lJj><~jl(TjlJj>.

(14)

= <~j

(]2)

(15)

which is again an equality between 2 x 2 supermatrices, from which we obtain

T(j,k) = ( J j IGj--*[Jj ) ( J j I6~,lYk>, (16) whence

=G]-~G; ~ (17) <-¢~lGkln~>. Eq. (14) and Eq. (17) constitute the formal exten• sion to the N-interface case of the standard SGFM results for N = 2 , as do Eq.(13) and Eq.(16) [3,12]. If we can calculate ~,, then Eq. (14) and Eq. (17) yield all desired elements for n and n' anywhere. Now, every G is a resolvent of a matrix which we shall formally denote as the Hamiltonlan matrix H. As indicated above, this could in practice be a discrete lattice dynamical matrix, for instance. This allows for a concise unified treatment of different physical problems, although some care must be exercised when applied to each particular case, in particular to phonons [3]. Then G~ is the inverse of (t2-//~) of the entire system under study, t2 being the eigenvalue matrix, for instance E1 or

L. FernSndez-Alvarez et al./Surface Science 369 (1996) 367-378

372

co2I. Consider the projectors Pj of the different constituent domains of the matched system (Fig. 1). Then the unit of the entire space of the total system is N

I=PI+

~_, Pj,

(18)

j=2

and we can write the definition of (;~ as

(~-n~)-i.a~=(a-Hs)-

el+ Y~ ej

.a~(19)

i=2

contributions not from the geometrical interface planes, but from the different atomic layers forming the different interfaces, in the framework of the physical model employed. Eq. (24) can be applied to perform real calcula' tions by using the transfer matrix algorithms in the same way as for the typical supeflattices [ 13,14 ] and quantum wells [ 15,16 ]. Then Eq. (24) reduces to N

~;1=aJ-~-~-

Y, bj,

(25)

j=2

N

=PI+ 2 ej,

where

i=2

/~i= [HL(02,1) 0 ].[/~L 0],

whence the projection onto J :

H~(1,2)

(26,

r~

1=2

Project Eq. (14) onto J j from the right, project Eq. (17) onto J 1 from the right and add up. We then obtain

PjG~J=PjGjJ;j .~j]-l.(;~j;.

(21)

Proceeding in a similar way for the medium 1 (L/R) we obtain

elG~J = P1GlJl "(7~ 1. (7~J.

(22)

As a consequence

PjGs.(7~-I=PjGjJj .(7[ 1,

(23)

elG~. (TUI= P1G1J1 • (7~ 1. By substituting Eq.(23), the identity (Eq.(20) becomes the matching formula

~ 1 =llo¢-- JH~el "elG, J l "(;~ ~

LI'

P1 =

zJ= TIJ lj/' where vj + 1 is the number of layers in the constituent slab j for j > 1. In order to obtain the density of states projected in the different layers of the system we can employ the following expressions

Gs(nj ,ni ) = Gj + [ T , f l,Tf~fl+l -'J] . l~j

(24)

sT, -1 ]

N

- ~ J n j ' j . e j c j j .~;~, j=2

which is a generalization of the matching formula for the case of two non-equivalent interfaces [2,3] and is obviously quite different from the equivalent extension for the case of continuous systems [12]. This formula differs from the analogous one for continuous media in the obvious absence of differential operators, which are substituted by the interface projectors o¢ and the projectors Pj of the different component media. In this way we have

~j = , ; 1 . ( j j ~ j j __Oj).G~,

o] o,, 6j =6j(m,m), ~J=

J lid'

(27)

L. Ferndndez-Alvarezet al./Surface Science 369 (1996) 367-378

/)~, vanish and these supermatrices are blockdiagonal. However, in the superperiodic structure of Fig. 3 the interfaces iN and i,, are physically identical, as are i, and i 1. Then the amplitudes at iN/i, are equal to those at i,n/i 1 except for the phase factor f = exp (iqd), where q is the superwavevector associated with the superperiod d. Then domains L and R appear as mathematically linked with the appropriate phase factor and the off-diagonal elements of the supermatrices G1 a n d / ) 1 no longer vanish. In fact they become [2,3]

Gs('L",)=6L+ Er L- ,0]

G.(nR,nR)=GR+ [O,T~-~] "(J~GJI-~O

O1=

o]

6~

--1

373

'

~1 = ~ f-X(rLlGl[Im> 1 '

l_f

(28)

I '

GL = GL(nL,nL), GR = GR(nR, nR).

0

/ ) 1 = [/-/1(02'1)

The polytype superlattice sketched in Fig. 3 can be equally studied by combining this analysis with the introduction of Bloch periodicity as in the binary superlattice [2,3,12]. When the extremes of the structure are open so L and R are semiinfinite, there are no cross terms between L a n d R. The off-diagonal elements, or blocks, of G~ and

Pl =

1-1

V)i

•

•

,

*

"

•

•

•

,YV

yv,m

• •

nZo •

1 l

)

I ,

, ~

r,~

•

• mmlo •

• vlnn •

• nnnio•

•

• •

•

1 1I

• vl,. •

•

oimiee*

-

• ..Io • u I I I I

i

y T ! I I "

•

.lOgo

a

, 1u

win

I

In

Om

•

• =Ioo

!

•

N

u

I

vim

•

'

I

vvl= •

•

i,!

/'1

/1 J

2 •

/-/1(1,2)1 "Pl"r~-l'

•

•

•

•

vv

•

•

.vvlmn

*

•

1

-nnmooo

•

•

•

•

1 1I •

• m[oo

• ol n

,v •

• vl= •

• =I. •

Ivv

vv~m x

= .~. •

IV •

•

•

1I • -I. •

• =io •

• .1

1 1I

u

,i I

•

rum

•

•

mno •

i

•

o i

VIII

•

WIlD •

Fig. 3. Sketch of a polytype superlattice. The period consists of the layers spanned by media 1-2-...-N. The interfaces l,,,rL are physically identical to lg,r,, respectively.Note that now the Land R domains are physicallyidentical.

L. Ferndndez-Alvarez et aL/Surface Science 369 (1996) 367-378

374

In the same way we obtain:

Gs(nl,nO=Gl + I ~ l - l , f f lT~l +x'"lll .la1

(29)

1 - 2 - 3 - 1 , which has three non-equivalent interfaces. This suffices to demonstrate the practical use of the formalism and to see how it can be extended to larger values of N. In the present case, Eq. (25) reduces to: ~21 = i 2 j _ i ~ r s _ b l _ / ~ 2 _ b3 '

Lf~l +l-,~A ,

where

#1 =~i -1" ( a s - g 0 " a;~,

These are the only differences with respect to the expressions obtained above.

"Q1

0

0

0

0

0

0

t9 2

0

0

0

0

0

0

Q2

0

0

0

0

0

0

Ds

0

0

0

0

0

0

Q3

0

0

0

0

0

0

Q1.

As a practical demonstration we shall apply the method presented here to the study of longitudinal phonons in systems formed by linear atomic chains and involving three or more interfaces. The model is not in itself realistic, but it is chosen because with it one can follow through the calculation in an analytic form and see how in practice one can use the method and what information can be obtained with it. In order to see analytically how the formalism works, we shall consider longitudinal phonons in linear atomic chains with only nearest-neighbor interactions [17]. For an infinite linear chain with lattice constant a and force constant K the Green's function elements are [17]

G(n,n') =

-4nK

f'~/" exp[i(n--n')qa] j -~/, cos (qa)- 4 + ie dq

1

t In-n'l+l

K

t2--1

where ~ = 1 -

Qj = Mj 0)2

"KI+~

I t= {4+i(1--42) m,

0

0

0

0

--~

K2+~

0

0

0

0

0

0

Kz+fl

--fl

0

0

0

--fl

K 3 + fl

0

0

0

0

0

Ks+~,

0

0

0

0

--7

-Katl 0 0 0 0 0

000

bl=

(30)

'

"0

0 0 0 -K~tll

000 000 000 0

0

m

0

d~2(2) 0 0 0

0 d~:(2) d~(2) 0 0 0 (31)

/

[ ~ + (~2_ 1)1/2,

000

0

~ < - 1.

We shall now consider a structure of the type

JD2 =

-7

°°iloO

000

0 dH(2)

1 < 4,

--1<4<1,

( j = 1, 2, 3),

-a

Mco2/2K and

~ -- (4 2 - 1)1/2,

(33)

&gJ =

3. Examples: one-dimensional atomic chains

a

(32)

0

0

0

0

0

0

0

0

0

0

0

0

.0

0

0

0

0

O_

K~ +7.1

L. Fern(mdez-Alvarez et aL/Surface Science 369 (1996) 367-378

375

6-

6-

A

~4-

O3

0 a2-

0 o_J2 -

d

0

0

11o

0.0

I

o15

0.0

1.0

e/e(Al)

¢~/~(AI) 0.6-

0.6-

.-~ 0.4-

.•_0.4-

tO 0 a 0.2..,.1

0

C~ -J 0.2-

0.0

i 015 OJ/~(AI)

0.0

0.0 1'.0

i 1.0

0.5

0.0

o/~(AI)

Fig. 4. Local density of states (LDOS) for the Ag-(A1)2-(Au)2-Ag system projected at the A1 and Au layers in arbitrary units versus frequency normalized to the A1 maximum frequency. Top row: A1 layers. The diagram on the left-hand side of the figure corresponds to the first A1 layer proceeding from left to right in the structure, and that on the right-hand side to the second A1 layer. Bottom row: same as above for the two Au layers. An imaginary part of 10 -3 has been added to the normalized frequency to broaden the peaks in the density of states.

"0 0 0

0

0

0"

0 0 0

0

0

0

0 0 0

0

0

0

0 0 0 dlt(3) d12(3 ) 0 0 0 0 d12(3) d11(3) 0 0 dl~(j) =

0 0

0

Kj

Kj d12(J) =

tj(1 --t2~ 0

0

B~ B'

0

0

0

0

B'

Ba

fl

0

0

0

0

fl

Ca

C'

0

0

0

0

C'

C~

y

0

0

0

0

~

A~_

0_

2 2(v . - - 1 ) t~(1--tj ~ ),

v. 2 tj~(1--tj),

0~,t , and y are the interactions at the interfaces between media 1/2, 2/3 and 3/1, respectively. They will be obtained as weighted averages of the bulk interactions of the media forming the different interfaces. In the full space, Eq. (32) has the form "A~ 0 0 0 0-

(j=2,3).

(~-1=

(34)

L. Ferndndez-Alvarezet al./Surface Science 369 (1996) 367-378

376

6-

6-

.•_4-

t-

..d O9 O t32' .-1

0

t'~ 2 . _.1

0 0.0

II 0.5 ~/¢~(AI)

110

o1

00.0

0t

0.8

" ~ 0.8

vlL "~ 0.6

~

,Io

0.5 ~/~{AI)

1

0.4

~° 4/

0.2

0.2

0.4

I

0.0 0.0

0.5

11o

1

0.0

0.0

0.5

1.0

o~/~(AI)

0J/oJ(AI)

Fig. 5. As Fig. 4 for the Au-(A1)2-(Ag)2-Ausystem. Top row: AI layers. Bottom row: Ag layers. An imaginary part of 10-3 has been added to the normalized frequency to broaden the peaks in the density of states. where A~ = I21 - (K1 + c~)+ K i t l ,

(35)

A~ = Q 1 - ( K 1 + 7) + Klti,

B~= f22--(K2 + ct)--dli( 2 ), Ba = O 2 - (K2 + fl)-- dll(2), B' = -- d12(2), C ' = -- di2(3),

C ~ = f 2 3 - ( K a + f l ) - d l i ( 3 ), Cr =~2 a - - ( K 3 + y ) - - dll(3). F r o m here it is very easy to see how the expressions can be generalized, with consequent increase in size of matrix ¢~-i.

We shall consider here the longitudinal phonons of this kind of structure by specializing our model to the metals A1, Ag and Au. These metals have a very good lattice-parameter matching (within 0.3%) and they can be grown forming good quality interfaces [ 18]. The force constants for bulk materials (KA1=4.416x 10 4 dyn cm -1, KAg=5.032x 104 dyn cm -1, KAu=7.854 x 104 dyn cm - i ) can be calculated from their elastic constants. For the interactions at the interfaces (~, r , 7) we shall employ the arithmetic mean of the corresponding bulk force constants. We shall illustrate here the cases Ag-(A1)2-(Au)2-Ag and Au-(A1)2-(Ag)2-Au. We have considered only two atomic layers for the elements forming the sandwich structure in order

377

L. Ferndmdez-Alvarez et aL/Surface Science 369 (1996) 367-378 0.5-

0.5-

0.4-

0.4-

e--

"~ 0.3-

0.3-

~2 O3 O.2-

cO O.2-

0 D _.J

0a

.._1

0.1-

0.1

0.0

i

0.0

0.5

1.0

~/(o(AI)

0.0

i

0.0

0.5

1.0

~/~(AI)

Fig. 6. L D O S projected at the two first Au layers at the right extreme for the Au-(A1)2-(Ag)2-Au system versus the frequency normalized to the A1 m a x i m u m frequency. An imaginary part of 10 -3 has been added to the normalized frequency to broaden the peaks in the density of states.

to reduce the number of local modes, thus enabling us to obtain simple pictures. Through inversion of Eq. (34) we obtain Gs, and from the imaginary part of its trace we obtain the local density of states (LDOS) of the system projected at the different atomic layers forming the interfaces. The positions of the peaks in the density of states give the eigenvalues. This is a more efficient numerical method than trying to find the zeros of the secular determinant of (~-1, which is usually a complex function of a non-polynomial type. Fig. 4 shows the L D O S projected at the A1 and Au layers for the Ag-A12-Auz-Ag system. It is very easy to see the peaks corresponding to the local modes of the A1 in the A1 layers, and their quick decay into the Au layers, due to the fact that the maximum frequency of Au is ~0.5 times the maximum frequency of A1. It is then clear that in this case we have only localization in the A1 layers. Fig. 5 shows the L D O S projected at the A1 and Ag layers for the system Au-A12-Ag2-Au. We see the same peaks corresponding to the local modes of the A1 in the A1 layers and their decay into the Ag layers, although the decay is not as strong as in the previous case for the Au layers. We see also

some peaks just beyond the maximum frequency of Au which corresponds to a local mode of the Ag, with frequency just beyond the maximum frequency of the Au. It can be seen that the penetration of these modes into the A1 layers is almost negligible, while they exhibit a reasonable penetration into the first Au layers at the right interface, as can be seen in Fig. 6. Thus in this way we can obtain information not only on the eigenmodes of the system, but also on the spectral properties and on the spatial distribution of the spectral strength of the different modes. If desired, the amplitudes can also be obtained, as in the general S G F M analysis I-3]. Thus, this type of structure shows a similar type of localised modes as one finds in a simple sandwich, but the situation is more complex on account of the greater complexity of the structure. These solutions can be obtained with great ease from the simultaneous S G F M formalism. A more realistic calculation for a 3D system described in terms of more complicated dynamical matrices would of course require a heavier numerical computation, but the general structure of the algebra involved is as straightforward as in the simple example studied here as an illustration.

378

L. Ferndndez-Alvarez et al./Surface Science 369 (1996) 367-378

4. Final comments

References

We have p r e s e n t e d here a n extension of the S G F M m e t h o d for discrete m e d i a with N - c o u p l e d n o n - e q u i v a l e n t interfaces. I t p r o v i d e s a concise a n d c o m p a c t a l g e b r a which effects the s i m u l t a n e o u s m a t c h i n g at all interfaces i n v o l v e d a n d allows for a n a l g o r i t h m with w h i c h p r a c t i c a l c a l c u l a t i o n s c a n be d o n e in simple m a t r i x form. T h e s u p e r m a t r i x ~ - a has a b l o c k - t r i d i a g o n a l structure which reflects the p h y s i c a l fact t h a t each slab is in c o n t a c t o n l y with its i m m e d i a t e neighbors, a n d is also a d v a n t a g e o u s w h e n it c o m e s to m a t r i x algebra, a n d in p a r t i c u l a r , i n v e r s i o n [ 19]. T h e m e t h o d c a n be r e a d i l y used in practice to s t u d y r a t h e r c o m p l i c a t e d systems of real p h y s i c a l interest such as t h o s e m e n t i o n e d in Section 1, a n d o t h e r s such as the m o d i f i e d m u l t i p l e q u a n t u m well often used for laser a c t i o n [ 2 0 - 2 2 ] , in which a sizable n u m b e r of interfaces can be involved. W o r k o n s o m e of these systems is c u r r e n t l y g o i n g o n in our laboratories.

[1] F. Garcia-Moliner and V.R. Yelasco, Surf. Sci. 175 (1986) 9. [2] F. Garcia-Moliner and V.R. Velasco, Phys. Scr. 34 (1986) 252. [3] F. Garcia-Moliner and V.R. Velasco, Theory of Single and Multiple Interfaces (World Scientific, Singapore, 1992). [4] L. Dobrzynski, Surf. Sci. 175 (1986) 1. [5] L. Dobrzynski, Surf. Sci. Rep. 6 (1986) 119. [6] L. Dobrzynski, Surf. Sci. 180 (1987) 489. [7] L. Dobrzynski, V.R. Velasco and F. Garcia-Moliner, Phys. Rev. B 35 (1987) 5872. [8] P. Masri and L. Dobrzynski, Surf. Sci. 198 (1988) 285. [9] A. Rodriguez, A. Noguer~t,T. Szwacka, J. Mendialdua and L. Dobrzynski, Phys. Rev. B 39 (1989) 12568. [10] B. Sylla, M. More and L. Dobrzynski, Surf. Sci. 213 (1989) 588. [11] A. Akjouj, E.H. E1 Boudouti, B. Sylla, B. Djafari-Rouhani and L. Dobrzynski, Solid State Commun. 97 (1996) 611. [12] R. Prrez-Alvarez, F. Garcia-Moliner and V.R. Velasco, J. Phys. Condens. Matter 7 (1995) 2037. [13] R.A. Brito-Orta, V.R. Velasco and F. Garda-Moliner, Phys. Rev. B 38 (1988) 9631. [14] M.C. Mufioz, V.R. Velasco and F. Garcia-Moliner, Phys. Rev. B 39 (1989) 1786. [ 15] D.A. Contreras-Solorio, V.R. Velasco and F. Garcia-Molinet, Phys. Rev. B 48 (1993) 12319. [16] L. Fern~ndez-Alvarez and V.R. Velasco, Phys. Scr. 52 (1995) 338. [17] B. Djafari-Rouhani, P. Masri and L. Dobrzynski, Phys. Rev. B 15 (1977) 5690. [18] T. Miller, A. Samsavar, G.E. Franklin and T.C. Chiang, Phys. Rev. Lett. 61 (1988) 1404. [19] S. Vlaev, V.R. Velasco and F. Garcia-Moliner, Phys. Rev. B 49 (1994) 11222. [20] M. Sundaram, A. Wixforth, R.S. Geels, A.C. Gossard and J.H. English, J. Vac. Sci. Technol. B 9 (1991) 1524. [21] A.C. Gossard, R.C. Miller and W. Wiegmann, Surf. SCI. 174 (1986) 131. [22] D.L. Mathine, G.M. Maracas, D.S. Gerber, R. Droopad, R.J. Graham and M.R. McCartney, J. Appl. Phys. 75 (1994) 4551.

Acknowledgements W e are grateful to P r o f e s s o r F. G a r c i a - M o l i n e r for his helpful advice a n d the critical r e a d i n g of the m a n u s c r i p t . This w o r k was p a r t i a l l y s u p p o r t e d b y the S p a n i s h D G I C Y T t h r o u g h G r a n t N o PB93-1251. G.M. thanks the European C o m m u n i t y for s u p p o r t t h r o u g h C o n t r a c t C I 1 " CT93-0225. L.F.-A. t h a n k s the S p a n i s h M i n i s t r y of E d u c a t i o n a n d Science for s u p p o r t . S.V. t h a n k s N A T O for s u p p o r t .