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Interface response theory of N-layered continuous superlattices
362
Surface
Science 182 (1987) 3622374 North-Holland. Amsterdam
INTERFACE RESPONSE THEORY OF N-LAYERED CONTINUOUS SUPERLATTICES L. DOBRZYNSKI Equipe Internutmude de D~vncrmrque des Interfaces, Luhorutorre de D_vnumrque de.? Cristoux Mol&ulcrwes ((/A No. 801) ussocih au Centre Natmnal de lu Recherche Scientifique, UFR de Phy.slque, Uniuersiti des Sciences et Techniques de Lille Flmdres - Artoa, F-59655 Vdlenewe d;4scq Cedex, Frunce and Imtituto de Fisico del Estudo Sblldo, CSIC, Serrano 123. .?a006 Mudrrd, Spun Received
17 July 1986: accepted
for publication
21 October
1986
N-layered continuous superlattices are formed out of a periodic repetition of N different slabs studied within a continuous space. A recent interface response theory is applied for the first time to these systems. General expressions for the response functions associated with a given operator are given. From them, all the physical properties of such materials can be derived, as for example the densities of states. The response functions for such semi-infinite materials, free and in contact with another medium are also given. This general theory applies as well to free electrons, elastic vibrations, long wavelength magnons, polaritons, etc.
1. Introduction The study of fundamental properties of superlattices formed out of two alternating different slabs attracted in the last decade a great deal of interest [l]. One aspect of these investigations is the determination of the band structure for different types of excitations: electronic, vibrational, magnetic, electrodynamic, etc. Many studies of two-layered superlattices use continuum models, as they are usually more simple than the atomic ones and give useful information in their domains of validity. N-layered superlattices will provide in the near future new artificial materials. It is in fact a logical development, from the point of view of the present standard technics used for the growth of the two-layered superlattice. The theoretical study of such new materials is also an exciting challenge, which was first faced for discrete models [2] with the help of the interface response theory [3]. This theory [4] enables one also to study the response function of any composite material in a d-dimensional continuous space Dcd) once the bulk response functions of all i (1 < i < N) subsystems domain Dcd). The application of this general theory superlattices is explicitly given here. 0039-6028/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
are known inside their to N-layered continuous
B.V.
L. Dohrrynski
/ Interface response theory of N-layered continuous superluttices
363
These N-layered superlattices are formed out of an infinite repetition of N different slabs labelled by the unit cell index n. Each slab of width 2ai is labelled by the index i (1 < i < N) within the unit cell n. All the interfaces are taken to be parallel to the (Xi, X,) plane. Taking advantage of this infinitesimal translational invariance in directions parallel to the interfaces, we Fourier analyze all operators according to, for example
where k,, is the propagation vector and X,, is the real space position vector, both parallel to the interfaces of the slabs. Bearing in mind that all operators which will appear throughout this text are functions of k,, , we will, for the sake of simplicity, not write any longer this dependence explicitly. The other remarkable consequence of this translational invariance is that we are left with a one-dimensional problem, as a function of X3. Rather than using X3 ranging from - cc to + co, we find it simpler in what follows to replace X3 by the three variables (n, i, z). We define within each slab (n, i) a reduced space coordinate z denoting the space position X,/a,; the origin being choosen in the middle of each slab. So -l
(2)
within the slab (n, i). Let us now outline, following ref. [4], the successive steps for the calculation of the response function of such a material. The knowledge of this response function enables one to study directly any physical property of these materials.
2. Bulk reference
response
function
(3ven a continuous (as function of X,) bulk operator H,,( X3), one defines the corresponding response function G,,( X3, X;) by H,,(X,)G,i(X,,X;)=“(X,-X,‘),
(3)
for - cc < X3 < + co. Remember that both operators are functions of k ,, and are, as well as the unit matrix I, in general (p x p) matrices. Define now a reference response function G for the N-layered superlattice by G( n, i, z; n’, i’, z’) = 6,,,J,,,G,,
(z,
z’),
(4)
where z and z’ are defined by eq. (2) and S,,,, and S,,, are the usual Kronecker symbols.
3. Surface
response
function
for one slab
Out of the infinite medium i, cut a free surface slab, such that -h, 6 X, < h,. The operator h,;( X,) corresponding to H,,( X,), may be written as s(z-z’)h,,(z)=S(z-z’)H~,(z)+[-S(z-1)+6(2+1)]V,,(z), where - 1 < z,z’ < 1 (eq. (2)) and L&(z) is the cleavage operator Calculate then the surface response operator for this slab
-
Vo,(z”)Gor(z”,z’)I;“=z=l,
A,,(z,z’) = And
finally
[4].
-l
(5)
+~~,(z”)G~,(z”,z’)I~,,=~=~~,~=+~.
the surface
response
function
corresponding
to h,,(z)
is given by
[41
gsrb’) = Go,(ZJ’) - [Goi(Z,‘)
1
A&z') Go,(z,l)I ALi
3
A(l,zf)
[
(6)
where - 1 < z, z’ 6 1; the second term is the product of a (p X 2~) rectangular matrix by a (2~ x 2~) and a (2~ X p) one and the (2~ X 2~) matrix A;: is the inverse of A sr2
I+ A,, (1J)
(7)
= A,,(Li)
Note also a useful particular g,‘(M,M,)
value [3] of eq. (6)
=A,,lG,?(M,M,),
where &;‘(M,M,)
gs,(M,M,>= Go,(M,M,>=
_ss,w g,,m I and G,;’
9,,03
Goi [ G&i) --
(M,M,)
(8) are the inverses
elements
Go,(iJ)
(10)
Go,04
of the N-layered
matrices
(9)
g,,04
Slab response functions were calculated ity theory [5-71 with other methods.
4. Interface
of the following
before, in particular
superlattice
response
within
elastic-
function
The interface space M of the N-layered superlattice is the infinite set of the discrete points (n,i, zM), where the integers take the following values: - cc < n
L. Dobrzynski
/ Inierfuce response theory of N-layered continuous superlaitices
365
< +co, l
+ Nn.
(11)
Remark that with these notations there are two different manners to label one and the same interface
(m,i>-(m - 1~).
(12)
The inverse g-‘(MM) within this discrete interface space M of the response function g(MM) of the N-layered superlattice is a block tridiagonal matrix whose elements are [4] 9-%,i;
d,i)
= ~,,~[9,-,‘(U)
+ 9;;_1(1,1)],
(13a)
9-‘(m
; m’J)
= S,,&7,-,l(Ll>
+ s,-r;+l(l,l)],
(13b)
g-l(m,i;m~,i)=8m,r~,-,'(i,i),
(13c)
g-l(m,i;
(13d)
d,i)=s,,,g,-,'(i,i).
Note also that the off-diagonal blocks of elements of this tridiagonal matrix gP ‘(MM) have the following property
gpQ&
m,i)= [g-l(m,i; m/j)]+,
(14)
which follows from the usual properties [5] of the surface response function -1
9 S”,
.
4.1. General methods for the calculation
of g(MA4)
The inversion of the block tridiagonal matrix gP’(MM) can be done with mathematical methods similar to those used before [2] in the discrete interface response theory of N-layered superlattices. As g-‘(MM)g(MM)
= I,
(15)
one may write
dm,l;
K(m)
m’,l)
(16)
g( m + 1,l ; m’,l)
where the following (2~ X 2~) matrices are used:
K(m)=
g-'(m+l,i;m+i,i)
gp'(m+l,i;m+l,l)
-I
0
[ H(m)
=
I1.
;m,i) 0
g-'(m,l
0
(17)
(18)
I is the (/A X p) unit matrix. P(m)
Let us define also the following
(2~ X 2~) matrices
= -KP’(m)H(m)
(19)
and R(m,m’)
= JQ,+,
In this last equation, R(m,m’)
= 0,
R(m,m)
m > m’.
P(m”)? the symbol
n means
(20)
a matrix
product,
such that
m’ > m,
(214
= I,
R(m,m
(2lb)
- 1) = P(m),
(2lc)
R(m,m-N)=P(m)P(m-l)...P(m-N+l).
(214
With these definitions, the calculation of the elements of g(MM) can now be done along the same lines as those calculated explicitly before, for discrete N-layered superlattices (section 2.5.2 of ref. [2]). Let us just stress that the central entity is the transfer matrix R(m,m - N) and that the bulk dispersion relations for the excitations (free electrons, elastic vibrations, long wavelength magnons, polaritons, etc.) under study, are given by detIR(m,m-N)-e’k3”“11 where k, parameter
Vm,
=O,
(22)
and a, are respectively the propagation perpendicular to the slabs
vector
0” =2ta,.
and
the lattice
(23)
r=l
4.2. Analytical expression
of &MM)
for simple models
In the particular cases when the matrices g(m,l ; m’,l) are scalars, the inversion of g-‘(MM) has an exact analytical solution, which can be written down as explained in section 3 of ref. [2]. First write R( N, 0) as the following (2 X 2) matrix
Then as in ref. [2], using g(n+
l,l,i;
[ g(n + l,l,l
n’,i’,l)
; n’,i’,l)
1
is(n,i,i;
N times eq. (16), one obtains
= R(N,
O),R(n,l’.l
n’,i’,l) ; n’,i’,l)
1
+ LJ,
(25)
L. Dobrzvnski / Interface response theory of N-lqwred
367
continuowsuperkuttices
where
B(i’)
=
= R(N,i’)K-‘(i’)
i
(26)
L .
21
[
[I
In this case, one has also [2] R
-Ii:,
jr-l,
-R12 R,,
(27)
1’
Using the above simple properties, one finds following the same demonstration as in section 3 of ref. [2], the following analytical results. The dispersion relation for the eigenvalues of the N-layered superlattice is given by 2 cos k3aU = R,, + R,,.
(28)
Then defining more generally 277 := R,, + R,,
(29)
and 7) -
t=
( v2 - 1)1’2,
7)+i(l-q2)1’2,
-l
/ 71-t (172- 1)1’2, one obtains g(n,l,i;
n’,i’,l)
[ g(n,l,l
; n’,i’,l)
n>l,
n<
(30)
-1,
_~~‘rIn-n’-ll)~( 1=_L(/‘n-n,’ (31)
Using several times eq. (16), one obtains also in closed form as in ref. [2] any other element of g(MM), namely g(n,i,i;
n’,i’,l)
[ g(n,i,l;
ri’,i’,l)
O)(R(N,i’)tl”-“‘I -R-‘(i’, 1[&R(i, O)tlnPn’-‘I)
=
+s.,.n(i,ij
KP(i’)[
;I.
(32)
4.3. Summary of this section In this section the general block tridiagonal form of gP1(MM) has been given, eqs. (13). Then the response function g(MM), within the interface space
368
M
L. Dobrzynski
of the
(section
/ Interface response theov
N-layered
continuous
4.1) using transfer
form expressions
of N-&wed
superlattice
matrix methods.
of g(MM)
continuous superlrttrces
can
be calculated
In simple cases (section
in general 4.2) closed
(eqs. (31) and (32)) are obtained.
The dispersion relations for the excitations under study are given in general by eq. (22) and for the simple models considered above by eq. (28). Numerous
former calculations
of the dispersion
continuous superlattices appeared before, A few of these studies give also interface
relations
within two-layered
with many different approaches [l]. elements of the response functions
for two-layered superlattices [8-lo]. The first two references [8,9] deal only with shear transverse elastic waves. The last one [lo] proposes a surface Green’s function matching theory of two-layered superlattices and can be applied as the present theory to any type of excitation studied within continuous models. The central result in this paper [lo] can be directly compared to a Fourier
transformed
entity, according
ular to the slabs of the present
block
to the propagation tridiagonal
vector k,, perpendic-
gP1(MM).
5. The complete response function of the N-layered superlattice Till
now
g(MM)
of
important N-layered answer
we have the
shown
response
only
function
to know the response superlattices for many
at (m’,z’)
to a stimulus
how g
to calculate of
function physical applied
the
the interface
N-layered
elements
superlattice.
It
is
g between any two points of problems, like for example: the at (m,z),
the interaction
between
defects, etc. The general expression giving the complete g for any composite continuous material was given before [4]. It is straightforward to write directly here its particular g(m,z;
form valid for N-layered m’,z’)
=a,,,
xg(m,z”’
continuous
superlattices
c G,,(z,~“)G,-~(z~~,z~~~ ZItli. 11,= + 1
; m’,z’lv’)G,,l(z’lv’,z’v’)G,,(z~v’,z’).
)G,(z”‘,z’
(33)
In this general result appear the elements of the bulk response function G,, = G,, as defined in eqs. (3) and (4) and the interface elements + 1) of g(MM) calculated in section 4. The g(m,z”’ ; m',z WI), (z f/t,z[I']= _ elements like G; ‘( z”,z “’ ) are the elements of the inverse of the matrix G,,(M,M,) as defined by eq. (lo), within the two surfaces space M,, of the m th slab (z”, z “’ = f 1). It will prove convenient in practical calculations to rewrite eq. (33) with the
L. Dobrzynski
/ Interfrrce response theory of N-luyered contznwus
superluttices
369
help of rectangular matrices in the following equivalent form g(m,z;
m’,z’)
+
[%(z,i) Gm(zJ)]G,-‘(M,M,)g(M,M,,)G~~(M,,M,,)
t
where G;‘(M,M,) is the matrix inverse of G,,,(M,M,) defined by eq. (lo) and g(M,M,,) is a similar square matrix between (m,l), (m,l) and (m’,l), (m’S). Finally let us note that, to the knowledge of the author, the elements of g were calculated before, by other methods, only for two points situated within the same slab i of a two-layered superlattice and only for shear horizontal elastic waves [8,9]. Let us now turn to two general physical problems which can be studied with the help of the above response functions.
6. Density of states in iv-layered superlattices The most usual operators for which response functions are defined are of the form Ha, = El - if,;
(35)
for the bulk operators (eqs. (3)) and also for the surface ones and those of the composite system. The total density of states of any composite system can then be obtained from [3] n(E)
= -(l/a)
ImTrg+(E),
(36)
where
g”( E) = fiiog(
E + ie).
(37)
The total density of states at a given value of k,, and per unit length in the & direction of the continuous N-layered superlattice is then given by n(k,,,E)=
-f
f I-1
:/+idz! u
-I
ImTrg+(k,,,EIn,i,z;n,i,z),
370
L. Dohrzynskt
/ Interface response theory of N-lqered
continuous superluttrce&
where a” =Zfa,
(39)
r=l
is the length of the unit cell of the superlattice. Similarly
the local density
n(k,,,E(n,i,z)=
of states at (n,i,z)
is given by
-(l/~)ImTrg+(k,,,EIn,i,z;n,i,z).
Another quantity An(E) as compared bulk materials.
(40)
of interest is the variation of the total density of states to the reference system formed out of the N different
This quantity
for any composite
system is given by [3,4]
(41) where n(E)
= -argdet
]g-l(MM)
In order to calculate it is helpful
to define
g--l(n,i,z;n’,i’,z’)
1.
the phase shift v( E,k ,,) for an N-layered the following
=
” z/*t;,, ”
dk,
Fourier
eikzuu(n~n’)g~l(E,k,,
,k, Ii,z;i’,z’),
+1. _
z,z’=
(42) shift
n( E,k ,,) per
arg det lgpl(E,k,, ,k, IMuM,) I,
(43)
This transformation enables one to calculate unit length in the i3 direction from
v(E,k,,) = -
superlattice
transform
&/:;;I dk,
the phase
II
where M,
is the interface
space (1 < i < N, z = k 1) within one unit cell. The
second term in eq. (41) can be calculated in the same manner as in eq. (38) the total density of states. And finally the variation of the density of states per unit length in the & direction
x /+‘dzi -1
ImTr
[G,(z,i) i
is
G,(z,l)]G,-‘(M,M,)
L. Dohrzynskr
/ Interface response themy
of N-lqwred continuous superluttices
371
These general results will be applied in forthcoming papers to practical calculations of density of states, and to calculations of local density of transverse elastic waves as in refs. [8,9]. This direction is opened widely for new investigations, for the density of states as well as for the thermodynamical functions. Of course practical calculations of the above variation of density of states in superlattices will lead to analytical results only for simple models and will in general lead to.numerical exploitation of the above equations. Let us stress that, to the knowledge of the author, nothing was published on this subject before. Another general problem we will consider in this paper is that of a semi-infinite N-layered superlattice. We will show how to obtain the corresponding response functions; knowledge of which enables one then to obtain all the physical properties of such a semi-infinite material.
7. Response function of a semi-infinite contact with another medium
N-layered
superlattice,
free and in
All real materials are finite; it is therefore important to understand what are the physical effects of a semi-infinite N-layered superlattice. The simplest boundary one can study is that of a surface parallel to the slab interfaces. Let us, however, assume that this surface is at an arbitrary position within one of the unit cells of the infinite N-layered superlattice. Let m,=(n,, i,) be the last slab before the superlattice’s surface and assume that this slab has a width 2/r, different (greater or smaller) of the corresponding width 2a, of the slabs situated inside the bulk of the material. This surface slab can also be of a nature different from that of the corresponding bulk i slab. One has then an adsorbed slab on a semi-infinite N-layered superlattice. As we will see in what follows, the same formalism applies also to this physical problem. So the surface plane is completely determined by m,=(n,,
i,),
z = -1,
a, # a;,,
(45)
and the nature of the surface slab. The material is situated at m > m,. Let us call d the response function associated with this physical situation. It will be shown that this new response function can be easily obtained once the interface elements g(MM) of the response function g of an infinite N-layered superlattice are known. First it is obvious within the interface response theory that d(m,z;
m’,z’)
= &,,[G,(z,z’)
- G,(z,M,>G,-‘(M,M,)G,(M,,z’)]
+ G,(z,M,)G,*(M,M,)d(M,M,JG~~‘(M,~M,,) xG,,,,(M,,,z’),
m,m’>
m,,
(46)
where we use a notation matrices
G,,,(z,M,)
appropriate
more compact
and G,,(M,,
than in eq. (34)
for the rectangular
z’). In eq. (46) one has of course
value of the bulk response
function
G,,,>(z,z’)
within
to use the the surface
(or the adsorbed) slab. So we see that the problem is down to the determination of the {d(M,M,,)}. This is an easy problem to solve within the frame of the discrete interface
theory [3] once the { g(M,M,,)}
of the infinite
N-layered
superlattice are known. It is indeed straightforward to write the surface elements of &‘(MM) using eqs. (13). Then one can see that one can obtain the same finite &‘(MM) matrix gP1(MM) in the superlattice
when starting from the infinite block tridiagonal following manner. First remove in the infinite
the slab (m, - 1). This
can be done
by the following
cleavage
operator
=-
I
l,i;m,-l.ilu,,)
9s'h
- l,l;m,-l,i]a,,)
9.:+,
I
gs’(m,-l,i;m,-l,lla,,) g~‘(m,-1,1;m,-l,lla,,)
’
(47)
where the parameter urs recalls that the g;‘(M,,_,M,,_,) 1h,,) are those of the corresponding bulk i, slab. This procedure leaves us with two independent semi-infinite
superlattices
having
their free surfaces
- 1) and at (m, - 2, z = 1). Construct
respectively
now the surface
by taking W,. g(MM) between the surface (m’,z’) from m’ > m, and z’ = k 1
(m,,?)
response
and
at (m,,
z =
operator
A,,
any other
interface
-g~‘(m,-1,1;m,-1,1(a,,)g(m,-1,1;m’,z’)
Asf(m,,i;m’,z’)=
-g~‘(m,-I,I;m,-l,iIu,,)g(m,-~l,i;m’,z’), z=
_tl.
(48)
Now if the surface slab has a width 2a,, different of the width 212,~ of the corresponding bulk slab, and (or) is of a different nature, one can consider that this perturbs the semi-infinite ideally cleaved gP’(MM) by the following perturbation V,(M,M,)
= g,‘(M,M,
I a,>- g;‘(MsW la,),
(49)
I a,) where M, = M,,,> and where although not explicitly indicated 9.; ‘(M,M, is the inverse of the matrix given by eq. (9) for the adsorbed slab m, of nature and width different of the corresponding bulk slab mrs. The complete interface response operator [3] for this problem A,(M,M,,)
= A,r(M,M,,)
is
m’ > m,,
+ V,(M,M,)g(M,M,,,),
(50)
where A,r(m,,
z; m’,z’)
=GZrlSziAsr(ms,i;
m’,z’),
z,z’=
kl,
m’>m,.
(51)
L. Dohrzynski
/ Interface response theoty of N-luyered continuous super-lattices
313
Define now A(M,M,)
= I(M,M,)
(52)
+ A,(M,M,).
And finally the interface response theory [3] provides us with d(M,M,,)
= g(M,,,M,,)
- g(M,M,)A-‘(M,M,)A,(M,M,,),
m,m’2m,.
(53)
This result once reported back into eq. (46) provides all the elements of the response function d of a semi-infinite N-layered superlattice with or without an adsorbed layer of arbitrary width. Another system of wide physical interest is the interface between the above defined semi-infinite (with or without adsorbate) N-layered superlattice and another semi-infinite medium. Let this other semi-infinite medium be defined by m = (m, - 1) and let its bulk response function be G,,ml(z,z’),
-cc
< z,z’ < 0.
(54)
The response function d of such a system > m, - 1, provided the above value (54) elements d(M,M,,) are still given by eq. (53) perturbation V, given by eq. (49) the following m,m’
VpI(msrz; m,,z’)
= ~,,J,i_g~‘(m,
is still given by eq. (46) for for G,S_l(:z,z’). The interface provided that one adds to the complementary term VP1
- 18; m, - 1,0),
(55)
where g;‘(m, - 1,O; m, - 1,O) is the inverse of the surface gs(ms - 1,O; m, 1,0) of the semi-infinite medium situated at - cc < z -C0. This modification of 5 is then included in eqs. (50)-(53). L.et us underline that the interface response theory provides directly the solution of such apparently complicated problems by mostly working within the interface space M.
8. Prospectives General solutions for the response functions of infinite and semi-infinite continuous N-layered superlattices are given, for the first time, in this paper, within the frame of the interface response theory. The case of an adsorbed layer on such a semi-infinite material is also considered. The knowledge of these response functions will enable study of all physical problems connected with these materials. Only one general application (namely the density of states) of these response functions was given here. These general new results will be used for the study in these materials of electromagnetic [ll] and elastic [12] waves, free electrons [13], magnons in the long wavelength limit, etc.
Acknowiedgments
The support of the Centre National de la Recherche Scientifique (France) during a sabbatical year in Madrid and the hospitality of the Instituto de Fisica de1 Estado S6lido of CSIC are gratefully acknowledged.
References [l] See, for example, Proc. Intern. Conf. on the Dynamics of Interfaces. J. Phys. (Paris). C5 (1984); Intern. School on Dynamical Phenomena at Surfaces, and Superlattices, Springer Series in Surface Science,Vol. 3 (Springer, Berlin, 1985). [2] L. Dobrzynski, Interface Response Theory of /V-Layered Discrete Superlattices (USTL Report, Villeneuve d’Ascq, 1986). [3] (a) L. Dobrzynski, Surface Sci. 17.5 (1986) 1: (b) L. Dobrzynski, Surface Sci. Rept. 6 (1986) 119. [4] L. Dobrzynski, Surface Sei. 180 (1987) 489. [S] See, for example: M.G. Cottam and A.A. Maradudin. in: Surface Excitations. Modern Problems in Condensed Matter Sciences. Vol. 9. Eds. V.M. Agranovich and R. Loudon (North-Holland, Amsterdam, 1986). [6] K. Portz and A.A. Maradudin, Phys. Rev. B16 (1977) 3535. [7] B. Djafari-Rouhani, 1981, private communication; see also ref. [5]. L. Dobmynski and A.A. Maradudin, Phys. Rev. B27 [8] R.E. Camley, B. Djafari-Rouhani. (1983) 7318. [9] M. Babiker, D.R. Tilley. E.L. Albuquerque and C.E.T. Goncalves da Silva. J. Phys. Cl8 (1985) 1269. [lo] F. Garcia-Moliner and V.R. Velasco, Surface Sci. 175 (1986) 9. [ll] L. Dobrzynski, to be published. [12] L. Dobrzynski, to be published. [13] L. Dobrzynski, to be published.
Surface
Science 182 (1987) 3622374 North-Holland. Amsterdam
INTERFACE RESPONSE THEORY OF N-LAYERED CONTINUOUS SUPERLATTICES L. DOBRZYNSKI Equipe Internutmude de D~vncrmrque des Interfaces, Luhorutorre de D_vnumrque de.? Cristoux Mol&ulcrwes ((/A No. 801) ussocih au Centre Natmnal de lu Recherche Scientifique, UFR de Phy.slque, Uniuersiti des Sciences et Techniques de Lille Flmdres - Artoa, F-59655 Vdlenewe d;4scq Cedex, Frunce and Imtituto de Fisico del Estudo Sblldo, CSIC, Serrano 123. .?a006 Mudrrd, Spun Received
17 July 1986: accepted
for publication
21 October
1986
N-layered continuous superlattices are formed out of a periodic repetition of N different slabs studied within a continuous space. A recent interface response theory is applied for the first time to these systems. General expressions for the response functions associated with a given operator are given. From them, all the physical properties of such materials can be derived, as for example the densities of states. The response functions for such semi-infinite materials, free and in contact with another medium are also given. This general theory applies as well to free electrons, elastic vibrations, long wavelength magnons, polaritons, etc.
1. Introduction The study of fundamental properties of superlattices formed out of two alternating different slabs attracted in the last decade a great deal of interest [l]. One aspect of these investigations is the determination of the band structure for different types of excitations: electronic, vibrational, magnetic, electrodynamic, etc. Many studies of two-layered superlattices use continuum models, as they are usually more simple than the atomic ones and give useful information in their domains of validity. N-layered superlattices will provide in the near future new artificial materials. It is in fact a logical development, from the point of view of the present standard technics used for the growth of the two-layered superlattice. The theoretical study of such new materials is also an exciting challenge, which was first faced for discrete models [2] with the help of the interface response theory [3]. This theory [4] enables one also to study the response function of any composite material in a d-dimensional continuous space Dcd) once the bulk response functions of all i (1 < i < N) subsystems domain Dcd). The application of this general theory superlattices is explicitly given here. 0039-6028/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
are known inside their to N-layered continuous
B.V.
L. Dohrrynski
/ Interface response theory of N-layered continuous superluttices
363
These N-layered superlattices are formed out of an infinite repetition of N different slabs labelled by the unit cell index n. Each slab of width 2ai is labelled by the index i (1 < i < N) within the unit cell n. All the interfaces are taken to be parallel to the (Xi, X,) plane. Taking advantage of this infinitesimal translational invariance in directions parallel to the interfaces, we Fourier analyze all operators according to, for example
where k,, is the propagation vector and X,, is the real space position vector, both parallel to the interfaces of the slabs. Bearing in mind that all operators which will appear throughout this text are functions of k,, , we will, for the sake of simplicity, not write any longer this dependence explicitly. The other remarkable consequence of this translational invariance is that we are left with a one-dimensional problem, as a function of X3. Rather than using X3 ranging from - cc to + co, we find it simpler in what follows to replace X3 by the three variables (n, i, z). We define within each slab (n, i) a reduced space coordinate z denoting the space position X,/a,; the origin being choosen in the middle of each slab. So -l
(2)
within the slab (n, i). Let us now outline, following ref. [4], the successive steps for the calculation of the response function of such a material. The knowledge of this response function enables one to study directly any physical property of these materials.
2. Bulk reference
response
function
(3ven a continuous (as function of X,) bulk operator H,,( X3), one defines the corresponding response function G,,( X3, X;) by H,,(X,)G,i(X,,X;)=“(X,-X,‘),
(3)
for - cc < X3 < + co. Remember that both operators are functions of k ,, and are, as well as the unit matrix I, in general (p x p) matrices. Define now a reference response function G for the N-layered superlattice by G( n, i, z; n’, i’, z’) = 6,,,J,,,G,,
(z,
z’),
(4)
where z and z’ are defined by eq. (2) and S,,,, and S,,, are the usual Kronecker symbols.
3. Surface
response
function
for one slab
Out of the infinite medium i, cut a free surface slab, such that -h, 6 X, < h,. The operator h,;( X,) corresponding to H,,( X,), may be written as s(z-z’)h,,(z)=S(z-z’)H~,(z)+[-S(z-1)+6(2+1)]V,,(z), where - 1 < z,z’ < 1 (eq. (2)) and L&(z) is the cleavage operator Calculate then the surface response operator for this slab
-
Vo,(z”)Gor(z”,z’)I;“=z=l,
A,,(z,z’) = And
finally
[4].
-l
(5)
+~~,(z”)G~,(z”,z’)I~,,=~=~~,~=+~.
the surface
response
function
corresponding
to h,,(z)
is given by
[41
gsrb’) = Go,(ZJ’) - [Goi(Z,‘)
1
A&z') Go,(z,l)I ALi
3
A(l,zf)
[
(6)
where - 1 < z, z’ 6 1; the second term is the product of a (p X 2~) rectangular matrix by a (2~ x 2~) and a (2~ X p) one and the (2~ X 2~) matrix A;: is the inverse of A sr2
I+ A,, (1J)
(7)
= A,,(Li)
Note also a useful particular g,‘(M,M,)
value [3] of eq. (6)
=A,,lG,?(M,M,),
where &;‘(M,M,)
gs,(M,M,>= Go,(M,M,>=
_ss,w g,,m I and G,;’
9,,03
Goi [ G&i) --
(M,M,)
(8) are the inverses
elements
Go,(iJ)
(10)
Go,04
of the N-layered
matrices
(9)
g,,04
Slab response functions were calculated ity theory [5-71 with other methods.
4. Interface
of the following
before, in particular
superlattice
response
within
elastic-
function
The interface space M of the N-layered superlattice is the infinite set of the discrete points (n,i, zM), where the integers take the following values: - cc < n
L. Dobrzynski
/ Inierfuce response theory of N-layered continuous superlaitices
365
< +co, l
+ Nn.
(11)
Remark that with these notations there are two different manners to label one and the same interface
(m,i>-(m - 1~).
(12)
The inverse g-‘(MM) within this discrete interface space M of the response function g(MM) of the N-layered superlattice is a block tridiagonal matrix whose elements are [4] 9-%,i;
d,i)
= ~,,~[9,-,‘(U)
+ 9;;_1(1,1)],
(13a)
9-‘(m
; m’J)
= S,,&7,-,l(Ll>
+ s,-r;+l(l,l)],
(13b)
g-l(m,i;m~,i)=8m,r~,-,'(i,i),
(13c)
g-l(m,i;
(13d)
d,i)=s,,,g,-,'(i,i).
Note also that the off-diagonal blocks of elements of this tridiagonal matrix gP ‘(MM) have the following property
gpQ&
m,i)= [g-l(m,i; m/j)]+,
(14)
which follows from the usual properties [5] of the surface response function -1
9 S”,
.
4.1. General methods for the calculation
of g(MA4)
The inversion of the block tridiagonal matrix gP’(MM) can be done with mathematical methods similar to those used before [2] in the discrete interface response theory of N-layered superlattices. As g-‘(MM)g(MM)
= I,
(15)
one may write
dm,l;
K(m)
m’,l)
(16)
g( m + 1,l ; m’,l)
where the following (2~ X 2~) matrices are used:
K(m)=
g-'(m+l,i;m+i,i)
gp'(m+l,i;m+l,l)
-I
0
[ H(m)
=
I1.
;m,i) 0
g-'(m,l
0
(17)
(18)
I is the (/A X p) unit matrix. P(m)
Let us define also the following
(2~ X 2~) matrices
= -KP’(m)H(m)
(19)
and R(m,m’)
= JQ,+,
In this last equation, R(m,m’)
= 0,
R(m,m)
m > m’.
P(m”)? the symbol
n means
(20)
a matrix
product,
such that
m’ > m,
(214
= I,
R(m,m
(2lb)
- 1) = P(m),
(2lc)
R(m,m-N)=P(m)P(m-l)...P(m-N+l).
(214
With these definitions, the calculation of the elements of g(MM) can now be done along the same lines as those calculated explicitly before, for discrete N-layered superlattices (section 2.5.2 of ref. [2]). Let us just stress that the central entity is the transfer matrix R(m,m - N) and that the bulk dispersion relations for the excitations (free electrons, elastic vibrations, long wavelength magnons, polaritons, etc.) under study, are given by detIR(m,m-N)-e’k3”“11 where k, parameter
Vm,
=O,
(22)
and a, are respectively the propagation perpendicular to the slabs
vector
0” =2ta,.
and
the lattice
(23)
r=l
4.2. Analytical expression
of &MM)
for simple models
In the particular cases when the matrices g(m,l ; m’,l) are scalars, the inversion of g-‘(MM) has an exact analytical solution, which can be written down as explained in section 3 of ref. [2]. First write R( N, 0) as the following (2 X 2) matrix
Then as in ref. [2], using g(n+
l,l,i;
[ g(n + l,l,l
n’,i’,l)
; n’,i’,l)
1
is(n,i,i;
N times eq. (16), one obtains
= R(N,
O),R(n,l’.l
n’,i’,l) ; n’,i’,l)
1
+ LJ,
(25)
L. Dobrzvnski / Interface response theory of N-lqwred
367
continuowsuperkuttices
where
B(i’)
=
= R(N,i’)K-‘(i’)
i
(26)
L .
21
[
[I
In this case, one has also [2] R
-Ii:,
jr-l,
-R12 R,,
(27)
1’
Using the above simple properties, one finds following the same demonstration as in section 3 of ref. [2], the following analytical results. The dispersion relation for the eigenvalues of the N-layered superlattice is given by 2 cos k3aU = R,, + R,,.
(28)
Then defining more generally 277 := R,, + R,,
(29)
and 7) -
t=
( v2 - 1)1’2,
7)+i(l-q2)1’2,
-l
/ 71-t (172- 1)1’2, one obtains g(n,l,i;
n’,i’,l)
[ g(n,l,l
; n’,i’,l)
n>l,
n<
(30)
-1,
_~~‘rIn-n’-ll)~( 1=_L(/‘n-n,’ (31)
Using several times eq. (16), one obtains also in closed form as in ref. [2] any other element of g(MM), namely g(n,i,i;
n’,i’,l)
[ g(n,i,l;
ri’,i’,l)
O)(R(N,i’)tl”-“‘I -R-‘(i’, 1[&R(i, O)tlnPn’-‘I)
=
+s.,.n(i,ij
KP(i’)[
;I.
(32)
4.3. Summary of this section In this section the general block tridiagonal form of gP1(MM) has been given, eqs. (13). Then the response function g(MM), within the interface space
368
M
L. Dobrzynski
of the
(section
/ Interface response theov
N-layered
continuous
4.1) using transfer
form expressions
of N-&wed
superlattice
matrix methods.
of g(MM)
continuous superlrttrces
can
be calculated
In simple cases (section
in general 4.2) closed
(eqs. (31) and (32)) are obtained.
The dispersion relations for the excitations under study are given in general by eq. (22) and for the simple models considered above by eq. (28). Numerous
former calculations
of the dispersion
continuous superlattices appeared before, A few of these studies give also interface
relations
within two-layered
with many different approaches [l]. elements of the response functions
for two-layered superlattices [8-lo]. The first two references [8,9] deal only with shear transverse elastic waves. The last one [lo] proposes a surface Green’s function matching theory of two-layered superlattices and can be applied as the present theory to any type of excitation studied within continuous models. The central result in this paper [lo] can be directly compared to a Fourier
transformed
entity, according
ular to the slabs of the present
block
to the propagation tridiagonal
vector k,, perpendic-
gP1(MM).
5. The complete response function of the N-layered superlattice Till
now
g(MM)
of
important N-layered answer
we have the
shown
response
only
function
to know the response superlattices for many
at (m’,z’)
to a stimulus
how g
to calculate of
function physical applied
the
the interface
N-layered
elements
superlattice.
It
is
g between any two points of problems, like for example: the at (m,z),
the interaction
between
defects, etc. The general expression giving the complete g for any composite continuous material was given before [4]. It is straightforward to write directly here its particular g(m,z;
form valid for N-layered m’,z’)
=a,,,
xg(m,z”’
continuous
superlattices
c G,,(z,~“)G,-~(z~~,z~~~ ZItli. 11,= + 1
; m’,z’lv’)G,,l(z’lv’,z’v’)G,,(z~v’,z’).
)G,(z”‘,z’
(33)
In this general result appear the elements of the bulk response function G,, = G,, as defined in eqs. (3) and (4) and the interface elements + 1) of g(MM) calculated in section 4. The g(m,z”’ ; m',z WI), (z f/t,z[I']= _ elements like G; ‘( z”,z “’ ) are the elements of the inverse of the matrix G,,(M,M,) as defined by eq. (lo), within the two surfaces space M,, of the m th slab (z”, z “’ = f 1). It will prove convenient in practical calculations to rewrite eq. (33) with the
L. Dobrzynski
/ Interfrrce response theory of N-luyered contznwus
superluttices
369
help of rectangular matrices in the following equivalent form g(m,z;
m’,z’)
+
[%(z,i) Gm(zJ)]G,-‘(M,M,)g(M,M,,)G~~(M,,M,,)
t
where G;‘(M,M,) is the matrix inverse of G,,,(M,M,) defined by eq. (lo) and g(M,M,,) is a similar square matrix between (m,l), (m,l) and (m’,l), (m’S). Finally let us note that, to the knowledge of the author, the elements of g were calculated before, by other methods, only for two points situated within the same slab i of a two-layered superlattice and only for shear horizontal elastic waves [8,9]. Let us now turn to two general physical problems which can be studied with the help of the above response functions.
6. Density of states in iv-layered superlattices The most usual operators for which response functions are defined are of the form Ha, = El - if,;
(35)
for the bulk operators (eqs. (3)) and also for the surface ones and those of the composite system. The total density of states of any composite system can then be obtained from [3] n(E)
= -(l/a)
ImTrg+(E),
(36)
where
g”( E) = fiiog(
E + ie).
(37)
The total density of states at a given value of k,, and per unit length in the & direction of the continuous N-layered superlattice is then given by n(k,,,E)=
-f
f I-1
:/+idz! u
-I
ImTrg+(k,,,EIn,i,z;n,i,z),
370
L. Dohrzynskt
/ Interface response theory of N-lqered
continuous superluttrce&
where a” =Zfa,
(39)
r=l
is the length of the unit cell of the superlattice. Similarly
the local density
n(k,,,E(n,i,z)=
of states at (n,i,z)
is given by
-(l/~)ImTrg+(k,,,EIn,i,z;n,i,z).
Another quantity An(E) as compared bulk materials.
(40)
of interest is the variation of the total density of states to the reference system formed out of the N different
This quantity
for any composite
system is given by [3,4]
(41) where n(E)
= -argdet
]g-l(MM)
In order to calculate it is helpful
to define
g--l(n,i,z;n’,i’,z’)
1.
the phase shift v( E,k ,,) for an N-layered the following
=
” z/*t;,, ”
dk,
Fourier
eikzuu(n~n’)g~l(E,k,,
,k, Ii,z;i’,z’),
+1. _
z,z’=
(42) shift
n( E,k ,,) per
arg det lgpl(E,k,, ,k, IMuM,) I,
(43)
This transformation enables one to calculate unit length in the i3 direction from
v(E,k,,) = -
superlattice
transform
&/:;;I dk,
the phase
II
where M,
is the interface
space (1 < i < N, z = k 1) within one unit cell. The
second term in eq. (41) can be calculated in the same manner as in eq. (38) the total density of states. And finally the variation of the density of states per unit length in the & direction
x /+‘dzi -1
ImTr
[G,(z,i) i
is
G,(z,l)]G,-‘(M,M,)
L. Dohrzynskr
/ Interface response themy
of N-lqwred continuous superluttices
371
These general results will be applied in forthcoming papers to practical calculations of density of states, and to calculations of local density of transverse elastic waves as in refs. [8,9]. This direction is opened widely for new investigations, for the density of states as well as for the thermodynamical functions. Of course practical calculations of the above variation of density of states in superlattices will lead to analytical results only for simple models and will in general lead to.numerical exploitation of the above equations. Let us stress that, to the knowledge of the author, nothing was published on this subject before. Another general problem we will consider in this paper is that of a semi-infinite N-layered superlattice. We will show how to obtain the corresponding response functions; knowledge of which enables one then to obtain all the physical properties of such a semi-infinite material.
7. Response function of a semi-infinite contact with another medium
N-layered
superlattice,
free and in
All real materials are finite; it is therefore important to understand what are the physical effects of a semi-infinite N-layered superlattice. The simplest boundary one can study is that of a surface parallel to the slab interfaces. Let us, however, assume that this surface is at an arbitrary position within one of the unit cells of the infinite N-layered superlattice. Let m,=(n,, i,) be the last slab before the superlattice’s surface and assume that this slab has a width 2/r, different (greater or smaller) of the corresponding width 2a, of the slabs situated inside the bulk of the material. This surface slab can also be of a nature different from that of the corresponding bulk i slab. One has then an adsorbed slab on a semi-infinite N-layered superlattice. As we will see in what follows, the same formalism applies also to this physical problem. So the surface plane is completely determined by m,=(n,,
i,),
z = -1,
a, # a;,,
(45)
and the nature of the surface slab. The material is situated at m > m,. Let us call d the response function associated with this physical situation. It will be shown that this new response function can be easily obtained once the interface elements g(MM) of the response function g of an infinite N-layered superlattice are known. First it is obvious within the interface response theory that d(m,z;
m’,z’)
= &,,[G,(z,z’)
- G,(z,M,>G,-‘(M,M,)G,(M,,z’)]
+ G,(z,M,)G,*(M,M,)d(M,M,JG~~‘(M,~M,,) xG,,,,(M,,,z’),
m,m’>
m,,
(46)
where we use a notation matrices
G,,,(z,M,)
appropriate
more compact
and G,,(M,,
than in eq. (34)
for the rectangular
z’). In eq. (46) one has of course
value of the bulk response
function
G,,,>(z,z’)
within
to use the the surface
(or the adsorbed) slab. So we see that the problem is down to the determination of the {d(M,M,,)}. This is an easy problem to solve within the frame of the discrete interface
theory [3] once the { g(M,M,,)}
of the infinite
N-layered
superlattice are known. It is indeed straightforward to write the surface elements of &‘(MM) using eqs. (13). Then one can see that one can obtain the same finite &‘(MM) matrix gP1(MM) in the superlattice
when starting from the infinite block tridiagonal following manner. First remove in the infinite
the slab (m, - 1). This
can be done
by the following
cleavage
operator
=-
I
l,i;m,-l.ilu,,)
9s'h
- l,l;m,-l,i]a,,)
9.:+,
I
gs’(m,-l,i;m,-l,lla,,) g~‘(m,-1,1;m,-l,lla,,)
’
(47)
where the parameter urs recalls that the g;‘(M,,_,M,,_,) 1h,,) are those of the corresponding bulk i, slab. This procedure leaves us with two independent semi-infinite
superlattices
having
their free surfaces
- 1) and at (m, - 2, z = 1). Construct
respectively
now the surface
by taking W,. g(MM) between the surface (m’,z’) from m’ > m, and z’ = k 1
(m,,?)
response
and
at (m,,
z =
operator
A,,
any other
interface
-g~‘(m,-1,1;m,-1,1(a,,)g(m,-1,1;m’,z’)
Asf(m,,i;m’,z’)=
-g~‘(m,-I,I;m,-l,iIu,,)g(m,-~l,i;m’,z’), z=
_tl.
(48)
Now if the surface slab has a width 2a,, different of the width 212,~ of the corresponding bulk slab, and (or) is of a different nature, one can consider that this perturbs the semi-infinite ideally cleaved gP’(MM) by the following perturbation V,(M,M,)
= g,‘(M,M,
I a,>- g;‘(MsW la,),
(49)
I a,) where M, = M,,,> and where although not explicitly indicated 9.; ‘(M,M, is the inverse of the matrix given by eq. (9) for the adsorbed slab m, of nature and width different of the corresponding bulk slab mrs. The complete interface response operator [3] for this problem A,(M,M,,)
= A,r(M,M,,)
is
m’ > m,,
+ V,(M,M,)g(M,M,,,),
(50)
where A,r(m,,
z; m’,z’)
=GZrlSziAsr(ms,i;
m’,z’),
z,z’=
kl,
m’>m,.
(51)
L. Dohrzynski
/ Interface response theoty of N-luyered continuous super-lattices
313
Define now A(M,M,)
= I(M,M,)
(52)
+ A,(M,M,).
And finally the interface response theory [3] provides us with d(M,M,,)
= g(M,,,M,,)
- g(M,M,)A-‘(M,M,)A,(M,M,,),
m,m’2m,.
(53)
This result once reported back into eq. (46) provides all the elements of the response function d of a semi-infinite N-layered superlattice with or without an adsorbed layer of arbitrary width. Another system of wide physical interest is the interface between the above defined semi-infinite (with or without adsorbate) N-layered superlattice and another semi-infinite medium. Let this other semi-infinite medium be defined by m = (m, - 1) and let its bulk response function be G,,ml(z,z’),
-cc
< z,z’ < 0.
(54)
The response function d of such a system > m, - 1, provided the above value (54) elements d(M,M,,) are still given by eq. (53) perturbation V, given by eq. (49) the following m,m’
VpI(msrz; m,,z’)
= ~,,J,i_g~‘(m,
is still given by eq. (46) for for G,S_l(:z,z’). The interface provided that one adds to the complementary term VP1
- 18; m, - 1,0),
(55)
where g;‘(m, - 1,O; m, - 1,O) is the inverse of the surface gs(ms - 1,O; m, 1,0) of the semi-infinite medium situated at - cc < z -C0. This modification of 5 is then included in eqs. (50)-(53). L.et us underline that the interface response theory provides directly the solution of such apparently complicated problems by mostly working within the interface space M.
8. Prospectives General solutions for the response functions of infinite and semi-infinite continuous N-layered superlattices are given, for the first time, in this paper, within the frame of the interface response theory. The case of an adsorbed layer on such a semi-infinite material is also considered. The knowledge of these response functions will enable study of all physical problems connected with these materials. Only one general application (namely the density of states) of these response functions was given here. These general new results will be used for the study in these materials of electromagnetic [ll] and elastic [12] waves, free electrons [13], magnons in the long wavelength limit, etc.
Acknowiedgments
The support of the Centre National de la Recherche Scientifique (France) during a sabbatical year in Madrid and the hospitality of the Instituto de Fisica de1 Estado S6lido of CSIC are gratefully acknowledged.
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