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Interface response theory of discrete superlattices
Progress in Surface Science, Vol. 26(1-4), pp. 103-116, 1987 Primed in the U.S.A.
0079-6816/87 $0.00 + .50 Copyright © 1988 Pergamon Press pie
INTERFACE RESPONSE THEORY OF DISCRETE SUPERLATTICES LEONARD DOBRZYNSKI Equipe Internationale de Dynarnique des Interfaces, Laboratoire de Dynamique des Cristaux Moleculaires, Unitd associee (No 801) au Centre National de la Recherche Scientifique, U.F.R. de Physique, Universite des Sciences et Techniques de Lille Flandres Artois, 59655 Villeneuve d'Ascq Cedex, France
Abstract Response function theories of discrete superlattices are reviewed. General expressions for the response functions of N-layered discrete superlattices are defined, with the help of a recent interface response theory. Such superlattices are formed out of a periodic repetition of N different crystalline slabs. I. Introduction Professor Koutecky [ I ] was the f i r s t for electrons after Rosenzweig's work for phonons [2] to use response functions (called also Green's functions) for the study of discrete materials limited by a free surface. This is now a well established method for the study of Surfaces and Interfaces. I t was also extended recent l y to the study of Superlattices. This mini review focuss on response functions of discrete models of superlattices. Response functions of continuous models of superlattices will be just briefly discussed at the end. Other mathematical methods for the study of superlattices are not reviewed here. This do not mean that these methods are not important, for example the transfer matrix method is very well designed for the calculation of the eigenvalues associated to physical properties of superlattices. This choice of focuss is mostly motivated by thepionnering interest of Professor Koutecky { I ] in Green's
103
104
L. Dobrzynski
functions for atomic models of surfaces. An N-layered s u p e r l a t t i c e is formed out of a periodic r e p e t i t i o n of N d i f f e r e n t c r y s t a l l i n e slabs. In most of the e x i s t i n g litterature
the word s u p e r l a t t i c e is used for 2-1ayered s u p e r l a t t i c e s .
Already N interfaces in an homogeneous crystal can be viewed as a simple superlatt i c e . A general expression for the corresponding response function was given [ 3 I as a function of the response function of the homogeneous c r y s t a l .
Its interface
elements were also derived in closed form. The formalism was seen to be generalizable to m u l t i - l a y e r systems, where the interfaces separate regions of d i f f e r e n t structure. Indeed one can construct [ 4 - 7 I the s u p e r l a t t i c e response function out of the free surface slab's response functions, using the coupling operators between the independant slabs and the Dyson equation.
However t h i s obliges in a f i r s t
step to
calculate the free surface slab's response functions out of the corresponding bulk response functions. This step was shown unnecessary, as i t
is possible to obtain
the s u p e r l a t t i c e response function d i r e c t l y from elements of the bulk response functions of each slab [8-91. This was achieved with the Surface Green's Function Method [ 8 ] which uses r e f l e x i o n and transmission operators through the slabs for 2-1ayered superlattices and with the Interface Response Theory of Composite Systems [ 9 ] which constructs any f i n a l elements. The f i r s t one [ l l I .
composite system out of i t s indeDendent
approach was recently reviewed [10I and compared to the second
So the remaining of this paper is mostly devoted to the interface res-
ponse theory of N-layered discrete s u p e r l a t t i c e s , which was only sketched in the original
paper [ 9 I . 2. General Interface Response Theory of Discrete Superlattices
A. D e f i n i t i o n of N-Layered Superlattice ( i ) Slab K of one homogeneous material.
Consider a slab labelled by the index K
of a homogeneous material and possessing translationnal
invariance in the space
directions XI and X2. This slab can be considered to be formed out of LK equivalent " p r i n c i p a l
layers" labelled by an index ~ such that I < ~ < LK. A principal
layer may consist of one or several atomic planes, depending on the range of the interatomic i n t e r a c t i o n s .
By d e f i n i t i o n ,
all
interactions
in the crystal slab are
described as interactions between the nearest neighbouring " p r i n c i p a l Due to t r a n s l a t i o n invariance p a r a l l e l
to the surfaces of the slab, one does the
usual Fourier transformation, which makes a l l where ~
is the propagation vector p a r a l l e l
the operators to be function of ~ ,
to the surfaces. In this paper a l l
the e n t i t i e s are function of k~//, although for s i m p l i c i t y , will
not be e x p l i c i t l y
w r i t t e n down.
layers".
this dependence on k~
Interface Response Theory of Discrete Superlattices l i i ) Slab of N Different Homogeneous Materials.
105
Consider now N different homoge-
neous slabs, limited each by ideally truncated free surfaces. This means that each of these homogeneous slabs can be considered as cut out of the corresponding i n f i nite material by a cleavage which leaves their bulk values to the interactions near the two free surfaces of the slab. Couple now together these N ideally cleaved slabs, (K = I, 2. . . .
N), by an
interface coupling operator called~V'i . We suppose that this coupling has matrix elements only between two interface adjacent "principal layers", (K, LK) and (K+I, 1) for example. However all what will follow can be easily generalized to interface coupling penetrating more deeply inside each adjacent slab by redefining the thickness of the "principal layers". This redefinition can also be easily done when the dimensions of the "principal layers" (due to the range of the bulk interactions) are not the same in each slab K. One obtains in this manner a N-layered slab, (we label by the index n), having ideal free surfaces at the principal (iii)
N-Layered Superlattice.
dically
layers (K = 1, ~ = I) and at (K : N, ~=LN).
In the same manner as above one can couple perio-
(-~ < n < +~) an i n f i n i t e
number of the above N layered slabs in order to
obtain another new bulk material, called here N-layered s u p e r l a t t i c e . In what follows,
will
the following more condensed notations
m ~ n N + K ~ (n ,K )
,
(1)
m' ~ n'N + K' ~ ( n ' , K ' )
,
(2)
prove convenient for l a b e l l i n g a given slab K (or K') in the unit cell n
(or n'). B. Response Functions
<--w
( i ) Bulk Response Function GK f o r Each Submaterial K.
For the i n f i n i t e
material
K, consider an operator~F~K . This bulk material can be considered as an i n f i n i t e stack of "principal
layers". The p e r i o d i c i t y
system to that of noninteracting
parallel
one-dimensional
to the layers reduces the
chains. There is one chain f o r
each wave vector ~U along the layer. The operator+FFK (function of k~H) w i l l general have the following tridiagonal
~K(OI)
"-F~K(O0 )
(oi)
form.
~-FFK(OI )
TFK
]
in
(3)
106
L. Dobrzynski
In this expression the intra and inter-principal layers interactions, respectively
+hrK(O0) ;+F~K(OI) and~-H~;(Ol) are square submatrices, whose dimension is function of the problem studied. For example, if+FCK is the dynamical matrix of a 3-dimensional monoatomic l a t t i c e with i n t e r a c t i o n s between f i r s t
nearest neighbor atomic planes,
the dimension of these submatrices w i l l be in general 3 x 3. If~rK is an Hamiltonian describing e l e c t r o n i c i n t e r a c t i o n s in t i g h t binding and i f there are p p r i n c i p a l layer o r b i t a l s (number of d i f f e r e n t e l e c t r o n i c o r b i t a l s m u l t i p l i e d by the number of atomic planes in the p r i n c i p a l l a y e r ) , the~H~K(O0) w i l l be u x u matrices. In what follows, we w i l l
refer in general to these operators as p x p matrices•
Define now the bulk response function GK by
where'*is
the u n i t y matrix.
There are many well known methods for c a l c u l a t i n g the p x p elements of GK, f o r the t r i d i a g o n a l operators given by Eq. (3). In what follows, we w i l l use the ones based on the notion of t r a n s f e r matrix, which provide [12-13] the following matrix relations +~K(~C') ="T'K(~-~') "-~K(II) ~+ GK(CC, ) = + f ~ L ' - ~ )
~K(II)
,
L b C'
,
(5)
,
L~ C'
,
(6)
for all the p x u inter layer matrix elements of KG function of GK(11) and of two u x u transfer matrices"-fK and+-fK. The calculation of these three submatrices has to be in general done by standard~umerical
methods [12-15], apart for simple
models for which these submatrices reduce to scalars and H~(01) = HK(01), then one may obtain the elements of GK in closed forms [16]
GK(Lg' ) (ii)
:
t~ ~-C'I GK(ll )
(7)
Reference Response Function~G+of the N-Layered Superlattice.
Define [ 9 ] for
the N-layered s u p e r l a t t i c e a reference response function+~as a block diagonal matrix formed out of only the elements of the bulk response functions GZK contained w i t h i n the space of d e f i n i t i o n of each slab, namely, with the notations given by Eqs ( I - 2 ) "-~(mL,m'~') = ~mm,~m(gL') , 1 ~ ~, L' ~ Lm = L K
,
(8)
where ~mm' is the usual Kronecker symbol. ( i i i ) Response F u n c t i o n ~ o f the N-Layered Superlattice. Out of each i n f i n i t e material K, construct f i r s t an i d e a l l y cleaved free surface slab K, by c u t t i n g
107
Interface Response Theory of Discrete Superlattices
out in Eq. (3) all interactions between the principal layers ~ = 0 and I on one hand and ~ = L K and LK + I on the other hand ; this procedure w i l l be described by a cleavage o p e r a t o r ~ K. Then couple all these ideally cleaved slabs by an interface coupling*i~ I, as defined in section 2.A. These procedures define out of the {~K}, {~K} andS"I, the operatorK~of the N-layered superlattice, to which one associates a response function+gby g . +~" = +1*
(9)
C. Interface Response Operator~' Corresponding to the above building procedures of the N-layered superlattice, introduce [ g ] f i r s t a free surface response o p e r a t o r ~ (K~ ; K~') for each slab K. This operator is formed out of the elements o f ~ K .~K) s t r i c t l y contained inside the domain of definition of the K slab (I < ~, 4' < LK), although the operator(~K ."V'K)has matrix elements in the i n f i n i t e space. As all the interactions are by def init ion only between nearest neighbor principal layers, this free surface response operator for the slab K can be written as
The interface response o p e r a t o r ~ ' of the N-layered superlattice is defined as the linear superposition of all these free surface response o p e r a t o r s ~ ( K ~ ; K~') and of~.~i~l), namely =
(11>
With the above definitions and due to the fact that all interactions are only between nearest neighbor principal layers, the elements of the interface response operator*~" of the N-layered superlattice are ~'(m~;m'~')
= ~m,{6~,l~'(m~;ml)
+ 6m+l,m' ~ ' i ~ ' ( m ~ ; m + l , l ) +
+ 6~,L+~"(m~; mLm)} + m am-l,m' ~ 'Lm_ I ~ ' ( m ~ ; m - I ,
Lm_l)
,
(12)
where the ~ x u matrix elements are (13)
"A~(m~; ml) =~G~m(~l){~K(11)+*~'l(m1; ml)}++~m(~O)~-~K(OI) "~'(m~ ; mLm) =+~m(ZLm){+~K(Lm Lm) +~l(m Lm; m Lm) } +<-~m(C,Lm+l) +V*K(Lm+I,Lm)
,
(14)
108
L. Dobrzynski
~'(mZ; m+l,l) :-G'm(L Lm)~l(m Lm; m+l,1 )
,
(15)
~'(mL; m-I, Lm_l) : ~ m ( ~ l ) ~ l ( m l ; m-I, Lm_1)
(16)
The interface response operator~#' is therefore entirely determined once one knows the ideal cleavage operators~i~K , the coupling operator~i~ I and the bulk response functions*G'K(~L' ) for I < L , L' < LK D. General Equation of the Interface Response Theory The elements of the N-layered superlattice response function g can be calculated from the general equation <--.
:
(17)
+C
Note that all the present study of N-layered superlattices can also be done with the equivalent general equation of the interface response theory given in Ref. [9]. The equation (17) can be rewritten with the help of Eq. (12) as g(m~; m'L') = 5mm'
(~L') Lm_l) +A~'(mL; ml) ] ~g+~(m-1,Lm_1 ; m'4 )-~
-[~'(mL;m-I,
I
L (ml;m -[
*#~'(m~;mLm)
~'(m~;m+l,1)l
)
~g+-*(mLm;m'L')
i
(is)
Lg+-~(m+1, 1; m'L') ...1 where the*~(m~;m'L') and G"-~m(~Z') are u x~matrices and the entities inside the brackets [ ] are rectangular matrices of dimensions respectively u x 2~ and 2u x ~ . E. Matrix Elements of the Response Function g (i) Relation be.tween g(mC;m'~'), g(m-l, Lm-l.;m'L') and g(ml;m'L'). (18), one obtains easily
K-~(m)Fg+-~(mLm;m'L')
~
[~9(m+1, 1;m'~)
=
-
~EF(m) F~(m-1 ' Lm-1;m'Z') ~ Lg(m Z ; m ~ ) _
+
From Eq.
!K-~m(IZ') I %m _~m(Lm~ ) -
.
(19)
109
Interface Response Theory of Discrete Superlattices In this expression, are used the following 2u x 2~ matrices
~
+~'(m) :
'(ml ; mLm)
~ ' ( m l ; m+l,l)
7
(2o)
L+~(mLm ; mLm)+~'(m Lm ; mLm) ~ ' ( m Lm ; m + l , l ) -
F~'(ml;m-1, H(m) = L ~ , ( m L m ; m _ l ,
Lm_l)
7+(ml;ml) + ~ ' ( m l ;
Lm_l)
ml)
7
~'(mLm; ml)
(21)
_
where the~+(mC; m~) are ~ x ~ unity matrices. Using the general relations
(5) and (6) in Eqs. (13-16) enables us to rewrite
Eq. (18) as ~(mC; m';t') : 5mm, <-G~m(~C') -<=~m(~-Lm) F-~h (mLm;m-l,Lm_l) ~-~' (mLm,ml)]~H~(-mI)
m'C')-
L g(mZ ; m ' ~ ' )
_~(I-~)
[ ~-~h(mI ; mLm) ~-~h(ml;m+l,l)
] ~-~'-1 ( m ) ~ ( m ) ~
±
g(mL m ; m' ')
i
-
22)
L g~-~(m+l,l ; m'C') The use of Eqs (19-21) and (5-6) in Eq. (22) gives f i n a l l y ~(m~;m'£
) : ~mm' e(C-C'-Z) . m-
-~m11-~)~m(9"-Lm) ]~(m)_ I+g(m-l,Lm-l;m'CI' _
~(ml;
m'~')
_
(23) '
where the last term has to be understood as the product of three matrices of successive dimension u x 2u, 2u x 2u and 2~ x u and O(~) is defined by @(Z - C' - i) : 0
,
C < ~' + I
@(~ - ~' - I) : I
,
~ ~ C' + I
,
(24a) (24b) w~
So from Eq. (23), one can obtain all the elements of g once one knows how to calculate the two elements of<-g~ forming the 2u x u matrix appearing on the r i g h t side of Eq. (23). (ii)
-~ Evaluation of K -9(m-l, Lm_l;m I I~ ) and o f ~ ( m l ; m ' ~ ' ) .
We w i l l
show now that
110
L. Dobrzynski
these two matrix elements can be calculated by the well known transfer matrix methods {12-15], starting from Eq. (19). Let us define for this purpose the 2~ x 2u matrices
<-P~(K) : <-#~(m) : and ~'(m,m')
:
-~-~'-l(m)+F~(m)
m 11 ~(m") m"=m'+l
,
(25) m ) m'
(26)
In this equation, the symbol ~ means a matrix product, such that, "-~(m,m') : ~
,
m' > m
,
(27a)
"-R~(m,m) = "-~
,
(27b)
"-~(m,m-Z) = ~-#~(m)
,
(27c)
~(m,m-N) =+#~(m)~(m-Z)....~(m-N+l)
(27d)
Using (m-m') times the Eq. (19), one obtains the following recurrence relations
Eg+(m+l, l;m'C' )_I --=R(m,m') _F g"--~(m"Lm' ;m'C')I~_+ g(m'+l,l;m'Z')_
, m > m'
g(m,Lm;m' ~') ] =,-~-l(m,_Z,m ) I g+-~(m'-l'Lm'-l;m'C')I I +~(m+Z,1;m'~') ~-~ R _ '~(m,l;m,C,) ,m < m -i
(28a)
(28b)
Considering the definition (26) of R(m,m'), one obtains easily as particular cases of the above recurrence relations [ ~(m'+(n-n' )N,Lm, ;m' ~.' ) I =~(m'+N,m'~ (n-n')E ~(m'+l+(n-n ' )N,I ;m' ~' ) '
*~(m' ,Lm, ;m'~' ) 1 , n > n' , (29a) "-~(m'+Z,1;m' C'
I g~-~(m'+(n-n')N'Lm';m'Z')I:~(m'+N(n-n'),m') I ~ I( m '. ''Lm'; _ ~ m ~') , n< n'. (29b) g~(m'+l+(n-n')N,1;m'Z') _ g(m'+l ; m'~' _
Let's add to these reccurence relations (29), the following particular value of Eq. (19)
Interface Response Theory of Discrete Superlattices
g(m , Lm, ;m'~.' ) "~(m'+1,1;m'~.' )
E
]
='~I~(m' )
[
++'~'-i (m')
111
"~(m,_l,Lm,_l; m, ) +~(m' I ;m' ~,' )
.~m,(L m ~.')
(29c)
The Eqs (29) are similar to the equation written in Ref. (15) for the calculation of the bulk Green's function of a homogeneous material. One sees that R(m'+N,m') is the transfer matrix between two equivalent planes of the N-layered superlattice. The solution of the system of Eqs. (29) can be obtained with the help of the eigenvalues and of the eigenvectors of the transfer matrix R(m'+N,m'). Let us recall briefly this method, as i t was proposed for bulk Green's functions in Ref. [15]. Half the eigenvalues of~'(m'+N,m') have modulus less than 1 and half have modulus greater than i. Consider now eB' B = 1,2 .... 2u to be the 2u normalized eigenvectors of"~'(m'+N,m') with respective eigenvalues ~B" Let us assume that IXBI < 1 when 1< B < ~ and [~B] > 1 for ~ + 1~ B < 2u. In order for the response function to satisfy (29a) and (29b) and remain normalizable, one must only retain the eigenvalues I~BI < lwhen~(m'+N,m') is diagonalized in (29a) and only retain IxBI > 1 in (29b). This implies that the (2~x2u) matrices appearing on the right side of Eqs. (29a) and (29b) can be expanded in terms of the eB's with I ~ B and u + 1 < B ~ 2u respectively. In particular, we may write g(m' Lm,,"m' ~' ) ~
~2 .~J~1
(30)
[+g(m'*l'l;m'~')-] = [~S*I +~'i ] and I +g(m'-1'Lm'-l;m'~'
"~(m'l;m'9.')
~4 "+~2
I:I~3
+'J~2 ]
(sl)
with
(SI)~y :
(~a " ~y)
($2)
(um
:
'
ey)
(S3)~y :
( ~ • ~y+~),
(S4)~y :
( ~ • ~+~),
y = 1,2 . . . . . ~ ,
(32)
112 where u
L. Dobrzynski and V
, m = 1,2 . . . . .
are 2~ - dimensional column vectors with compo-
nents given by (~)i
=
5~i
(33a)
(#m)i
=
~m+~,i,
and i = 1,2 . . . . . 2u .
(33b)
The ~ x u matrices+~l a n d ~ 2 represent the unknown expansion c o e f f i c i e n t s .
Substi
t u t i n g Eqs. (30) and (31) in Eq. (29c) enables to determine Jl and+J~2 and then from Eq. (30) and (31), the
and L ~ ( m ' + 1 , 1 ; m'L ) _ The recurrence relations
_
~(m'l;
)
m'L')
(28) and (29) provide then
~-g(m-1, Lm_1;m'L') _! ~ ( m l ; m, 4') And f i n a l l y
g(m -1,Lm,_1 ,m
-~
I
Eq. (23) enables to calculate any element of the response function g
of N-layered s u p e r l a t t i c e . Let us recall
also a general property of the t r a n s f e r matrices. They are symplec-
t i c matrices and t h e i r eigenvalues can be associated in couples such that i f ~g is one eigenvalue o f ~ ( m ' + N ,
m'), then ~ I
is also an eigenvalue o f ~ ( m ' + N , m ' ) .
From this property follows immediately that det I+R~(m'+N,m') I : i ,
Vm'
(34)
F. Physical Properties of N-Layered Superlattices All the physical properties of N-layered s u p e r l a t t i c e
can be studied once the
response function g is known, in the same manner as f o r usual bulk s o l i d s . Let us stress only here that the bulk dispersion r e l a t i o n s
for the e x c i t a t i o n s
(electrons, phonons, magnons . . . . ) under study, can be easily obtained from Eqs. (29)and (25) to be given by detl~(m'+N,m' ) - e ik3au ~T~I = 0
,
where k3 and a u are respectively the propagation vector and the l a t t i c e
(35) parameter
(distance between two equivalent planes) perpendicular to the slabs of the N-
Interface Response Theory of Discrete Superlattices
113
layered superlattice. Note that these bulk dispersion relations are, of course, independent of the value of m' used in Eq. (39), in particular one may take m ' : O . Rather than expending more on the above general results, let us now turn to a simple particular case, for which all the above equations can be solved in closed form. 3. Exact Solutions for Simple Models In the particular cases, for which the matrices*~(n K S; n' K' ~') are scalars, one easily sees from Eqs. (34) that ~*'i
:
I R22 _ -R21
-RI2 I RII
(36)
Then using N times the Eq. (19) one obtains !g(nNLN;n'K'~')+g(n-2,N,LN;n'K'~.')
I
g ( n + l , l , l ; n ' K' ~' ) + g ( n - l , l , l ; n ' K ' ~ ' ) _
= (RII+R22)*I *.
I
o(n-I,N,LN;n'K'~.') I ~ g(nl l ; n ' K ' ~ ' )
+ 6nn' ~ - 5n-l,n ' ~ - I
~
(37)
where
: R(N,K')
G K, (1~') l
I
I<- (K')
4 . - -),-
GK'
(LK' ~')_]
And from Eq. (35), the dispersion relation for the eigenvalues of the N-layered sup erlattice is obtained to be 2 cos k3a u :
RII + R22
(38)
Remember that the solution of h(n+l,n') + h ( n - l , n ' ) - 2nh(n,n') = F1 6n_l, n ' + F2 ~nn' + F3 an+l,n' is for - ~ < n,n' < + h(n,n') : ~
t
(F I t Jn-n'-ll + F2 tln-n'[
+ F3 t
[n-n'+l[)
(39)
(4o)
+
114
L. Oobrzynski
where
1/2 t :
Then define here
I
n -
(n 2 -
I)
1/2
,
q + i(I q2) q + (n 2 - I)1/2
n >
I
, -1 < n < I , n < -I
(41)
(4z)
2n = Rll + R22 and write the general solution of Eq. (37) as
I g(n-I'N'LN;n'K'z') I t In-n'[ _*-~,-1 tln-n'-ll) g(nl I;n'K'L') = tT-1 (*T*t
(43)
or with the help of the definitions given by Eqs. (25-27) as
I g(n-I)'N'LN;n'K'~') 1 g(n l l;n'K'~')
t
(tln-n'l~--~N,K,)_tln-n'-ll y-1 (K',O))
: ~
~-I(K')
i GK'(IL') 1 GK,(LK,L,)
(44)
Using Eq. (19), one obtains also
I g(n K LK;n'K'~' ) =I ~ t
~(KO) ~(N K' ) t ]n-n'[ - ~,-1 (K' ,0) tin-n'-II )
g(n,K+I,I;n'K'L') + 6nn,~(KK, )
]
I GK'(I~')
]
+~-I(K,) -GK'(LK' L , ) j
,
for i < K < N, and where"-R~(KK') is the matrice product defined by Eqs. remembering that~(K,K') =+0~, for K' > K.
(45) (25-27),
The results given by Eqs. (44) and (45) provide all the elements of~necessary in order to calculate from Eq. (23), all the other matrix elements of the response function g of a N-layered superlattice, for which these elements g(nKL;n'K'~') are scalars. So the problem under study here is completly solved in closed form. These solutions were used for the study of s electrons in N-layered superlattices [1~,within the tight-binding approximation.
Interface Response Theory of Discrete Superlattices
115
4. Discussions Although this paper is focussed only on discrete models, i t would be unfair not to mention the continuous response function theories of superlattices. The Ninterface problem was represented [18] as a set of N-scattering centres. A multiple-scattering technique was required because the i t e r a t i v e procedure did not lead to a closed expression of the response function. A Green's function formulation of an N-interface system was also proposed [19] and closed form expression for the inverse of the interface projected Green's function elements were given. The so-called direct method, using the eigenvalues of the equations and the corresponding Wronskian, was used [20-21] to derive some elements of the response functions of 2-1ayered superlattices. Other elements of such response functions describing the coupling of i t s different Fourier components were given for specific simple models [22]. The Surface Green's Function Matching Theory was applied recently to 2-1ayered superlattices [10,23] and a closed expression of the interface projected elements of the Green's function was given in a Fourier representation. An Interface Response Theory of N-layered Continuous Superlattices giving all the elements of the response function in real space was also developped [24] and offers together with the theory presented in this paper an unified method to study discrete and continuous models of superlattices. Finally and in order to stress the practical interest of N-layered superlattices l e t us recall that the idea of polytype (ABC) superlattices was recently proposed [25] together with an application to InAs- GaSb- AISb multiheterojunctions. The dispersion relation of polytype superlattices was also already discussed using the envelope-function approxi~w~tion {26]. The miniband structure of a GaAs-Gal. x Al x As superlattice consisting of double layers of GaAs and Gal_x Al x As mater i a l s per period was also studied [27]. Energy levels and electron wave functions in semiconductor quantum wells having superlattice alloy like material (GaAs/ Al Ga As) as barrier layers were also studied both experimentally and theoretically [28]. Compared with binary (AB) superlattices, a polytype superlattice provides additional degrees of freedom and more complex properties. References 1. 2. 3. 4.
J. Koutecky, Phys. Rev., 108, 13 (1957). L.N. Rosenzweig, Uch. Z ap~ar k. Gos. Univ. Tr. Fiz. Mat. Otdel., 2, 19 (1950). S.G. Davison and K.W. Sulston, Phys. stat. sol. (b~, 120, 415 (198~). L. Dobrzynski, B. Djafari-Rouhani and O. Hardouin Dupa'r-c, J. Electron. Spectrosc. Relat. Phenom., 30, 119 (1983) and Phys. Rev. B, 29, 3138 (1984). 5. B. Djafari-Rouhani, L. ITobrzynski and P. Masri, Phys. Rev__B, 31, 7739 (1985). 6. P. Masri, L. Dobrzynski, B. Djafari-Rouhani and J.O.A. I d i o d i , ~ u r f . Sci., 166, 301 (1986). 7. L. Dobrzynski, B. Djafari-Rouhani and H. Puszkarski, Phys. Rev. B, 33, 3251 (1986)
116 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
L. Dobrzynski F. Garcia-Moliner and V.R. Velasco, Physica Scripta, 34, 252 (1986). L. Dobrzynski, Surf. Sci. Reports, 6, 119 (1986). F. Garcia-Moliner and V.R. Velasco,-Prog. in Surf. Sci., 21, ~3 (1986) L. Dobrzynski, V. Velasco and F. Garcia-Moliner, Phys. Rev., 8, 35, 5872 (1987) L.M. Falicov and F. Yndurain, J. Phys. C, 8, 147 (1975). K.S. Dy, Shi-Yu Wu and T. S p r a t l i n , Phys. Rev. B, 20, ~237 (1979). M.P. Lopez Sancho, J.M. Lopez Sancho and J. R u b i o , ~ . Phys. F 14, 1205 (1984). D.H. Lee and J.D. Joannopoulos, Phys. Rev. B 23, 4997 (1981) an--a ~ B 24, 6899 (1981) and references enclosed. See for example, L. Dobrzynski, Ann. Phys. (Paris) 4, 637 (1969). P. Masri and L. Dobrzynski, to be published. I. Bartos, Phys. Stat. Sol. (b) 85, K 127 (1978). H. Ueba and S.G. Davison, J. Phy~ C. 13, 1175 (1980). R.E. Camley, B. Djafari-Rouhani, L. DoFzynski and A.A. Maradudin, Phys. Rev. B 27, 7318 (1983) and J. Vac. Sci. Tech. B1, 371 (1983). T.--Bartos and S.G. Davison, Phys. Rev. B -SrZ, 7863 (1985). M. Babiker, D.R. T i l l e y , J. Phys. C 17, L--~29 (1984) ; M. Babiker, D.R. T i l l e ~ and E.L. Albuquerque, J. Phys. C 18,--i-285 (1985). F. Garcia-Moliner and V.R. VelascT,, Surface Sci. 175, 9 (1986). L. Dobrzynski, Surf. Sci. 182, 362 (1987). L. Esaki, L.L. Chang and E.T. Mendez, Jpn. J. Appl. Phys. 2__0_0,L 529 (1981). G. Bastard, Phys. Rev. B 25, 7584 (198Z). H.X. Jiang and J.Y. Lin, l~-~ys. Rev. B 33, 5851 (1986). H. Sakaki, M. Tsuchiya and J. Yoshino,-~ppl. Phys. Lett. 47, 295 (i985).
0079-6816/87 $0.00 + .50 Copyright © 1988 Pergamon Press pie
INTERFACE RESPONSE THEORY OF DISCRETE SUPERLATTICES LEONARD DOBRZYNSKI Equipe Internationale de Dynarnique des Interfaces, Laboratoire de Dynamique des Cristaux Moleculaires, Unitd associee (No 801) au Centre National de la Recherche Scientifique, U.F.R. de Physique, Universite des Sciences et Techniques de Lille Flandres Artois, 59655 Villeneuve d'Ascq Cedex, France
Abstract Response function theories of discrete superlattices are reviewed. General expressions for the response functions of N-layered discrete superlattices are defined, with the help of a recent interface response theory. Such superlattices are formed out of a periodic repetition of N different crystalline slabs. I. Introduction Professor Koutecky [ I ] was the f i r s t for electrons after Rosenzweig's work for phonons [2] to use response functions (called also Green's functions) for the study of discrete materials limited by a free surface. This is now a well established method for the study of Surfaces and Interfaces. I t was also extended recent l y to the study of Superlattices. This mini review focuss on response functions of discrete models of superlattices. Response functions of continuous models of superlattices will be just briefly discussed at the end. Other mathematical methods for the study of superlattices are not reviewed here. This do not mean that these methods are not important, for example the transfer matrix method is very well designed for the calculation of the eigenvalues associated to physical properties of superlattices. This choice of focuss is mostly motivated by thepionnering interest of Professor Koutecky { I ] in Green's
103
104
L. Dobrzynski
functions for atomic models of surfaces. An N-layered s u p e r l a t t i c e is formed out of a periodic r e p e t i t i o n of N d i f f e r e n t c r y s t a l l i n e slabs. In most of the e x i s t i n g litterature
the word s u p e r l a t t i c e is used for 2-1ayered s u p e r l a t t i c e s .
Already N interfaces in an homogeneous crystal can be viewed as a simple superlatt i c e . A general expression for the corresponding response function was given [ 3 I as a function of the response function of the homogeneous c r y s t a l .
Its interface
elements were also derived in closed form. The formalism was seen to be generalizable to m u l t i - l a y e r systems, where the interfaces separate regions of d i f f e r e n t structure. Indeed one can construct [ 4 - 7 I the s u p e r l a t t i c e response function out of the free surface slab's response functions, using the coupling operators between the independant slabs and the Dyson equation.
However t h i s obliges in a f i r s t
step to
calculate the free surface slab's response functions out of the corresponding bulk response functions. This step was shown unnecessary, as i t
is possible to obtain
the s u p e r l a t t i c e response function d i r e c t l y from elements of the bulk response functions of each slab [8-91. This was achieved with the Surface Green's Function Method [ 8 ] which uses r e f l e x i o n and transmission operators through the slabs for 2-1ayered superlattices and with the Interface Response Theory of Composite Systems [ 9 ] which constructs any f i n a l elements. The f i r s t one [ l l I .
composite system out of i t s indeDendent
approach was recently reviewed [10I and compared to the second
So the remaining of this paper is mostly devoted to the interface res-
ponse theory of N-layered discrete s u p e r l a t t i c e s , which was only sketched in the original
paper [ 9 I . 2. General Interface Response Theory of Discrete Superlattices
A. D e f i n i t i o n of N-Layered Superlattice ( i ) Slab K of one homogeneous material.
Consider a slab labelled by the index K
of a homogeneous material and possessing translationnal
invariance in the space
directions XI and X2. This slab can be considered to be formed out of LK equivalent " p r i n c i p a l
layers" labelled by an index ~ such that I < ~ < LK. A principal
layer may consist of one or several atomic planes, depending on the range of the interatomic i n t e r a c t i o n s .
By d e f i n i t i o n ,
all
interactions
in the crystal slab are
described as interactions between the nearest neighbouring " p r i n c i p a l Due to t r a n s l a t i o n invariance p a r a l l e l
to the surfaces of the slab, one does the
usual Fourier transformation, which makes a l l where ~
is the propagation vector p a r a l l e l
the operators to be function of ~ ,
to the surfaces. In this paper a l l
the e n t i t i e s are function of k~//, although for s i m p l i c i t y , will
not be e x p l i c i t l y
w r i t t e n down.
layers".
this dependence on k~
Interface Response Theory of Discrete Superlattices l i i ) Slab of N Different Homogeneous Materials.
105
Consider now N different homoge-
neous slabs, limited each by ideally truncated free surfaces. This means that each of these homogeneous slabs can be considered as cut out of the corresponding i n f i nite material by a cleavage which leaves their bulk values to the interactions near the two free surfaces of the slab. Couple now together these N ideally cleaved slabs, (K = I, 2. . . .
N), by an
interface coupling operator called~V'i . We suppose that this coupling has matrix elements only between two interface adjacent "principal layers", (K, LK) and (K+I, 1) for example. However all what will follow can be easily generalized to interface coupling penetrating more deeply inside each adjacent slab by redefining the thickness of the "principal layers". This redefinition can also be easily done when the dimensions of the "principal layers" (due to the range of the bulk interactions) are not the same in each slab K. One obtains in this manner a N-layered slab, (we label by the index n), having ideal free surfaces at the principal (iii)
N-Layered Superlattice.
dically
layers (K = 1, ~ = I) and at (K : N, ~=LN).
In the same manner as above one can couple perio-
(-~ < n < +~) an i n f i n i t e
number of the above N layered slabs in order to
obtain another new bulk material, called here N-layered s u p e r l a t t i c e . In what follows,
will
the following more condensed notations
m ~ n N + K ~ (n ,K )
,
(1)
m' ~ n'N + K' ~ ( n ' , K ' )
,
(2)
prove convenient for l a b e l l i n g a given slab K (or K') in the unit cell n
(or n'). B. Response Functions
<--w
( i ) Bulk Response Function GK f o r Each Submaterial K.
For the i n f i n i t e
material
K, consider an operator~F~K . This bulk material can be considered as an i n f i n i t e stack of "principal
layers". The p e r i o d i c i t y
system to that of noninteracting
parallel
one-dimensional
to the layers reduces the
chains. There is one chain f o r
each wave vector ~U along the layer. The operator+FFK (function of k~H) w i l l general have the following tridiagonal
~K(OI)
"-F~K(O0 )
(oi)
form.
~-FFK(OI )
TFK
]
in
(3)
106
L. Dobrzynski
In this expression the intra and inter-principal layers interactions, respectively
+hrK(O0) ;+F~K(OI) and~-H~;(Ol) are square submatrices, whose dimension is function of the problem studied. For example, if+FCK is the dynamical matrix of a 3-dimensional monoatomic l a t t i c e with i n t e r a c t i o n s between f i r s t
nearest neighbor atomic planes,
the dimension of these submatrices w i l l be in general 3 x 3. If~rK is an Hamiltonian describing e l e c t r o n i c i n t e r a c t i o n s in t i g h t binding and i f there are p p r i n c i p a l layer o r b i t a l s (number of d i f f e r e n t e l e c t r o n i c o r b i t a l s m u l t i p l i e d by the number of atomic planes in the p r i n c i p a l l a y e r ) , the~H~K(O0) w i l l be u x u matrices. In what follows, we w i l l
refer in general to these operators as p x p matrices•
Define now the bulk response function GK by
where'*is
the u n i t y matrix.
There are many well known methods for c a l c u l a t i n g the p x p elements of GK, f o r the t r i d i a g o n a l operators given by Eq. (3). In what follows, we w i l l use the ones based on the notion of t r a n s f e r matrix, which provide [12-13] the following matrix relations +~K(~C') ="T'K(~-~') "-~K(II) ~+ GK(CC, ) = + f ~ L ' - ~ )
~K(II)
,
L b C'
,
(5)
,
L~ C'
,
(6)
for all the p x u inter layer matrix elements of KG function of GK(11) and of two u x u transfer matrices"-fK and+-fK. The calculation of these three submatrices has to be in general done by standard~umerical
methods [12-15], apart for simple
models for which these submatrices reduce to scalars and H~(01) = HK(01), then one may obtain the elements of GK in closed forms [16]
GK(Lg' ) (ii)
:
t~ ~-C'I GK(ll )
(7)
Reference Response Function~G+of the N-Layered Superlattice.
Define [ 9 ] for
the N-layered s u p e r l a t t i c e a reference response function+~as a block diagonal matrix formed out of only the elements of the bulk response functions GZK contained w i t h i n the space of d e f i n i t i o n of each slab, namely, with the notations given by Eqs ( I - 2 ) "-~(mL,m'~') = ~mm,~m(gL') , 1 ~ ~, L' ~ Lm = L K
,
(8)
where ~mm' is the usual Kronecker symbol. ( i i i ) Response F u n c t i o n ~ o f the N-Layered Superlattice. Out of each i n f i n i t e material K, construct f i r s t an i d e a l l y cleaved free surface slab K, by c u t t i n g
107
Interface Response Theory of Discrete Superlattices
out in Eq. (3) all interactions between the principal layers ~ = 0 and I on one hand and ~ = L K and LK + I on the other hand ; this procedure w i l l be described by a cleavage o p e r a t o r ~ K. Then couple all these ideally cleaved slabs by an interface coupling*i~ I, as defined in section 2.A. These procedures define out of the {~K}, {~K} andS"I, the operatorK~of the N-layered superlattice, to which one associates a response function+gby g . +~" = +1*
(9)
C. Interface Response Operator~' Corresponding to the above building procedures of the N-layered superlattice, introduce [ g ] f i r s t a free surface response o p e r a t o r ~ (K~ ; K~') for each slab K. This operator is formed out of the elements o f ~ K .~K) s t r i c t l y contained inside the domain of definition of the K slab (I < ~, 4' < LK), although the operator(~K ."V'K)has matrix elements in the i n f i n i t e space. As all the interactions are by def init ion only between nearest neighbor principal layers, this free surface response operator for the slab K can be written as
The interface response o p e r a t o r ~ ' of the N-layered superlattice is defined as the linear superposition of all these free surface response o p e r a t o r s ~ ( K ~ ; K~') and of~.~i~l), namely =
(11>
With the above definitions and due to the fact that all interactions are only between nearest neighbor principal layers, the elements of the interface response operator*~" of the N-layered superlattice are ~'(m~;m'~')
= ~m,{6~,l~'(m~;ml)
+ 6m+l,m' ~ ' i ~ ' ( m ~ ; m + l , l ) +
+ 6~,L+~"(m~; mLm)} + m am-l,m' ~ 'Lm_ I ~ ' ( m ~ ; m - I ,
Lm_l)
,
(12)
where the ~ x u matrix elements are (13)
"A~(m~; ml) =~G~m(~l){~K(11)+*~'l(m1; ml)}++~m(~O)~-~K(OI) "~'(m~ ; mLm) =+~m(ZLm){+~K(Lm Lm) +~l(m Lm; m Lm) } +<-~m(C,Lm+l) +V*K(Lm+I,Lm)
,
(14)
108
L. Dobrzynski
~'(mZ; m+l,l) :-G'm(L Lm)~l(m Lm; m+l,1 )
,
(15)
~'(mL; m-I, Lm_l) : ~ m ( ~ l ) ~ l ( m l ; m-I, Lm_1)
(16)
The interface response operator~#' is therefore entirely determined once one knows the ideal cleavage operators~i~K , the coupling operator~i~ I and the bulk response functions*G'K(~L' ) for I < L , L' < LK D. General Equation of the Interface Response Theory The elements of the N-layered superlattice response function g can be calculated from the general equation <--.
:
(17)
+C
Note that all the present study of N-layered superlattices can also be done with the equivalent general equation of the interface response theory given in Ref. [9]. The equation (17) can be rewritten with the help of Eq. (12) as g(m~; m'L') = 5mm'
(~L') Lm_l) +A~'(mL; ml) ] ~g+~(m-1,Lm_1 ; m'4 )-~
-[~'(mL;m-I,
I
L (ml;m -[
*#~'(m~;mLm)
~'(m~;m+l,1)l
)
~g+-*(mLm;m'L')
i
(is)
Lg+-~(m+1, 1; m'L') ...1 where the*~(m~;m'L') and G"-~m(~Z') are u x~matrices and the entities inside the brackets [ ] are rectangular matrices of dimensions respectively u x 2~ and 2u x ~ . E. Matrix Elements of the Response Function g (i) Relation be.tween g(mC;m'~'), g(m-l, Lm-l.;m'L') and g(ml;m'L'). (18), one obtains easily
K-~(m)Fg+-~(mLm;m'L')
~
[~9(m+1, 1;m'~)
=
-
~EF(m) F~(m-1 ' Lm-1;m'Z') ~ Lg(m Z ; m ~ ) _
+
From Eq.
!K-~m(IZ') I %m _~m(Lm~ ) -
.
(19)
109
Interface Response Theory of Discrete Superlattices In this expression, are used the following 2u x 2~ matrices
~
+~'(m) :
'(ml ; mLm)
~ ' ( m l ; m+l,l)
7
(2o)
L+~(mLm ; mLm)+~'(m Lm ; mLm) ~ ' ( m Lm ; m + l , l ) -
F~'(ml;m-1, H(m) = L ~ , ( m L m ; m _ l ,
Lm_l)
7+(ml;ml) + ~ ' ( m l ;
Lm_l)
ml)
7
~'(mLm; ml)
(21)
_
where the~+(mC; m~) are ~ x ~ unity matrices. Using the general relations
(5) and (6) in Eqs. (13-16) enables us to rewrite
Eq. (18) as ~(mC; m';t') : 5mm, <-G~m(~C') -<=~m(~-Lm) F-~h (mLm;m-l,Lm_l) ~-~' (mLm,ml)]~H~(-mI)
m'C')-
L g(mZ ; m ' ~ ' )
_~(I-~)
[ ~-~h(mI ; mLm) ~-~h(ml;m+l,l)
] ~-~'-1 ( m ) ~ ( m ) ~
±
g(mL m ; m' ')
i
-
22)
L g~-~(m+l,l ; m'C') The use of Eqs (19-21) and (5-6) in Eq. (22) gives f i n a l l y ~(m~;m'£
) : ~mm' e(C-C'-Z) . m-
-~m11-~)~m(9"-Lm) ]~(m)_ I+g(m-l,Lm-l;m'CI' _
~(ml;
m'~')
_
(23) '
where the last term has to be understood as the product of three matrices of successive dimension u x 2u, 2u x 2u and 2~ x u and O(~) is defined by @(Z - C' - i) : 0
,
C < ~' + I
@(~ - ~' - I) : I
,
~ ~ C' + I
,
(24a) (24b) w~
So from Eq. (23), one can obtain all the elements of g once one knows how to calculate the two elements of<-g~ forming the 2u x u matrix appearing on the r i g h t side of Eq. (23). (ii)
-~ Evaluation of K -9(m-l, Lm_l;m I I~ ) and o f ~ ( m l ; m ' ~ ' ) .
We w i l l
show now that
110
L. Dobrzynski
these two matrix elements can be calculated by the well known transfer matrix methods {12-15], starting from Eq. (19). Let us define for this purpose the 2~ x 2u matrices
<-P~(K) : <-#~(m) : and ~'(m,m')
:
-~-~'-l(m)+F~(m)
m 11 ~(m") m"=m'+l
,
(25) m ) m'
(26)
In this equation, the symbol ~ means a matrix product, such that, "-~(m,m') : ~
,
m' > m
,
(27a)
"-R~(m,m) = "-~
,
(27b)
"-~(m,m-Z) = ~-#~(m)
,
(27c)
~(m,m-N) =+#~(m)~(m-Z)....~(m-N+l)
(27d)
Using (m-m') times the Eq. (19), one obtains the following recurrence relations
Eg+(m+l, l;m'C' )_I --=R(m,m') _F g"--~(m"Lm' ;m'C')I~_+ g(m'+l,l;m'Z')_
, m > m'
g(m,Lm;m' ~') ] =,-~-l(m,_Z,m ) I g+-~(m'-l'Lm'-l;m'C')I I +~(m+Z,1;m'~') ~-~ R _ '~(m,l;m,C,) ,m < m -i
(28a)
(28b)
Considering the definition (26) of R(m,m'), one obtains easily as particular cases of the above recurrence relations [ ~(m'+(n-n' )N,Lm, ;m' ~.' ) I =~(m'+N,m'~ (n-n')E ~(m'+l+(n-n ' )N,I ;m' ~' ) '
*~(m' ,Lm, ;m'~' ) 1 , n > n' , (29a) "-~(m'+Z,1;m' C'
I g~-~(m'+(n-n')N'Lm';m'Z')I:~(m'+N(n-n'),m') I ~ I( m '. ''Lm'; _ ~ m ~') , n< n'. (29b) g~(m'+l+(n-n')N,1;m'Z') _ g(m'+l ; m'~' _
Let's add to these reccurence relations (29), the following particular value of Eq. (19)
Interface Response Theory of Discrete Superlattices
g(m , Lm, ;m'~.' ) "~(m'+1,1;m'~.' )
E
]
='~I~(m' )
[
++'~'-i (m')
111
"~(m,_l,Lm,_l; m, ) +~(m' I ;m' ~,' )
.~m,(L m ~.')
(29c)
The Eqs (29) are similar to the equation written in Ref. (15) for the calculation of the bulk Green's function of a homogeneous material. One sees that R(m'+N,m') is the transfer matrix between two equivalent planes of the N-layered superlattice. The solution of the system of Eqs. (29) can be obtained with the help of the eigenvalues and of the eigenvectors of the transfer matrix R(m'+N,m'). Let us recall briefly this method, as i t was proposed for bulk Green's functions in Ref. [15]. Half the eigenvalues of~'(m'+N,m') have modulus less than 1 and half have modulus greater than i. Consider now eB' B = 1,2 .... 2u to be the 2u normalized eigenvectors of"~'(m'+N,m') with respective eigenvalues ~B" Let us assume that IXBI < 1 when 1< B < ~ and [~B] > 1 for ~ + 1~ B < 2u. In order for the response function to satisfy (29a) and (29b) and remain normalizable, one must only retain the eigenvalues I~BI < lwhen~(m'+N,m') is diagonalized in (29a) and only retain IxBI > 1 in (29b). This implies that the (2~x2u) matrices appearing on the right side of Eqs. (29a) and (29b) can be expanded in terms of the eB's with I ~ B and u + 1 < B ~ 2u respectively. In particular, we may write g(m' Lm,,"m' ~' ) ~
~2 .~J~1
(30)
[+g(m'*l'l;m'~')-] = [~S*I +~'i ] and I +g(m'-1'Lm'-l;m'~'
"~(m'l;m'9.')
~4 "+~2
I:I~3
+'J~2 ]
(sl)
with
(SI)~y :
(~a " ~y)
($2)
(um
:
'
ey)
(S3)~y :
( ~ • ~y+~),
(S4)~y :
( ~ • ~+~),
y = 1,2 . . . . . ~ ,
(32)
112 where u
L. Dobrzynski and V
, m = 1,2 . . . . .
are 2~ - dimensional column vectors with compo-
nents given by (~)i
=
5~i
(33a)
(#m)i
=
~m+~,i,
and i = 1,2 . . . . . 2u .
(33b)
The ~ x u matrices+~l a n d ~ 2 represent the unknown expansion c o e f f i c i e n t s .
Substi
t u t i n g Eqs. (30) and (31) in Eq. (29c) enables to determine Jl and+J~2 and then from Eq. (30) and (31), the
and L ~ ( m ' + 1 , 1 ; m'L ) _ The recurrence relations
_
~(m'l;
)
m'L')
(28) and (29) provide then
~-g(m-1, Lm_1;m'L') _! ~ ( m l ; m, 4') And f i n a l l y
g(m -1,Lm,_1 ,m
-~
I
Eq. (23) enables to calculate any element of the response function g
of N-layered s u p e r l a t t i c e . Let us recall
also a general property of the t r a n s f e r matrices. They are symplec-
t i c matrices and t h e i r eigenvalues can be associated in couples such that i f ~g is one eigenvalue o f ~ ( m ' + N ,
m'), then ~ I
is also an eigenvalue o f ~ ( m ' + N , m ' ) .
From this property follows immediately that det I+R~(m'+N,m') I : i ,
Vm'
(34)
F. Physical Properties of N-Layered Superlattices All the physical properties of N-layered s u p e r l a t t i c e
can be studied once the
response function g is known, in the same manner as f o r usual bulk s o l i d s . Let us stress only here that the bulk dispersion r e l a t i o n s
for the e x c i t a t i o n s
(electrons, phonons, magnons . . . . ) under study, can be easily obtained from Eqs. (29)and (25) to be given by detl~(m'+N,m' ) - e ik3au ~T~I = 0
,
where k3 and a u are respectively the propagation vector and the l a t t i c e
(35) parameter
(distance between two equivalent planes) perpendicular to the slabs of the N-
Interface Response Theory of Discrete Superlattices
113
layered superlattice. Note that these bulk dispersion relations are, of course, independent of the value of m' used in Eq. (39), in particular one may take m ' : O . Rather than expending more on the above general results, let us now turn to a simple particular case, for which all the above equations can be solved in closed form. 3. Exact Solutions for Simple Models In the particular cases, for which the matrices*~(n K S; n' K' ~') are scalars, one easily sees from Eqs. (34) that ~*'i
:
I R22 _ -R21
-RI2 I RII
(36)
Then using N times the Eq. (19) one obtains !g(nNLN;n'K'~')+g(n-2,N,LN;n'K'~.')
I
g ( n + l , l , l ; n ' K' ~' ) + g ( n - l , l , l ; n ' K ' ~ ' ) _
= (RII+R22)*I *.
I
o(n-I,N,LN;n'K'~.') I ~ g(nl l ; n ' K ' ~ ' )
+ 6nn' ~ - 5n-l,n ' ~ - I
~
(37)
where
: R(N,K')
G K, (1~') l
I
I<- (K')
4 . - -),-
GK'
(LK' ~')_]
And from Eq. (35), the dispersion relation for the eigenvalues of the N-layered sup erlattice is obtained to be 2 cos k3a u :
RII + R22
(38)
Remember that the solution of h(n+l,n') + h ( n - l , n ' ) - 2nh(n,n') = F1 6n_l, n ' + F2 ~nn' + F3 an+l,n' is for - ~ < n,n' < + h(n,n') : ~
t
(F I t Jn-n'-ll + F2 tln-n'[
+ F3 t
[n-n'+l[)
(39)
(4o)
+
114
L. Oobrzynski
where
1/2 t :
Then define here
I
n -
(n 2 -
I)
1/2
,
q + i(I q2) q + (n 2 - I)1/2
n >
I
, -1 < n < I , n < -I
(41)
(4z)
2n = Rll + R22 and write the general solution of Eq. (37) as
I g(n-I'N'LN;n'K'z') I t In-n'[ _*-~,-1 tln-n'-ll) g(nl I;n'K'L') = tT-1 (*T*t
(43)
or with the help of the definitions given by Eqs. (25-27) as
I g(n-I)'N'LN;n'K'~') 1 g(n l l;n'K'~')
t
(tln-n'l~--~N,K,)_tln-n'-ll y-1 (K',O))
: ~
~-I(K')
i GK'(IL') 1 GK,(LK,L,)
(44)
Using Eq. (19), one obtains also
I g(n K LK;n'K'~' ) =I ~ t
~(KO) ~(N K' ) t ]n-n'[ - ~,-1 (K' ,0) tin-n'-II )
g(n,K+I,I;n'K'L') + 6nn,~(KK, )
]
I GK'(I~')
]
+~-I(K,) -GK'(LK' L , ) j
,
for i < K < N, and where"-R~(KK') is the matrice product defined by Eqs. remembering that~(K,K') =+0~, for K' > K.
(45) (25-27),
The results given by Eqs. (44) and (45) provide all the elements of~necessary in order to calculate from Eq. (23), all the other matrix elements of the response function g of a N-layered superlattice, for which these elements g(nKL;n'K'~') are scalars. So the problem under study here is completly solved in closed form. These solutions were used for the study of s electrons in N-layered superlattices [1~,within the tight-binding approximation.
Interface Response Theory of Discrete Superlattices
115
4. Discussions Although this paper is focussed only on discrete models, i t would be unfair not to mention the continuous response function theories of superlattices. The Ninterface problem was represented [18] as a set of N-scattering centres. A multiple-scattering technique was required because the i t e r a t i v e procedure did not lead to a closed expression of the response function. A Green's function formulation of an N-interface system was also proposed [19] and closed form expression for the inverse of the interface projected Green's function elements were given. The so-called direct method, using the eigenvalues of the equations and the corresponding Wronskian, was used [20-21] to derive some elements of the response functions of 2-1ayered superlattices. Other elements of such response functions describing the coupling of i t s different Fourier components were given for specific simple models [22]. The Surface Green's Function Matching Theory was applied recently to 2-1ayered superlattices [10,23] and a closed expression of the interface projected elements of the Green's function was given in a Fourier representation. An Interface Response Theory of N-layered Continuous Superlattices giving all the elements of the response function in real space was also developped [24] and offers together with the theory presented in this paper an unified method to study discrete and continuous models of superlattices. Finally and in order to stress the practical interest of N-layered superlattices l e t us recall that the idea of polytype (ABC) superlattices was recently proposed [25] together with an application to InAs- GaSb- AISb multiheterojunctions. The dispersion relation of polytype superlattices was also already discussed using the envelope-function approxi~w~tion {26]. The miniband structure of a GaAs-Gal. x Al x As superlattice consisting of double layers of GaAs and Gal_x Al x As mater i a l s per period was also studied [27]. Energy levels and electron wave functions in semiconductor quantum wells having superlattice alloy like material (GaAs/ Al Ga As) as barrier layers were also studied both experimentally and theoretically [28]. Compared with binary (AB) superlattices, a polytype superlattice provides additional degrees of freedom and more complex properties. References 1. 2. 3. 4.
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