- Email: [email protected]
Variation of tight-binding parameters near surfaces
~ " ' ~ Solid State Communications, ~',8~_~:~Printed in Great Britain.
VARIATION
OF
Vol.65,No.8,
pp.775-777,
TIGHT-BINDING
1988.
0038-I098/88 $3.00 + .00 Pergamon Journals Ltd.
PARAMETERS
NEAR
SURFACES
R. A. Barrio Instituto de Investlgaciones en Materiaies, UNAM, Apartado Postal 70-360, 04510, Mdxico, D.F. and M. del Castillo-Mussot Instituto de Fisica, UNAM, Apartado Postal 20-364, 01000, Mdxico, D.F. (Revised Manuscript received 20/06/87 by A.A. Maradudin) Surface effects on the tight-binding parameters for electrons are examined in a very simple system that allows the analytical calculation of all the atomic integrals needed. In spite of the simplicity of the model various qualitative results are obtained. It is seen that the inclusion of electron-electron interaction is essential to characterize the variation of the tight-binding parameters near defects. Effects of surface relaxation are also modelled and useful conclusions are drawn.
The tight-binding method (TBM) has been a useful tool in order to study the electronic properties of solids [1-4], because of its simplicity and the intuitive physical meaning of the site energies (a) and interaction parameters between different sites (/3). However, when the TBM is applied to calculations near defects, like vacancies and surfaces, the is no clear connection between the tight binding parameters and the real interactions between particles in the material and the results obtained are not completely satisfying. Indeed there have been debates [5-6] about its validity because one does not know how these parameters vary around a defect or surface. As an extreme example of the drawbacks of the TBM is the attempt by Sakurai and Sugano [7] to describe defect states in the interface between Si and Si02: The defect states in the Si gap could be at any desired energy by just varying the TBM parameters on the atoms near the defect. There are methods like the ab initio [8] and the semiempirical [9] techniques that have been applied to the same sort of defects and interfaces [10], and give useful local information about the electronic behaviour but it is difficult to connect the results with the TBM. It is the purpose of this paper to investigate this connection. In order to proceed, a very simple system is examined, in which a minimum of approximations should be made: a semi-infinite linear chain of hydrogen atoms with one s-orbital per atom in which one limits the interactions to nearest neighbours only. The advantages of doing this lie in the fact that the presence of a surface atom represents the maximum perturbation of defect in this system and that all fundamental quantities can be calculated analytically. In the TBM one generally assumes an orthonormai basis, therefore, the first difficulty in solving this system to find the appropriate orthonormal basis located around different lattice sites (Wannier functions). The simplest system that contains information about the "bulk" and ~surface~ atoms is a linear chain of 5 atoms in which the functions can be calculated exactly as follows: Consider a group of atomic orbitals l%bi)centered in every site Ri, that are not orthogonal, that is (¢i1%') = SO, (1)
as in a straight LCAO approximation. One may calculate the unitary matrix S -1/2 such that S -112 • S . S -I/2 -~- I,
(2)
and then
(3) J form the desired orthonormai basis. The method used to calculate S -1/2 is straightforward, first one finds the transformation U that diagonaiizes S and therefore
U -I.S'U=S d
(4)
is diagonal and
S-ID
=-
v . s ~ '12 • U -I
(5)
is the desired matrix used in (3). It is clear that this method is difficult to apply to an infinite system, like a solid, but it can be done in the simple chain chosen here. Furthermore in the 5 atom chain, w h e n hydrogen like s-functions are used, the matrix elements of S are calculated analytically [11] in terms of the constant distance between nelghbours R. The one-electron part of the Hamiltonian ~i. =
(¢i,I/'/~°"1¢i.)
(7a)
and /~id,. = (¢i,[Hc°relc]¢),
(7b)
where/_/core = V2 _ ~ i =5 1 1/It - Ri[, r is t he e l e c t ron coord i n a t e and a is the spin coordinate. In the one orthogonal basis we define, accordingly, &~a, &ha, fi~, •b, where s and b refer to ~surface" and ~bulk" respectively. So far the electron-electron interaction has been neglected, these interactions can be calculated in an entirely similar way exactly [11], using the atomic, bulk and surface states. These two-electron integrals have the form
(ABICD) = f ¢ ~ t ( r l ) ¢ b ( r l ) ~ J
775
r12
¢ c ( r 2 ) ¢ D ( r , ) d r l d r 2 (9)
776
V A R I A T I O N OF T I G H T - B I N D I N G P A R A M E T E R S N E A R SURFACES
where A, B, G', D refer to the basis orbitals in the atoms. One can calculate these integrals in the orthogonal base as well. The results for various interatomic distances are given in Table I. There are to i m p o r t a n t results from this table. First one notices t h a t , after orthogonalization the Coulomb integrals ( A A / A A ) and ( A A / B B ) do not vary very much from their atomic values. The second result is t h a t orthogonalization makes the direct Coulomb integrals much larger t h a n the exchange ones ( A A / A B ) and ( A B / A B ) and therefore the latter can be neglected for practical purposes. All the other combinations are relatively small due to overlap. In order to calculate the TB p a r a m e t e r s from the previous results, one can perform a self-consistent Hartree-Fock calculation in the o r t h o n o r m a l basis, retaining only the direct Coulomb two-electron part, t h a t is, using an extended H u b b a r d Hamiltonian, trusting t h a t the results will be physically sound and t h a t no i m p o r t a n t real interactions are being neglected. The connection between the H u b b a r d interactions and the TB p a r a m e t e r s is as follows: Consider the Hubbard-like Hamiltonian,
: E
,,c,t + Z
i,,~
err
%
-.\
leeee~e~'oeeea oae eeHe e ~
,':"...................... 6""
-0.5 0,0 2.0
2.0
4.0
4.0
6.0
Rs(a,u.) (lO)
t
~e. t
cqa eierci*r + Z Pij,~ci~ cier
i,~
Figure 1. Site energies (alphas) and hoping integrals (betas) for a bulk a t o m (b) and the surface a t o m (s) of a linear chain w i t h neighbour distance R = 3.0 a.u. as a function of the separation of the surface a t o m Rs. Matrix elements between atomic orbitals are shown as dashed lines, between o r t h o n o r m a l states in a 5 a t o m chain are shown in solid lines and the tight binding parameters, after self-consistent calculation in a semi-infinite chain, are shown in dotted lines. The origin for the self-consistent alphas has been shifted by 2.2 a.u.
(11)
ij,er
which bears the tight-binding form and where the tight binding p a r a m e t e r s are
~
\~ ............. b
%
-1,0
+
OLitr
-2.0
>n~ - 1 . 5 I.~ Z w
\
%
e*e.
~1,-
=
-
3
here i and j are nearest neighbours and eia (eia) and nia are the creation (annihilation) and n u m b e r operators for electrons in the orthogonal orbital labeled i with spin a. I n the Hartree-Foek approximation this can be w r i t t e n
eft Otio-
/11-., \
O
BETA
ALPHA
% % %
-2.5
+ Y]~(iilii)nianio + Z Z (ii/jj)nianj~' i,er id ~rx,,~
HH-F = Z
b
% %
-3.0
t't3,o" itr 3 a
as
Vol. 65, No. 8
+ (ii/ii)(nie) + y ~ ( i i / j j ) ( n i a )
shows the results for R = 3 a.u. in the chain. The parameters a s and fls in the atomic basis exhibit a decay with Rs according to the analytical formulae a s = - [ 1 + (1 + Rs) exp(-Rs)]/Rs and fls = - ( l + R s ) e x p ( - R s ) . The decay of fls after orthogonalization does not change much, whereas the a s shows a totally different behaviour, in particular, it shows a m i n i m u m at a r o u n d Rs = 3.5 a.u. The bulk values are not affected by orthogonalization and are constant for Rs > 3.5 a.u., although for short distances they vary strongly. The TB self-energies calculated self-consistently, show t h a t ~b ~ c~s, in agreement with the i m p o r t a n t result mentioned above, however the dependence on Rs is damped: c~s is now constant for large Rs and inherits a hint of the mini m u m at 3.5 a.u. The effective fi's show an exponential decay at s h o r t distances and show a crossover at around 3.5 a.u., ~b becoming practically constant and approaching the atomic value and fls becoming similar to the atomic value as well. The same calculation, but for Rs : 5 a.u., is shown in Fig. 2. The same trends are found except t h a t now the
(12)
j,o" and eft
~i1,o" = hij,a - 2(ii/jj)
(13)
In the linear chain there are undesirable spin instabilities t h a t produce spurious magnetic m o m e n t s near the surface. In order to avoid these difficulties one can restrict the spin coordinate to closed shell. The m o s t i m p o r t a n t result is t h a t self-consistency makes the effective interactions very similar in the bulk and surface sites. This result justifies n u m e r o u s calculations made in defects using the same TB p a r a m e t e r s in all sites. One can further simulate relaxation near the surface, or adsorption of a t o m s by changing the distance of last a t o m (Rs) in the semi-infinite chain and calculating all integrals, including the self consistent effective interactions. Fig. 1
TABLA 1. Atomic (a), bulk (b) and surface (s) values of the two electron integrals in the equally spaced 5 atom chain (in a.u.)
R
{AA/AA) a
5.0 4.5 4.0 3.5 3.0
0.625 0.625 0.625 0.625 0.625
b
0.631 0.636 0.644 0.659 0.624
(AA/BB) s
0.628 0.630 0.635 0.642 0.657
a
0.199 0.221 0.247 0.280 0.320
b
0.201 0.225 0.257 0.302 0.375
(AA/AB) s
0.200 0.222 0.256 0.286 0.339
a
0.035 0.052 0.077 0.112 0.161
(AB/AB)
b
s
a
b
s
-0.005 -0.006 -0.006 -0.007 -0.004
-0.005 -0.006 -0.007 -0.010 -0.017
0.004 0.008 0.016 0.031 0.059
0.001 0.001 0.002 0.001 -0.006
0.001 0.001 0.002 0.002 0.000
VARIATION OF TIGHT-BINDING
Vol. 65, No.
PARAMETERS NEAR SURFACES
777
3.
s~ -3.0 •
ALPHA
s
x
I".
-2.5
~
BETA
',
-2.0
S.-'"'...-
....
~ ;
- e°°.. "''" ,°oee
-1,5
~t "%o ~t %e
." b
~
l~ i~1tl
-t .0
"...
tlltl 1114
-0,5
0,0
t--
2.0
4.0
2.0
4.0
6.0
Rs(O u) Figure 2. Same as Fig. 1, but for R --- 5.0 a.u. ab and #fibin the orthogonal basis do not change very much at short distances, and that the effective site energies decay for long distances in very different ways; c~b ~ 1/In Ra and a 6 N 1/vrR-;. This last result is important since it is frequent to relax surfaces by assuming that a 8 decays linearly with
R, [7]. Summarizing, from the orthogonalization process the results are: 1. The one-electron hoping integral between neighbours is reduced in the orthogonal basis (see figures for Ra =
R). 2.
These integrals are less affected in the surface than in the bulk.
Direct Coulomb two electron integrals are practically the same in the bulk and in the surface and they are not much different from atomic value. 4. Exchange integrals are negligible after orthogonalization. This justifies the use of the Extended Hubbard approximation but not simple Hubbard calculations. From the self-consistent calculation two important conclusions can be drawn: 1. In a tight-binding calculation the parameters near a defect do not vary substantially with respect to the bulk value. This is due to compensation of various effects during the self-consistent procedure. 2. The TB hoping parameters ~ vary little in the surface, but they generally become more negative. From the surface relaxation simulation one can say that: 1. The self energies of the surface do not vary linearly with distance. 2. In general the TB parameters in the surface are not very different from the bulk for large limits of relaxation. It is true that the results presented here can not be directly extrapolated to real solids. However, we belive that they are indicative of what may be happening in an infinite system. An obvious extension of this work is to implement the ingredients of the present calculation to a real 3 dimensional lattice with a real surface, taking advantage of the simple mapping of real crystals into linear chains [13]. In particular, we believe that important problems as surface magnetism and interfaces could be tackled using the elements given here. This work is currently in progress. Acknowledgments - - We are indebted to Prof. F. Ynduraln for suggesting interesting points about the tight-binding method, to E. Martfnez and J. Tagfiefia-Martinez for useful discussions and enthusiasm. One of us (R.A.B.) is grateful to the Universidad Auton6ma de Madrid, where part of this work was carried out. This work was supported in part by Consejo Nacional de Ciencia y Tecnologia (Mdxico) through Grant No. PCEXNA-040428.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
F.M.D. Haldane & P.W. Anderson, Phys. Rev. B 18, 2553 (1956). R.A. Barrio, K I N A M 5, 337 (1983). E.P. O'Reilly & J. Robertson, Phys. Rev. B 27, 3786 (1983). M.J. Pinto & B. Koiller, J. Phys. C 15, 7229 (1982). J.B. Krieger & P.M. Lanfter, Phys. Rev. B 23, 4063 (1981). J.A. Vergds, Phys. Rev. B 26, 1059 (1982). T. Sakural & T. Sugano, J. Appl. Phys. 52, 2889 (1980).
8. 9. 10. 11. 12. 13.
P. Durand & J.C. Barthelat, Theor. Chim. Acta 88, 283 (1975). R.C. Bingham, M.J.S. Dewar & P.H. Lo, J. Am. Chem. Soc. 97, 1285 (1975). A.H. Edwards, J. Electron. Mater. 14a, 491 (1981). P.W. Atkins, "Molecular Quantum Mechanics% Second Edition, Oxford Univ. Press (1983). R. Baquero, V.R. Velasco & F. Garcla-Moliner (to be published in Phys. Rev. Lett.) L. Falicov & F. Yndurain, J. Phys. C 8, 1563 (1975).
VARIATION
OF
Vol.65,No.8,
pp.775-777,
TIGHT-BINDING
1988.
0038-I098/88 $3.00 + .00 Pergamon Journals Ltd.
PARAMETERS
NEAR
SURFACES
R. A. Barrio Instituto de Investlgaciones en Materiaies, UNAM, Apartado Postal 70-360, 04510, Mdxico, D.F. and M. del Castillo-Mussot Instituto de Fisica, UNAM, Apartado Postal 20-364, 01000, Mdxico, D.F. (Revised Manuscript received 20/06/87 by A.A. Maradudin) Surface effects on the tight-binding parameters for electrons are examined in a very simple system that allows the analytical calculation of all the atomic integrals needed. In spite of the simplicity of the model various qualitative results are obtained. It is seen that the inclusion of electron-electron interaction is essential to characterize the variation of the tight-binding parameters near defects. Effects of surface relaxation are also modelled and useful conclusions are drawn.
The tight-binding method (TBM) has been a useful tool in order to study the electronic properties of solids [1-4], because of its simplicity and the intuitive physical meaning of the site energies (a) and interaction parameters between different sites (/3). However, when the TBM is applied to calculations near defects, like vacancies and surfaces, the is no clear connection between the tight binding parameters and the real interactions between particles in the material and the results obtained are not completely satisfying. Indeed there have been debates [5-6] about its validity because one does not know how these parameters vary around a defect or surface. As an extreme example of the drawbacks of the TBM is the attempt by Sakurai and Sugano [7] to describe defect states in the interface between Si and Si02: The defect states in the Si gap could be at any desired energy by just varying the TBM parameters on the atoms near the defect. There are methods like the ab initio [8] and the semiempirical [9] techniques that have been applied to the same sort of defects and interfaces [10], and give useful local information about the electronic behaviour but it is difficult to connect the results with the TBM. It is the purpose of this paper to investigate this connection. In order to proceed, a very simple system is examined, in which a minimum of approximations should be made: a semi-infinite linear chain of hydrogen atoms with one s-orbital per atom in which one limits the interactions to nearest neighbours only. The advantages of doing this lie in the fact that the presence of a surface atom represents the maximum perturbation of defect in this system and that all fundamental quantities can be calculated analytically. In the TBM one generally assumes an orthonormai basis, therefore, the first difficulty in solving this system to find the appropriate orthonormal basis located around different lattice sites (Wannier functions). The simplest system that contains information about the "bulk" and ~surface~ atoms is a linear chain of 5 atoms in which the functions can be calculated exactly as follows: Consider a group of atomic orbitals l%bi)centered in every site Ri, that are not orthogonal, that is (¢i1%') = SO, (1)
as in a straight LCAO approximation. One may calculate the unitary matrix S -1/2 such that S -112 • S . S -I/2 -~- I,
(2)
and then
(3) J form the desired orthonormai basis. The method used to calculate S -1/2 is straightforward, first one finds the transformation U that diagonaiizes S and therefore
U -I.S'U=S d
(4)
is diagonal and
S-ID
=-
v . s ~ '12 • U -I
(5)
is the desired matrix used in (3). It is clear that this method is difficult to apply to an infinite system, like a solid, but it can be done in the simple chain chosen here. Furthermore in the 5 atom chain, w h e n hydrogen like s-functions are used, the matrix elements of S are calculated analytically [11] in terms of the constant distance between nelghbours R. The one-electron part of the Hamiltonian ~i. =
(¢i,I/'/~°"1¢i.)
(7a)
and /~id,. = (¢i,[Hc°relc]¢),
(7b)
where/_/core = V2 _ ~ i =5 1 1/It - Ri[, r is t he e l e c t ron coord i n a t e and a is the spin coordinate. In the one orthogonal basis we define, accordingly, &~a, &ha, fi~, •b, where s and b refer to ~surface" and ~bulk" respectively. So far the electron-electron interaction has been neglected, these interactions can be calculated in an entirely similar way exactly [11], using the atomic, bulk and surface states. These two-electron integrals have the form
(ABICD) = f ¢ ~ t ( r l ) ¢ b ( r l ) ~ J
775
r12
¢ c ( r 2 ) ¢ D ( r , ) d r l d r 2 (9)
776
V A R I A T I O N OF T I G H T - B I N D I N G P A R A M E T E R S N E A R SURFACES
where A, B, G', D refer to the basis orbitals in the atoms. One can calculate these integrals in the orthogonal base as well. The results for various interatomic distances are given in Table I. There are to i m p o r t a n t results from this table. First one notices t h a t , after orthogonalization the Coulomb integrals ( A A / A A ) and ( A A / B B ) do not vary very much from their atomic values. The second result is t h a t orthogonalization makes the direct Coulomb integrals much larger t h a n the exchange ones ( A A / A B ) and ( A B / A B ) and therefore the latter can be neglected for practical purposes. All the other combinations are relatively small due to overlap. In order to calculate the TB p a r a m e t e r s from the previous results, one can perform a self-consistent Hartree-Fock calculation in the o r t h o n o r m a l basis, retaining only the direct Coulomb two-electron part, t h a t is, using an extended H u b b a r d Hamiltonian, trusting t h a t the results will be physically sound and t h a t no i m p o r t a n t real interactions are being neglected. The connection between the H u b b a r d interactions and the TB p a r a m e t e r s is as follows: Consider the Hubbard-like Hamiltonian,
: E
,,c,t + Z
i,,~
err
%
-.\
leeee~e~'oeeea oae eeHe e ~
,':"...................... 6""
-0.5 0,0 2.0
2.0
4.0
4.0
6.0
Rs(a,u.) (lO)
t
~e. t
cqa eierci*r + Z Pij,~ci~ cier
i,~
Figure 1. Site energies (alphas) and hoping integrals (betas) for a bulk a t o m (b) and the surface a t o m (s) of a linear chain w i t h neighbour distance R = 3.0 a.u. as a function of the separation of the surface a t o m Rs. Matrix elements between atomic orbitals are shown as dashed lines, between o r t h o n o r m a l states in a 5 a t o m chain are shown in solid lines and the tight binding parameters, after self-consistent calculation in a semi-infinite chain, are shown in dotted lines. The origin for the self-consistent alphas has been shifted by 2.2 a.u.
(11)
ij,er
which bears the tight-binding form and where the tight binding p a r a m e t e r s are
~
\~ ............. b
%
-1,0
+
OLitr
-2.0
>n~ - 1 . 5 I.~ Z w
\
%
e*e.
~1,-
=
-
3
here i and j are nearest neighbours and eia (eia) and nia are the creation (annihilation) and n u m b e r operators for electrons in the orthogonal orbital labeled i with spin a. I n the Hartree-Foek approximation this can be w r i t t e n
eft Otio-
/11-., \
O
BETA
ALPHA
% % %
-2.5
+ Y]~(iilii)nianio + Z Z (ii/jj)nianj~' i,er id ~rx,,~
HH-F = Z
b
% %
-3.0
t't3,o" itr 3 a
as
Vol. 65, No. 8
+ (ii/ii)(nie) + y ~ ( i i / j j ) ( n i a )
shows the results for R = 3 a.u. in the chain. The parameters a s and fls in the atomic basis exhibit a decay with Rs according to the analytical formulae a s = - [ 1 + (1 + Rs) exp(-Rs)]/Rs and fls = - ( l + R s ) e x p ( - R s ) . The decay of fls after orthogonalization does not change much, whereas the a s shows a totally different behaviour, in particular, it shows a m i n i m u m at a r o u n d Rs = 3.5 a.u. The bulk values are not affected by orthogonalization and are constant for Rs > 3.5 a.u., although for short distances they vary strongly. The TB self-energies calculated self-consistently, show t h a t ~b ~ c~s, in agreement with the i m p o r t a n t result mentioned above, however the dependence on Rs is damped: c~s is now constant for large Rs and inherits a hint of the mini m u m at 3.5 a.u. The effective fi's show an exponential decay at s h o r t distances and show a crossover at around 3.5 a.u., ~b becoming practically constant and approaching the atomic value and fls becoming similar to the atomic value as well. The same calculation, but for Rs : 5 a.u., is shown in Fig. 2. The same trends are found except t h a t now the
(12)
j,o" and eft
~i1,o" = hij,a - 2(ii/jj)
(13)
In the linear chain there are undesirable spin instabilities t h a t produce spurious magnetic m o m e n t s near the surface. In order to avoid these difficulties one can restrict the spin coordinate to closed shell. The m o s t i m p o r t a n t result is t h a t self-consistency makes the effective interactions very similar in the bulk and surface sites. This result justifies n u m e r o u s calculations made in defects using the same TB p a r a m e t e r s in all sites. One can further simulate relaxation near the surface, or adsorption of a t o m s by changing the distance of last a t o m (Rs) in the semi-infinite chain and calculating all integrals, including the self consistent effective interactions. Fig. 1
TABLA 1. Atomic (a), bulk (b) and surface (s) values of the two electron integrals in the equally spaced 5 atom chain (in a.u.)
R
{AA/AA) a
5.0 4.5 4.0 3.5 3.0
0.625 0.625 0.625 0.625 0.625
b
0.631 0.636 0.644 0.659 0.624
(AA/BB) s
0.628 0.630 0.635 0.642 0.657
a
0.199 0.221 0.247 0.280 0.320
b
0.201 0.225 0.257 0.302 0.375
(AA/AB) s
0.200 0.222 0.256 0.286 0.339
a
0.035 0.052 0.077 0.112 0.161
(AB/AB)
b
s
a
b
s
-0.005 -0.006 -0.006 -0.007 -0.004
-0.005 -0.006 -0.007 -0.010 -0.017
0.004 0.008 0.016 0.031 0.059
0.001 0.001 0.002 0.001 -0.006
0.001 0.001 0.002 0.002 0.000
VARIATION OF TIGHT-BINDING
Vol. 65, No.
PARAMETERS NEAR SURFACES
777
3.
s~ -3.0 •
ALPHA
s
x
I".
-2.5
~
BETA
',
-2.0
S.-'"'...-
....
~ ;
- e°°.. "''" ,°oee
-1,5
~t "%o ~t %e
." b
~
l~ i~1tl
-t .0
"...
tlltl 1114
-0,5
0,0
t--
2.0
4.0
2.0
4.0
6.0
Rs(O u) Figure 2. Same as Fig. 1, but for R --- 5.0 a.u. ab and #fibin the orthogonal basis do not change very much at short distances, and that the effective site energies decay for long distances in very different ways; c~b ~ 1/In Ra and a 6 N 1/vrR-;. This last result is important since it is frequent to relax surfaces by assuming that a 8 decays linearly with
R, [7]. Summarizing, from the orthogonalization process the results are: 1. The one-electron hoping integral between neighbours is reduced in the orthogonal basis (see figures for Ra =
R). 2.
These integrals are less affected in the surface than in the bulk.
Direct Coulomb two electron integrals are practically the same in the bulk and in the surface and they are not much different from atomic value. 4. Exchange integrals are negligible after orthogonalization. This justifies the use of the Extended Hubbard approximation but not simple Hubbard calculations. From the self-consistent calculation two important conclusions can be drawn: 1. In a tight-binding calculation the parameters near a defect do not vary substantially with respect to the bulk value. This is due to compensation of various effects during the self-consistent procedure. 2. The TB hoping parameters ~ vary little in the surface, but they generally become more negative. From the surface relaxation simulation one can say that: 1. The self energies of the surface do not vary linearly with distance. 2. In general the TB parameters in the surface are not very different from the bulk for large limits of relaxation. It is true that the results presented here can not be directly extrapolated to real solids. However, we belive that they are indicative of what may be happening in an infinite system. An obvious extension of this work is to implement the ingredients of the present calculation to a real 3 dimensional lattice with a real surface, taking advantage of the simple mapping of real crystals into linear chains [13]. In particular, we believe that important problems as surface magnetism and interfaces could be tackled using the elements given here. This work is currently in progress. Acknowledgments - - We are indebted to Prof. F. Ynduraln for suggesting interesting points about the tight-binding method, to E. Martfnez and J. Tagfiefia-Martinez for useful discussions and enthusiasm. One of us (R.A.B.) is grateful to the Universidad Auton6ma de Madrid, where part of this work was carried out. This work was supported in part by Consejo Nacional de Ciencia y Tecnologia (Mdxico) through Grant No. PCEXNA-040428.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
F.M.D. Haldane & P.W. Anderson, Phys. Rev. B 18, 2553 (1956). R.A. Barrio, K I N A M 5, 337 (1983). E.P. O'Reilly & J. Robertson, Phys. Rev. B 27, 3786 (1983). M.J. Pinto & B. Koiller, J. Phys. C 15, 7229 (1982). J.B. Krieger & P.M. Lanfter, Phys. Rev. B 23, 4063 (1981). J.A. Vergds, Phys. Rev. B 26, 1059 (1982). T. Sakural & T. Sugano, J. Appl. Phys. 52, 2889 (1980).
8. 9. 10. 11. 12. 13.
P. Durand & J.C. Barthelat, Theor. Chim. Acta 88, 283 (1975). R.C. Bingham, M.J.S. Dewar & P.H. Lo, J. Am. Chem. Soc. 97, 1285 (1975). A.H. Edwards, J. Electron. Mater. 14a, 491 (1981). P.W. Atkins, "Molecular Quantum Mechanics% Second Edition, Oxford Univ. Press (1983). R. Baquero, V.R. Velasco & F. Garcla-Moliner (to be published in Phys. Rev. Lett.) L. Falicov & F. Yndurain, J. Phys. C 8, 1563 (1975).