Cunningham, S. L.

Tapilin, V. M.

Weinberg, W. H.

English

Caltech Authors - Main

Abstract: Technique for studying chemisorption on substrates 
with complex band structuresa> 
S. L. Cunningham and W. H. Weinbergbl 
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 
V. M. Tapilin 
Institute of Catalysis, Novosibirsk, 630090 USSR 
(Received 19 September 1977; accepted 22 November 1977) 
PACS numbers: 68.45.Da 
Chemisorption calculations of single adatoms on infinitely 
periodic surfaces using the Green's function approach have 
been limited heretofore to simple model systems due to the 
difficulty in obtaining the appropriate Green's functions. 1 In 
this paper a new technique is presented which has a number 
of numerical advantages making it possible to obtain the 
necessary Green's functions for materials with realistic and 
complex band structures. 
The problem is approached by determining the Green's 
functions for four subsystems. Whereas only the last of the 
four contains the desired informaticm regarding the chemi-
sorbed system, the method relies on the fact that the first 
Green's function is easy to calculate, and through a series of 
steps, the other three can be found sequentially. The method 
first determines G0 , the Green's function for a single two-
dimensional layer of atoms having the structure of a layer in 
the bulk crystal. Second, the Green's function G for the bulk 
crystal is calculated; this is the difficult step. Third, the Green's 
function cs for a crystal with a surface is found. Fourth, the 
Green's function ca which contains the desired information 
regarding the chemisorbed system, is obtained. 
For the first step, interactions between layers are ignored. 
The Schrodinger matrix equation in the LCAO formalism 
is 
L HiJ(k)a1 = e(k)a;, 
j 
(1) 
where k is the two-dimensional wave vector for the layer, and 
the Hamiltonian matrix is (N X N), where N is the number 
of orbitals on each atom. Even for complex systems, this ma-
trix is not very large. The Green's function can be calculated 
directly as 
(2) 
where E is the energy variable, K labels the eigenvalue and 
associated eigenvector, and the indices t and t' refer to layer 
number. This matrix is infinite in dimension due to the t and 
t' indices, but it is made up of identical (N X N) blocks along 
the diagonal. The first step, then, requires finding the ei-
genvalues and eigenvectors of the (N X N) matrix H. 
For the second step, the Green's function for the bulk is 
obtained by writing a Dyson equation 
G;1(0t) = Gf1(0t) 
+ L: Gfr(OO)Vq(Ot')Gn(t' t), (3) 
i'j'f' 
where the first layer index is set to zero and where the per-
turbation matrix V is that part of the bulk crystal Hamiltonian 
that connects layers 0 and t' ~ 0. Define M to be the largest 
integerfor which Vi1(0M) ~ 0. Then Eq. (3) breaks into two 
types of equations depending upon the value of the index 
e. 
Fort> M, Eq. (3) becomes simply 
f+M 
ci1(ot) = L: L: Gf;.(OO)Vq(t' t)Gn(t'O). (4) 
i'j' f'=f-M 
This equation can be solved by defining a stepping operator 
q such that 
(5) 
Upon substituting Eq. (5) into (4), we obtain an eigenvalue 
equation (for the case M = 1) 
where 
and 
L: (Aw- Qow)Gn(OO) = o, 
j' 
Aij' = L Gf;.(OO)Vi'j'(01) 
i' 
(6) 
(7) 
(8) 
There are 2N solutions for q from Eqs. (6)-(8). By choosing 
theN solutions which satisfy lql :::::; 1, we have the set of ho-
mogeneous solutions to Eq. (3). 
For the case oft< Min Eq. (3) we obtain a more complex 
equation. The general solution is constructed from a linear 
combination of the homogeneous solutions with arbitrary 
coefficients. This gives the bulk Green's function as 
N 
G;j(Of) = L hajq~G~(OO), (9) 
a=l 
where the values of qa and G~(OO) are the ath eigenvalue 
and eigenvector from Eq. (6) and where the coefficients hat 
are determined from the set of linear equations 
572 J. Vac. Set. Techno!. 15(2), March/April 1978 0022-5355/7811502-0572$01.00 © 1978 American Vacuum Society 572 
573 Cunningham, Weinberg, and Tapilin: Technique for studying chemisorption 573 
N 
L B;abaj = G7j(OO), (10) 
a=l 
where 
B;a = 2:: (ow- 2qaAw)Gj\(OO). (11) 
j' 
The second step, then, requires finding the eigenvalues and 
eigenvectors of the (N X N) matrix A and solving N times the 
(N X N) set of linear equations in Eq. (10). This is the difficult 
step. The bulk Green's function is constructed from Eq. 
(9). 
For the third step, the Green's function for the surface is 
simply found by inverting the Dyson equation 
G~1(00) = G;1(00)- 2:: Gu,(01)Vi'j'(10)Gj'j(OO), (12) 
i'j' 
where the minus sign is due to the fact that the surface can be 
formed merely by subtracting all interactions across an 
imaginary plane. Hence, we obtain 
(13) 
where 
(14) 
The third step, then, requires inverting an (N X N) matrix. 
The fourth step of finding the Green's function for the 
chemisorbed system ca is standard1·2 and will not be repeated 
here. 
All of the Green's functions mentioned depend implicitly 
on k, the wave vector parallel to the surface. To obtain the 
J. Vac. Sci. Technol. Vol. 15, No.2, Mar./ Apr. 1978 
total Green's function, a numerical sum over the special points 
in the surface Brillouin zone is performed.3 
Previous methods for finding the surface Green's functions 
involve either solving a continued fraction4•5 or performing 
an integration over the z component of the wave vector. 6 Both 
of these methods encounter numerical difficulties and become 
prohibitively time consuming when applied to systems with 
complex band structures and multiple neighbor interactions. 
The method presented here, while appearing to be more 
complex for model systems, does not have the same drawbacks 
as these previous methods when applied to realistic sys-
tems. 
To further elucidate the procedure, we have applied the 
formalism to examine the chemisorption of a one-level adatom 
in the on-site configuration on the (100) surface of a two-band 
substrate with the fcc structure. This problem has not been 
considered previously. It is shown that when a single level 
interacts with a multiple band substrate, increasing the in-
teraction between the adatom and only one band significantly 
affects the electron density in the other bands. 
•)Complete paper to appear in Physical Review. Supported by National 
Science Foundation Grant No. CHE77-10629. 
h)Alfred P. Sloan Foundation Fellow, and Camille and Henry Dreyfus 
Foundation Teacher-Scholar. 
lSee, for example, T. B. Grimley, Prog. Surf. Membr. Sci. 9, 71 (1975). 
2W. Ho, S. L. Cunningham, and W. H. Weinberg, Surf. Sci. 54,139 (1976); 
Surf. Sci. 62, 662 (1977). 
ss. L. Cunningham, Phys. Rev. B 10,4988 (1974). 
4R. Haydock, V. Heine, and M. J. Kelly, J. Phys. C 5, 2845 (1972). 
sF. Cyrot-Lackmann, M. C. Desjonqueres, and J.P. Gaspard, J. Phys. C 7, 
925 (1974). 
6W. Ho, S. L. Cunningham, W. H. Weinberg, and L. Dobrzynski, Phys. Rev. 
B 12, 3027 (1975). 


Technique for studying chemisorption on substrates

with complex band structures

Abstract: Technique for studying chemisorption on substrates 
with complex band structuresa> 
S. L. Cunningham and W. H. Weinbergbl 
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 
V. M. Tapilin 
Institute of Catalysis, Novosibirsk, 630090 USSR 
(Received 19 September 1977; accepted 22 November 1977) 
PACS numbers: 68.45.Da 
Chemisorption calculations of single adatoms on infinitely 
periodic surfaces using the Green's function approach have 
been limited heretofore to simple model systems due to the 
difficulty in obtaining the appropriate Green's functions. 1 In 
this paper a new technique is presented which has a number 
of numerical advantages making it possible to obtain the 
necessary Green's functions for materials with realistic and 
complex band structures. 
The problem is approached by determining the Green's 
functions for four subsystems. Whereas only the last of the 
four contains the desired informaticm regarding the chemi-
sorbed system, the method relies on the fact that the first 
Green's function is easy to calculate, and through a series of 
steps, the other three can be found sequentially. The method 
first determines G0 , the Green's function for a single two-
dimensional layer of atoms having the structure of a layer in 
the bulk crystal. Second, the Green's function G for the bulk 
crystal is calculated; this is the difficult step. Third, the Green's 
function cs for a crystal with a surface is found. Fourth, the 
Green's function ca which contains the desired information 
regarding the chemisorbed system, is obtained. 
For the first step, interactions between layers are ignored. 
The Schrodinger matrix equation in the LCAO formalism 
is 
L HiJ(k)a1 = e(k)a;, j (1) 
where k is the two-dimensional wave vector for the layer, and 
the Hamiltonian matrix is (N X N), where N is the number 
of orbitals on each atom. Even for complex systems, this ma-
trix is not very large. The Green's function can be calculated 
directly as 
(2) 
where E is the energy variable, K labels the eigenvalue and 
associated eigenvector, and the indices t and t' refer to layer 
number. This matrix is infinite in dimension due to the t and 
t' indices, but it is made up of identical (N X N) blocks along 
the diagonal. The first step, then, requires finding the ei-
genvalues and eigenvectors of the (N X N) matrix H. 
For the second step, the Green's function for the bulk is 
obtained by writing a Dyson equation 
G;1(0t) = Gf1(0t) 
+ L: Gfr(OO)Vq(Ot')Gn(t' t), (3) 
i'j'f' 
where the first layer index is set to zero and where the per-
turbation matrix V is that part of the bulk crystal Hamiltonian 
that connects layers 0 and t' ~ 0. Define M to be the largest 
integerfor which Vi1(0M) ~ 0. Then Eq. (3) breaks into two 
types of equations depending upon the value of the index 
e. 
Fort> M, Eq. (3) becomes simply 
f+M 
ci1(ot) = L: L: Gf;.(OO)Vq(t' t)Gn(t'O). (4) 
i'j' f'=f-M 
This equation can be solved by defining a stepping operator 
q such that 
(5) 
Upon substituting Eq. (5) into (4), we obtain an eigenvalue 
equation (for the case M = 1) 
where 
and 
L: (Aw- Qow)Gn(OO) = o, 
j' 
Aij' = L Gf;.(OO)Vi'j'(01) 
i' 
(6) 
(7) 
(8) 
There are 2N solutions for q from Eqs. (6)-(8). By choosing 
theN solutions which satisfy lql :::::; 1, we have the set of ho-
mogeneous solutions to Eq. (3). 
For the case oft< Min Eq. (3) we obtain a more complex 
equation. The general solution is constructed from a linear 
combination of the homogeneous solutions with arbitrary 
coefficients. This gives the bulk Green's function as 
N 
G;j(Of) = L hajq~G~(OO), (9) 
a=l 
where the values of qa and G~(OO) are the ath eigenvalue 
and eigenvector from Eq. (6) and where the coefficients hat 
are determined from the set of linear equations 
572 J. Vac. Set. Techno!. 15(2), March/April 1978 0022-5355/7811502-0572$01.00 © 1978 American Vacuum Society 572 
573 Cunningham, Weinberg, and Tapilin: Technique for studying chemisorption 573 
N 
L B;abaj = G7j(OO), (10) 
a=l 
where 
B;a = 2:: (ow- 2qaAw)Gj\(OO). (11) 
j' 
The second step, then, requires finding the eigenvalues and 
eigenvectors of the (N X N) matrix A and solving N times the 
(N X N) set of linear equations in Eq. (10). This is the difficult 
step. The bulk Green's function is constructed from Eq. 
(9). 
For the third step, the Green's function for the surface is 
simply found by inverting the Dyson equation 
G~1(00) = G;1(00)- 2:: Gu,(01)Vi'j'(10)Gj'j(OO), (12) 
i'j' 
where the minus sign is due to the fact that the surface can be 
formed merely by subtracting all interactions across an 
imaginary plane. Hence, we obtain 
(13) 
where 
(14) 
The third step, then, requires inverting an (N X N) matrix. 
The fourth step of finding the Green's function for the 
chemisorbed system ca is standard1·2 and will not be repeated 
here. 
All of the Green's functions mentioned depend implicitly 
on k, the wave vector parallel to the surface. To obtain the 
J. Vac. Sci. Technol. Vol. 15, No.2, Mar./ Apr. 1978 
total Green's function, a numerical sum over the special points 
in the surface Brillouin zone is performed.3 
Previous methods for finding the surface Green's functions 
involve either solving a continued fraction4•5 or performing 
an integration over the z component of the wave vector. 6 Both 
of these methods encounter numerical difficulties and become 
prohibitively time consuming when applied to systems with 
complex band structures and multiple neighbor interactions. 
The method presented here, while appearing to be more 
complex for model systems, does not have the same drawbacks 
as these previous methods when applied to realistic sys-
tems. 
To further elucidate the procedure, we have applied the 
formalism to examine the chemisorption of a one-level adatom 
in the on-site configuration on the (100) surface of a two-band 
substrate with the fcc structure. This problem has not been 
considered previously. It is shown that when a single level 
interacts with a multiple band substrate, increasing the in-
teraction between the adatom and only one band significantly 
affects the electron density in the other bands. 
•)Complete paper to appear in Physical Review. Supported by National 
Science Foundation Grant No. CHE77-10629. 
h)Alfred P. Sloan Foundation Fellow, and Camille and Henry Dreyfus 
Foundation Teacher-Scholar. 
lSee, for example, T. B. Grimley, Prog. Surf. Membr. Sci. 9, 71 (1975). 
2W. Ho, S. L. Cunningham, and W. H. Weinberg, Surf. Sci. 54,139 (1976); 
Surf. Sci. 62, 662 (1977). 
ss. L. Cunningham, Phys. Rev. B 10,4988 (1974). 
4R. Haydock, V. Heine, and M. J. Kelly, J. Phys. C 5, 2845 (1972). 
sF. Cyrot-Lackmann, M. C. Desjonqueres, and J.P. Gaspard, J. Phys. C 7, 
925 (1974). 
6W. Ho, S. L. Cunningham, W. H. Weinberg, and L. Dobrzynski, Phys. Rev. 
B 12, 3027 (1975). 


http://authors.library.caltech.edu/32774/1/CUNjvst78.pdf

Technique for studying chemisorption on substrates with complex band structures

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main