- Email: [email protected]
Numerical studies of surface-ion neutralization
Chemical Physics 124 (1988) North-Holland, Amsterdam
NUMERICAL
41 I-415
STUDIES
K.W. SULSTON
OF SURFACE-ION
NEUTRALIZATION
‘,‘, A.T. AMOS
Department OfMathematics, University ofNottingham, Nottingham NG7 2RD, UK
and S.G. DAVISON
‘.3
Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA Received
20 November
1987; in final form 18 April 1988
A molecular-orbital approach is used to obtain exact numerical solutions to the equations of motion for the time-dependent Anderson-Newns Hamiltonian describing the surface-ion-neutralization process in one, two and three dimensions. The effect of dimensionality, and the presence of surface states, on the ion occupancy are discussed in detail.
procedure quite difficult to apply to complicated systems. These drawbacks were avoided by Sebastian et al. [ 5 1, who instead utilized the Euler method to solve the integro-differential equation for the time-evolution operator of the system. Although they only studied a one-dimensional system, generalization of the method to higher dimensions appears to be fairly straightforward. In the first of two numerical treatments, McDowell [ 6 ] used the molecular orbitals of a one-dimensional tight-binding chain to develop a formalism essentially equivalent to that of ref. [ 5 1, but which utilized a different method of numerical solution. Secondly, a Langevin approach [ 61 was developed, which separated the system into a primary zone and a memory kernel. Although the Langevin technique was later extended to two-dimensional models [ 7 1, its application to three-dimensional systems appears to be computationally quite expensive, though tractable. The purpose of the present article is to describe a molecular-orbital approach, whereby exact numerical results for the SIN problem can be obtained for one-, two- and three-dimensional tight-binding models. In addition, the effects on the SIN process of surface states in higher dimensions (d) are compared with those found in one-dimensional systems
1. Introduction
The extensive theoretical interest in the process of surface-ion neutralization (SIN) during recent years has led to the development of a number of approximate methods to treat the phenomenon, most of them based upon the time-dependent Anderson-Newns model [ 11. Although an exact analytic solution to the equations of motion does not seem possible at the present time, accurate numerical techniques are available, which allow quantitative investigation of the SIN process, and provide a standard for assessing the validity of the various approximations [2,3]. Muda and Hanawa [4] computed numerical solutions of the set of coupled first-order differential equations for the quantities (cf(t)c,(t)), ct (c;) being the creation (annihilation) operator for the atomic orbital 1i) centered on site i. Unfortunately, the large number of coupled equations to be solved, in addition to the slow convergence of the solution with the number of atoms in the solid, renders the ’ Work supported by the Natural Sciences and Engineering Research Council of Canada. * NATO Science Fellow. ’ On the leave from the Applied Mathematics and Physics Departments, University of Waterloo, Ontario, Canada N2L 3Gl.
0301-0104/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
B.V.
412
K. W. Sulston et al. /Surface-ion
[ 8 ] using the single-orbital
approximation
(SOA )
19,101.
2.Formalism The Hamiltonian for the ion-solid interaction tem is written as a sum of three terms
+V(t)~~*V*(lXk)~~l+l0)(X*/)~
sys-
(1)
The first term represents the originally empty orbital IO), of energy eO,on the positive ion before it interacts with the solid. The second, which represents the band of the isolated solid with which the ion interacts, is written in terms of the band energies ek and the associated one-electron wavefunctions (molecular orbitals) I&) . These are expressed in terms of the atomic orbitals I n) of the solid as
neutralization
argued that the set { I&)} is already complete, so that the mixed set is overcomplete and non-orthogonal. Effectively, this is the same problem which arises in chemisorption theory, and has been discussed, in particular, by Grimley [ 11, who suggested the use of a pseudo-Hamiltonian for the crystal states and a pseudo-interaction, which couples the adatom to the crystal. He then showed that, replacing the pseudoHamiltonian by the actual Hamiltonian for the semiinfinite crystal, was a reasonable approximation, so this approach is adopted here. From (5), it follows that the occupancy of the ion orbital at any time t can be written as n(t)=2
2
l4l,(t) I2 9
(6)
JZI
where the N orbitals in the solid of lowest energy are assumed to be originally doubly occupied. After substituting (5) into (4), the equations of motion for u,(t) (i=O, l,..., 2N:j=l,..., N) canbeshowntobe &j(t)=-i
y
,
VkV(t) exp(io@)a,(t)
(7a)
k=l Ixk>=~cknIn>
>
(2)
n
&kj(t)=-ivkI/(t)
with vk= cki = ( 1 I &), where I 1) represents the occupied target atom orbital in the surface layer of the solid. The third term represents the time-dependent interaction between the ion and the solid and is assumed to be the product of a time-independent term depending on vk and a time-dependent interaction potential of the form
Ut)=Voexp(-lltl),
(3)
where V0 is the interaction strength at t =O and ;I is proportional to the ion velocity. In atomic units (fi= 1), the solution to the time-dependent Schriidinger equation .d 1% Iw,(~)>=~(~)l~Vi(O)
9
exp(-iwkt)&j(t)
&j(-~)=o,
(7c)
akj(-a)=dkjs
The first step in the numerical solution of the N sets of 2N+ 1 equations (7) is the evaluation of the energies ek and coefficients vk, which requires the time-independent Hamiltonian H, of the pre-interaction solid to be specified. Within the tight-binding approximation, it is assumed that
(mlf& In>=a’,
if m = n is a surface atom;
=(Y,
if m = n is a bulk atom;
=P,
if m, n are neighbouring
=o,
otherwise.
ek and vk are then computed the eigenvalue problem
I!fi(t)>=aOj(z) ew(-k&IO)
(7b)
with ok= Q,- ek, subject to the initial conditions
(4)
has the general form
,
by numerically
atoms
solving
ffsIXk)=~kIXk),
+ ‘c
akj(l)
exp(-iekt)
ixk>
,
(5)
kc1
on the assumption that the mixed set of orbitals { I&) , I0 ) } is complete and orthogonal. It can be
with vk= ( 1 I&) . Knowing vk and ek, it is possible to integrate eqs. (7 ) using, in our case, the Numerical Algorithm Group library routine based on a RungeKutta-Merson method. The size of the model solid
K. W. Sulston et al. /Surface-ion neutralization
needed to ensure satisfactory convergence of the results is parameter dependent, the largest number of atoms being required for a slow-moving ion interacting with a wide-band solid. However, for most parameters employed here, it was found sufficient to use 8 atoms for one-dimensional chain, 20 atoms for the two-dimensional case (5 x 4 lattice) and 36 atoms for the three-dimensional case (3 x 3 x 4 lattice).
3. Results and discussion For investigating the dependence of n (co) on e. for systems with and without surface states, the parameters from fig. 3 of ref. [ 8 1, for a one-dimensional system within the SOA, were adopted for comparison purposes. Those curves are virtually identical with the exact one-dimensional curves of fig. 1 herein; thus the validity of the conclusions drawn in ref. [ 81 is con-
1.50
(d=1J
1.25
I
413
firmed. The main implication is that surface states play a potentially dominant role in the SIN process. In the absence of surface states (surface perturbation parameter CX’= 0), the graph of n( 03) versus e. is broad, low and centered about the middle of the occupied portion of the bulk band (energy values - 1 to 0), indicating that a significant chance of charge transfer occurs when to is aligned with one of the bulk energy levels. As a surface state is detached from the bottom of the band, the position of the peak in the curve shifts to a lower energy to match that of the surface state. Additionally, the peak becomes higher and thinner as the intensity of the state increases. Indeed, for some values of (Y’, n (co) exceeds unity, indicating it is possible that a second electron may be transferred to the ion. Moreover, for (Y’= - 2.0, a small satellite peak appears in the bulk-band region. The results for two and three-dimensional models (figs. 2 and 3, respectively) corroborate the conclusion that electrons in surface states are more likely to be transferred to the ion, by reason of their localization at the surface, as opposed to the delocalized nature of bulk electrons. In the case where no surface states exist ( CX’= 0.0)) the curves of IZ(m) versus e. are very much the same for all d, although there is some decrease in n(m) with increasing d. This observation suggests that the effect of dimensionality on charge transfer from bulk states may not be too large.
0.75
0.50
! 0.50
: 0.25
-3.0
0.25
-2.0
Fig. I. Final ion occupancy n ( GO)versus e,, in one dimension for LU=O.O,p~O.5, V,=O.2 and 1=0.4. Surface state energies are - 1.0, - 1.25 nd -2.125. There is no surface state for (Y’=O.O.
01//M -3.0
-2:o
-l.'O
Ah 6
1:o
60
Fig. 2. Final ion occupancy n(m) versus to in two dimensions forcu=0.0,,9=0.25, V,=0.2nd1=0.4. Curvesareasintig. 1.
414
K. W Sulston et al. /Surface-ion
-3.0
-2.0
-1.0
0
1.0
Lo
Fig. 3. Final ion occupancy n(oo) versus t0 in three dimensions forcu=O.O, p=O.l67, V0=0.2and1=0.4. Curvesareas in fig. 1.
On the other hand, when surface states exist ((Y’ #0), the differences among the curves for various d are immediately noticeable. They are broader and shorter, although the latter feature is not always present. Indeed, the overall shape of the curves more closely resembles those for the bulk band (a’ ~0.0) than for the one-dimensional model with a surface state, reflecting the fact that the ion is interacting with
neutralization
a band of surface states, rather than just a single one. Furthermore, the peaks for d=2 and 3 are centered about (Y’, the middle of the surface band, rather than the surface state energy, as ford= 1. Another point to be noted is the disappearance of the satellite peak in the curve for cr’ = - 2.0 in one dimension, apparently due to the widening of the primary peak. The dependence of n(a) on I, for solids both without and with surface states, is examined in table 1, upper and lower part respectively. The expected oscillation of n (CD) with 2 is seen to exist in all cases, and the maxima and minima appear to occur at about the same points, although not without exception. As I increases, the dimensionality differences in the results diminish, because, during a shorter interaction, there is less time for the charge transfer to be affected by the details of the electronic structure and geometric configuration of the surface. It is also clear that the effect of the solid’s dimensionality is more pronounced in the presence of surface states, because, between d= 1 and d=2, this represents the greatest increase in localized charge available. When surface
2.c
1.6
Table 1 Final ion occupancy n= (co) for various i, and rf, with (Y= -0.1, V,=O.S and band-width ~~0.4. Surface state at -0.35. 1.2
d=l (PzO.1)
d=2 (8=0.05)
d=3 (PcO.033)
0.941 0.942 0.840 0.462 0.899 0.698
0.915 0.937 0.797 0.376 0.884 0.701
0.825 0.928 0.761 0.303 0.871 0.703
e!J= -0.35 0.805 1.395 0.804 0.387 1.650 1.299
1.652 1.556 1.327 0.603 1.695 1.362
1.397 1.610 1.313 0.514 1.693 1.382
i
a’=--O.l,t,=-0.1 0.1 0.2 0.25 0.3333 0.5 1.0 cY’= -0.3, 0.1 0.2 0.25 0.3333 0.5 1.0
E c 0.8
0.4
Fig. 4. Ion occupancy n(t) versus time t fora= -0.1, co= -0.35, V,=O.S, LO.2 and band width 0.4.
a’ = -0.3,
L W. S&ton et al. I Surfkce-ion neutr~l~zai~on
states exist, increasing d tends to increase n(co), because more electrons are localized in the surface layer, and are thus available for interaction with the ion. However, because of the complexity of the chargetransfer process, other factors play important roles, so the increase in n( GO)with dis not universal. In the opposing situation, where surface states are absent, an increase in d generally produces a corresponding decrease in n( co), because, in a higher dimension, the electrons can become more delocalized. Further light can be shed on the effects of dimensionality by examining the time dependence of the ion occupancy n(t) (fig. 4). In the depicted situation, the qualitative behaviour of n (2) is the same in each dimension, although this is not always the case. Quantitatively, it can be seen that, although II(cc, ) is different in each case, the values of n ( t ) are virtually identical up until the time of closest approach (t=0). Thereafter, the occupancies differ from one other, each eventually converging to a final value n (00). Thus, although the probability of the first charge transfer(s) to the ion is about the same in each dimension, the probability of reionization, etc., can vary considerably, because solids of different dimensionality interact differently with ions during the scattering process.
4. Conclusion In summary, we have described the procedure whereby we have solved numerically the equations of motion for the time-dependent Anderson-Newns
415
Hamiltonian, neglecting the Coulomb repulsion term on the ion. We have examined the effects on the ion occupancy of the model solid’s dimensionality and the existence of surface states. It was concluded that dimensionality is of rather minor significance when surface states are absent, but when they exist, the quantitative effect of dimensionality becomes nonnegligible. Consequently, it is clear that the degree of localization of electrons in the surface layer of the solid plays a fundamental role in determining the SIN probability. Although one-dimensional bulk-band models have provided much useful information about the SIN process, it is important that they be applied with care, so as not to draw conclusions which are unwarranted, because of the omission of the effects of dimensionality and surface states. References [ 1] A. Yoshimori and K. Makoshi, Prog. Surface Sci. 2 1 ( 1986 ) 251. f2] S. Shindo and R. Kawai, Surface Sci. 165 (1986) 477. [3] K.W. Sulston, A.T. Amos and S.G. Davison, Surface Sci. 197 (1988) 555. [4] Y. Muda and T. Hanawa, Surface Sci. 97 ( 1980) 283. [ 51K.L. Sebastian, V.C. Jyothi Bhasu and T.B. Grimley, Surfacesci. 110(1981)L571. [6] H.K. McDowell, J. Chem. Phys. 77 (1982) 3263. [7] H.K. McDowell, Chem. Phys. 72 (1982) 451. [K] K.W. Sulston, SC. Davison and A.T. Amos, Solid State Commun. 62 (1987) 781. [9] S.G. Davison, K.W. Sulston and A.T. Amos, J. Electroanal. Chem. 204 (1986) 173. [ lo] A.T. Amos, S.G. Davison and K.W. Sulston, Phys. Letters A118(1986)471. [ 1l] T.B. Grimley, J. Phys. C 3 (1970) 1934.
NUMERICAL
41 I-415
STUDIES
K.W. SULSTON
OF SURFACE-ION
NEUTRALIZATION
‘,‘, A.T. AMOS
Department OfMathematics, University ofNottingham, Nottingham NG7 2RD, UK
and S.G. DAVISON
‘.3
Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA Received
20 November
1987; in final form 18 April 1988
A molecular-orbital approach is used to obtain exact numerical solutions to the equations of motion for the time-dependent Anderson-Newns Hamiltonian describing the surface-ion-neutralization process in one, two and three dimensions. The effect of dimensionality, and the presence of surface states, on the ion occupancy are discussed in detail.
procedure quite difficult to apply to complicated systems. These drawbacks were avoided by Sebastian et al. [ 5 1, who instead utilized the Euler method to solve the integro-differential equation for the time-evolution operator of the system. Although they only studied a one-dimensional system, generalization of the method to higher dimensions appears to be fairly straightforward. In the first of two numerical treatments, McDowell [ 6 ] used the molecular orbitals of a one-dimensional tight-binding chain to develop a formalism essentially equivalent to that of ref. [ 5 1, but which utilized a different method of numerical solution. Secondly, a Langevin approach [ 61 was developed, which separated the system into a primary zone and a memory kernel. Although the Langevin technique was later extended to two-dimensional models [ 7 1, its application to three-dimensional systems appears to be computationally quite expensive, though tractable. The purpose of the present article is to describe a molecular-orbital approach, whereby exact numerical results for the SIN problem can be obtained for one-, two- and three-dimensional tight-binding models. In addition, the effects on the SIN process of surface states in higher dimensions (d) are compared with those found in one-dimensional systems
1. Introduction
The extensive theoretical interest in the process of surface-ion neutralization (SIN) during recent years has led to the development of a number of approximate methods to treat the phenomenon, most of them based upon the time-dependent Anderson-Newns model [ 11. Although an exact analytic solution to the equations of motion does not seem possible at the present time, accurate numerical techniques are available, which allow quantitative investigation of the SIN process, and provide a standard for assessing the validity of the various approximations [2,3]. Muda and Hanawa [4] computed numerical solutions of the set of coupled first-order differential equations for the quantities (cf(t)c,(t)), ct (c;) being the creation (annihilation) operator for the atomic orbital 1i) centered on site i. Unfortunately, the large number of coupled equations to be solved, in addition to the slow convergence of the solution with the number of atoms in the solid, renders the ’ Work supported by the Natural Sciences and Engineering Research Council of Canada. * NATO Science Fellow. ’ On the leave from the Applied Mathematics and Physics Departments, University of Waterloo, Ontario, Canada N2L 3Gl.
0301-0104/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
B.V.
412
K. W. Sulston et al. /Surface-ion
[ 8 ] using the single-orbital
approximation
(SOA )
19,101.
2.Formalism The Hamiltonian for the ion-solid interaction tem is written as a sum of three terms
+V(t)~~*V*(lXk)~~l+l0)(X*/)~
sys-
(1)
The first term represents the originally empty orbital IO), of energy eO,on the positive ion before it interacts with the solid. The second, which represents the band of the isolated solid with which the ion interacts, is written in terms of the band energies ek and the associated one-electron wavefunctions (molecular orbitals) I&) . These are expressed in terms of the atomic orbitals I n) of the solid as
neutralization
argued that the set { I&)} is already complete, so that the mixed set is overcomplete and non-orthogonal. Effectively, this is the same problem which arises in chemisorption theory, and has been discussed, in particular, by Grimley [ 11, who suggested the use of a pseudo-Hamiltonian for the crystal states and a pseudo-interaction, which couples the adatom to the crystal. He then showed that, replacing the pseudoHamiltonian by the actual Hamiltonian for the semiinfinite crystal, was a reasonable approximation, so this approach is adopted here. From (5), it follows that the occupancy of the ion orbital at any time t can be written as n(t)=2
2
l4l,(t) I2 9
(6)
JZI
where the N orbitals in the solid of lowest energy are assumed to be originally doubly occupied. After substituting (5) into (4), the equations of motion for u,(t) (i=O, l,..., 2N:j=l,..., N) canbeshowntobe &j(t)=-i
y
,
VkV(t) exp(io@)a,(t)
(7a)
k=l Ixk>=~cknIn>
>
(2)
n
&kj(t)=-ivkI/(t)
with vk= cki = ( 1 I &), where I 1) represents the occupied target atom orbital in the surface layer of the solid. The third term represents the time-dependent interaction between the ion and the solid and is assumed to be the product of a time-independent term depending on vk and a time-dependent interaction potential of the form
Ut)=Voexp(-lltl),
(3)
where V0 is the interaction strength at t =O and ;I is proportional to the ion velocity. In atomic units (fi= 1), the solution to the time-dependent Schriidinger equation .d 1% Iw,(~)>=~(~)l~Vi(O)
9
exp(-iwkt)&j(t)
&j(-~)=o,
(7c)
akj(-a)=dkjs
The first step in the numerical solution of the N sets of 2N+ 1 equations (7) is the evaluation of the energies ek and coefficients vk, which requires the time-independent Hamiltonian H, of the pre-interaction solid to be specified. Within the tight-binding approximation, it is assumed that
(mlf& In>=a’,
if m = n is a surface atom;
=(Y,
if m = n is a bulk atom;
=P,
if m, n are neighbouring
=o,
otherwise.
ek and vk are then computed the eigenvalue problem
I!fi(t)>=aOj(z) ew(-k&IO)
(7b)
with ok= Q,- ek, subject to the initial conditions
(4)
has the general form
,
by numerically
atoms
solving
ffsIXk)=~kIXk),
+ ‘c
akj(l)
exp(-iekt)
ixk>
,
(5)
kc1
on the assumption that the mixed set of orbitals { I&) , I0 ) } is complete and orthogonal. It can be
with vk= ( 1 I&) . Knowing vk and ek, it is possible to integrate eqs. (7 ) using, in our case, the Numerical Algorithm Group library routine based on a RungeKutta-Merson method. The size of the model solid
K. W. Sulston et al. /Surface-ion neutralization
needed to ensure satisfactory convergence of the results is parameter dependent, the largest number of atoms being required for a slow-moving ion interacting with a wide-band solid. However, for most parameters employed here, it was found sufficient to use 8 atoms for one-dimensional chain, 20 atoms for the two-dimensional case (5 x 4 lattice) and 36 atoms for the three-dimensional case (3 x 3 x 4 lattice).
3. Results and discussion For investigating the dependence of n (co) on e. for systems with and without surface states, the parameters from fig. 3 of ref. [ 8 1, for a one-dimensional system within the SOA, were adopted for comparison purposes. Those curves are virtually identical with the exact one-dimensional curves of fig. 1 herein; thus the validity of the conclusions drawn in ref. [ 81 is con-
1.50
(d=1J
1.25
I
413
firmed. The main implication is that surface states play a potentially dominant role in the SIN process. In the absence of surface states (surface perturbation parameter CX’= 0), the graph of n( 03) versus e. is broad, low and centered about the middle of the occupied portion of the bulk band (energy values - 1 to 0), indicating that a significant chance of charge transfer occurs when to is aligned with one of the bulk energy levels. As a surface state is detached from the bottom of the band, the position of the peak in the curve shifts to a lower energy to match that of the surface state. Additionally, the peak becomes higher and thinner as the intensity of the state increases. Indeed, for some values of (Y’, n (co) exceeds unity, indicating it is possible that a second electron may be transferred to the ion. Moreover, for (Y’= - 2.0, a small satellite peak appears in the bulk-band region. The results for two and three-dimensional models (figs. 2 and 3, respectively) corroborate the conclusion that electrons in surface states are more likely to be transferred to the ion, by reason of their localization at the surface, as opposed to the delocalized nature of bulk electrons. In the case where no surface states exist ( CX’= 0.0)) the curves of IZ(m) versus e. are very much the same for all d, although there is some decrease in n(m) with increasing d. This observation suggests that the effect of dimensionality on charge transfer from bulk states may not be too large.
0.75
0.50
! 0.50
: 0.25
-3.0
0.25
-2.0
Fig. I. Final ion occupancy n ( GO)versus e,, in one dimension for LU=O.O,p~O.5, V,=O.2 and 1=0.4. Surface state energies are - 1.0, - 1.25 nd -2.125. There is no surface state for (Y’=O.O.
01//M -3.0
-2:o
-l.'O
Ah 6
1:o
60
Fig. 2. Final ion occupancy n(m) versus to in two dimensions forcu=0.0,,9=0.25, V,=0.2nd1=0.4. Curvesareasintig. 1.
414
K. W Sulston et al. /Surface-ion
-3.0
-2.0
-1.0
0
1.0
Lo
Fig. 3. Final ion occupancy n(oo) versus t0 in three dimensions forcu=O.O, p=O.l67, V0=0.2and1=0.4. Curvesareas in fig. 1.
On the other hand, when surface states exist ((Y’ #0), the differences among the curves for various d are immediately noticeable. They are broader and shorter, although the latter feature is not always present. Indeed, the overall shape of the curves more closely resembles those for the bulk band (a’ ~0.0) than for the one-dimensional model with a surface state, reflecting the fact that the ion is interacting with
neutralization
a band of surface states, rather than just a single one. Furthermore, the peaks for d=2 and 3 are centered about (Y’, the middle of the surface band, rather than the surface state energy, as ford= 1. Another point to be noted is the disappearance of the satellite peak in the curve for cr’ = - 2.0 in one dimension, apparently due to the widening of the primary peak. The dependence of n(a) on I, for solids both without and with surface states, is examined in table 1, upper and lower part respectively. The expected oscillation of n (CD) with 2 is seen to exist in all cases, and the maxima and minima appear to occur at about the same points, although not without exception. As I increases, the dimensionality differences in the results diminish, because, during a shorter interaction, there is less time for the charge transfer to be affected by the details of the electronic structure and geometric configuration of the surface. It is also clear that the effect of the solid’s dimensionality is more pronounced in the presence of surface states, because, between d= 1 and d=2, this represents the greatest increase in localized charge available. When surface
2.c
1.6
Table 1 Final ion occupancy n= (co) for various i, and rf, with (Y= -0.1, V,=O.S and band-width ~~0.4. Surface state at -0.35. 1.2
d=l (PzO.1)
d=2 (8=0.05)
d=3 (PcO.033)
0.941 0.942 0.840 0.462 0.899 0.698
0.915 0.937 0.797 0.376 0.884 0.701
0.825 0.928 0.761 0.303 0.871 0.703
e!J= -0.35 0.805 1.395 0.804 0.387 1.650 1.299
1.652 1.556 1.327 0.603 1.695 1.362
1.397 1.610 1.313 0.514 1.693 1.382
i
a’=--O.l,t,=-0.1 0.1 0.2 0.25 0.3333 0.5 1.0 cY’= -0.3, 0.1 0.2 0.25 0.3333 0.5 1.0
E c 0.8
0.4
Fig. 4. Ion occupancy n(t) versus time t fora= -0.1, co= -0.35, V,=O.S, LO.2 and band width 0.4.
a’ = -0.3,
L W. S&ton et al. I Surfkce-ion neutr~l~zai~on
states exist, increasing d tends to increase n(co), because more electrons are localized in the surface layer, and are thus available for interaction with the ion. However, because of the complexity of the chargetransfer process, other factors play important roles, so the increase in n( GO)with dis not universal. In the opposing situation, where surface states are absent, an increase in d generally produces a corresponding decrease in n( co), because, in a higher dimension, the electrons can become more delocalized. Further light can be shed on the effects of dimensionality by examining the time dependence of the ion occupancy n(t) (fig. 4). In the depicted situation, the qualitative behaviour of n (2) is the same in each dimension, although this is not always the case. Quantitatively, it can be seen that, although II(cc, ) is different in each case, the values of n ( t ) are virtually identical up until the time of closest approach (t=0). Thereafter, the occupancies differ from one other, each eventually converging to a final value n (00). Thus, although the probability of the first charge transfer(s) to the ion is about the same in each dimension, the probability of reionization, etc., can vary considerably, because solids of different dimensionality interact differently with ions during the scattering process.
4. Conclusion In summary, we have described the procedure whereby we have solved numerically the equations of motion for the time-dependent Anderson-Newns
415
Hamiltonian, neglecting the Coulomb repulsion term on the ion. We have examined the effects on the ion occupancy of the model solid’s dimensionality and the existence of surface states. It was concluded that dimensionality is of rather minor significance when surface states are absent, but when they exist, the quantitative effect of dimensionality becomes nonnegligible. Consequently, it is clear that the degree of localization of electrons in the surface layer of the solid plays a fundamental role in determining the SIN probability. Although one-dimensional bulk-band models have provided much useful information about the SIN process, it is important that they be applied with care, so as not to draw conclusions which are unwarranted, because of the omission of the effects of dimensionality and surface states. References [ 1] A. Yoshimori and K. Makoshi, Prog. Surface Sci. 2 1 ( 1986 ) 251. f2] S. Shindo and R. Kawai, Surface Sci. 165 (1986) 477. [3] K.W. Sulston, A.T. Amos and S.G. Davison, Surface Sci. 197 (1988) 555. [4] Y. Muda and T. Hanawa, Surface Sci. 97 ( 1980) 283. [ 51K.L. Sebastian, V.C. Jyothi Bhasu and T.B. Grimley, Surfacesci. 110(1981)L571. [6] H.K. McDowell, J. Chem. Phys. 77 (1982) 3263. [7] H.K. McDowell, Chem. Phys. 72 (1982) 451. [K] K.W. Sulston, SC. Davison and A.T. Amos, Solid State Commun. 62 (1987) 781. [9] S.G. Davison, K.W. Sulston and A.T. Amos, J. Electroanal. Chem. 204 (1986) 173. [ lo] A.T. Amos, S.G. Davison and K.W. Sulston, Phys. Letters A118(1986)471. [ 1l] T.B. Grimley, J. Phys. C 3 (1970) 1934.