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Calculation of the dynamical image potential at metal surfaces
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Surface Science 283 (1993) 84-86 North-Holland
surface science
Calculation of the dynamical image potential at metal surfaces H. Sakai, Y.H. Ohtsuki Department of Physics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169, Japan
and M. Kitagawa Department of Electronics and Information Technology, North Shore College, SONY Institute of Higher Education, Atsugi 243, Japan
Received 21 April 1992; accepted for publication 12 May 1992
We derived the integral formula for the dynamical image potential of a system in which the electron density changes one-dimensionally. We used the generalized dielectric function in inhomogeneous many-electron systems under the high frequency approximation. We calculated the dynamical image potential for a fast ion taking account of the real electron distribution (a jellium model) at the surface.
1. Introduction
An ion beam incident on solid surfaces at a small angle is a powerful probe to characterize surfaces. Since ions move near surfaces for a long time in grazing-incidence surface scattering, they return with detailed information about the surface structure. The theory of the dynamical potential of the surface is much more complex than that of the bulk, because the uniformity of the electron gas is broken at the surface where the variation of the electron density or the boundary conditions must be taken into account. The dynamical image force and the stopping power for ions near the surface have been studied theoretically [l-3]. However, to avoid mathematical difficulties, the surface potential has been derived using a step function for the electron density and the continuity of the potentials at the surface for the boundary conditions. The simplicity of the formulae of previous works [l-31 is strongly based on the assumption 0039-6028/93/$06.00
of the step density. But this assumption is not good for the problems of grazing-angle ionsurface scattering. Newns studied the dielectric function of a more realistic electron density, but the formula was very complicated [4]. A simpler formula for the dielectric function was desired. Recently, Kitagawa [5,6] derived a simple dielectric function of a nonuniform electron gas under the high frequency approximation. Using this dielectric function, Kitagawa [6] also derived the surface stopping power formula for a system, in which the electron density changes one-dimensionally. In this paper, we calculate the dynamical image force and its potential for ions near a realistic surface by using Kitagawa’s dielectric function, which is valid when ions are fast and the high frequency approximation is applicable.
2. Derivation We will look at a point charge Ze constrained to move with uniform velocity I/ parallel to the
0 1993 - Elsevier Science Publishers B.V. All rights reserved
H. Sakai et al. / Calculation of the dynamical image potential at metal surfaces
.-Fig. 1. Geometry of the problem. The ion moves at a distance b from the surface in the direction of the z-axis with a velocity V.
surface and at distance b from the surface (fig. 1). Using the electromagnetic theory, the image force F is given as follows
Q’(l)
F(b)
= -Zey
1
4’(l)
=/+(13)[e-‘(32)
b,
=
-km
dkJd
I’
-
b I k,
k2
+
w2(
b),v2
P
+“( dq-:m
-6(32)]#“(2)
xd(23),
(2)
@=‘(2)=S(r27J,
of density of the external charge if we consider a classically straight trajectory of the projectile. The image force F is the gradient of the polarization potential normal to the surface (eq. (1)). In contrast to this, the stopping power is the gradient of the potential parallel to the surface when the ion moves parallel to the surface (see eq. (14) in ref. [6]). We will apply l-l of eq. (11) in ref. [61 to the derivation of the polarization potential. The local plasma frequency w,(x) = (4~n(x)e*/m)‘/* is needed for the calculation, where n(x) is the electron density at a distance x from the surface. We have the polarization potential from eq. (2) as follows
6(‘,
r,=rb,
85
IX-xl+lX-bl)k]
xdX&[(
rb=(b,0,Vf2),
(3) where the abbreviated notation (1) means (rr, tr). In eqs. (l)-(3), b’(l) indicates the potential from the polarization cloud, in which the second term denotes the subtraction of the pure Coulomb part because such a part does not act on the ion itself. uc denotes the Coulomb interaction between two charged particles. ~~“‘(2) is the number
1/v*
k*
’
k* + w;(
x
b-X
lb-XI
X)/V*
k* + o;(
b)/v*
&J;(x)
ax
.
(4)
In the above, we take y = 0, z = Vt.
(a)
-,: 0
I 2
I 4
I
I 6
I
I 8
,
I 10
Distance from the surface (A)
-0 Distance from the surface (i)
Fig. 2. The dynamical image force (a) and its potential (b) for the proton which moves parallel to Al surfaces with the velocity V= 2Vr (VP is the Fermi velocity of aluminum). The dashed curve shows the force for the step density, and the solid curve the force for the jellium model.
H. Sakai et al. / Calculation of the dynamical image potential at metal surfaces
86
We used the Thomas-Fermi electron distribution n(x) as a jellium model to take account of the realistic electron distributions 171.It has a tail outside the surface as follows n(x)
inoexp(Px), i inoexp( -Px>, n0
=
-
(x CO)
(5)
(x20),
where no is the electron density in the bulk and p is a parameter that determines the shape of the electron tail. For example, p = 2.34 A- ’ and oP = (47wzoe2/m) ‘I2 = 15.8 eV for aluminum [71. If we assume the step density n(x) = no@ -x>, eq. (4) reduces to the results of refs. [1,2] as follows +(x7 b) = -jmdk~o[(l~l+lbi)k]k2~/w~vz 0 /
mdkJo(lx-blk)
0
x e( -b)
+ jrn dkJo[( 0
W;/V2 X
k* + w2,/V2
where W, = or,/ fi quency.
e( -x)0(
s
Up2 k* + o;/V2
O( -x)
Ix I + I b I)k] -b),
is the surface
at larger distances from the surface (a range between 2 and 10 A), the magnitude of the force for the jellium is about twice as large as that for the step. This result is very important when we consider the force acting on the ions moving near surfaces. In fig. 2b, the potentials of the image forces are shown, which are calculated by using the following equation: (We tssume that both potentials are the same at 10 A, because the potential for the jellium cannot be obtained analytically.) U(x) = - /*F( x’) dx’. m
(7)
The potential for the jellium is deep compared to that for the step, though they are not so different from each other exactly on the surface. The electron tail of the jellium model has the effect on the trajectory of ions moving near the surface. In summary, we have shown that the electrons outside the surface have a great contribution to the dynamical image force at a distance from the surface. This is an improvement on the theories of dynamical image force which have neglected the electron tail.
(6) plasma fre-
Acknowledgement
We are grateful to Dr. H. Nitta and Dr. T. Iitaka for helpful suggestions.
3. Results and discussion
In fig. 2a, by using eq. Cl), the results of the numerical differentiations of eq. (4) with eq. (5) for the jellium and those of eq. (6) for the step are compared. We can see that, as a function of the distance from the surface, the image force for the jellium model decreases more slowly than that for the step-densi model. In the vicinity of the surface (within 1 A), the magnitude of the force for the jellium is smaller than that for the step. However,
References
111K.
Suzuki, M. Kitagawa and Y.H. Ohtsuki, Phys. Status Solidi (b) 82 (1977) 643. 121Y.H. Ohtsuki, Charged Beam Interaction with Solids (Taylor and Francis, London, 1983) sect. 7.5. [31 R. Kawai, N. Itoh and Y.H. Ohtsuki, Surf. Sci. 114 (1982) 137. [41D.M. Newns, Phys. Rev. B 1 (1970) 3304. [51M. Kitagawa, Nucl. Instrum. Methods B 13 (1986) 133. M. Kitagawa, Nucl. Instrum. Methods B 33 (1988) 409. ['31 [71J.R. Smith, Phys. Rev. 181 (1969) 522.
..:.:>;
jji+.:.
Surface Science 283 (1993) 84-86 North-Holland
surface science
Calculation of the dynamical image potential at metal surfaces H. Sakai, Y.H. Ohtsuki Department of Physics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169, Japan
and M. Kitagawa Department of Electronics and Information Technology, North Shore College, SONY Institute of Higher Education, Atsugi 243, Japan
Received 21 April 1992; accepted for publication 12 May 1992
We derived the integral formula for the dynamical image potential of a system in which the electron density changes one-dimensionally. We used the generalized dielectric function in inhomogeneous many-electron systems under the high frequency approximation. We calculated the dynamical image potential for a fast ion taking account of the real electron distribution (a jellium model) at the surface.
1. Introduction
An ion beam incident on solid surfaces at a small angle is a powerful probe to characterize surfaces. Since ions move near surfaces for a long time in grazing-incidence surface scattering, they return with detailed information about the surface structure. The theory of the dynamical potential of the surface is much more complex than that of the bulk, because the uniformity of the electron gas is broken at the surface where the variation of the electron density or the boundary conditions must be taken into account. The dynamical image force and the stopping power for ions near the surface have been studied theoretically [l-3]. However, to avoid mathematical difficulties, the surface potential has been derived using a step function for the electron density and the continuity of the potentials at the surface for the boundary conditions. The simplicity of the formulae of previous works [l-31 is strongly based on the assumption 0039-6028/93/$06.00
of the step density. But this assumption is not good for the problems of grazing-angle ionsurface scattering. Newns studied the dielectric function of a more realistic electron density, but the formula was very complicated [4]. A simpler formula for the dielectric function was desired. Recently, Kitagawa [5,6] derived a simple dielectric function of a nonuniform electron gas under the high frequency approximation. Using this dielectric function, Kitagawa [6] also derived the surface stopping power formula for a system, in which the electron density changes one-dimensionally. In this paper, we calculate the dynamical image force and its potential for ions near a realistic surface by using Kitagawa’s dielectric function, which is valid when ions are fast and the high frequency approximation is applicable.
2. Derivation We will look at a point charge Ze constrained to move with uniform velocity I/ parallel to the
0 1993 - Elsevier Science Publishers B.V. All rights reserved
H. Sakai et al. / Calculation of the dynamical image potential at metal surfaces
.-Fig. 1. Geometry of the problem. The ion moves at a distance b from the surface in the direction of the z-axis with a velocity V.
surface and at distance b from the surface (fig. 1). Using the electromagnetic theory, the image force F is given as follows
Q’(l)
F(b)
= -Zey
1
4’(l)
=/+(13)[e-‘(32)
b,
=
-km
dkJd
I’
-
b I k,
k2
+
w2(
b),v2
P
+“( dq-:m
-6(32)]#“(2)
xd(23),
(2)
@=‘(2)=S(r27J,
of density of the external charge if we consider a classically straight trajectory of the projectile. The image force F is the gradient of the polarization potential normal to the surface (eq. (1)). In contrast to this, the stopping power is the gradient of the potential parallel to the surface when the ion moves parallel to the surface (see eq. (14) in ref. [6]). We will apply l-l of eq. (11) in ref. [61 to the derivation of the polarization potential. The local plasma frequency w,(x) = (4~n(x)e*/m)‘/* is needed for the calculation, where n(x) is the electron density at a distance x from the surface. We have the polarization potential from eq. (2) as follows
6(‘,
r,=rb,
85
IX-xl+lX-bl)k]
xdX&[(
rb=(b,0,Vf2),
(3) where the abbreviated notation (1) means (rr, tr). In eqs. (l)-(3), b’(l) indicates the potential from the polarization cloud, in which the second term denotes the subtraction of the pure Coulomb part because such a part does not act on the ion itself. uc denotes the Coulomb interaction between two charged particles. ~~“‘(2) is the number
1/v*
k*
’
k* + w;(
x
b-X
lb-XI
X)/V*
k* + o;(
b)/v*
&J;(x)
ax
.
(4)
In the above, we take y = 0, z = Vt.
(a)
-,: 0
I 2
I 4
I
I 6
I
I 8
,
I 10
Distance from the surface (A)
-0 Distance from the surface (i)
Fig. 2. The dynamical image force (a) and its potential (b) for the proton which moves parallel to Al surfaces with the velocity V= 2Vr (VP is the Fermi velocity of aluminum). The dashed curve shows the force for the step density, and the solid curve the force for the jellium model.
H. Sakai et al. / Calculation of the dynamical image potential at metal surfaces
86
We used the Thomas-Fermi electron distribution n(x) as a jellium model to take account of the realistic electron distributions 171.It has a tail outside the surface as follows n(x)
inoexp(Px), i inoexp( -Px>, n0
=
-
(x CO)
(5)
(x20),
where no is the electron density in the bulk and p is a parameter that determines the shape of the electron tail. For example, p = 2.34 A- ’ and oP = (47wzoe2/m) ‘I2 = 15.8 eV for aluminum [71. If we assume the step density n(x) = no@ -x>, eq. (4) reduces to the results of refs. [1,2] as follows +(x7 b) = -jmdk~o[(l~l+lbi)k]k2~/w~vz 0 /
mdkJo(lx-blk)
0
x e( -b)
+ jrn dkJo[( 0
W;/V2 X
k* + w2,/V2
where W, = or,/ fi quency.
e( -x)0(
s
Up2 k* + o;/V2
O( -x)
Ix I + I b I)k] -b),
is the surface
at larger distances from the surface (a range between 2 and 10 A), the magnitude of the force for the jellium is about twice as large as that for the step. This result is very important when we consider the force acting on the ions moving near surfaces. In fig. 2b, the potentials of the image forces are shown, which are calculated by using the following equation: (We tssume that both potentials are the same at 10 A, because the potential for the jellium cannot be obtained analytically.) U(x) = - /*F( x’) dx’. m
(7)
The potential for the jellium is deep compared to that for the step, though they are not so different from each other exactly on the surface. The electron tail of the jellium model has the effect on the trajectory of ions moving near the surface. In summary, we have shown that the electrons outside the surface have a great contribution to the dynamical image force at a distance from the surface. This is an improvement on the theories of dynamical image force which have neglected the electron tail.
(6) plasma fre-
Acknowledgement
We are grateful to Dr. H. Nitta and Dr. T. Iitaka for helpful suggestions.
3. Results and discussion
In fig. 2a, by using eq. Cl), the results of the numerical differentiations of eq. (4) with eq. (5) for the jellium and those of eq. (6) for the step are compared. We can see that, as a function of the distance from the surface, the image force for the jellium model decreases more slowly than that for the step-densi model. In the vicinity of the surface (within 1 A), the magnitude of the force for the jellium is smaller than that for the step. However,
References
111K.
Suzuki, M. Kitagawa and Y.H. Ohtsuki, Phys. Status Solidi (b) 82 (1977) 643. 121Y.H. Ohtsuki, Charged Beam Interaction with Solids (Taylor and Francis, London, 1983) sect. 7.5. [31 R. Kawai, N. Itoh and Y.H. Ohtsuki, Surf. Sci. 114 (1982) 137. [41D.M. Newns, Phys. Rev. B 1 (1970) 3304. [51M. Kitagawa, Nucl. Instrum. Methods B 13 (1986) 133. M. Kitagawa, Nucl. Instrum. Methods B 33 (1988) 409. ['31 [71J.R. Smith, Phys. Rev. 181 (1969) 522.