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Position-dependent charge-exchange and energy-loss processes for MeV He ions near a crystal surface
Nuclear Instruments and Methods in Physics Research B 90 (1994) 212-215 North-Holland
NOMB
Beam Interactions with Materials&Atoms
Position-dependent charge-exchange and energy-loss processes for MeV He ions near a crystal surface Yoshikazu Fujii, Shinsuke Fujiwara, Kazumasa Narumi, Kenji Kimura and Michi-hiko Mannami * Department of Engineering Science, Kyoto University, Kyoto 606-01, Japan
Inelastic interaction of fast ions at glancing angles of incidence on a low index crystal surface is described by the probabilities, which depend on the distance of the ion from the surface; the charge state and energy distributions of the scattered ions are found to result from position-dependent interactions along their trajectories in vacuum. A stochastic equation to describe the charge exchange and energy loss of specularly reflected ions is proposed. Taking account of ion scattering at surface steps, the theory is applied to derive the position-dependent probabilities of electron loss and electron capture of MeV ions from the observed charge state and energy distributions of the scattered ions at glancing angle incidence on the (100) surface of SnTe.
1. Introduction Specular reflection of fast ions from a low index crystal surface occurs at glancing angles of incidence on the surface. The reflection is sensitive to irregularities on the surface, and the angular and energy distributions and the charge state distribution of the scattered ions are affected by steps on the crystal surface [1,2]. Recently we have demonstrated that the yield of scattered ions at glancing angles of incidence for 3 keV He+ ions on the (001) surface of GaAs is sinusoidally modulated during homo-epitaxial growth of the GaAs surface [3]. The modulation period is equal to the time to grow one GaAs molecular mono layer on the surface, which corresponds to the change in step density during layer-by-layer growth of the surface. Interaction of an ion with a surface at glancing angles of incidence depends on the distance of the ion from the surface, since the ion trajectory can be approximated by a path parallel to the surface. Thus, the energy loss and charge state of the ion along its trajectory are described with the use of relevant probabilities which depend on the distance of the ion from the surface: energy loss of the ion is obtained by integrating the position-dependent stopping power along the ion trajectory [4-61, and the charge state distribution of the ion along its trajectory is obtained by solving the rate equations for charge state fractions, where the charge exchange probabilities are position dependent
L&71. * Corresponding 771 7286.
author, tel. +81 75 753 5196, fax +81 75
0168-583X/94/$07.00
In the present paper, we describe a new method to derive the position-dependent charge-exchange probabilities of MeV ions from the observed energy losses and charge-state distribution of scattered ions scattering at glancing angle from a surface. As a demonstration, the method is applied to the glancing angle scattering of MeV He ions from the (100) surface of SnTe.
2. Effects of surface step on the traljectories of ions Angular and energy distributions of scattered ions were observed at glancing angle incidence of MeV He+ ions on the (100) surface of SnTe which had been prepared by vacuum epitaxial growth on the (100) surface of KC1 under UHV conditions [2]. The surfaces of the SnTe single crystals had many small pyramidal hillocks as observed by AFM (atomic force microscopy), where the widths of the terraces were a few tens of nm and the step heights were one and two atomic layers. Now we express the trajectory of an ion by z(x) at its glancing angle incidence on an atomically flat surface. The ion is specularly reflected from the surface, where the scattering angle, 8,, is twice the angle of incidence, 13~.We choose coordinates where the x-axis is normal to the surface and the z-axis is parallel to the surface, and the origin of the coordinates on the surface atomic plane is shown in Fig. 1. When there is a step on the surface, the trajectories of ions deviate from z(x) at the step edge, where z(x) is shown by broken lines in Fig. 1. The deviation angle depends on the distance, zO, of the step from the apex of the trajectory and on the type of step. Ion trajectories
0 1994 - Elsevier Science B.V. All rights reserved
SSDI 0168-583X(93)E0657-3
Y Fujii et al. /Nucl. Instr. and Meth. in Phys. Res. B 90 (1994) 212-215
A
D
I
;
energy. In deriving Eqs. (1) and (21, we retained only the term containing CY~and & in the Moliere approximation to the Thomas-Fermi screening function [S], because the distance of closest approach of the ions to the surface is larger than 4a,, for the present experimental conditions. Due to this approximation, the error to the trajectories in the cases discussed here is less than 2% [4]. On type A trajectories, the distances of the ions from the surface are larger than monolayer height (0.314 nml after they have passed over the step edge, and the interaction between the ions and the surface is negligibly small at z > zo, since most of the interaction probabilities decrease exponentially from the surface. The charge state and energy of the ion at the step are frozen on the following trajectories. Thus, the charge state fraction, F$Bi; e,), and the energy, Ej(Bi; 0,). of ions with charge +je scattered at an angle es at incidence with glancing angle, Bi, are approximately related to those at the point z = z. on the trajectory z(x) via
I/
, N\ .\‘,<,\~‘\\\\\\+J\\\ \\X
\
\‘
\\’ \\\\’ __>_--L___--,
’
\\‘\
Fj(ei;
2,’
step. The broken lines show the trajectories of specular reflection. Only type A and C trajectories are possible at pyramidal hillocks on the (100) surface of SnTe, since the terraces second from the top are not wide enough for other trajectories.
passing near surface steps are classified into 4 types, as shown in Fig. 1. On the SnTe surfaces with pyramidal hillocks, however, only the trajectories of type A and C are possible [2]. Consider an ion of type A trajectory which is deflected to a scattering angle BS(< 28,). On the trajectory, the distance of the ion from the surface is larger than one atomic height (0.314 nm) after it has passed over the step. The surface continuum planar potential is negligibly small at large distances, and the trajectory is approximated by a straight line for z >z,. The distance, xS, of the trajectory z(x) at the step and the position zc, of a down-step from the apex of the trajectory z(x) as defined in Fig. 1 are related to the scattering angle BSand the angle of incidence Bi by the following equations:
[%~/&l ln((a3/&)+Z/[4(2~i -%>I}7 z. = %/(Wi) ln{~~/[%W - es)l)T
XS=
2,Z,e*n,~,,/E]~~~, Z, and of the ion and target atom elementary charge, its is the surface atomic plane, aTF screening distance, and E is
4) =Fj(zo),
Ej(0i;
Fig. 1. Four types of ion trajectories passing near a surface
where $, = [ZP atomic numbers tively, e is the density of the Thomas-Fermi
213
(1) (2)
Z, are respecatomic is the the ion
0,) =E,-AE,(z,),
(3)
where Fj(zo) and AEj(zo) are the charge fraction and energy loss for ions of charge +je at z. on the trajectory of specularly reflected ions, and E, is the incident energy of the ion. Since zo, Bi and es are related by Eqs. (1) and (2), it is possible to derive the charge state fraction and energy losses, Fj(z) and AEj(z), for ions on z(x) at z > 0.
3. Stochastic
model of charge exchange and energy loss of specularly reflected ions
The charge state and energy of ions at specular reflection from a crystal surface change along the tra-
rEzoz !! g,l5LL Y w 10-
0
I
5
I
I
10
15
20
ANGLE (mrad 1
Fig. 2. Energy losses of scattered He+ and He’+ ions for 0.67 MeV He+ incident of the (100) surface of SnTe with 19~= 3.3 and 6.5 mrad. III. SURFACE
PHENOMENA
?I Fgjii et al. /Nucl. k&r. and Meh. in Phys.Res. 3 90 (1994) 212-215
214
jectory, z(x). We define the probability distribution, fj (z(x), E), for ions with charge +je and energy E at a point (x, z) on the trajectory z(x) of specular reflection. The distribution Jcl(z(x), E) satisfies the following stochastic equation: fj(z+Sz~
3
I
B-
LOSS of He’
E)
= Qjj( I) Szkm d(GE)fj(z,
+
I
0.67 MeV He* on Snfe (100)
C Q,j( Z) izj
xw,(z;
SE)
E + SE)wj( z; SE) Sr
Szimd(SE)fi( Z, E + SE) sz,
(4) 0
where wj(z(n); SE) is the probability (per unit path length) for energy loss SE of an ion of charge +je at distance x from the surface, Qji(z(n)) is the transition probability (per unit length) of ionic charge from tje to +ie at a distance x from the surface. We now neglect the dependence of wj(z(x); 6E) and Qji(z(x)) on ion energy, E. Expanding fj in Eq. (4) into a Taylor series, and neglecting the terms containing (8~)~ and (SE)” (n > l), we obtain a set of differential equations for fifz(x), E):
The charge state fraction, Ij;(z), and energy loss, AEj(z(x)), at z(x) are derived from fi(t, E) using
vjt z, E)Pz
5:( z(x))
=Sj(z(x)>afi(z, + C[
DISTANCE
:ROM
SURFACE
x (
A i2
Fig. 4. Position-dependent probabilities, Qrz(x) and Q,,(x), of electron loss and electron capture of 0.67 MeV He ions near the (109) surface of SnTe. Solid lines are the probabilities derived from Bohr and Bohr-tindhard models. Broken lines are drawn to guide the eye.
= km&(z, E) dE,
(7)
E)/aE
Qij(z)fi(Z, E)-Qji(Z>.&~El], (5)
AEj(z(X))
=Es-~~Efi(z~
E) dE/imfj(z,
E) dE,
i#j
(8)
where Sj(z(x)) is the position-dependent stopping power for an ion of charge +je. Sj(z(x)) is related to wj(z; SE) via s,(z(~))=~~(x)=~~d(SE)[~~(z;
SE) SE].
(6)
where E, is the energy of the incident ion. For the scattering of MeV He ions from the (100) surface of SnTe, we consider only He’ and He2+ ions, since the He0 fraction is less than 1%. Thus, using Eqs. (5), (7) and (8), we obtain the electron-loss and electron-capture probabilities, QIZ(x) and Q,,(x),
Qji(x)=I;l(z)/[4:Fl(z)+q22F*(z>] x(l/[AEj(z)-A&(z)]
o.7i
-q;AE,(z)],‘dz
d[q;AEi(z) +q$‘F;.(z)
d4(z)/dz], (9)
where qje is the effective charge of the Hei+ ion and s&x) is the stopping power for an isotachic proton and we assume sj(x) = q,3Jx).
4. Derivation of electron-transfer I
I
0
15
5
20
ANGLE ‘fmrad)
Fig. 3. Ratio of the fraction FI (0,; 0,) of He+ ions and the fraction Fz(Bi; 0,) of He’+ ions in the scattered beam for 0.67 MeV He+ ions incident on the (100) surface of SnTe with fIi = 3.3 and 6.5 mrad.
probabilities
The position-dependent electron-loss and electroncapture probabilities, Q&x> and Q,,(x), are obtained from the experimental E;(ei; @,) and Ej(Bi; 0,) using Eqs. (3) and (9). With the observed angular dependence of energy loss and charge state fractions of the scattered He ions
Y. Fujii et al. /Nucl. Instr. and Meth. in Phys. Res. B 90 (1994) 212-215
such as those shown in Figs. 2 and 3, the electron-loss and electron-capture probabilities are calculated. Examples of the probabilities obtained in this way are shown in Fig. 4 for 0.67 MeV He ions near the (100) surface of SnTe, where we assumed that qj = j. The probabilities are smaller than those calculated with Bohr and Bohr-Lindhard models, which are shown by solid lines in the figure. With the derived charge-exchange probabilities and with the position-dependent stopping powers, which can be obtained from experimental energy losses of the scattered ions [S-7], we can now study details of the ion-surface interaction processes at glancing angle scattering from a clean surface.
Acknowledgements We are grateful to K. Yoshida, K. Norisawa and the staff members of the Department of Nuclear Engineering of Kyoto University for the availability of the 4 MV
215
Van de Graaff accelerator. The work is partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education.
References [l] M. Mannami, Y. Fujii and K. Kimura, Surf. Sci. 204 (1988) 213. [2] Y. Fujii, S. Fujiwara, K. Kimura and M. Mannami, Nucl. Ins&. and Meth. B 58 (1991) 18. [3] Y. Fujii, K. Narumi, K. Kimura, M. Mannami, T. Hashimoto, K. Ogawa, F. Ohtani, T. Yoshida and M. Asari, Appl. Phys. Lett. 63 (1993) 2070. [4] K. Kimura, M. Hasegawa and M. Mannami, Phys. Rev. B 36 (1987) 7. [5] Y. Fujii, S. Fujiwara, K. Narumi, K. Kimura and M. Mannami, Surf. Sci. 277 (1992) 164. [6] Y. Fujii, K. Kishine, K. Narumi, K. Kimura and M. Mannami, Phys. Rev. A 47 (1993) 2047. [7] K. Kimura, Y. Fujii, M. Hasegawa, Y. Susuki and M. Mannami, Phys. Rev. B 38 (1988) 1052. [8] D.S. Gemmell, Rev. Mod. Phys. 46 (19741 129.
III. SURFACE PHENOMENA
NOMB
Beam Interactions with Materials&Atoms
Position-dependent charge-exchange and energy-loss processes for MeV He ions near a crystal surface Yoshikazu Fujii, Shinsuke Fujiwara, Kazumasa Narumi, Kenji Kimura and Michi-hiko Mannami * Department of Engineering Science, Kyoto University, Kyoto 606-01, Japan
Inelastic interaction of fast ions at glancing angles of incidence on a low index crystal surface is described by the probabilities, which depend on the distance of the ion from the surface; the charge state and energy distributions of the scattered ions are found to result from position-dependent interactions along their trajectories in vacuum. A stochastic equation to describe the charge exchange and energy loss of specularly reflected ions is proposed. Taking account of ion scattering at surface steps, the theory is applied to derive the position-dependent probabilities of electron loss and electron capture of MeV ions from the observed charge state and energy distributions of the scattered ions at glancing angle incidence on the (100) surface of SnTe.
1. Introduction Specular reflection of fast ions from a low index crystal surface occurs at glancing angles of incidence on the surface. The reflection is sensitive to irregularities on the surface, and the angular and energy distributions and the charge state distribution of the scattered ions are affected by steps on the crystal surface [1,2]. Recently we have demonstrated that the yield of scattered ions at glancing angles of incidence for 3 keV He+ ions on the (001) surface of GaAs is sinusoidally modulated during homo-epitaxial growth of the GaAs surface [3]. The modulation period is equal to the time to grow one GaAs molecular mono layer on the surface, which corresponds to the change in step density during layer-by-layer growth of the surface. Interaction of an ion with a surface at glancing angles of incidence depends on the distance of the ion from the surface, since the ion trajectory can be approximated by a path parallel to the surface. Thus, the energy loss and charge state of the ion along its trajectory are described with the use of relevant probabilities which depend on the distance of the ion from the surface: energy loss of the ion is obtained by integrating the position-dependent stopping power along the ion trajectory [4-61, and the charge state distribution of the ion along its trajectory is obtained by solving the rate equations for charge state fractions, where the charge exchange probabilities are position dependent
L&71. * Corresponding 771 7286.
author, tel. +81 75 753 5196, fax +81 75
0168-583X/94/$07.00
In the present paper, we describe a new method to derive the position-dependent charge-exchange probabilities of MeV ions from the observed energy losses and charge-state distribution of scattered ions scattering at glancing angle from a surface. As a demonstration, the method is applied to the glancing angle scattering of MeV He ions from the (100) surface of SnTe.
2. Effects of surface step on the traljectories of ions Angular and energy distributions of scattered ions were observed at glancing angle incidence of MeV He+ ions on the (100) surface of SnTe which had been prepared by vacuum epitaxial growth on the (100) surface of KC1 under UHV conditions [2]. The surfaces of the SnTe single crystals had many small pyramidal hillocks as observed by AFM (atomic force microscopy), where the widths of the terraces were a few tens of nm and the step heights were one and two atomic layers. Now we express the trajectory of an ion by z(x) at its glancing angle incidence on an atomically flat surface. The ion is specularly reflected from the surface, where the scattering angle, 8,, is twice the angle of incidence, 13~.We choose coordinates where the x-axis is normal to the surface and the z-axis is parallel to the surface, and the origin of the coordinates on the surface atomic plane is shown in Fig. 1. When there is a step on the surface, the trajectories of ions deviate from z(x) at the step edge, where z(x) is shown by broken lines in Fig. 1. The deviation angle depends on the distance, zO, of the step from the apex of the trajectory and on the type of step. Ion trajectories
0 1994 - Elsevier Science B.V. All rights reserved
SSDI 0168-583X(93)E0657-3
Y Fujii et al. /Nucl. Instr. and Meth. in Phys. Res. B 90 (1994) 212-215
A
D
I
;
energy. In deriving Eqs. (1) and (21, we retained only the term containing CY~and & in the Moliere approximation to the Thomas-Fermi screening function [S], because the distance of closest approach of the ions to the surface is larger than 4a,, for the present experimental conditions. Due to this approximation, the error to the trajectories in the cases discussed here is less than 2% [4]. On type A trajectories, the distances of the ions from the surface are larger than monolayer height (0.314 nml after they have passed over the step edge, and the interaction between the ions and the surface is negligibly small at z > zo, since most of the interaction probabilities decrease exponentially from the surface. The charge state and energy of the ion at the step are frozen on the following trajectories. Thus, the charge state fraction, F$Bi; e,), and the energy, Ej(Bi; 0,). of ions with charge +je scattered at an angle es at incidence with glancing angle, Bi, are approximately related to those at the point z = z. on the trajectory z(x) via
I/
, N\ .\‘,<,\~‘\\\\\\+J\\\ \\X
\
\‘
\\’ \\\\’ __>_--L___--,
’
\\‘\
Fj(ei;
2,’
step. The broken lines show the trajectories of specular reflection. Only type A and C trajectories are possible at pyramidal hillocks on the (100) surface of SnTe, since the terraces second from the top are not wide enough for other trajectories.
passing near surface steps are classified into 4 types, as shown in Fig. 1. On the SnTe surfaces with pyramidal hillocks, however, only the trajectories of type A and C are possible [2]. Consider an ion of type A trajectory which is deflected to a scattering angle BS(< 28,). On the trajectory, the distance of the ion from the surface is larger than one atomic height (0.314 nm) after it has passed over the step. The surface continuum planar potential is negligibly small at large distances, and the trajectory is approximated by a straight line for z >z,. The distance, xS, of the trajectory z(x) at the step and the position zc, of a down-step from the apex of the trajectory z(x) as defined in Fig. 1 are related to the scattering angle BSand the angle of incidence Bi by the following equations:
[%~/&l ln((a3/&)+Z/[4(2~i -%>I}7 z. = %/(Wi) ln{~~/[%W - es)l)T
XS=
2,Z,e*n,~,,/E]~~~, Z, and of the ion and target atom elementary charge, its is the surface atomic plane, aTF screening distance, and E is
4) =Fj(zo),
Ej(0i;
Fig. 1. Four types of ion trajectories passing near a surface
where $, = [ZP atomic numbers tively, e is the density of the Thomas-Fermi
213
(1) (2)
Z, are respecatomic is the the ion
0,) =E,-AE,(z,),
(3)
where Fj(zo) and AEj(zo) are the charge fraction and energy loss for ions of charge +je at z. on the trajectory of specularly reflected ions, and E, is the incident energy of the ion. Since zo, Bi and es are related by Eqs. (1) and (2), it is possible to derive the charge state fraction and energy losses, Fj(z) and AEj(z), for ions on z(x) at z > 0.
3. Stochastic
model of charge exchange and energy loss of specularly reflected ions
The charge state and energy of ions at specular reflection from a crystal surface change along the tra-
rEzoz !! g,l5LL Y w 10-
0
I
5
I
I
10
15
20
ANGLE (mrad 1
Fig. 2. Energy losses of scattered He+ and He’+ ions for 0.67 MeV He+ incident of the (100) surface of SnTe with 19~= 3.3 and 6.5 mrad. III. SURFACE
PHENOMENA
?I Fgjii et al. /Nucl. k&r. and Meh. in Phys.Res. 3 90 (1994) 212-215
214
jectory, z(x). We define the probability distribution, fj (z(x), E), for ions with charge +je and energy E at a point (x, z) on the trajectory z(x) of specular reflection. The distribution Jcl(z(x), E) satisfies the following stochastic equation: fj(z+Sz~
3
I
B-
LOSS of He’
E)
= Qjj( I) Szkm d(GE)fj(z,
+
I
0.67 MeV He* on Snfe (100)
C Q,j( Z) izj
xw,(z;
SE)
E + SE)wj( z; SE) Sr
Szimd(SE)fi( Z, E + SE) sz,
(4) 0
where wj(z(n); SE) is the probability (per unit path length) for energy loss SE of an ion of charge +je at distance x from the surface, Qji(z(n)) is the transition probability (per unit length) of ionic charge from tje to +ie at a distance x from the surface. We now neglect the dependence of wj(z(x); 6E) and Qji(z(x)) on ion energy, E. Expanding fj in Eq. (4) into a Taylor series, and neglecting the terms containing (8~)~ and (SE)” (n > l), we obtain a set of differential equations for fifz(x), E):
The charge state fraction, Ij;(z), and energy loss, AEj(z(x)), at z(x) are derived from fi(t, E) using
vjt z, E)Pz
5:( z(x))
=Sj(z(x)>afi(z, + C[
DISTANCE
:ROM
SURFACE
x (
A i2
Fig. 4. Position-dependent probabilities, Qrz(x) and Q,,(x), of electron loss and electron capture of 0.67 MeV He ions near the (109) surface of SnTe. Solid lines are the probabilities derived from Bohr and Bohr-tindhard models. Broken lines are drawn to guide the eye.
= km&(z, E) dE,
(7)
E)/aE
Qij(z)fi(Z, E)-Qji(Z>.&~El], (5)
AEj(z(X))
=Es-~~Efi(z~
E) dE/imfj(z,
E) dE,
i#j
(8)
where Sj(z(x)) is the position-dependent stopping power for an ion of charge +je. Sj(z(x)) is related to wj(z; SE) via s,(z(~))=~~(x)=~~d(SE)[~~(z;
SE) SE].
(6)
where E, is the energy of the incident ion. For the scattering of MeV He ions from the (100) surface of SnTe, we consider only He’ and He2+ ions, since the He0 fraction is less than 1%. Thus, using Eqs. (5), (7) and (8), we obtain the electron-loss and electron-capture probabilities, QIZ(x) and Q,,(x),
Qji(x)=I;l(z)/[4:Fl(z)+q22F*(z>] x(l/[AEj(z)-A&(z)]
o.7i
-q;AE,(z)],‘dz
d[q;AEi(z) +q$‘F;.(z)
d4(z)/dz], (9)
where qje is the effective charge of the Hei+ ion and s&x) is the stopping power for an isotachic proton and we assume sj(x) = q,3Jx).
4. Derivation of electron-transfer I
I
0
15
5
20
ANGLE ‘fmrad)
Fig. 3. Ratio of the fraction FI (0,; 0,) of He+ ions and the fraction Fz(Bi; 0,) of He’+ ions in the scattered beam for 0.67 MeV He+ ions incident on the (100) surface of SnTe with fIi = 3.3 and 6.5 mrad.
probabilities
The position-dependent electron-loss and electroncapture probabilities, Q&x> and Q,,(x), are obtained from the experimental E;(ei; @,) and Ej(Bi; 0,) using Eqs. (3) and (9). With the observed angular dependence of energy loss and charge state fractions of the scattered He ions
Y. Fujii et al. /Nucl. Instr. and Meth. in Phys. Res. B 90 (1994) 212-215
such as those shown in Figs. 2 and 3, the electron-loss and electron-capture probabilities are calculated. Examples of the probabilities obtained in this way are shown in Fig. 4 for 0.67 MeV He ions near the (100) surface of SnTe, where we assumed that qj = j. The probabilities are smaller than those calculated with Bohr and Bohr-Lindhard models, which are shown by solid lines in the figure. With the derived charge-exchange probabilities and with the position-dependent stopping powers, which can be obtained from experimental energy losses of the scattered ions [S-7], we can now study details of the ion-surface interaction processes at glancing angle scattering from a clean surface.
Acknowledgements We are grateful to K. Yoshida, K. Norisawa and the staff members of the Department of Nuclear Engineering of Kyoto University for the availability of the 4 MV
215
Van de Graaff accelerator. The work is partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education.
References [l] M. Mannami, Y. Fujii and K. Kimura, Surf. Sci. 204 (1988) 213. [2] Y. Fujii, S. Fujiwara, K. Kimura and M. Mannami, Nucl. Ins&. and Meth. B 58 (1991) 18. [3] Y. Fujii, K. Narumi, K. Kimura, M. Mannami, T. Hashimoto, K. Ogawa, F. Ohtani, T. Yoshida and M. Asari, Appl. Phys. Lett. 63 (1993) 2070. [4] K. Kimura, M. Hasegawa and M. Mannami, Phys. Rev. B 36 (1987) 7. [5] Y. Fujii, S. Fujiwara, K. Narumi, K. Kimura and M. Mannami, Surf. Sci. 277 (1992) 164. [6] Y. Fujii, K. Kishine, K. Narumi, K. Kimura and M. Mannami, Phys. Rev. A 47 (1993) 2047. [7] K. Kimura, Y. Fujii, M. Hasegawa, Y. Susuki and M. Mannami, Phys. Rev. B 38 (1988) 1052. [8] D.S. Gemmell, Rev. Mod. Phys. 46 (19741 129.
III. SURFACE PHENOMENA