Dependence of electronic stopping power on incident angle for fast ions moving near solid surfaces

Nuclear Instruments and Methods in Physics Research B 135 (1998) 164±168

Dependence of electronic stopping power on incident angle for fast ions moving near solid surfaces You-Nian Wang *, Xin-Lu Deng, Teng-Cai Ma The State Key Laboratory for Materials Modi®cation by Laser, Electron, and Ion Beams, Dalian University of Technology, Dalian 116023, People's Republic of China

Abstract The stopping power of fast ions moving near solid surfaces at glancing incidence is studied using the specular-re¯ection model and the dielectric response theory. A local frequency-dependent dielectric function is used to describe surface electronic excitations. General formulas expressing the dependence of the stopping power on the incident angle are given for incoming trajectories and outgoing trajectories, respectively. Numerical results show that the stopping power increases for the incoming trajectory and decreases for the outgoing trajectory with the increasing of incident angle, respectively. Ó 1998 Elsevier Science B.V. PACS: 34.50.Bw; 61.85.+p Keywords: Solid surfaces; Charged particles; Electronic excitations; Surface wake potential; Surface stopping power

1. Introduction The problem of charged particles interacting with solid surfaces has received much attention during the past years mainly related to the surface wake potential, the energy loss and the ion-induced electron emission. By using the specular-re¯ection model and the dielectric response theory, several authors [1±5] have studied the electronic stopping power (i.e., the energy loss per unit length due to electronic excitations) for an ion moving near the solid surface. It was found that the stopping power increases rapidly as the distance of the *

Corresponding author. E-mail: [email protected].

ion from the surface decreases. All of the above investigations, however, are almost limited to the case of the ion trajectory parallel to the solid surface. In fact, when an ion is incident onto a solid surface at a glancing angle smaller than a critical angle, it does not penetrate into the solid and is re¯ected specularly into the vacuum region. Recent experiments [6±9] showed that there is a tendency that the energy loss depends on the incident angle. The purpose of the present work is to study the dependence of the stopping power on the incident angle for a fast ion using the dielectric response theory and the specular-re¯ection model. A local frequency-dependent dielectric function will be used to describe the bulk solid. Explicit expressions of the stopping power as a function of the

0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 8 - 5 8 3 X ( 9 7 ) 0 0 6 4 2 - 3

Y.-N. Wang et al. / Nucl. Instr. and Meth. in Phys. Res. B 135 (1998) 164±168

glancing incident angle are obtained for the incoming trajectory and the outgoing trajectory, respectively. 2. General formulas We consider that a point particle with charge number Z1 and velocity v is incident onto a solid surface at a glancing angle and is re¯ected specularly into the vacuum when it is close to the surface. The surface is de®ned by the z ˆ 0 plane. The solid lies in the region z < 0 and the vacuum is in the region z > 0. The trajectory of the particle, whose recoil is assumed to be negligible, is described by the equation r ˆ vt. In the following discussions, we shall use the notations r ˆ …R; z†; k ˆ …Q; kz †, and v ˆ …vjj ; vz †. R; Q, and vjj represent components parallel with the surface. When the charged particle approaches the surface the electronic excitations and the surface wake potential will be induced in the neighbourhood of the external charge. The surface electronic excitation can be described by the well-known specular-re¯ection model (SRM) introduced by Ricthie and Marusak [10]. The SRM assumes that the solid is described by an electron gas with the density n0 in which the surface is assumed to be abruptly terminated, the electrons of the solid are considered to be specularly re¯ected at the surface, and thus, the electronic charge density vanishes outside the surface. Although more sophisticated treatments are necessary to describe a real surface, this mode has found wide applications involving the interaction of charges with plane-bounded solids. In the SRM, the surface wake potential is determined by the external charge, its image and a surface charge ®xed by the boundary conditions. By solving the Poisson equation, one can obtain the potential Z Z1 e jvz jeÿQjzj Wind …r; t† ˆ 2 dQ dx 2 2 2p …x ÿ Q
vjj † ‡ …Qvz †  F …Q; x†ei…Q
Rÿxt† ; where F …Q; x† ˆ



 es …Q; x† ÿ 1 ; es …Q; x† ‡ 1

…1†

…2†

165

R1 es …Q; x† ˆ …Q=p† ÿ1 dkz =‰k 2 e…k; x†Š is the surface dielectric function [11], and e…k; x† is the bulk dielectric function. The wake potential is asymmetrical in the direction of the projectile motion. As a result, the retarding force @Wind =@r acts on the ion and causes it to lose its energy. The stopping power can be written as   Z 1 @Wind …3† S ˆ Z1 e dr ÿ : v @t rˆvt

Using Eqs. (1) and (3), we get 2Z …Z1 e† ixjvz jeÿQjz0 j dQ dx Sˆ 2 2p v …x ÿ Q
vjj †2 ‡ …Qvz †2  F …Q; x†ei…Q
vjj ÿx†t ;

…4†

where z0 ˆ jvz tj is the distance of the ion with respect to the surface. In the case of the parallel trajectory, after taking jvz j ! 0‡ and performing the x integration in Eq. (4), the stopping power can be reduced to that [5] S 2…Z1 e† ˆ pv

2

Z1

ZQv dQ

0

0

x dx p Im‰F …Q; x†Šeÿ2Qz0 : Q2 v2 ÿ x2 …5†

Eq. (4) gives us the stopping power when the ion moves in the vacuum for an arbitrary orientation of its trajectory with respect to the surface. 3. Local dielectric function In order to get easily surveyed analytical results of the stopping power, here we use a local frequency-dependent dielectric function [12] e…k; x† ˆ 1 ÿ

x2p ; x…x ‡ ic† 1=2

…6†

where xp ˆ …4pn0 e2 =me † is the plasma frequency of the electron gas and c is the damping factor of the plasma mode. The local frequency-dependent dielectric function gives realistic values of the energy losses only for fast particles and points in the vacuum far p from the surface …z0  vjj =xs †, where xs ˆ xp = 2 is the surface plasma frequency.

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However, it can reproduce the qualitative properties of the stopping power that are obtained with some more complex dielectric functions. For a glancing incident ion, its motion history can be divided into two parts: the incoming (IN) trajectory and the outgoing (OUT) trajectory corresponding to vz < 0 and vz > 0, respectively. Substituting Eq. (6) into Eq. (4) and performing the x integration, the stopping power can be put in the forms, for the IN trajectory, 2

…Z1 e† x2s ˆ p vvjj

SIN

Z1

Zq dq

du ÿq

0

GIN …q; u; a; b† ÿ2sq p e q2 ÿ u2

…7†

the incident angle for the IN trajectory and the exit angle for the OUT trajectory, respectively. Comparing the stopping powers for the IN trajectory and that for the OUT trajectory, we observe the stopping powers depend on the history of the particle, which can be easily understood: when the particle approaches the solid surface and gets re¯ected, it creates a surface response accompanied by an induced ®eld and the ®eld will act on the particle only when it is travelling out from the surface due to the retarded character of the response.

and for the OUT trajectory

4. Numerical results and conclusions

SOUT

For a given value of the dimensionless distance s ˆ 5 and various values of the dimensionless damping factor b, Figs. 1 and 2 show the dependence of the stopping power ratio SIN …a†=S…0† [or SOUT …a†=S…0†] on the incident angle for the IN trajectory and on the exit angle for OUT trajectory, respectively. Here, S…0† is the stopping power for the parallel trajectory (vz ! 0). One observes from the ®gures that for larger values of the dimensionless damping factor b, the stopping power is almost independent of the incident angle and the exit angle for both of the IN trajectory and the OUT trajectory. For smaller value of b, however, the stopping power increases for the IN trajectory

2

ˆ

…Z1 e† x2s p vvjj ‡

Z1

Zq dq

0

ÿq

du ÿ2sq p ‰G…1† OUT …q; u; a; b†e q2 ÿ u2

…2† GOUT …q; u; s; a; b†eÿs…qÿb=2a† Š;

…8†

where GIN …q; u; a; b† ˆ

u2 …2aq ‡ b† ÿ aq…u2 ÿ 1 ÿ a2 q2 ÿ baq† 2

…u2 ÿ 1 ÿ a2 q2 ÿ baq† ‡ u2 …2aq ‡ b†

2

;

…9†

…1†

GOUT …q; u; a; b† ˆ

ÿu2 …2aq ÿ b† ‡ aq…u2 ÿ 1 ÿ a2 q2 ‡ baq† 2

…u2 ÿ 1 ÿ a2 q2 ‡ baq† ‡ u2 …2aq ÿ b†

2

;

…10†

2aq k …k cos H ÿ 12 b sinH†X2r ‡ …k sin H ‡ 12 b cos H†X2i

…2†

GOUT …q; u; s; a; b† ˆ 

X4r ‡ X4i

;

…11†

1 X2r ˆ …u ‡ k†2 ‡ a2 q2 ÿ b2 ; 4

…12†

X2i ˆ b…u ‡ k†; …13† q k ˆ 1 ÿ b2 =4, H ˆ s…k ‡ u†=a, b ˆ c=xs , s ˆ xs jz0 j=vjj , and a ˆ jvz j=vjj . Generally, the values of jvz j=vjj are much less than 10ÿ2 in most of the experiments, so a can be regarded approximately as

Fig. 1. The dependence of the stopping power ratio SIN …a†=S…0† on the incident angle for the IN trajectory and for a given s ˆ 5.

Y.-N. Wang et al. / Nucl. Instr. and Meth. in Phys. Res. B 135 (1998) 164±168

167

Fig. 2. The dependence of the stopping power ratio SOUT …a†=S…0† on the exit angle for the OUT trajectory and for a given s ˆ 5.

Fig. 3. The dimensionless stopping power W as a function of the dimensionless distance s of the ion from the surface for the IN trajectory and for a given incident angle a ˆ 0:005.

with the increasing of the incident angle and decreases for the OUT trajectory with the increasing of the exit angle. In addition to that, Fig. 2 shows that the stopping powers will become negative for the OUT trajectory, which can be explained as follows. In this trajectory, the electric ®eld acting on the particle comes from two parts: the ®eld induced directly in the OUT trajectory and the retarded ®eld induced in the IN trajectory. The direction of the former is same as that of the particle's motion and the latter is an oscillatory ®eld (see Eq. (8)), so the total ®eld may accelerate the particle (i.e., the stopping power becomes negative) for large exit angles. A similar result was obtained by N un~ez et al. [2] for the perpendicular trajectory (vjj ˆ 0). For a given value of the angle a ˆ 0:005 and various values of the dimensionless damping factor b, we plot the diemensionless stopping power 2 W ˆ S‰pvvjj =…Z1 exs † Š as a function of the dimensionless distance s of the ion from the surface in Figs. 3 and 4 for the IN trajectory and the OUT trajectory, respectively. The stopping power increases rapidly as the distance s from the surface decreases. The surface stopping power cannot be directly measured in experiments, however, one can measure the energy loss DE…a† which relates to both of the IN trajectory and the OUT trajectory. It

has been shown from experimental results [6±9] that the energy loss DE…a† decreases monotonously with increasing incident angle a. In the future work, we will calculate the energy loss, given by integrating the stopping power along both of the IN trajectory and the OUT trajectory, and expect that the theoretical results can reproduce the experimental data. In conclusion, we have studied the stopping power for a swift ion being onto a solid surface

Fig. 4. The dimensionless stopping power W as a function of the distance s of the ion from the surface for the OUT trajectory and for a given exit angle a ˆ 0:005.

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Y.-N. Wang et al. / Nucl. Instr. and Meth. in Phys. Res. B 135 (1998) 164±168

at a glancing angle by using the specular-re¯ection model and the local frequency-dependent dielectric function. The e€ects of the incident angle on the stopping power have been analysed for the IN trajectory and the OUT trajectory, respectively. It is found that the stopping power increases for the IN trajectory with the increasing of the incident angle and decreases for the OUT trajectory with the increasing of the exit angle. The local frequency-dependent dielectric function used in the present work is a very crude approximation. We will use a more realistic dielectric function such as the hydrodynamic approximation dielectric function to calculate the stopping power. Acknowledgements This work is supported by the National Natural Science Foundations of China (No.19575008 and No.69493502).

References [1] T.L. Ferrell, P.M. Echenique, R.H. Ritchie, Solid State Commun. 32 (1979) 419. ~ez, P.M. Echenique, R.H. Ritchie, J. Phys. C 13 [2] R. N un (1980) 4229. [3] G. Gumbs, M.L. Glasser, Phys. Rev. B 37 (1988) 1391. [4] M. Kitagawa, Nucl. Instr. and Meth. B 33 (1988) 409. [5] Y.N. Wang, W.K. Liu, Phys. Rev. A 54 (1996) 636. [6] K. Narumi, Y. Fujii, K. Kishine, K. Kimura, M.H. Mannami, Surf. Sci. 293 (1993) 152. [7] M. Fritz, K. Kimura, H. Huroda, M. Mannami, Phys. Rev. A 54 (1996) 3139. [8] Y. Fujii, S. Fujiwara, K. Narumi, K. Kimura, M. Mannami, Surf. Sci. 277 (1992) 164. [9] R. Pfandzelter, F. St olzle, Nucl. Instr. and Meth. B 72 (1992) 163. [10] R.H. Ritchie, A.L. Marusak, Surf. Sci. 4 (1966) 234. [11] D.M. Newns, Phys. Rev. B 1 (1970) 3304. [12] E. Fermi, Phys. Rev. 57 (1940) 485.