Memory effect on energy losses of charged particles moving parallel to solid surface

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Memory eﬀect on energy losses of charged particles

moving parallel to solid surface

C.M. Kwei

a,*

, Y.H. Tu

a

, Y.H. Hsu

a

, C.J. Tung

b

a_{Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan} b_{Department of Nuclear Science, National Tsing Hua University, Hsinchu 300, Taiwan}

Received 16 June 2005; received in revised form 22 August 2005 Available online 26 October 2005

Abstract

Theoretical derivations were made for the induced potential and the stopping power of a charged particle moving close and parallel to the surface of a solid. It was illustrated that the induced potential produced by the interaction of particle and solid depended not only on the velocity but also on the previous velocity of the particle before its last inelastic interaction. Another words, the particle kept a mem-ory on its previous velocity, v0, in determining the stopping power for the particle of velocity v. Based on the dielectric response theory,

formulas were derived for the induced potential and the stopping power with memory effect. An extended Drude dielectric function with spatial dispersion was used in the application of these formulas for a proton moving parallel to Si surface. It was found that the induced potential with memory effect lay between induced potentials without memory effect for constant velocities v0and v. The memory effect

was manifest as the proton changes its velocity in the previous inelastic interaction. This memory eﬀect also reduced the stopping power of the proton. The formulas derived in the present work can be applied to any solid surface and charged particle moving with arbitrary parallel trajectory either inside or outside the solid.

PACS: 78.20.Bh; 78.70.g

Keywords: Memory eﬀect; Surface excitation; Inelastic interaction

1. Introduction

When a charged particle moves close and parallel to the surface of a solid, induced potential is produced due to the interaction of particle and solid. This potential is then acted on the particle resulting to a stopping power. Theoretical derivations of the induced potential and the stopping power were previously made[1–4]for a constant velocity, v0, of the

particle until it experienced an inelastic interaction. After the interaction, the particle changed its velocity to v and continued to interact with the solid. For a second inelastic interaction, it was generally assumed that a new induced

potential, dependent only on v but not on v0, was generated.

This new potential then determined the stopping power acting on the particle of velocity v. In the present work, the induced potential and stopping power for the second inelastic interaction were derived using image charges and dielectric response functions. It was found that the particle previous velocity v0had also an eﬀect on the second inelastic

interaction. Another words, the particle kept a memory on its previous velocity, v0, in determining the stopping power

for the particle of velocity v.

The response of solid to a charged particle moving close and parallel to the surface may be characterized by its sur-face loss-function, Im[1/(e + 1)], where e is the dielectric function of the solid and Im[ ] denotes the imaginary part. A sum-rule-constrained extended Drude dielectric function with spatial dispersion [5]was established with parameters determined from optical data. Previously, this dielectric

* _{Corresponding author. Tel.: +886 3 5712121x54136; fax: +886 3}

5727300.

E-mail addresses:[email protected],[email protected] (C.M. Kwei).

www.elsevier.com/locate/nimb Nuclear Instruments and Methods in Physics Research B 243 (2006) 293–298

NIM

B

Beam Interactions with Materials & Atoms

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function was applied to the interface [6–8] and overlayer systems [3,9,10] for a charged particle without memory effect. With the consideration of memory effect, we applied this dielectric function in this work. The induced potential was then derived by solving Poisson equations in Fourier space by a method of image charges which satisfied the boundary conditions. The stopping power was then con-structed from the derivative of the induced potential at par-ticle position. Calculations were made using these formulas for a proton moving parallel to Si surface. Results were ana-lyzed for the dependences of the induced potential and stop-ping power on proton velocities before and after the last inelastic interaction, the distance from surface, and the dis-tance from previous inelastic interaction. Finally, the calcu-lated results with memory effect were compared with the corresponding results without memory effect.

2. Theory

Fig. 1illustrates the problem studied in the present work. A particle of charge q, velocity ~v0and energy E0moves

par-allel to the interface of two media of dielectric functions e1ðk

*

;xÞ and e2ðk *

;xÞ. The interface is located at z = 0, with z-axis perpendicular to the interface plane and directed from e1ðk

*

;xÞ to e2ðk *

;xÞ. The particle is moving along y-direction at a distance D above the interface. At the moment t = 0, the particle experiences an inelastic interaction which changes particle velocity and energy to ~vand E. Assuming the particle continues to move along the same direction, the induced potential at t > 0 is of special interest here. For z > 0, the scalar potential is produced by the particle and a ﬁctitious charge at z < 0 near the interface. For

z < 0, the potential is produced by a fictitious charge at par-ticle position and by another fictitious charge at z > 0 near the interface. These fictitious charges should be established using boundary conditions that are satisfied at the interface. Thus the Poisson equations in Fourier space are

u1ðk * ;xÞ ¼ 4p k2e1ðk * ;xÞ qðk*;xÞ þ qfðQ * ;xÞ h i ð1Þ for z < 0 and u2ðk * ;xÞ ¼ 4p k2e2ðk * ;xÞ qðk*;xÞ qfðQ * ;xÞ h i ð2Þ for z > 0, where k*¼ ðkx; ky; kzÞ ¼ ðQ * ; kzÞ is the momentum

transfer and x is the energy transfer. The Fourier trans-form of the charge density distribution of the particle qð~r; tÞ ¼ qdðxÞdðz DÞ½dðy v0tÞHðtÞ þ dðy vtÞHðtÞ ð3Þ is given by qðk*;xÞ ¼ qeikzD Z 0 1 eiðxkyv0Þs_ds_þ Z 1 0 eiðxkyvÞs_ds ; ð4Þ

where d( ) and H( ) are the delta- and step-functions, respec-tively. To satisfy the boundary conditions at the interface, the ﬁctitious charge in Fourier space is given by

qfðQ * ;xÞ ¼ Z 1 1 qðk*;xÞ k2 1 e2ðk * ;xÞ 1 e1ðk * ;xÞ 2 4 3 5dkz Z 1 1 1 k2 1 e2ðk * ;xÞ þ 1 e1ðk * ;xÞ 2 4 3 5dkz . ð5Þ

Combining Eqs.(4) and (5), one gets qfðQ * ;xÞ ¼ q Z 0 1 eiðxkyv0Þs_ds_þ Z 1 0 eiðxkyvÞs_ds 1 e2ðD;Q * ;xÞ 1 e1ðD;Q * ;xÞ 1 e2ðQ * ;xÞþ 1 e1ðQ * ;xÞ 2 4 3 5; ð6Þ

where the eﬀective dielectric function is given by 1 eLðD; Q * ;xÞ ¼ 1 2p Z 1 1 eikzD k2eLðk * ;xÞ dkz ð7Þ

for L = 1 and 2 and eLðQ *

;xÞ ¼ eLð0; Q *

;xÞ.

Substituting Eqs. (4)–(7) into Eqs. (1) and (2), one obtains the scalar potentials in Fourier space, i.e. u1

ðk*;xÞ and u2ðk *

;xÞ. The induced potentials in Fourier space, uind 1 ðk * ;xÞ and uind 2 ðk *

;xÞ, are then obtained by removing the vacuum potential of the particle from scalar potentials. One gets

uind 1 ðk * ;xÞ ¼4p k2 1 e1ðk * ;xÞ 1 0 @ 1 Aqðk*;xÞ þ 4p k2e1ðk * ;xÞ q_fðQ*;xÞ ð8Þ

Fig. 1. A sketch of the problem studied in the present work. A particle of charge q, velocity ~v0 moves parallel to the interface of two media of

dielectric functions e1ðk *

;xÞ and e2ðk *

;xÞ. The interface is located at z = 0 and the particle is moving along y-direction at a distance D above the interface. At time t = 0, the particle experiences an inelastic interaction which changes particle velocity to v. Special interest is on the induced potential and the stopping power at t > 0.

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for z < 0 and uind 2 ðk * ;xÞ ¼4p k2 1 e2ðk * ;xÞ 1 0 @ 1 Aqðk*;xÞ 4p k2e2ðk * ;xÞ qfðQ * ;xÞ ð9Þ for z > 0. If the particle is moving in vacuum, i.e. e2= 1,

one gets uind 2 ðk * ;xÞ ¼ 4pq k2 Z 0 1 eiðxkyv0Þs_ds_þ Z 1 0 eiðxkyvÞs_ds 1 e2ðD;Q * ;xÞ 1 e1ðD;Q * ;xÞ 1 e2ðQ * ;xÞþ 1 e1ðQ * ;xÞ 2 4 3 5 ð10Þ

after substituting Eq. (6) into Eq. (9). Since e is weakly dependent on kzthan the rest of k

*

components, one may assume eðk*;xÞ ¼ eðQ*;xÞ. This assumption was previously adopted by Yubero et al.[11,12]in the analyses of REELS spectra and by Kwei et al.[9,13]in the calculations of elec-tron elastic backscattering spectra. Using this assumption, Eq.(7) becomes 1 eLðD; Q * ;xÞ e Dj jQ 2Q 1 eLðQ * ;xÞ . ð11Þ

Applying the relation[14]for the product of the step-func-tion and the delta-funcstep-func-tion, i.e. HðsÞdðsÞ ¼1

2dðsÞ, Eq.(10) may be written as uind 2 ðk * ;xÞ ¼ 4p 2_q k2 dðx kyv0Þ þ dðx kyvÞ e Dj jQe1ðQ * ;xÞ 1 e1ðQ * ;xÞ þ 1 2 4 3 5. ð12Þ

Applying eðQ*;xÞ ¼ eðQ; xÞ and e(Q, x) = e*_{(Q, x), the}

induced potential in real space is obtained by an inverse Fourier transform as uind₂ ð~r; tÞ ¼q p2 Z 1 0 dx Z 1 0 dk Z p 0 sin h dh Z 2p 0 d/ dðx kyv0Þe Dj jQ cos kð zzÞRe e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 ei kðxxþkyyxtÞ þq p2 Z 1 0 dx Z 1 0 dk Z p 0 sin h dh Z 2p 0 d/ dðx kyvÞe Dj jQ cos kð zzÞRe e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 ei kðxxþkyyxtÞ . ð13Þ

Expanding the d-function according to dðx kv sin h sin /Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðkv sin hÞ2 x2 q d / sin1 x kvsin h n þd / p sin1 x kvsin h h io ; ð14Þ and applying the conservation relations of energy and momentum, it gives uind 2 ð~r; tÞ ¼ 2q p2 Z E 0 dx Z kmax kmin dk Z k x v0 dQQ k cos zpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 Q2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 Q2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQv0Þ 2 x2 q e Dj jQ cos x v0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQv0Þ 2 x2 q Re e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 cos x v0 ðy v0tÞ Im e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 sin x v0 ðy v0tÞ þ2q p2 Z E 0 dx Z kmax kmin dk Z k x v dQQ k cos zpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 Q2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 Q2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQvÞ2 x2 q e Dj jQ cos x v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQvÞ2 x2 q Re e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 cos x vðy vtÞ Im e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 sin x vðy vtÞ ; ð15Þ

where Q¼ k sin h, kmax¼

ffiffiffiffiffiffiffiffiffiffi 2ME p þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ME 2Mx, kmin¼ ffiffiffiffiffiffiffiffiffiffi 2ME p

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ME 2Mxand M is the mass of the particle. The stopping power is related to the derivative of uind

2 ð~r; tÞ at the position of particle, i.e. ~rp¼ ðxp; yp; zpÞ ¼

ð0; vt; DÞ, for t > 0. One finds dW ds yp ¼2q 2 p2_v 0 Z E 0 dx Z kmax kmin dk Z k x v0 dQQ k xcosj jDpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 Q2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 Q2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQv0Þ 2 x2 q e Dj jQ Re e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 sin x v0v ðv v0Þyp þIm e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 cos x v0v ðv v0Þyp þ2q 2 p2_v Z E 0 dx Z kmax kmin dk Z k x v dQQ k xcos D j jpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 Q2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 Q2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQvÞ2 x2 q e Dj jQIm e1ðQ; xÞ 1 e1ðQ; xÞ þ 1 . ð16Þ

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Letting v = v0, Eqs.(15) and (16)reduce to the same

for-mulas for the induced potential and the stopping power that were derived without memory eﬀect by Kwei et al. [3]. Note that the stopping power in Eq.(16) is yp

depen-dent. Thus the stopping power with memory eﬀect may be obtained by an average over all particles paths

dW ds ¼ R1 0 yp ke yp=k dW ds yp dyp R1 0 yp ke yp=k_dy p ; ð17Þ where yp ke

yp=k _{is the probability that the particle} encoun-ters an inelastic interaction in the distance yp [11,15] and

k is the inelastic mean free path. 3. Results and discussion

Fig. 2shows plots of the real and imaginary parts of the surface response function, i.e. [e1(0, x) 1]/[e1(0, x) + 1],

for vacuum–Si interface as a function of energy transfer x. The solid curves are ﬁtting results obtained in the pres-ent work. The dotted curves are corresponding data deduced from optical data[16]. It is seen that the calculated results are in good agreement with measured data.

The induced potential for a proton moving parallel to the surface of Si was calculated using Eq. (15). Results at the position of proton for yp= 5 a.u., v0= 10 a.u. and

D = 1 a.u. are plotted in Fig. 3(solid curve) as a function of proton velocity v. These results are compared with cor-responding data without the memory eﬀect (dotted curve), where the abscissa v here may also be interpreted as v0. It

reveals that both curves show a dip around v = 1.5 a.u. The existence of a dip was also shown for a proton moving parallel to Al surface in the plasmon-pole dielectric func-tion model [2]. As indicated in the ﬁgure, the magnitude of induced potential decreases with increasing velocity for

velocities larger than the dip velocity and increases with increasing velocity for velocities smaller than the dip veloc-ity. Note that there is a signiﬁcant diﬀerence between solid and dotted curves. At v = 2 a.u., for instance, solid and dotted curves correspond to v0= 10 a.u. (with memory

eﬀect) and v0= 2 a.u. (without memory eﬀect), respectively.

Since the velocity change, v0 v, in the solid curve is large

so that the diﬀerence between solid and dotted curves is also large. At v = 8 a.u., on the other hand, solid and dot-ted curves correspond to v0= 10 a.u. and 8 a.u.,

respec-tively. In this case, the velocity change in the solid curve is small so that the difference is also small. Thus the differ-ence in induced potentials calculated with and without the memory effect increases with increasing velocity change in the previous inelastic interaction when v P 1.5 a.u. When v < 1.5 a.u., however, the difference decreases with increas-ing velocity change.

The induced potential shown inFig. 3is at proton posi-tion, i.e. y = yp.Fig. 4plots the induced potential at a

posi-tion, along the trajectory of proton, with a distance y yp

from the proton for yp= 5 a.u. and D = 1 a.u. The solid

curve is results (with memory eﬀect) of the induced poten-tial for v0= 5 a.u. and v = 3 a.u. The dotted and dashed

curves are corresponding results (without memory eﬀect) for v = v0= 3 a.u. and 5 a.u., respectively. In all cases,

the induced potential exhibits an oscillation behavior over the distance from the proton, a behavior which was also observed by Arista [4]. Note that the induced potential for v0= 5 a.u. and v = 3 a.u. (with memory eﬀect) lies

between induced potentials for v = v0= 3 a.u. and 5 a.u.

(without memory eﬀect). This indicates that the induced potential carries a memory eﬀect on proton previous veloc-ity before its last inelastic interaction. A similar plot is made in Fig. 5 for a proton moving at distances D = 1

Fig. 2. A plot of the real and imaginary parts of the surface response function, [e1(0, x) 1]/[e1(0, x) + 1], for vacuum–Si interface as a

func-tion of energy transfer x. The solid curves are ﬁtting results using the extended Drude dielectric function. The dotted curves are results deduced from the optical data[16].

Fig. 3. The induced potential for a proton moving parallel to the surface of Si. Results (solid curve) are plotted at proton position for yp= 5 a.u.,

v0= 10 a.u. and D = 1 a.u. as a function of proton velocity v.

Corre-sponding results without the memory eﬀect are plotted (dotted curve) for a comparison.

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(solid curve), 2 (dotted curve) and 3 a.u. (dashed curve) from Si surface with yp= 5 a.u., v0= 5 a.u. and v = 3 a.u.

It is seen that as D increases the induced potential (absolute value) decreases. This reveals that the induced potential is greater for a proton moving closer to the solid surface.

Fig. 6shows results of the stopping power for a proton moving parallel to and with at a distance D = 2 a.u. from the Si surface. Solid and dotted curves are for v0= 10 a.u. (with memory eﬀect) and v0= v (without

mem-ory effect), respectively. The existence of a maximum stop-ping power at a velocity around 1.5 a.u. was shown. The existence of a maximum was also found for some semi-infi-nite solids by Arista[4]. A notable difference between solid and dotted curves was shown in the figure. Without the

memory eﬀect, stopping power is solely determined by v without reference to v0. With memory eﬀect, however, the

stopping power is aﬀected by both v and v0. The memory

eﬀect reduced the stopping power for the proton. 4. Conclusions

A theoretical treatment was developed to account for the memory eﬀect on the induced potential and the stop-ping power for a charged particle moving close and parallel to the surface of a solid. It was illustrated that the particle had a memory on its velocity before the previous inelastic interaction, v0, in determining the induced potential and

the stopping power for the particle of velocity v. Using the method of image charges, analytical formulas have been derived for the induced potential and the stopping power for particle–surface interactions. An extended Drude dielectric function with spatial dispersion was used in the calculations of the induced potential for a proton moving parallel to Si surface. It was found that the mem-ory effect affected the induced potential and the stopping power. The induced potential with memory effect lay between induced potentials without memory effect for con-stant velocities v0and v. The stopping powers with memory

effect were lower than those without memory effect. For a small difference between v0and v, which was more probable

in an interaction, the memory effect on both induced poten-tial and stopping power was unobvious. Thus, any Monte Carlo simulation using inelastic cross sections without the memory effect in the particle transport near the surface was practically valid. The memory effect, however, should be detected for rare consecutive surface interactions involv-ing large difference between v0and v. The formulas derived

in the present work can be applied to any solid surface and

Fig. 4. The induced potential at a distance y ypfrom the proton with

yp= 5 a.u. and D = 1 a.u. from Si surface. The solid curve is results with

the memory eﬀect for v0= 5 a.u. and v = 3 a.u. The dotted and dashed

curves are results without the memory eﬀect for v = v0= 3 a.u. and 5 a.u.,

respectively.

Fig. 5. The induced potential at a distance y ypfrom the proton with

D = 1 (solid curve), 2 (dotted curve) and 3 a.u. (dashed curve) from Si surface. Here yp= 5 a.u., v0= 5 a.u. and v = 3 a.u.

Fig. 6. Results of the stopping power for a proton moving parallel to with a distance D = 2 a.u. from the Si surface. Solid and dotted curves are for v0= 10 a.u. (with memory eﬀect) and v0= v (without memory eﬀect),

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charged particle moving with arbitrary parallel trajectory either inside or outside the solid.

Acknowledgement

This research was supported by the National Science Council of the Republic of China under contract no. NSC92-2215-E-009-063.

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