Oscillating surface effect in reflection-electron-energy-loss spectra

N. Pauly*Université Libre de Bruxelles, Service de Métrologie Nucléaire (CP 165/84), 50 avenue F. D. Roosevelt, B-1050 Brussels, Belgium

S. TougaardPhysics Institute, University of Southern Denmark, DK-5230 Odense M, Denmark

F. YuberoInstituto de Ciencia de Materiales de Sevilla, Isla de la Cartuja, E-41092 Sevilla, Spain

�Received 20 December 2004; revised manuscript received 6 September 2005; published 3 January 2006�

Development of models describing the processes undergone by an electron when it travels in the surfaceregion of solids is of great importance to acquire a more complete understanding of electron spectroscopies asx-ray photoelectron spectroscopy, Auger electron spectroscopy, or reflection-electron-energy-loss spectroscopy.Based on their previous semiclassical dielectric response model, and Tougaard and Yubero �Surf. InterfaceAnal. 36, 824 �2004�� recently developed the software QUEELS �QUantitative analysis of Electron EnergyLosses at Surfaces�, which allows to easily perform electron energy loss calculations within a dielectricresponse theory for different cases of electrons moving near surfaces in general geometries. We present resultsobtained within this model for an incoming electron that is backscattered in the surface region. The separateinteractions that take place while the electron moves in the vacuum and in the medium are also studied. Weshow that coupling effects between the incoming and outgoing trajectories occur with a magnitude thatdepends on the trajectory angle to the surface and on the backscattering depth. We find that this results in anoscillating behavior of the effective inelastic electron scattering. By studying the spatial and time dependenceof this phenomenon, we can conclude that it originates from charge fluctuations of surface plasmons excitedwhen the electron enters the solid. Moreover, we find that the main oscillatory behavior is damped and thecharacteristic damping energy is linearly correlated to the damping of the oscillator in the dielectric functionand to the energy of the plasmon.

DOI: 10.1103/PhysRevB.73.035402 PACS number�s�: 79.20.Uv, 73.20.Mf

I. INTRODUCTION

Quantitative interpretation of electron spectroscopies asx-ray photoelectron spectroscopy �XPS�, Auger electronspectroscopy �AES�, transmission-electron-energy-loss spec-troscopy �TEELS�, or reflection-electron-energy-loss spec-troscopy �REELS� requires a clear understanding of the in-elastic scattering processes undergone by an electrontraveling within the surface region.

Theories that describe these experiments were developedwithin the dielectric response theory for different cases ofelectrons moving near surfaces and for various geometries.Lindhard1 described the energy losses of electrons moving inan infinite medium; TEELS calculations based on the hydro-dynamical model were carried out by Ritchie;2 REELS cal-culations were done for normal incidence and exit angles inRef. 3 and for general angles in Ref. 4, whereas a model forXPS was analyzed in Ref. 5 taking into account the presenceof a static core hole. In order to easily perform these calcu-lations, a software including all these models namedQUEELS �quantitative analysis of electron energy losses atsurfaces� was recently developed by two of us.6 The modelsin Refs. 4 and 5 have previously been tested by comparisonto both REELS �Ref. 7� and XPS �Ref. 8� experiments andthe agreement was found to be good. As an example, weshow in Fig. 1 the effective cross sections modeled for a1000 eV electron traveling a fixed distance �30 � in alumi-num corresponding to a TEELS, REELS, or XPS geometry;

we also show the cross section for an electron traveling in ainfinite medium. Figure 1 shows that the amount of surfaceexcitation depends not only on the depth traveled by theelectron in the medium but also on the considered phenom-enon.

In the case of REELS or XPS experiments, electronsreach different depths in the solid. This implies that an inte-gration of the effective cross sections over all involved elec-tron path lengths must be performed in order to comparetheoretical results with experiments. Inelastic scattering crosssections determined from an average over all electrons pathshave already been calculated3,4 and have shown a goodagreement with experiments �see also discussion below inSec. II�. From the inelastic scattering cross sections we arethus able to calculate the electron inelastic mean free path �for different geometries and to compare our � results to pre-vious values obtained theoretically or experimentally. In thecase of REELS calculations, for instance, we expect � resultsclose to the values given in the literature,9,10 displaying onlyweak variations as a function of the incidence and exitangles, because of the different surface excitations. We show,in Fig. 2, these � values calculated within the formalismdeveloped in Ref. 4 for an aluminum target as a function ofthe exit angle �out for an angle of incidence �in=0° both withrespect to the surface normal and for two electron energies�E=1000 eV and E=3400 eV�. For exit angles larger thanabout 15°, we obtain, as expected, a good agreement be-tween our results and the values given in Refs. 9 and 10.

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However, for �out�15°, a large deviation between the tworesults appears.

The physical origin of this variation is a coupling betweenthe incoming and outgoing trajectories; this coupling effect isa consequence of the interaction of the electron charge in itsoutgoing trajectory with the electric field excited in the me-dium by the electron in its incoming trajectory. This interac-tion has already been reported in Ref. 3 and was also ob-served by other groups11–14 in calculations of energy lossesas a function of the penetration depth for an incoming elec-tron in a specular reflection geometry. However, up to nowthis interaction effect was either regarded as an artifact,11 ornot studied in detail. No experimental evidence of oscillationin a REELS geometry has been found in the literature, but itshould be possible to verify the variation observed in Fig. 2by experiment.

It is the aim of this paper to study, in greater detail, thiscoupling effect for a REELS geometry. In order to obtain agood understanding of this phenomenon, we will consider inthe following single trajectories and fixed paths traveled by

the electron in the medium rather than the average over alltrajectories as in Fig. 2 because the effects are then moreclearly observed. This also makes it is possible to study thedevelopment of the phenomenon both in space and time.Thus, in the following, when we will talk about our REELScalculated results and will always imply results obtained fora fixed path length traveled by the electron in the solid.

We will also compare the REELS results to those obtainedin a TEELS geometry for the same path length traveled. Thiscomparison will allow us to clearly identify the effects of theinteraction process that is missing in the TEELS geometrycalculations.

We also note that the coupling effect treated in this paperis a purely classical effect as opposed to a well-knownquantum-mechanical interference effect that has been studiedin low-energy high-resolution REELS.15

In the following, we will briefly describe the method usedto obtain the inelastic electron-scattering cross sections for ageneral REELS geometry.4

FIG. 1. Calculated cross sections for a 1000 eV electron traveling a path of 30 Å inside an Al sample for �a� TEELS, �b� REELS, and�c� XPS geometries; �d� shows the cross section for an infinite medium.

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II. THEORY

The present REELS formalism is based on a semiclassicaldescription of the interaction of charged particles with solidinterfaces in terms of the dielectric properties of the bulkmaterial. It was first calculated for normal entrance and exitangles.3 Based on the so-called surface reflection model,12,16

the theory was later extended to a formalism for generalincidence and exit angles.4 The validity of this rather simplemodel was thoroughly studied by comparison to experi-ments. These studies have shown that the model is surpris-ingly accurate. In particular, it was shown to be able to re-produce, quantitatively, both the angular and the energydependence of the differential inelastic scattering cross sec-tions of electrons obtained by experimental REELS �seeRefs. 7 and 17–19 for instance�. Besides, it has been used tosuccessfully determine the dielectric function of several ma-terials from analysis of REELS spectra and these were foundto be in good agreement with existing data based on opticalmeasurements.17–20 A similar semiclassical dielectric re-sponse model for photoelectron spectroscopy, which includesthe effect of the static core hole, was also developed.5 Thevalidity of this was also tested recently8 and found to be ingood quantitative agreement with experimental spectra.

We study, thus, the problem shown in Fig. 3�a�. An inci-dent electron travels in vacuum ��=1� with a velocity vi

toward a solid at an angle �i with respect to the surfacenormal. It enters the solid specified by its dielectric function��k ,�� and is elastically backscattered at a depth x0 �surfacemedium at x=0�. Then it changes direction and leaves thesolid with a velocity vo at an outgoing angle �o with respectto the surface normal. It is assumed that the energy �� lostby the electron is much smaller than the primary electronenergy E.

We want to calculate the effective inelastic electron-scattering cross section Keff�E ,�� ,x0 ,�i ,�o�, defined as theaverage probability that the electron will lose an energy ��per unit energy loss and per unit path length traveled by theelectron in the solid xt=x0�1/cos �i+1/cos �o�. It is found4

that Keff�E ,�� ,x0 ,�i ,�o� can be obtained from the followingexpression:

Keff�E,��,x0,�i,�o� = Re� − 2i

�2��4�2�x0� dk� dr

�� dtei�kr−�t�kve�r,t�ind�k,��� ,

�1�

where e�r , t� is the charge density of the electron andind�k ,�� is the potential induced by the electron.

ind�k ,�� is obtained within the surface reflectionmodel12,16 in which the potential of a system of movingcharges in a semi-infinite medium is solved by consideringtwo infinite pseudomedia M �medium� and V �vacuum�. Inthe pseudomedia M and V, we consider the electron and itsimage charge traveling while they are, in the real case, insidethe medium or inside the vacuum, respectively. The pseudo-medium M is characterized by the dielectric function ��k ,��and by a surface charge density � at x=0, while V is definedby �=1 and by a surface charge density �V at x=0. Thefictitious surface charges � and �V are introduced to fulfillthe boundary conditions. They are determined by the require-ment that the potentials, and the normal components of thedisplacement vectors in each pseudomedium must be con-tinuous at the surface.4

With these definitions, the effective inelastic scatteringcross sections can be expressed4 as a sum of four contribu-tions, which correspond to the energy losses of the electronwhile traveling in vacuum Keff

Vi and in the medium KeffMi for the

incoming trajectory and in vacuum KeffVo and in the medium

KeffMo for the outgoing trajectory

Keff = KeffVi + Keff

Mi + KeffMo + Keff

Vo . �2�

The complete expressions for the different KeffM,Vi,o can be

found in Ref. 4.Now, we consider �see Fig. 3�b�� the particular case of an

electron of velocity ve=ve1� and energy E coming perpen-dicularly from the vacuum to a semi-infinite medium. Thiselectron is elastically backscattered at the depth x=x0 insidethe solid. It reverses its direction of motion and leaves themedium along the same path. The complete expression forKeff�E ,�� ,x0 ,0 ,0�=Keff�E ,�� ,x0� is again given in Ref. 4

FIG. 2. Electron inelastic mean free path � for Al calculatedwithin the formalism developed in Ref. 4 ��� and found in theliterature �straight line� as a function of the exit angle �out for anangle of incidence �in=0 degree and for two electron energies: �a�E=1000 eV and �b� E=3400 eV.

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with an average made over the total path length xt=2x0. Anexpression for Keff�E ,�� ,x0� has already been calculated inRef. 3, with a different mathematical model description forthe evaluation of the induced potential ind�k ,��. These twodifferent expressions are identical for the limits x0→0 andx0→�. For other values of x0, they differ by �5%. Thesesmall deviations comes from the different mathematicalmodel descriptions for the evaluation of �ind�k ,��. We havechosen to show here the results calculated by the expressiongiven in Ref. 4, with an induced potential determined by thesurface reflection model; but, of course, this choice has onlyan insignificant influence on our final results.

To evaluate Keff�E ,�� ,x0 ,�i ,�o� or Keff�E ,�� ,x0�, theenergy-loss function �ELF� Im�−1/��k ,�� must be known.Expressions for the dielectric function ��k ,�� and thus forthe ELF can be developed based on the Lorentz oscillatormodel for the electrons.21,22

In the Lorentz model23,24 the electrons are imagined to bebound in harmonic potentials. The equation of motion for theoscillator is

md2r

dt2 + idr

dt+ �0i

2 r� = − eE0e−i�t, �3�

where m is the electron mass, r is the coordinate of theelectron, E0 the strength of an external electric field of angu-lar frequency �, �0i the oscillator frequency of the ith oscil-lator and i a damping constant �i.e., the width of the ithoscillator�. In this formalism, the ELF can be defined as22

Im�−1

��k,��� = �i=1

nAi� i��

��2�0ik2 − �2�2�2 + �2 i

2�2�2

����� − EG� , �4�

where

��0ik = ��0i + �i�2k2

2m. �5�

Here Ai is the ith oscillator strength. The Lorentz approxima-tion for the ELF was originally expressed without kdependence.2 In accordance with previous works,21 a k de-pendence is introduced in Eq. �5�. In general, the dependenceof �0ik on k is unknown, but Eq. �5� is generally acceptedwith the dispersion coefficient �i as an adjustable parameter�in this work, �i has been considered as a constant for eachmaterial, i.e., independent of the oscillator�. The step func-tion ����−EG� is included to describe the effect of an en-ergy band gap EG in semiconductors and insulators. The ELFhas also to fulfill the optical sum rule25

2

��

0

�

Im�−1

��k,���d�

�= 1 −

1

n2 , �6�

where n is the refractive index of the material in the opticallimit ��→0.

The coefficients of Eqs. �4� and �5� are finally selected�with the constraint given by Eq. �6�� to obtain the bestagreement with experimental data or are taken from opticaldata, the agreement between the results given by the twomethods being remarkably good.17 They are listed in Table Ifor Al,26 Si,17 SiO2,17 and Cu.22

A quantity that is representative for the electron-solid in-teractions is Keff�E ,�� ,x0� integrated over �� or equiva-lently, the effective inelastic mean free path �eff�E ,x0� de-fined according to the following expression3,27 �a similarexpression is of course valid for Keff�E ,�� ,x0 ,�i ,�o��:

�eff�E,x0� = �0

�

Keff�E,��,x0�d���−1

. �7�

In this paper, we will therefore focus the study on varia-tions in the effective mean free path �eff with the total pathlength xt traveled by the electron in the solid, with the char-acteristics of the solid and with the geometry of the electron

FIG. 3. Geometries considered in our models. �a� General REELS geometry: The electron coming from −� with a velocity vi iselastically backscattered at depth x0 and leaves the solid with a velocity vo. �b� REELS perpendicular geometry: The electron coming from−� with a velocity ve=ve1� is specularly backscattered at depth x0=xt /2. �c� Thin film �i.e., TEELS� geometry: The electron with a velocityve=ve1� passes through the solid of thickness xt.

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path. As we shall see, �eff is sensitive to the variations incharge fluctuations induced in the solid by the incomingelectron.

III. RESULTS AND DISCUSSION

A. Normal incidence and specular reflection

We first consider the geometry shown in Fig. 3�b�. Thisparticular REELS problem was already discussed in Ref. 3,where an interaction between the incoming and outgoingparts of the electron trajectory has been recorded. Indeed, aneffect of this coupling can be seen as a damped oscillatingbehavior of �eff on the total path xt.

3,17 These results for �effare shown in Fig. 4 for a 1000 eV electron incident on alu-minum, silicon dioxide, and copper targets, as well as resultsfor TEELS �Fig. 3�c�� for the same path length traveled inthe medium �film thickness xt�. This comparison allows toclearly highlight the coupling effect, which is present inREELS but absent in TEELS.

We note an oscillating behavior for aluminum and silicondioxide �largest for Al� and no oscillation for copper. Forlarge xt, �eff converges, both for REELS and TEELS geom-etries, to its bulk value �� in quite good agreement withvalues available in the literature �see, e.g., Ref. 28�.

We consider now an aluminum target, for which theelectron loss function is the simplest, involving only oneoscillator �see Eq. �4� and Table I�. It turns out thatfor xt�10 Å, the damped oscillating behavior of the REELSspectrum in Fig. 4 can be fitted by the function

�eff = �� − A exp�− xt/B�cos2�xt

D+ � , �8�

where A, B, and D are parameters expressed in angstroms.The phase �in rad� turns out to be very close to 0 in allcases, and we will use =0 in the rest of the paper. Theempirical fit for xt�10 Å to Eq. �8� is introduced to facilitatethe discussion of the results. The result of the fit is shown inFig. 5 for three different energies of the incident electron,

i.e., 600, 1000, and 5000 eV. We note that these fits areperformed by fixing the values of the first maximum andminimum of Eq. �8�. Table II shows the values of the differ-ent parameters obtained by fitting Eq. �8� to calculated �efffor five energies. We do not attempt to fit our results forxt�10 Å, where they are obviously not well described byEq. �8�. The reason why �eff decreases monotonically forxt�10 Å is that, for the smallest xt, the excitations that takeplace when the electron moves in the vacuum form a largerrelative part of the excitations on the total path �recall thatKeff for a given trajectory is defined as the average probabil-ity for energy loss per unit path length traveled in the solid�.

We note that A /���0.7 and that B and D are approxi-mately proportional to the square root of the energy �with Ein electron volts and B and D in angstroms�

B = 1.1�E and D = 2.1�E , �9�

and are thus proportional to the electron velocity. This obser-vation incite to express the dependence as a function of thetime t=xt /v between entrance and exit of the electron. Thus,Eq. �8� becomes

�eff = �� − A exp�− �t�cos��t� , �10�

where �=v /B s−1 and �=2�v /D s−1. These latter quantitiescan also be expressed in unit of energy by multiplying by thePlanck constant � �see Table II�. We observe that the quantity�� is quite constant with the mean ���11.8 eV, i.e., veryclose to the energy of the surface plasmon. For simple met-als, it is well known29,30 that ��S=��B /�2 and for Al, weobtain ��S=11.0 eV, with ��B=��0i=15.6 eV the bulkplasmon energy.

On the other hand, �, the damping parameter in Eq. �10�,is also quite constant �with ���3.5 eV�. This damping con-stant � must be related to the input parameters of the ELF. Todetermine this dependence, we have studied several hypo-thetical media for which we have systematically varied thedifferent parameters ��0i, � i, and �i one by one, the otherparameters being equal to the Al constants. Thus, for each setof �0i, i, �i, we calculate a �eff curve as in Fig. 5. By fitting

TABLE I. Parameters used in the dielectric loss function for Al, Si, SiO2, and Cu.

Medium i ��0i �eV� Ai �eV2� � i �eV� �i

Al �EG=0 eV� 1 15.6 225 0.8 1

Si �EG=1.12 eV� 1 10.0 6 5.0 0.5

2 14.0 30 5.0 0.5

3 16.8 210 3.8 0.5

SiO2 �EG=9.3 eV� 1 15.0 9 5.0 0.02

2 23.1 320 14.0 0.02

Cu �EG=0 eV� 1 4.5 1 2.5 1

2 8.3 5 4.0 1

3 12.8 20 6.9 1

4 21.4 141 10.8 1

5 29.2 35 5.4 1

6 34.8 380 32.1 1

7 65.0 355 50 1

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Eq. �8� to this, B and D �and thus � and �� are then deter-mined. We show in Fig. 6 the dependence of � on and�S /2. We clearly observe that � is linearly dependent onboth and �S /2 with slopes equal to 0.5. However, � and �were found to be essentially independent of the plasmon dis-persion �i. We can then formulate a general rule for Al validto a good approximation

�� �1

2� +

��S

2� . �11�

The same procedure was used to study the coupling phe-nomenon in Si. For Si, three oscillators simulate the lossfunction �see Table I�, a strong oscillator at 16.8 eV for the

bulk plasmon peak and two oscillators at 10 and 14 eV forthe interband transitions which are significantly weaker thanthe plasmons contributions.17 In this case, Eq. �10� was againfound to fit the calculated data quite well. We note that whenmore than one oscillator is used in the dielectric loss functionmodel, we define the width as the mean width at halfmaximum of the energy loss function and ��B as the maxi-mum in the ELF curve plotted with the values given in TableI. This gives � �Si�=4.5 eV and ��B�Si�=16.8 eV. Weshow, for example, in Fig. 7�a� the result of our fit for1000 eV electrons incident on silicon obtained with ��=12.0 eV and ��=5.5 eV. The oscillation frequency is inperfect agreement with the surface plasmon energy for sili-

FIG. 4. Effective inelastic mean free path �eff as a function ofthe total path xt traveled by a 1000 eV electron in different media:�a� Al, �b� SiO2, and �c� Cu. The solid lines are for the REELSgeometry and the dashed lines for the TEELS geometry.

FIG. 5. Effective inelastic mean free path �eff as a function ofthe total path xt traveled in aluminum targets by the electron withenergies of �a� 600 eV, �b� 1000 eV, and �c� 5000 eV. Solid linesare the calculated values, and dashed lines are the fits obtainedaccording to Eq. �10�.

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con, ��S�Si�=11.9 eV. The damping value ��=5.5 eV isalso in good agreement with Eq. �11�.

In the case of copper, we find similarly from a plot of theELF ��B�Cu�=36.0 eV and � �Cu�=43.0 eV. Here, thedamping is so huge that the oscillating part of Eqs. �8� and�10� is completely hidden. A further complication for Cu isthat the expression ��S=��B /�2 is not valid for solids thatexhibit strong interband or intraband transitions. For Cu, wecan therefore not make a direct comparison of the deter-mined � from a fit by Eq. �10� to the value obtained from Eq.�11�. We show in Fig. 7�b� the fit obtained for copper �1000 eV incident electrons� with only an exponential com-ponent in the second term of Eq. �10�. Consequently, depend-ing on the solid, the coupling effect appears in two differentways. Solids with a huge parameter �as copper, iron, gold,etc.� are characterized by an exponential aperiodic behavior,whereas the materials with a small �aluminum, silicon,etc.� exhibit oscillations.

Considering now SiO2 ���B�SiO2�=23.1 eV and� �SiO2�=14.0 eV�, the situation is a little bit more compli-cated. The best fit to Eq. �10� is obtained with ��=16.7 eV and ��=15.9 eV. Although the frequency of the

oscillation is again very close to �S ���S�SiO2�=16.3 eV�,the damping constant ��=15.9 eV is much larger than thevalue given by Eq. �11�. This might be due to the large gapof SiO2. Furthermore, similar as for Cu, the surface plasmonfor SiO2 is also not well defined from the bulk plasmonfrequency.

This general behavior and, particularly, this oscillating ef-fective inelastic mean free path in the case of media such asAl, Si, or SiO2, is due to the fact that the electron againcrosses a region where it has excited plasmons in its incom-ing trajectory. More precisely, our analysis shows that �eff

TABLE II. Parameters used in Eq. �8� for energies of 300, 600, 1000, 3000, and 5000 eV in the case ofan Al target.

E �eV� �� �� A �� B �� �v /B=�� �eV� D �� 2��v /D=�� �eV�

300 9.7 6.6 18.5 3.7 36 11.8

600 16.2 11.3 28.1 3.4 52 11.6

1000 24.1 15.2 38.0 3.3 64 12.1

3000 58.9 38.7 60.2 3.6 117 11.5

5000 90.4 61.0 78.6 3.5 146 11.9

FIG. 6. Damping parameter �� calculated for hypothetical me-dia as a function of ���: � ���S being constant and chosen tocorrespond to Al�; ��� ��S /2 with � =0.8 eV �lines are a guide tothe eye�.

FIG. 7. Effective inelastic mean free path �eff as a function ofthe total path xt traveled by 1000 eV electrons in �a� a silicon targetand �b� a copper target. Solid lines are the calculated values anddashed lines are fits by Eq. �10�.

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oscillates at the frequency ���S of the surface plasmons,which are thus responsible for the coupling effect. Surfaceplasmons are longitudinal waves of charge density that travelalong the surface.29,30 The electrostatic potential at the planarsurface induced by the surface plasmon is of the formcos�q�z−�St�exp�−q�x� �where q� and z are the componentsof the wave vector and of the position along the surface,respectively� and thus decreases exponentially with the depthx. Thus, the origin of the coupling effect is the following.When the electron exits at a time T=2n� /�S �where n is aninteger� after it entered the solid, the charge density at thesurface is at a maximum of the plasmon wave, implying amaximum in the interaction probability and thus a minimumfor the effective inelastic mean free path �eff. When T= �2n+1�� /�S, the charge density and, thus, the probability ofinteraction are minimum and �eff reaches a maximum.

Concerning now the behavior of the damping parameter �expressed by Eq. �11�, the dependence of � on the width ofthe ELF is as expected. On the contrary, the dependence onthe surface plasmon energy �S is more surprising. Up tonow, this point stays unsolved. In the future, we will inves-tigate this question in more detail.

We will now consider the different contributions KeffVi , Keff

Mi,Keff

Mo, and KeffVo �see Eq. �2�� to the total excitation probability

Keff. Since these are intermediate contributions to the effec-

tive inelastic mean free path, we consider here the integratedvalues

KintA �E,x0� =� Keff

A �E,��,x0�d�� , �12�

where A=Vi ,Mi ,Mo ,Vo. For 1000 eV electrons incident onAl, we show in Fig. 8�a� the different contributions, Kint

Vi , KintMi,

KintMo, and Kint

Vo as a function of xt and in Fig. 8�b�, the totalcontributions of the vacuum, Kint

Vi +KintVo, and of the medium,

KintMi +Kint

Mo. We observe, in Fig. 8�a�, that KintVi and Kint

Vo con-verge both to zero and that Kint

Mi and KintMo converge to the

same value, Kint� /2, which is half of the value obtained within

the theory of Lindhard1,22 for an infinite medium. KintMo and

KintVo show oscillations due to the coupling at the surface. The

charge fluctuations of the surface plasmon can thus influencethe electron scattering in both the vacuum and the medium.As expected, Kint

Mi and KintVi do not show oscillations. The os-

cillating behavior is only observed for the outgoing part ofthe electron trajectory, which proves that coupling occursbecause the outgoing electron interacts with the electric fieldfrom the charge fluctuations induced when it enters the solid.

The total contributions from the electron trajectory in thevacuum or medium, shown in Fig. 8�b�, exhibit stronger os-

FIG. 8. For aluminum and E=1000 eV, �a� Different contribu-tions, Kint

Vi , KintMi, Kint

Mo, and KintVo to the total excitation probability Kint,

as a function of xt. �b� Total contributions from the trajectory in thevacuum, Kint

Vi +KintVo, and from the trajectory in the medium, Kint

Mi

+KintMo to Kint, as a function of xt. We have also indicated in the

figure the constant value of Kint� , calculated for an infinite medium.

FIG. 9. For copper and E=1000 eV, �a� Contributions, KintVi , Kint

Mi,Kint

Mo, and KintVo to the total excitation probability Kint as a function of

xt. �b� Total contributions from the trajectory in the vacuum, KintVi

+KintVo, and from the trajectory in the medium, Kint

Mi +KintMo to Kint as a

function of xt. We have also indicated in the figure the constantvalue of Kint

� calculated for an infinite medium.

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cillations, and they both converge to their respective bulkvalues, 0 and Kint

� . We also observe a range of xt for whichthe contribution from the vacuum trajectory is negative. Thismeans that for these xt, the electron gains energy while it ismoving in the vacuum. This is a consequence of the couplingeffect. However, the sum of the vacuum and medium effectsis always positive, as expected.

Figures 9�a� and 9�b� show the results obtained for thesame quantities in the case of 1000 eV electrons incident onCu. The same general behavior as for Al is observed, but thedifferent curves of Kint for Cu show no oscillation because of the large value of the damping parameter �. Consequently,

the possible energy gains in the vacuum observed for Al areabsent in the case of a Cu target.

B. Angular dependence

We examine now a REELS geometry with angles differ-ent from zero as shown in Fig. 3�a�. We consider the particu-lar case of a specular reflection for which the electron leavesthe solid with an exit angle equal to the entrance angle, �i=�0=�. We have chosen this configuration to keep the sym-metry, but another geometry would give similar results. Inthis geometry, we can introduce an effective inelasticelectron-scattering cross section Keff�E ,�� ,x0 ,�� and an ef-fective inelastic mean free path �eff�E ,x0 ,��, depending onthe angle �. In the rest of our discussion, we will only con-sider aluminum targets and incident energy E=1000 eV.

Figure 10 shows the results obtained for �eff�E ,x0 ,�� as afunction of the total path traveled by the electron in the solidxt=2x0 /cos � for different angles �. Note that, for large x0,�eff�E ,x0 ,�� should be independent of � and be equal to thevalue obtained for electrons traveling in an infinite medium.The small deviations from this ��10% � are due to the ap-proximations made in the analytical expression of Keff for��0, as already pointed out in Ref. 4. However, this devia-tion in the absolute value does not influence the conclusionsbelow about the oscillating behavior.

FIG. 10. �eff as a function of the total path xt traveled by a1000 eV electron in an aluminum target for �=0° ,5° ,10° ,15° ,20° ,30° ,40°.

FIG. 11. Amplitude Amax−min of the oscillations defined as thedifference between �eff at the first maximum and the followingminimum as a function of the angle � �calculated for a 1000 eVelectron incident on an aluminum target�. The solid line is a guideto the eye.

FIG. 12. For aluminum, �a� KintVi , Kint

Mi, KintMo, and Kint

Vo and �b�Kint

Vi +KintVo and Kint

Mi +KintMo as a function of xt for �=40° �calculated

for a 1000 eV electron incident on Al�.

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Note that for increasing �, the maximum in �eff is shiftedto larger xt, which implies that there is a delay in the onset ofthe interactions. This is attributed to the limited velocity ofthe expansion of the surface plasmon wave along the surface.We also note a decrease in the amplitude of the oscillationswith increasing �. This effect is also expected, because thechange in the surface charge density is damped as the exitpoint of the electron in the medium moves away from theentrance point and, consequently, the coupling effect de-creases. Figure 11 shows the decrease of the amplitude of theoscillations in �eff represented by the difference Amax−min be-tween the first maximum of the oscillation and the followingminimum as a function of the angle. For about ��10°, thecoupling effect becomes considerable in agreement with re-sults shown previously in Fig. 2. For � larger than �30° �andthus an angle of 60° between the incoming and outgoingtrajectories�, the oscillations completely disappear.

We show also in Fig. 12, for �=40°, the values of KintVi ,

KintMi, Kint

Mo �Fig. 12�a�� and KintVo and Kint

Vi +KintVo, and Kint

Mi +KintMo

�Fig. 12�b�� as a function of xt. The oscillations, which arepronounced for �=0° �Fig. 8�, have disappeared in agree-ment with the above arguments.

IV. CONCLUSION

In a REELS geometry, we have studied the effect ofcharge fluctuations in the surface plasmon, which is excitedas the electron enters the solid. The outgoing electron isshown to interact with the corresponding electric field as itexits the surface. We have probed the surface excitationswith the effective electron mean free path �eff calculated ac-cording to the dielectric theory. For materials exhibiting asmall damping, this coupling process arises as an oscillatingbehavior of �eff as a function of the depth traveled by theelectron in the medium. For normal entrance and exit angles

where the effect is largest, we have demonstrated that, ingeneral, �eff oscillates as a function of the difference betweenthe time where the electron enters the surface and the timewhere it exits the surface. By systematic investigations ofdifferent materials, path lengths, and energies, we found thatthe characteristic frequency of these oscillations is equal tothe surface plasmon frequency. This implies that the cou-pling effect is mainly a surface effect induced by the chargefluctuations excited by the incoming electron.

We also calculated the separate contributions to the in-elastic scattering for the trajectories in the vacuum and in thesolid for the incoming and the outgoing trajectories. It isfound that the oscillations are only seen in the contributionfrom the outgoing trajectory. This gives for the support to theconclusion that the effect is due to the coupling of the out-going electron charge to the charge fluctuations excited bythe incoming electron.

Finally, we studied this effect as a function of the spaceseparation along the surface of the electron entrance and exitpoints. We found that oscillations in �eff are observed also asa function of this distance. This variation is attributed to theexpansion of the surface plasmon wave along the surface.We find that the coupling effect decreases with increasingangle and disappears completely for angles exceeding �30°�and, thus, angles of 60° between the incoming and outgoingtrajectories�. This is because of a decreasing amplitude in thesurface plasmon for increasing distance between the entranceand exit points at the surface.

ACKNOWLEDGMENTS

N. Pauly acknowledges the Université Libre de Bruxellesand the Fonds National de la Recherche Scientifique �FNRS�for their financial support, and S. Tougaard the Danish Natu-ral Science Research Council �SNF�.

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