From: REFLECTIONHIGH ENERGY ELECTRON DIFFRACTIONAND REFLECTION ELECTRON IHAGING OF SURFACES

Edited by P. K. Larsen and P. ~. Dobson

(Plenum Pub11sh1ng Corporatlon, 1988)

EXCITATION OF DIELECTRIC SPHERES BY ELECTRON BEAMS

P. M. Echenique

Departamento de Físcia. Facultad de QuimicasEuska1 Herriko Unibertsitatea, Apdo. 1072San Sebastián 20080, Euskadi, Spain

Recent new deve10pments and app1ications of the scanning transmissionmicroscope(STEM) have stimu1ated renewed interest in the interaction ofhigh-energy e1ectron beams with sma11 partic1es and surfaces[I-7]. Usua11ythe c1assica1 theory of energy 10ss has been emp10yed to ana1yze the exper-imental energy-10ss spectra for p1anar, spherica1 or cy1indrica1 geo-metries[8-14]. A new general approach. based on the se1f-energy forma1ismof the many body prob1em has been presented recent1y[15]. In this ap-

proach. the mean potentia1 energy Eo of the incoming e1ectron, in a statedefined by a wave function ~o(r) and energy Eo is written as the averageof an effective local potentia1 Veff(r), representing the comp1ex inter-action of the incoming e1ectron with the many body target

*E = Jw (r)Vff(r)w (r)dr.o o e o (1)

We use atomic units (e2 = h = m) throughout this paper. The real part of

LQgives us the 10wering of the energy of the partic1e due to virtual excit-ations of the medium, and the imaginary part is direct1y re1ated to theprobabi1ity of energy 10ss due to creation of real excitations. The ef-fective potentia1 is written as a spatia1 integral of the non local se1f-

1nergy[16], E (r. r'. Eo)' which in turn can be expressed in terms of theureen's function G(r, r'. W+Eo) and the causal screened interaction W(r,r'. w)[16]. The Green's function is given as a sum over a complete set

~f(r) of final states with energy Ef. After some a1gebra, we get

I Wf*(r)w (r)Wf(r)W*(r)E = _ E Jo o _R

o 11 f w + llE+ i'" x Im[W--Cr,r' ,w)] x dwdrdr' (2)

where llE = Eo - Ef'stands for retarded

e1ectron microscopydirect1y re1ated to

o is an infinitesima11y sma11 positive constant and WRresponse function. A quantity of practica1 uti1ity inis the probabi1ity of 10sing energy w. Pw, which isthe energy 10ss rate Y. experienced by the partic1e

Y = v J P dww (3)

where v is the ve10city of the incident e1ectron. The energy loss rate, y,is in turn. given in terms of the imaginary part of the incident e1ectronse1f-energy

91

y 2 1m L .

l' o(4)

Now we use Equation (2) .for the case of an electron of velocity v incidentupon a dielectric sphere, radius a, characteri~edby a dielectric responsefunction e(w). The screened interaction is obtained from the Poissonequation[15,17].

To proceed further, we need to specify the initial state Vo' If wehave a broad-beam geometry, i.e. if the width of the incident electron beamis much greater than the size of the sphere, we use plane waves to describethe incident electron states. The expression one obtains[15] is valid for

any dielectric response function e(w). When the further approximation ofthe free electron gas model is made, it reproduces the results of Fujimotoand Komaki[18] (when recoil is completely neglected) and the results ofBarberán and Bausells[19] (when only the forward scattering recoil isneglected). In STEM experiments we look for the probability of energy lossfor a well-focused beam incident upon a sphere of radius a at a fixedimpact parameter b. A physical description can be obtained, if we neglect

beam size effects, by taking, in Equation (2) a o function in the trans- ~

verse direction and plane waves in the direction of motion to describe tbelectron[20]. When the incident electron moves outside the sphere, the

probability of energy loss obtained from Equation (2) agrees with theresult first obtained by the use of classical electrodynamics by Ferrelland Echenique[lO], and is given in terms of the usual angular momentumcomponents ~, m by

4 ~ (2- o ) 2~P (b) = - 'f L Om x [~ ]w 7Iv2a2~=O m=O (~+m)!U - m)! v

x K2 [ wb ]m v

x Im[ (e(w)- l)a3E(w) + (~+ I/~) ].

(5)

Here 00m is the Kronecker delta, and Km is the modified Bessel funct-ion of order m. In Equation (5) b> a. For ~ = 1, the dipole contributiongoes over into the usual result; in this case, the energy loss is pro-portional to the square of the w Fourier component of the external field,if such a field is taken to be constant over tbe volume of tbe sphere[17].

The usual result for a harmonically bound charge replacing the sphere isthen recovered when one takes a unit volume containing a single chargereplacing the sphere and if e(w) is taken to be descriptive of such asituation.

In general, many (~,m) terms are necessary to describe adequately theinteraction process. The dipole (~=l) approximation is only valid eitherwhen the electron is far away from the sphere, or for very small spheres,which is not the case in most experimental situations. To illustrate thispoint, we first consider a free electron response e(w) and calculate theenergy-loss probability as a function of impact parameter for a 50-keVelectronincidentupon a sphere of radius 100 A. In Fig. l. we show thebulk losses, the dipole mode contribution and the total contribution of the

first twenty modes. The dipole contribution, however, does not show theprobability enhancement near the surface, which also can be seen in theexperimental results[2l]. In Fig. 2 we show, for a grazing incident elect-ron of energy 50 keV the relative weight of the contribution to the totalenergy loss of the ~tb mode, in the free-electron gas model, for a one

electron radius of rs = 2, for different sphere radii. As it is clear fromthe figure for large enough radii, the dipole contribution is negligible.Also we show, on the right-hand side the relative contributions of thefirst ten ~ modes as function of the spbere radius. In Fig. 3, we show the

probability of losing energy w for a 50 keV electron at grazing incidence

92

5

50keV Al4

~

22::1 3~g:o.g 2"ii:

100

b (1)

Fig. l. Energy-10ss probabi1ity [ca1cu1ated in the free e1ectron gas mode1,rs = 2] as a function of impact parameter b for a 50 keV e1ectronincident upon an a1uminium sphere of radius 100 A. Solid 1ine,bu1k mode; dashed 1ine, total contribution of the first 20 surfacemodes (x 6); dashed-dotted 1ine, dipo1e (~ = 1) surface mode (x6).

oo

/"./ \\\\\\ ,

"-.......-

~ ::::-50 150

0.4 \0í:R 1.0I L

5

t

Fig. 2. Re1ative weight R~ of the ~thmode contribution to the total energy1055 [ca1cu1ated in the free-e1ectron gas mode1 for rs = 2 as afunction of the mode number for an e1ectron of energy 50 keV atgrazing incidence (b:a)for differentsphere radii]. A1so, thecontribution to the 10ss of the first ten~modes is shown as a

function of the radius of the sphere.

1 10

93

10W (eV)

liS!5

Fig. 3. ProbabilityPw losing energy w for a 50 keV electron moving atgrazing incidence on an aluminium sphere of radius a = 10 nm...dipole mode t =1; -- t = 1 + t = 2; -.- contribution of the firstten tmodes; - contribution up to t = 100 included.

......

on a sphere of radius a = 10 nm. We show the contributions of the dipolemode, the first two t's, and ten t's, and the total probability. For smallradii and low energy losses, the dipole contribution dominates; while evenat as a small radius as 10 nm, it is inadequate to describe the high energylosses. We have used experimental optical data for E(W) in these calcu-lations[22].

CONCLUSIONS

We have presented a general formula to calculate the probability-lossfunction in electron microscopy, and in RHEEDspectroscopy, and apply it sosome specific cases. Our analysis is val id for any w-dependent dielectricresponse function E(W). We have shown that, for most cases of practicalinterest in electron microscopy, or even more in RHEED, many multipoles(t,m) should be included to describe adequately the experimental data overthe relevant energy range. The free electron model for spherical targets,originally used by Fujimoto and Komaki[18] in broad-beam geometry, has beenapplied to the focused-beam case by Schmeits[23] and Kohl[24] but only forthe case of dipole or quadrapole excitations. The case of a sphere coatedwith an oxide layer can be analyzed easily within our formalismo Calcu-lations using the classical theory of energy loss valid for any E(w)[11,12,25] or within the free electron gas model[26] have been presented recent17~No account of beam profile has been given, but beam size effects can beintroducedin a naturalway into our formalismo Ritchie and Howie haveproved that if all inelastically scattered electronsare collected,thequantum mechanical excitation probability is obtained by convoluting theclassical, impact parameter dependent result over the distribution ofimpact parameter determined by the probe intensity function[27]. Spatialdispersion and relativistic effects have been discussed, within the specu-lar reflectionmodel of Ritchieand Marusak[28]by Echenique[29]. Retard-ation effectsfor the case of planar geometries have been analyzed byGarcia-Molinaet al.[30]

Acknowledgement

It is a pleasureto thank ProfessorsA. Howie and R.H. Ritchie formany stimulatingconversationsand IberdueroS.A. for help and support.Support for the Spanish CAYCIT, Eusko Jaurlaritza and Gipuzkoako ForuAldundia is gratefully acknowledged.

94

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