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ELECTRON ENERGY LOSS IN STEM SPECTRA..J,~1-

P.M. Echenique*, A. Rivacoba*,N. Zabala** and R.H. Ritchie***

(*) Dpto. de Fisica de Materiales, Universidad del Pais Vasco, Facultad de Química.Apdo. 1072. San Sebastian 20080. Euskadi. Spain(**) Dpto. Electricidad y Electrónica. Universidad del Pais Vasco. Facultad deCiencias. Apdo 644. Bilbao 48080. Euskadi. Spain(***) Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge TN 37831-6123.USAand Deparunentof Physics, University olf Tennessee, Knoxville. TN 39996, USA.

Abstraet. The interaction of fast electrons incident at a fixed impact parameter ontargets of different geometries is studied. The connection between classical andquantum descriptions of the probe electrons is discussed. The relative contribution,of surface and bulk modes is studied as a function of the impact parameter.

1. Introduetion.

t:,'-""

Recent developments in the scanning transmission electron microscope(STEM) have made it possible to study electronic excitations of inhomogeneoussystems in highly localized regions. This is achieved by recording changes inenergy 10ss distributions measured when a well-focused -0.5 nm probe of swift(-100 KeV) electrons is scanned slowly across a specimen. The interpretation ofthese data raises a number of questions about the description and localization ofexcitations in inhomogeneous systems [1,3] and involves a wave-particle duality inan interesting context. Published treatments of excitations in periodic crystallinemedia have used broad-beam wave mechanical descriptions of the fast electronstates [4] whereas studies of excitations in other systems bombarded by swiftcharged particles particles [5.11] have usually employed a classical description ofthe fast electron together with dielectric response theory for the solido

In this paper we begin by establishing the connection between classical andwave mechanical description in STEM [12] and later discuss the application ofclassical dielectric theory to materials of different geometries.

2. Broad eoherent irradiation.

Using time dependent perturbation theory to evaluate the cross section fortransitions from the ground state (energy roO) to the nth excited state (ron) of thetarget caused by an incident electron plane wave basis set one finds[l]

(""''- 4

f2

cr = - dq 2 qnO V - I p O(q) I 8

'

( v.q - _2 - (00 - 00 )).4 n., n O

- q

(1)

where

N

PnO(q) =(nl L, eiq.1j10).4j=l

(2)

is the matrix element oí the density operator, and we use atomic units throughout:h=e2=m=1. Note that q = kO - kr: where kO and kf are the initial and final electronmomenta and

21 2 2 q-(k -k) =v q --2 O r -' 2

(3)

----

To introduce an impact parameter conjugate to momentum transfer, we firstneglect recoil of the incident electron; that is, we drop, for the moment, the termproportional to q2 in the argument of the delta function. The larger v becomes, theless important will be the neglected termo We take v in the direction of thez-coordinate axis and rewrite equation (1) in the equivalent forro

- 4

f d fd ' *anO - v ~ p O(q) ~ P O(q') 0((0 0- v q ) o(q -q')2 n 2 n n z

q q'(4)

We now make use of the identity

o (Q -Q) =~ fdb e-i b.(Q-Q'). (5)(21t)2

where q = (Q ,qz)' Q is the variable conjugate to b [13,14]. Then equation (4) may bewri tten

(-~"":-.)anO = fdb I anO(b),2.

(6)

'--'where

2

r~~o(b)f = ~ Ifdq e- i b.q P O(q) 0((0 0- v.q) I .n 2 2 n n z

1t q(7)

'-

Note that ~O(b) is precisely the probability amplitude that the many-electronsystem experiences a transition under the influence of the Coulomb field of aclassical point electron travelling with constant velocity v along a path specifiedby the impact paranieter b beginning at z=- 00 arid ending at z=+oo.

In the early days of quantum mechanics Frame [15] and Mott [16] using firstorder time-independent perturbation theory showed that for a swift incidentproton identical results are obtained when an infinite-plane-wave representationis chosen and when a classical trajectory is assumed in computing excitation of atarget..-:-'..-,>

So we have shown that in the case of broad beam coherent irradiation of thetarget, energy an-alysis of swift imaged electrons that have generated localizedelectronic excitations in the target gives nearly the same result as if the electronswere classical projectiles moving on rectilinear trajectories and with a uniformdistribution in impact parameter.This equivalence is more accurately satisfied thelarger the electrons speed. In reference 1 quantal corrections to this wereevaluated for various velocities and for different targets.

¡[O".; "",

3. Excitation of electronic .transitions by a rnicroprobe electron

Now we consider excitation of the target by an electron prepared in the formof a narrow beam centered at the impact parameter b relative the target. Theelectron may be represented by the wave packet

i leoze

'Vo(r)=<I>(p-b) ..[L

(8)

---

By analogy with the above development Ritchie and Howie [12] showed that whenall inelasticaly scattered electrons are included, the transition probability is givenby

P~(b) = fdP I<I>(p-b) 12Pnc(p)(9)

fi.~>"

where Pn e is the classical probability for excitation at impact parameter p.This shows that when all inelastically scattered electrons are collected the

measured probability of exciting a given transition may be computed theoreticallyas if the microprobe consisted of an incoherent superposition of classicaltrayectories distributed laterally to the beam direction according to the probabilitydensity 1 <I>(P-b) 12.

Saying this in another way, provided that the spectrometer aperture in theSTEM is large enough, classical excitation functions are correet when averagedover a range of impact parameters corresponding to the current distribution inthe probe. The classieal approximation could therefore fail when the speetrometersemi-angle of aeeptanee am is very small. For typical values in a STEM experimentam » 1 mrad. In practice, the tendency is to maximize detection efficiency bytaking a m 2 a a ==8 mrad, w h e re a a is the maximun probe convergencehalf-angle. Thus this condition is well satisfied in practice.

In view of the above it is of interest to review work on the local dielectrictreatment of the interaction with various targets of the electrons that are assumedto "move on classical trajectories.

4. A classical electron moving parallel to aplane.

One may compute the retarding field, and hence the rate of energy lossexperienced by a fast electron travelling parallel to and at distance b from asurface using ordinary local dielectric theory of excitation by a moving charge.We take the charge density due to the swift electron to be

p(r,t) =-8(z-b) 8(x) 8(y-vt) . (10)

From the force acting on the particle we define a probability of energy lossP( (0) in terms of the probability of the momentum transfer kx and energy transfer00 as [8,17-20]

f'-

'-dW =J P(oo) 00 doody O

P(oo)=Jdk P(k ,00) .x xO

(11)

,

where

2 -2 Q b

[ 1 2 -2 Q b

[ 2e e- e -P(k ,00)=- lID -] =- lID -]

x JT:V2 Q e+l JT:V2 Q e+l(12)

where Q2 =kx2 + CI)2v-2 . P( (0) is given as

P(oo) = d W(b) 2-rm [~]Ko(2OOb)dz doo rcv2 e+1 v

(13)

---

The properties of the KO Bessel function give rise to interesting behaviour inthe spectra. For large values of its argument x = 2 rob/v, it falls off like exp( -x)/v xso that the probability of interaction will be appreciable out to distances of theorder vIro and which are typically 50 Á in the valence loss region. On the otherhand, at small values of x, the function has a more rapid logarithmic variation sothe losses at larger ro, in particular, can decrease rapidly with impact parameter.It is important to realize that vIro is an upper estimate of the impact parametercorresponding to zero scattering angle and thus zero momentum transferperpendicular to the initial velocity. Larger values of momentum transfer areasociated with smaller interaction distances. The function KO(2 rob/v) varies quiterapidly for small arguments thus allowing a high spatial resolution [21] that mayexplain recent experimental fíndings[ 22,23]. The error incurred by using a localE(ro) has been evaluated [24,27]. It is found that for distances greater than 5, 10 Á noappreciable errors may be expected at typical STEM conditions. Generalization ofEqs. ( 12,13) to take account of relativistic effects have been made [28]. Therelativistic corrections are expected to be small unless Re(E(ro» becomes largeenough that the criteron for the emission of Cherenkov photons ( EV2/c2 >1) issatisfied for a substantial range of frequencies.

Equation (13) can be adapted to deal with glancing angle trajectories byputting dy=e-1dz and integrating over z. The probability of exciting a surface modein such a trajectory is given by

-(;.:.-~

2

f1 [ -2 ]Q(e) =- - 1m - dro

ve o ro e+1

=...1L in the free electton case.ve

(14)

When z < O and the electron travels in the dielectric

2 { [ -1 ] 'lev ( [ -2 ] [-1])

2rob}P(ro)=- 1m- ln(-)+ 1m- -1m - Ko(-) .

rr;v2 e(ro) ro e+1 e v(15)

f~

The logarithmic term yields the ordinary loss rate to volume excitations, qc isa cut-off wave number; and the terms containing the KO function describeboundary corrections to these losses.

The effect of the boundary as fírst pointed out by Ritchie [29,30] is twofold: asurface mode appears via the term -2/(1 + E( ro» while on the other hand, areduction in loss due to excitation of bulk modes is introduced via the term -l/E(ro).Howie and Milne [9] from their experience in applying the dielectric theory to thesilicon-silica and other interfaces have conc1uded that the experimental resultscan be a critical check of the d,ielectric function used since they must reproduceboth features of the individual media and the interface features.

For the case of aplane of thickness tia" having a thin coating of a differentdielectric material it can easily be shown(see Eq. 12) that for small values of a,when the thin coating might correspond to one or two anomalous atomic layersnear the interface the result only is sensitive to tbis surface layer for Qa >0.1, Le.kx a >0.1 [31]. The conventiona1 axial spectroscopy technique is rather insensitiveto thin surface or interfacial layers because tipically ro/v ==0.01. Larger values ofro/v and hence Q are important in low energy electron spectroscopy, which _ is, forexample, able to detect changes in the spectra due to surface reconstruction [32]when these cannot be detected by axial experiments with 100KeV electrons [33].

The spatial resolution ofaxial high energy spectroscopy improves at higherloss values. In the case of the silicon-silica interface, Howie and Milne [9] havebeen able to shown that the Si L edge, which occurs at 100eV in Si and is shifted to-107eV in Si02' has a value of -105eV at the interface.

----

5. Spherical particles.

The probability function for this case is given by [34]

4a ~~ 2-0mO coa2~2 rob [ L(e(co)-1) ]P(co)= ¡r;v2 6~ (L+m)!.(L-m)! (7) m(y) 1m Le(co)+L+l

(16)

f: ,

Here 30m is the Kroneeker delta, and Km is the modifíed Bessel funetion of thesecond order of order m.

Equation (16) allows us to understand the relative importance of a given modein the loss proeess. For small values Ko - -ln(x) and Km- 0.5r(m)(xl2)-m, while forlarge values Km - V (TC/2X). e-x. When roa!v«l, the L=1 term dominates,and weregain the dipole approximation [35]. When roa/v «1 but rob/v »1, no m termprevails and Pco(b) goes like (roa Iv )[36]. If roa Iv »1 many L'S are necessary. Evenif roa Iv »1 when rob/v »1 only small ro« vlb are appreciably excited. The essentialdipole condition roa/v «1 therefore still applies provided that the sphere hassuffieient dielectric loss at these frequencies.In many experimental situations b - a and roa/v - 1 and one therefore needs toinclude many L values in Eq.(16) [36,40]. The most important charaeteristicfrequencies involved in an energy 10ss proeess will vary both with L and with thedielectric funetion e(ro). Since the excitation of the various modes also depends onroa/v for a given electron energy, spheres of different materials can also showsubstantial differenees in energy 10ss spectra as well as in the number and natureof the modes excited.

To illustrate these points we show in Fig. 1 the probability of losing energy rofor a 50KeV electron at grazing incidence on a sphere of radius a=lOnm. We displaythe eontribution of the dipole mode, the fírst two LiS, the ten L's, and the totalprobability. For small radii and low energy losses, the dipole contributiondominates, while even at small radius, it is inadequate to describe the high energylosses. Experimental optical data [41] has been used in the calculations of theresults shown in Fig 1. ..

When the electron penetrates the spheres, the expression for the probabilitybecomes more eomplicated. Explicit express ion for the energy los s probabilityvalid for any e(ro) can be found in the literature[42]. For a free eleetron gas

f:~o:-)

2a" (L-m)!{ [

o i]

2

P(co) = -;¡L.,¡ (2-0m~ (L+m) , roLO(CO-~) ALm+ALm -vL.m .

-COp()(CO-COp)[A~+A~]A~ } (17)

where roL = ropV L/(2L+l) and

00 Jl_b2i 1

f. L-mZ coz

A Lm=- dzr PL (-)g Lm(-)'L+l r v

a o(18)AO =aLfdz..2...pmeZ) (coz);1m L+l L r gLm v

r:¡-:¡ r ·va--b-

where PL m stands for the Legendre functions, r=v z2+b2 and the functions gLm(x)are sin(x) if (L+m) is odd, or cos(x) otherwise.

Figure 2 shows for the case of a free electron gas model the probability oísurface and bulk loss for axial trajectory as a function of the radius of the sphere.The planar limits of TC/v and TC/(2v) respectively fírst derived by Ritchie areclearly obtained for big radiL .

Excitation functions for dielectric bodies bounded by other coordinate systemshave been found. These include some allowance for the effeet of the support of a

spherical particle [43]. The interactions between closely-spaced pairs of sphericalparticles has been analyzed as well [38]. A spheroidal dielectric has been studied[44]. Solutions found for a cylindrical wedge have relevance to the case of a fastelectron passing near a comer of a cube [4S,46]. Results for excitation of adielectric by a fast electron passing through a cylindrical cavity in the mediumhave also been obtained [47-49].. It has been possible to interpret in considerabledetail the energy loss spectra obtained experimentally in this geometry [SO].

6. Deflection effect

STEM observations have indicated a deflection or flaring effect when a 100Ke V _ lnm diameter electron beam travels parallel to, but at a distance x of up to 5nm outside of a surface face of a _100nm MgO cube[Sl] or an Au [S2] particle. For a100 KeV electron moving parallel to and outside the surface of MgO or Au thepotential well generating the image force has a depth of about O.leV and extendsabout 10nm into the vacuum. The deflection angle for an electron travellingoutside an Al cube of edge lS0nm can be easily estimated as

V 2óS = 2- F1t (1)s.length---

v v - . 4V

-6::= 10 rads. (19)

~:;;;:: wh~ch is about three orders of magnitude less than the experimental resulto Moreelaborate calculations by Echenique and Howie [S3], corraborate this resultoCalculations by other authors in different geometries also confirm this result[54,55]. For instance Echenique and Howie [S3] conclude that although the classicaldielectric model is able to account for the energy losses observed in the STEMexperiments, the image force it yields is much too small to explain the beamdeflection effects observed by Cowley [S2] and that the explanation must be sougbtelsewhere. In the case of MgO particle charging should perhaps be considered butthis may be somewhat less likely in the case of Au particles.

Acknowled gements

The authors gratefulIy acknowledge Iberduero S.A., Gipuzkoako Foro Aldundiaand the Education Department of the Basque Goverment for help and support. TheOffice of Health and Environmental Research of the U.S. Departament of Energyhas given partial support for this research.

fA

.

References

w':-..'

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,~

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4~D. De Zutler andD. De Vleeschauwer. J. appl. Phys. 59, 4146 (1986)49N. Zabala. A. Rivacoba and P. M. Echenique. Surf. Sci. 209. 465 (1988)50M.G. Walls. Electron Energy Loss Spectroscopy 01 surfaces and interfaces. Ph.D.Thesis. University of Cambridge (unpublished) (1988)SIJ.M. Cowley. Ultramicroscopy 9. 231 (1982)52C.S. Tan and J.M. Cowley. Ultramicroscopy 12. 333 (1983)53p.M. Echenique and A. Howie. Ultramicroscopy 16. 269 (1985)54A. Rivacoba and P.M. Echenique. Ultramicroscopy 26. 389 (1988)55 A. Maninez-Torregrosa. R. Garcia-Molina and A. Gras Marti. to be published.

{;j,--'~

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------

Figure captions

Fig. 1 Energy loss spectra oí 50 eV electrons moving at grazing incidence on analuminum sphere (a=lO nm).

Fig. 2 Dependence oí the bulk and surface losses on the radius oí the sphere foraxial electron° trajectories. Curve (a) corresponds to -P(rop) while curve (b) is theZ peroL). A free electron response function e(ro) has been used.

~~.

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