Energy-loss profiles of transmitted electrons incident on dielectric spheroids

Ultramicroscopy 35 (1991) 1-10 North-Holland

1

Energy-loss profiles of transmitted electrons incident on dielectric spheroids B.L. Illman a, V.E. Anderson, R.J. Warmack and T.L. Ferrell Health and Safety Research Division, Oak Ridge National Laboratory *, Oak Ridge, TN 37831-6123, USA

Received 15 October 1990

We have obtained expressions for the differential energy-loss probability for electrons passing through small condensedmatter spheroids, parallel to the symmetry axis, at arbitrary impact parameter. The expressions, appropriate to oblate, prolate, and spherical targets, are a sum over all multipole contributions. They are expressed as functions of the local dielectric function c(to), which is obtained from experimental bulk data as it is used here. Numerical evaluations have been carried out for silver and aluminum oblate and prolate targets, including multipole modes up to order (l, m) = (15, 15). The energy-loss functions have been averaged over impact parameter and presented graphically as functions of plasmon energy for a variety of parameters.

1. Introduction The use of electron beams as probes for studying various materials provides the basis for information about the materials and the nature and results of the interaction between electrons and targets. Today, with the scanning transmission electron microscope (STEM) capable of delivering high-energy, well-placed narrow beams, and with the availability of high resolution electron spectrometers, the field of electron energy-loss spectroscopy (EELS) supports research in basic physics and applied fields, and its expanded use in various technical fields appears inevitable. Recently, m u c h attention has been focused on b e a m probing of small (submicron) targets [1-7]. These structures occur as particulates in biological samples [8], voids in bulk materials [9], models of roughened surfaces [7], and as " p l a s m o n generators" in various devices [10,11]. In some cases

i Also Department of Physics, West Virginia Institute of Technology, Montgomery, WV, USA. * Sponsored by the Office of Health and Environmental Research, US Department of Energy, under contract DE-AC0584OR21400 with Martin Marietta Energy Systems, Inc.

E E L S provides composition and geometrical characterization of particles, while in other situations devices m a y be fashioned from a knowledge of b e a m - t a r g e t interactions. Beam losses naturally divide into two categories: those associated with the target bulk and those with its surface. In large target probing, only bulk losses m a y be significant. However, when a b e a m penetrates a small target or is configured at near grazing incidence the collective excitations engendered on the surface (surface plasmons) become significant energy-loss vehicles. Indeed, only the surface losses can provide geometrical characterization of the target. I n this paper we confine our attention solely to the surface excitations. In an analytical treatment the energy-loss function is expressed as a multipole sum of excitation modes; each m o d e amplitude being dependent on the t a r g e t - b e a m configuration, plasmon energy, and multipole order. In general, a determination of m o d e amplitudes is a formidable task. Without reasonable levels of s y m m e t r y the analysis is often intractable b e y o n d a dipole approximation. Recently Ferrell and Echenique [12] provided an exact calculation, including all multipole orders, for n a r r o w - b e a m losses at near-grazing incidence

0304-3991/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

2

B.L. lllman et al. / Energy-loss profiles of transmitted electrons

to a submicron sphere. Echenique et al. [13] later expanded this to include beam penetration. The problem of non-spherical targets was partially addressed by Illman et al. [14] with a derivation appropriate for beams at near-gazing incidence to oblate and prolate spheroids, and recently Ouyang and Isaacson [15] cast the loss function in a form that, in principle, could be applied to targets of arbitrary geometry, again for the external beam. In this paper we present a derivation of the energy-loss function for beams incident on spheroidal targets at arbitrary impact parameter, but parallel to the symmetry axis. The resulting expression includes all multipole orders and may be used to model near-gazing beams, narrow penetrating beams or full-beam coverage of the spheroidal target. In section 2 we define the target-beam configuration and proceed to derive the differential loss probability for a beam incident on a spheroid. A table is provided to adapt this expression to oblate, prolate and spherical targets. In section 3 numerical examples are given for aluminum and silver for both oblate and prolate geometries. In these examples, values for the dielectric functions are taken from an experimental file on optical data. The loss functions are calculated by summing to (l, m) = (15, 15) and then averaging over impact parameter. Higher values of l and m give no significant contribution. We show the results graphically. In section 4 we discuss the problem of resolving energy-loss peaks and suggest monitoring radiative plasmon decay as a supplement to beam spectroscopy when higher resolution is needed. 2. Theory Appropriate coordinates for spheroidal targets are the orthogonal spheroidal coordinates (~/, ~, q,) for oblate geometry, (~, ~, q~) for prolate geometry and (r, 0, q~) for spherical geometry. We shall denote any one of the coordinate sets by (~, fl, q~). The coordinate ~ labels the set of confocal spheroids (spheres) and the coordinate 13 labels the family of confocal hyperboloids (cones) orthogonal to the spheroids (spheres). The coordinate 4~ is the usual azimuthal coordinate.

We begin with a point charge q constrained to a non-relativistic trajectory parallel to the symmetry axis (z axis) of the target spheroid. The particle trajectory is specified by (b, 0, vt) in rectangular and by (~'(t), fl'(t), 0) in spheroidal coordinates. Choosing the trajectory in the xz plane is a non-compromising simplification. With origin at the center of the spheroid, the coordinate b denotes the impact parameter and the particle enters and exits the target at (b, 0, -Y-v~-) or (~0, -Y-fl, 0). The response of the spheroid to the passing charge is obtained by matching boundary values for the appropriate fields at the target's surface. In our considerations both the target and the surrounding medium are characterized by the complex, frequency-dependent, dielectric functions e(~0) and i(to) respectively, and boundary conditions must be applied to Fourier components of the fields. Only local dielectric functions are used, and a more general case must await considerably more complicated analysis. From our experience with metal-island films bombarded in a scanning electron microscope, useful results can be obtained strictly from optical data [18]. For the two regions (inside and outside the target) of interest we write the Fourier component potentials

C~o(r, ~) = 1 G ( r - r ' , o~) + Vo(r, o~)

(1)

for outside the target, and

~i(r, o))=l G(r-rt, o))-~ Vi(r, to)

(2)

inside, where the Green function traces the incident electron as

G(r-r', o~) I f - ' G ( r - r', t) +

exp(i~0t) at

f~G(r-r , t)

If_ 7

'

G(r- r', t)

exp(irot) dt

exp(i~t) dt

(r r'),

"r

(3b)

B.L. lllman et al. / Energy-loss profiles of transmitted electrons

and where G(r-r', ~) is the Fourier frequency component of the Green function. When the fields are expressed in terms of multipole moments, we have

Vo(r , ~o) = Y'At,"(o~)R(o)t,"(,~)Yl,"(fl, ,), Vi(r , ~o) = ZBt,"(~o)R(i)t,"(l~)Yt,"(fl, ~), G ( r - r ' , ~o)=qY" Ct,"Ht,"(~, ~o)Yt,"(fl, ¢?),

3

We note here that the case for an external beam is obtained from the above expression by setting r = 0. The resulting integral (and Green function) can then be found analytically [12,14]. The coefficients Az,"(to ) and Btm(¢O) a r e determined by requiring the potential and the normal component of the displacement field to be continuous across the spheroidal surface ~ = ~0We express the results as

(4) (5) (6)

where Atm(°~)

H,m(L

qCt,"

R(o)/," (~o)

[ 1 - a/,"] Ht,"(,o, ¢o) '

(8) = R(i,l," (~g)

))

qQ,"

Bt,"(°~)-

× exp(i~ot) dt

R(i)t,"(~o) [ l-°ttm]nt,"('o, t°), (9)

+ R(o)t,"(l~)f~R(i)t,"(U(t))Yl,"(fl'(t)) where

× exp(io~t) dt

1 - - ~lm

R(o)t,"(~'(t))Yl,"(fl'(t))

+ R(i)t,"(~)f~ × exp(itot) d t.

Ollm

(10)

C - - ~Clm '

and the ¢l," are the geometry dependent surface plasmon eigenvalues given by

(7)

In these expressions R(o)t,"(~ ) represents the generalized radial solution of Poisson's equation, for the external field. Its specific form depends on the spheroidal geometry (oblate, prolate, spherical) to be used. R(i)t,"(~ ) represents the analogous internal field. These functions along with the generalized spherical harmonics, Yt,"(fl, q~), are given by Smythe [16] and listed in table 1. The quantities At," and Bt,. are superposition coefficients determined by the boundary conditions below, and the Ct,. are known numerical constants (see table 1).

R~o)t,"('~o)R(i)tm( ~O) Qm = R(o)tm(~o)R~i)lm(~;o ) •

(11)

The prime denotes differentiation with respect to the argument. The coefficients (8) and (9) are expressed as a sum of two parts. The part containing the ~-1 and e-1 dependence will provide energy losses characteristic of losses to the bulk and are typically considered as corrections to the infinite bulk losses. In the second part there is a similar bulk correc-

Table 1 Functions and generalized spherical harmonics Oblate

Prolate

Spherical

Rio)tin(P)

Qtm(ivl)

Qtm(~)

r-l-1

R(i)lm(~)

Pt,,(in)

Ptm(~)

rt

YIm(fl, ~ )

Ptm(I.t) coS mt~

Ptm(~ ) COS m~

Plm(COS0) cos m~

G,.

a ( 2 - 8ore"(2l) + 1)N2

i2m+l

i2m+2 a (2 - 3o.,) (21 + 1)N 2

RAt

Pt,,, and Qt,,, are Legendre functions of the first and second kind, respectively; a is the focal length, R is the spherical radius, and N = (2 - 80.)(1 - m)!/(l + m)!.

4

B.L. lllman et al. / Energy-loss profiles of transmitted electrons

tion in a0o, due to the eigenvalue c~01 = 0. Since our interest here is only with surface losses, distinct from bulk losses, we obtain a degree of simplification by disregarding the bulk terms and write

from which it follows (in Hartree atomic units)

Alm( W)R(o)lm(,f;o ) = Blm( w)R(i)lm(,~o )

This is the sought result. It is adapted to the desired spheroidal geometry by using table 1.

= qCimoQmHlm(~o, t.,o),

(12)

for (l, m) > (1, 0). The energy-loss function is obtained by considering the work done on the incident charge by the target spheroid: oo

W=

f'_

(13)

qEz, dz'.

Here E z, is the plasmon field, induced by the passing charge and evaluated along its trajectory. It is obtained from the homogeneous parts of (1) and (2): q

oo d

q

. - o , d z ' ~V° -~zeXp(-i-~z')

d' f

dz' OViozz, exp(-i-~ z')

q of.o d~fo~dz'OV°z, exp(-i-~z'). 8z

27r ~_~

(14) Integrating by parts, we find that as a consequence of our simplified form of the coefficients (12), the integrated part vanishes and the unintegrated part can be fashioned in the form of (7) to give

q2

oo

W = --~-Y'~ ctmfo d~OR(o)tm(,~o)R(i)#,,(,~o ) x I Hr.,(,%, w)12 Im a,m-

(15)

The differential energy loss probability is defined by

W = fo~ ( h OJ) d ~

) d ( h o~) ,

(16)

dP

1

1

dw - ~ y'~ Clm R(o)lm(,~o)R(i)lm( ~o )

× I Htm(~ 0, w)12 Im Otlm.

(17)

3. Numerical evaluations

We have evaluated the loss function (17) for both oblate and prolate targets as a function of energy loss for a variety of parameters. Our targets were aluminum and silver spheroids in vacuum ( i = 1). In all of our evaluations the loss function was summed to (l, m ) = (15, 15) and evaluated numerically using experimental data for the aluminum and silver dielectric functions [17]. The number of modes contributing was never found to be greater than 15 for the accuracy presented by the graphs we obtained. Also, in all of our evaluations the loss function was averaged by dividing the circular cross section of the target, and its immediate surroundings, into concentric annular regions of equal width. The loss function was evaluated within each region and multiplied by the area. This process was carried out (except where noted) into a small region beyond the particle, nominally taken to have an annular area equal to 50% of the target cross sectional area. This represents an external impact range of 1.22 times the semi-major (semi-minor) axis of the oblate (prolate) target. The extent of the external region is somewhat arbitrary, but all interactions drop off rapidly with impact parameter, and averaging out too far only dilutes the energy-loss spectrum. The results of the averaged loss function are shown graphically as a function of energy loss. The evaluation of the energy-loss function is a lengthy process of numerical integrations. To reduce the duration of these calculations mesh sizes were somewhat larger than might otherwise be desirable. In some cases the graphs have lost a degree of sharpness as a consequence of the numerical process. The graphs are categorized as follows: panels (a) in figs. 1-5 refer to aluminum

B.L. Illman et a L / Energy-loss profiles of transmitted electrons

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,~ (~v) ,,, (ev) Fig. 1. Average differential energy-loss probability ( d P / d w ) versus energy loss w for 50 keV electrons incident on: (a) aluminum oblate spheroids of different shape (710 = 0.3; solid curve and 70 = 1.5; broken curve) but constant volume of 1600 nm3; (b) silver oblate spheroids of different shape (710 = 0.3; solid curve and 70 = 1.5; broken curve) but constant volume of 1600 nm3; (c) aluminum prolate spheroids of different shape (~10 = 1.04; solid curve and ~0 = 1.8; broken curve) but constant volume of 1600 nm3; (d) silver prolate spheroids of different shape (% = 1.04; solid curve and ~0 = 1.8; broken curve) but constant volume of 1600 nm 3.

oblate targets, panels (b) in figs. 1-5 to silver oblate targets, panels (c) in figs. 1-3 refer to aluminum prolate targets and panels (d) in figs. 1-3 to silver prolate targets. In fig. 1 we contrast the energy-loss profiles of nearly spherical to spheroidal targets. The oblate comparisons for silver (fig. lb) and aluminum (la) show the general broadening of the energy-loss profile as the target becomes more oblate. The "highly" oblate (~/=0.3) targets (solid curves)

cause significant red shifting of the (1, 1) dipole and higher modes, while shifting the (1, 0) dipole to slightly higher energy. For aluminum, which is stimulated in a higher energy range, the (1, 1) mode is discernible even in the spherical limit. The oblateness amplifies this peak, and causes other modes to be resolved. With more extreme oblateness the shift increases, thus allowing resolution of higher modes. For prolate targets (figs. lc and ld) it is the axially symmetric (1, 0) mode which is red

6

B.L. lllman et aL / Energy-loss profiles of transmitted electrons

shifted. ( H e r e the " h i g h l y " prolate, ~ = 1.04, is the solid curve.) I n b o t h m a t e r i a l s this is a n intense isolated mode. The e n h a n c e d intensity of the energy-loss function in the p r o l a t e targets c o m p a r e d to the o b l a t e is a consequence of b e a m - t a r g e t geometry. T h e b e a m engulfing a p r o l a t e target is aligned with the m a j o r axis a n d m o r e of the b e a m is closer to the target in c o m p a r i s o n to a l i g n m e n t with the m i n o r axis of the o b l a t e target. The (1, 0) m o d e is p a r ticularly affected b y these configurations. In all of

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Fig. 2. Average differential energy-loss probability versus energy loss w for 50 keY electrons incident on: (a) aluminum oblate spheroids of shape ri0 = 0.3 and volume of 1600 nm3; (b) silver oblate spheroids of shape rl0 = 0.3 and volume of 1600 nm3; (c) aluminum prolate spheroids of shape gl0 = 1.04 and volume of 1600 nm3; (d) silver prolate spheroids of shape ~0 = 1.04 and volume of 1600 nm3. Broken curves show only the dipole contribution. Approximation locations of some multipole contributions are shown by the explicit (l, m) values.

B.L. Illman et aL / Energy-loss profiles of transmitted electrons

mode is sufficiently shifted to fully account for the low-energy peak. The comparable high-energy peak, however, is the cumulative effect of many modes, with the (1, 1) only moderately responsible for the energy losses in this range. In the oblate case of silver the (1, 1) mode is a reasonable approximation to the low-energy peak, but the dipole approximation significantly underestimates energy losses at the higher energies. When the oblate target is aluminum, the dipole approximation is naturally valid at the dipole energies of 6.5

b

eV and 12.5 eV, but completely misses the very comparable losses that occur between these extremes, due to higher-order modes. The locations of a few of the multipole contributions are shown explicitly in this group of figures. Since the locations are only affected by geometry they can be used in all figures (2-5). In fig. 3 the losses incurred by an internal beam are compared to those of an external beam. Essentially, we have taken the averaged loss function used in the other figures (for a 50 keV beam and

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Fig. 3. Average differential energy-loss probability (dP/d¢o) versus energy loss w for 50 keV electrons incident on: (a) aluminum oblate spheroid of shape 710 = 0.3; (b) silver oblate spheroid of shape r/0 = 0.3; (c) aluminum prolate spheroid of shape ~10 = 1.04; (d) silver prolate spheroid of shape ~0 = 1.04. Volume is 1600 nm3; solid curve shows losses due to beam coverage of particle only; broken curve shows losses due to external grazing beam.

8

B.L. lllman et a L / Energy-loss profiles of transmitted electrons 0.016

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Fig. 4. A v e r a g e d i f f e r e n t i a l e n e r g y - l o s s p r o b a b i l i t y ~d P/d¢o) versus e n e r g y loss w f o r t w o d i f f e r e n t b e a m energies ( E = 15 keV, solid curve; a n d E = 50 keV, b r o k e n curve) i n c i d e n t o n : (a) a l u m i n u m o b l a t e s p h e r o i d o f s h a p e ~0 = 0.3 a n d v o l u m e 1600 nm3; a n d (b) silver o b l a t e s p h e r o i d o f s h a p e ~0 = 0.3 a n d v o l u m e 1600 n m 3.

1600 nm 3 spheroids with 3 / 1 ratios) and have shown separately the internal contribution (solid curve) and the external contribution (broken curve). These results show clearly the importance of the b e a m / t a r g e t geometry. The prolate targets are comparably stimulated across the entire spectrum by both the internal and external beams.

0.012

That the external beam can significantly affect the axial modes is due to its relative proximity (5-6 nm) to the axis over a vertical range of 35 nm. In contrast, the external b e a m incident on the oblate targets is 11-13 nm from the axis, over a vertical range of 6.5 nm, and its effect on the axial modes is minimal. Conversely, the azimuthal modes,

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(a (eV)

Fig. 5. A v e r a g e d i f f e r e n t i a l e n e r g y - l o s s p r o b a b i l i t y ( d P / d t o ) versus e n e r g y loss f o r 50 keV e l e c t r o n s i n c i d e n t o n (a) a l u m i n u m a n d (b) silver o b l a t e s p h e r o i d s o f d i f f e r e n t v o l u m e s (43 200 n m 3, solid c u r v e ; a n d 1600 n m 3, b r o k e n c u r v e ) b u t of e q u a l s h a p e ~/0 = 0.3.

B.L. lllman et al. / Energy-loss profiles of transmitted electrons

which are more sensitive to near-surface beams, are comparably stimulated by both the external and internal beams for both geometries. Figs. 4a and 4b compare the loss function for two different beam energies (15 keV - solid curve and 50 keV - broken curve) on 1600 nm 3 oblate spheroids with 3 / 1 ratios. Both materials exhibit the same general behavior; lower beam energy promotes increased loss probability across the spectrum. Notice the dramatic increase in the aluminum peak about the 10.5 eV range. This is particularly interesting because it is the cumulative effect of modes higher than the quadrupole modes. Figs. 5a and 5b compare the loss function on oblate targets of equal shape (3/1 ratios) but of different volumes, the larger volume (solid curve) being 27 times the 1600 nm 3 volume represented by the broken curve. For the silver spheroids there is a greater loss probability across the spectrum for the larger target. In aluminum we see a larger target causes a decrease in the (1, 1) peak, has no effect on the (1, 0) mode and has only minimal effect on the quadrupole modes. Modes higher than the quadrupole combine to produce a substantial enhancement in the 10.5 eV range.

4. Discussion and conclusions

An intriguing aspect of spheroidal excitations is their potential for resolution. For most materials the spherical eigenmodes are clustered within a narrow energy band, and resolving even a few modes is experimentally impossible. For spheroids, however, resolution of these modes may be possible. Still this is a difficult task as typical separations between eigenmodes are generally of the order of less than 1 eV. Although electron energy spectroscopy is the preferred approach, since it is a direct measurement of beam losses, we suggest some consideration might usefully be given to the radiative decay of the plasmons. The incident electron beam would be a convenient way to stimulate the full plasmon spectrum, but the radiation pattern and wavelength of photons emitted from these

9

plasmons might be monitored. A full exploitation of this approach for absolute intensities would require knowledge of the correlation between the incident beam and the subsequent radiation. Although a complete correlation theory is not available, one aspect of this problem has been solved by Little et al. [18] in their work on radiative decay of oblate eigenmodes. By applying a transformation technique to the wave vector of the radiation field they were able to circumvent the use of spheroidal wave functions and obtain an expression for the radiative decay rate of the (l, m)th eigenmode in terms of spherical harmonics and Bessel functions. Their calculation provides valuable information about the angular dependence of the radiation field associated with a particular spheroidal (oblate) eigenmode. It is possible to correlate the absolute intensity of the radiation pattern with the beam and target parameters. Considering the higher resolution instrumentation available for optical radiation analysis, this may be a fruitful approach. In conclusion, we have provided an expression for the differential energy-loss probability for an electron beam incident on spheroidal targets parallel to the symmetry axis at arbitrary impact parameter. As non-retarded classical expressions, their limits of validity are obviously restricted to appropriate target dimensions. However, within these limits a wide variety of target geometries, from needles to discs, may be modeled as some type of spheroid. Our numerical evaluations have demonstrated that spheroidal targets have significantly different energy-loss profiles than spheres and that several multipole modes might be required to adequately account for these profiles.

References

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B.L. lllman et al. / Energy-loss profiles o] transmitted electrons

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