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Spatial resolution in electron energy loss spectroscopy
~ll~,,~m~t,,~r~r~r~n
Ultramlcroscopy 47 (1992) 133-144 North-Holland
Spatial resolution in electron energy loss spectroscopy P E Batson IBM Thomas J Watson Research Center, Yorktown Helghts, N Y 10598, USA Recewed at Edltorml Office 8 January 1992
Within the semi-classical theory, energy loss scattering may be calculated in an electrostatic hmlt The range of interaction is then dominated by the response of the probed system and not by the Coulomb potentml of the swift electron General quantum mechamcal considerations lead to the conclusion that the semi-classical approach Is very good for high-energy electron scattering, and prowde a good framework for evaluation of scattering using small probes Excitation of a locahzed electron state, or an electron-hole pair, is highly modified by the presence of dielectric screening within a sohd Th~s screening may be wsuahzed as a response field within the bulk, and is closely related to the charge density wake whLch trads the swift electron The spatml resolution for th~s sttuahon ~s less than 0 5 nm
1. Introduction In the past, inelastic scattering has been treated as an incoherent process for purposes of image formation This view assumes that a swift electron loses phase information during an inelastic scatterlng event, and so cannot subsequently participate in interference effects This approach forbids an analysis of image formation similar to that which is commonly used for elastic scatterlng, and has led to much confusion over the years as to the ultimate limits on spatial resolution in electron energy loss spectroscopy (EELS) This discussion summarizes our present understanding of the Inelastic scattering process, attempting to place it within an intuitively reasonable physical basis I start by briefly discussing probe spreading within the specimen This has been treated incoherently in the past, but recent work on field emission systems has shown the need for coherent treatments, especially in zone-axis orientations I next discuss energy-filtered imaging in general, using plasmon scattering as an example A discussion of the semi-classical treatment of finite impact parameter scattering follows I contmue with the perhaps surprising result that general quantum mechanical considerations provide
a justification for the semi-classical approach This justification also suggests a close connection between energy-filtered Imaging and high-resolution annular dark field STEM imaging Finally, I will discuss the on-going efforts to deduce impact parameter information from inelastic scattering, and point out similarities to the problem of generation of the plasmon wake by a swift charged particle
2. Probe spreading When a narrow electron beam traverses bulk or thin specimens, the diameter of the illuminated region becomes larger as electrons within the probe are scattered through relatively large angles This process has been treated largely in an incoherent manner, because the probe ts assumed to be made up of many, Independently propagating, electrons In fig 1 I show a typical illustration of this analysis [1] done for energy dispersive X-ray (EDS) production A result of this analysts is that the specimen thickness must be closely controlled to minimize analytical signal from regions that are far removed from the initial probe position at the top of the foil Since this
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P E Batson /Spattal re~olution m electron energ) lo~s"~pectroscopy
134
tOpic IS covered In detail elsewhere in this proceedings [2], I will not go into great depth here However, it is useful to point out one difference in the EELS case Usually, EELS is obtained from a limited angular range near the central beam, suggested by the small aperture in the figure Since the worst probe spreading is the result of large-angle scattering, most of the EELS information originates from a confined region near the center of the probed volume As EELS instrumentation improves with time, and wider scattering angles are utilized, this situation will be less true For the probe sizes m the sub-nm range that are now routinely available, the general consideratlons which lead to flg 1 are less vahd In order to form a sub-nm spot s~ze, a high-brightness source, usually a field emission gun, is required The high-brightness source produces a probe that is best characterized as a coherent superposlt~on of waves, forming a wavepacket When we consider the propagation of this wavepacket within a crystalline solid, we find that the simple probe spreading described above is no longer valid As shown in fig 2, a probe which is incident on a crystal in a low-order zone ax~s rather quickly resolves into distinct filaments which propagate without spreading down the atom columns [3,4] The intensity of each filament is directly proportIonal to the intensity at that locahon of the probe Fig 2 summarizes the results for a 1 nm
i incldent I ] Probe
Spec,men:///////~/~'~//////~^,7 .///]l'/////////r///X//~ "x-Rays //
", EELS ~ Aperture
FJg 1 Representahon of probe spread,ng within a thick sohd assuming incoherent scattering X-ray generation occurs throughout the dlummated volume Electron energy loss scattermg, which is collected, occurs within a more confined region near the center of the dlumlnated volume
probe incident along the (111) zone axis in sihcon Within 10 nm, the probe has broken up into thin filaments As the probe propagates through the crystal, these filaments vary m intensity, but do not significantly spread Elsewhere in this volume, this behavior is discussed in detail [5] In addition to the behavior In fig 2 of the probe intensity, the probe probability amplitude has a well defined value at particular lateral positions Thus, the phase of the swift electron is a well defined quantity As a result, the symmetry of the swift electron wavefunction may be defined relative to the position of an atom column This is trivially assumed for traditional image formation using plane waves, but, perhaps, needs to be re-emphasized within the context of the propagation of a small probe It is emerging that this coherent view is also appropriate when we wish to obtain EELS information and energy-filtered crystal lattice images at the ultimate hmlts that the present instrumentation and physical processes will allow
3. Energy-filtered imaging In the early 1980's, we found that the STEM was very good at obtaining images using electrons which have lost a particular amount of energy within the sample [6] These are obtained by positioning energy-selecting detector Silts on a peak in the energy loss scattering, and then scanning the probe across the sample, point-by-point, to generate a map of the scattering probability as a function of poSltlOn An example is reproduced from ref [6] in fig 3 There, I show an image of an oxide-coated aluminum sphere using slits centered at 23 eV, on the aluminum oxide bulk plasmon Clearly, very local spaUal Information is present m this image The 4 nm oxide layer on the sphere surface is very well defined We interpret this image by realizing that the projected thickness of the oxide layer is largest near the edges, and so the oxide plasmon scattering probablhty is also largest there The localized nature of the scattering, demonstrated in fig 3, is puzzling, when we realize that the wavelength, Ap, of the bulk plasmon for small scattering angles is con-
P E Batson / Spaual resolution m electron energy loss spectroscopy
trolled by the energy loss, hoJp, a n d the swift electron velocity, v 0 Ap - 27rVo/OJ p
(1)
This is 30 n m for a 23 eV loss using 100 keY electrons Thus, it Is n o t clear (1) w h e t h e r a p l a s m o n of that w a v e l e n g t h can exist wtthm the 4 n m v o l u m e that ts available, a n d (2) if it does exJst, how the swift electron scatters to preserve the very htgh spattal r e s o l u t t o n reformation whtch ts e v i d e n t in fig 3
135
Fig 4 shows results which lead to different conclusions T h e r e , I r e p r o d u c e results from ref [6] for a two-sphere system where the lme j o m l n g the two spheres xs p e r p e n d i c u l a r to the swift electron trajectory I show two tmages, o b t a m e d with 4 eV a n d 7 eV energy loss peaks I also show energy loss spectra o b t a i n e d from two different positrons relative to the two-sphere system First, tt ts k n o w n that the two spheres are both alum m u m s u r r o u n d e d with oxide layers T h e r e f o r e , the changes m signal, moving from fig 4a to fig
Fig 2 Intensity distribution for a Gausslan-shaped probe incident m a zone-axis orientation on a single crystal at a depth of about 9 nm The probe current rather quickly resolves into free filaments which propagate down the atom columns The intensity of each filament ~s proportional to the intensity at that lateral posmon within the modent probe Photo from ref [3] prowded courtesy of J Sdcox
136
P E Bat~on /Spattal resolutton m electron energy loss spectroscopy
4b, are not due to changes m local compos~hon In fact, it can be verified that the spectra (c) and (d) are obtained without hitting any sphere at all It turns out that the signals m fig 4 are a result of surface plasmon scattering [7] The 7 eV peak results from the normal A I / A I 2 0 ~ interface The 4 eV peak is due to a coupled surface plasmon mode having bl-spherxcal symmetry involving both spheres This collective mode has predominantly d~polar symmetry, oriented parallel to the hne which joins the centers of the two spheres Thts experiment, therefore, illustrates energy-filtered imaging which is sensitive to the global structure of the object Thus, we can give examples of energy-filtered imaging which show eaher local or global information Explanation of this behavior reqmres a detailed theory which takes strong account of the particular morphology of the specimen, and the impact parameter of the probe
4. Semi-classical treatment
A simple way to investigate the spatial variation of EELS is to consider a semi-classical model, summarized in fig 5 The swift electron is assumed to be a point charge, and to travel with a large constant velocity, v 0, passing some finite distance, b, away from a bounded object The problem can be considered in an adiabatic, electrostatic hmlt provided some conditions are met First, the electron is assumed to be swift, to allow the assumption of constant velocity, but not relativistic, so that the transverse part of the electric field due to the swlft electron may be neglected Second, b should not be too large, otherwise propagation times of response electric fields become large and retardation must be considered Th~s cond~ton is eqmvalent to assuming that a large angular range of scattering is utlhzed As the swift electron passes, the small object will be polarized by the Coulomb electric field of the electron At any instant of time, therefore, a s~mple boundary value problem in electrostatics may be solved to obtain the the response field due to the polarized object This response field produces a retarding force on the swift electron, leading to energy loss The energy loss probability can then be written as an Integral along the swift electron path of the force parallel to the electron velocity This is summarized [8] Pc(w, A, b) -
e ~ hoJ f_" ~¢~xx{4,o(x, A, b) -¢hex,(X, A, b)}~=,,,, dx,
Fig 3 Energy-filtered Image of an oxide-coated AI sphere using the oxide bulk plasmon scattering at 23 eV The spatial resolution is better than 1 nm in sp~te of the expected 30 nm wavelength of the bulk plasmon From ref [6]
(2)
where h w 1s the energy loss, qS0(x, b) the solution for the total electric potential in the system formed by the swift electron and the object, ¢hext(x, b) the potential due to the swift electron alone, h defines a momentum transfer given up to the object during the energy loss process, and b IS the impact parameter The differential is evaluated at the position of the swift electron as it moves past the object The electrostatics problem xs usually solved within a co-ordinate system which slmphfles specification of the boundary conditions at the object
137
P E Batson / S p a t t a l resolutton m electron energy loss spectroscopy
0
5 Energy
10 Loss
0
5
10
(eV)
Fig 4 (a) Fdtered image of a two-sphere system using the single-sphere surface plasmon at 7 eV (b) Image of the same system using the 4 eV coupled bl-sphencal surface plasmon (c) Energy spectrum obtained from position C, on the plane of symmetry which bisects the hne joining the centers of the two spheres (d) Energy spectrum obtained from position D From ref [6]
Homogenous solutions of Laplace's equahon then prowde sets of elgenfunct~ons approprmte to the object Expansion of the swift electron external field m the same co-ordinate system facdltates solutions of Po~sson's equation for the total field Then the energy loss may be computed as above
Appl)ed Field / "
A simple example of this procedure has been gwen for surface plasmons on a flat surface [9] In typical instruments, a collection aperture rotegrates over the m o m e n t u m transfer, A, producing a scattering probabdlty which ~s dependent only on position and energy loss For the case of the flat surface, th~s aperture integration produces an impact p a r a m e t e r dependence [10]
l Resp°nse 'Field
1--E P¢(oJ, b) = --7--~Ko\
I
Fig 50uasl-electrostahc boundary value problem used to evaluate the ~mpact-parameter-dependent scattering m the semi-classical hmlt
(3)
where K 0 is a Bessel function and E describes the dielectric constant of the object material The Bessel function describes the flat surface response to the potential apphed by the swift electron evaluated at the posmon of the swift electron It ~s ~mportant to reahze that ~t does not describe the spatial behavior of the surface plasmon field for a single p o s m o n of the swift elec-
1 ~8
P E Batson / Spatial resolution m electron energy loss spectroscopy
•
I
I
I
--
I
, Aluminum Surface Scattenn J21
a~ -4 0)
}
L I
-6-~ c~
-10 0
~ >~
i 100
b
". Ko(2
b)
..
i _ t i 200 300 400 Energy Loss (eV)
i00
Fig 6 C o m p a r i s o n o f r e s u l t s f r o m e q (3) w i t h e x p e r i m e n t T h e i m p a c t - p a r a m e t e r v a r m t ~ o n c o n f o r m s to t h e B e s s e l f u n c -
hon K0 From ret [6]
tron Instead, it describes the variation of the field, evaluated at the swift electron, as the impact parameter is changed The function Ko(2wb/z' o) falls off rapidly as b is increased Fig 6 reproduces the results from ref [6] for a flat surface of aluminum covered with a than oxide layer The dastance from the surface where the scattering intensity has reduced to about l / e is about 3 nm This is a surprising result considering that the surface plasmon wavelength for small-angle scattering is of order 100 nm The reason that it is so short IS that the surface plasmon response field, which dominates the energy loss process, decays exponentmlly away from the surface Integration over the fairly large-angle collection conditions then produces the Bessel function behavior The singularity at the surface predicted by eq (3) is not observed in the experiment [11] Recently, an analysas has shown that inclusion of the non-local (q-dependent) nature of the dielectric constant removes this singularity [12] The results ot fig 4 may be understood within the framework of the semi-classical theory by considering the symmetry of the response field Notice that spectrum (c), obtained from a pos1taon close to the plane of symmetry between the two spheres m fig 4a, does not show any scatterlng peak at 4 eV Results of the electrostatic
boundary value problem in bl-spherlcal coordinates show that the 4 eV peak is produced by a collective mode which has strong dipolar symmetry oriented along the hne joining the centers of the two spheres Thus, the response electric field is oriented perpendicular to the plane which bisects the line joining the sphere centers Spectrum (c) was obtained by posmonlng the probe so that ~ts trajectory hes within this central plane Thus the response field for the blspherlcal mode was required to be perpendicular to the trajectory of the swift electron Therefore, the work done on the swift electron, defined above in eq (2), must be zero On the other hand, at the position in fig 4b, the response field has a strong component which is parallel to the electron trajectory Energy transfer is therefore possible - a force is available to slow the electron down - and the 4 eV peak appears
5. Wave mechanical treatment
In angle-resolved scattering experiments, electron energy loss scattering is treated within the framework of plane wave scattering Thus, we obtain a scattering probability as a function of energy loss and deflection angle, using Ferml's Golden Rule [13]
Pq(q, w)=
2~-I V,q I
2
~,l(P,i+ ),,o126(~° -oJ,,o) II
(4) Thls expresses the quantum mechanical scattering probability as a sum over final states of a squared matrix element times a densaty of states, weighted by the swift electron Coulomb interaction potential, Vq We refer to a dynamical form factor, S(q, w), which is the extension to energy loss of the well known, static form factor for elastic scattering
Pq(q, oJ) =
2~-1V,,
12S(q, w)
(5)
The problem with this formulation is that there ~s no spatial information preserved within it The initial and final states of the swift electron are assumed to be plane wave states, which extend to
139
P E Batson /Spattal resolution m electron energy loss spectroscopy
mfimty m all directions In 1983, Kohl [14] used an idea lmtlally introduced by Rose [15] as a way of recovering some spatml reformation from the standard formulation They used a mixed dynamical form factor, S(q, q', to), to descnbe the scattermg when the incident electron ~s represented not by a single plane wave, but by two plane waves Impinging in shghtly different directions, and having a well defined phase relationship to each other The mixed factor results from products of matrix elements for transitions involving different values of q Thus, for an lnodent probe which consists of two plane waves having wavevectors (k, k'), inelastic scattering into a small, axially located aperture may be described + 2 Pq(q, to)=2rrE{[Vq] 2I(Pq),,o[ n
+]Vq, 12l(p~).o[: + 2VqVq,(p+) .o(Pq') .o} 6 ( to -
W.o)
(6) Now we can see that, w~th two initial plane waves, we obtain two squared terms plus a cross term The cross term expresses interferences among spatial f r e q u e n o e s w~thln the sample Thus S(q, q', to) may be wewed as a simple product of matrix element amphtudes for two different spatlal frequencies The scattering probaNhty for the cross terms in eq (6) then may be written
Pq(q, q', to) = 27rVqVq,S(q, q', to) 87r 2 -- q e q , e S ( q ,
q ' , to),
(7)
where I have inserted the appropriate Fourier transform for the Coulomb interaction potential for the last equality Thin expression summarizes the response m reciprocal space of the two correlated spatml frequencies, (q, q ' ) The result may be evaluated in real space (p) by weighting the (q, q ' ) component above with the appropriate probe spatial frequency coefficients, A(k), A(k'), for the particular probe specimen geometry which
IS relevant, and then Fourier transforming to obtain
Pq(p, to)= f fA+(k) e 'k" XPq(kf-k,
kf-k',
w) d2k dZk ',
(8) where we have now replaced (q, q ' ) with the appropriate scattering from the lmtlal swift electron states (k, k ' ) to the single final state, k) Kohl showed how the A ( k ) ' s can include the microscope aperture and aberration factors, and evaluated scattering for dipolar surface plasmons using a small probe initial state, and a single plane wave (k 0 final state In particular, referring to fig 4 above, we can use eq (8) to make an interesting observation Kohl pointed out that because there exist two degenerate &polar surface plasmon modes on a single sphere, there will be two contributions to the scattering, (All, A ± ) As the notation indicates these are just the collective dipole resonance oriented parallel and perpendicular to the incident beam direction If the probe is located at the center of the sphere, and kf is confined to the optic axis, then excitation of A .~ is zero This experimental situation is very close to the blspherical e x p e n m e n t above when the probe is positioned on the central plane of symmetry of the two spheres At that position, no scattering to the coupled surface plasmon at 4 eV was obtained (fig 4c) In semi-classical terms, we observed that the dipolar response field was always perpendicular to the swift electron trajectory, and so no work could be done on the swift electron In the language of eq (8), transitions involving A ± are dipole forbidden Thus, the scattering integral ~s precisely zero because its lntegrand is antl-symmetnc upon reflection through the central plane The lntegrand is anti-symmetrlc because the initial and final states of the swift electron are both symmetric about the central plane, and the dipole surface plasmon is obviously antlsymmetrlc If the small probe is moved away from the central plane, then the initial swift electron state is no longer symmetric about the
140
P E Batson / S p a t t a l resolutton m electron energy loss spectroscopy
particle center and the scattering involving A ± is no longer zero This describes the case for the spectrum in fig 4d Thus, although it was not generally reahzed at the time, the experiment described above [7] was the first to show conclusively that energy loss scattering with a small probe must be treated as a coherent process which can therefore be sensitive to the relative symmetries of the lmtlal and final swift electron states
specimen response, which supports the statement made in eq (9) The need for collection of all scattered waves may be explored by examining the basis for eq (9) in more detail Following the approach of Rltchle and Howle [16], we may evaluate the quantum mechanical scattering probabIhty, for excitation to an excited state n
Pq( b, to.o)
fkmd2kf
fa d2p¢(
p -b)
2,
6. Quantum mechanical treatment in real space In the wave mechanical treatment above, the final state of the swift electron was confined to the optic axis for simplicity and because it accurately reflected the scattering geometry which was then attainable Today, electron spectrometers routinely collect wide angular ranges of scatterlng intensity Therefore, it is appropriate to integrate eq (8) over many possible final states, kf, to obtain the total scattering This has been done for the general case by Rltchle and Howle [16] They found, for a swift electron characterlzed by a wavepacket having a lateral extent described by the probablhty [ ¢ ( p - b)[ 2, Pq(to, b) = f d2plg)(p-b)12 p¢(p),
(9)
where the quantum result, Pq, is related to the classical result of eq (2) above, Pc, by a convolution over all possible classical trajectories within an extended probe, weighted by the quantum mechanical probability of finding an electron at that particular position within the probe This result is valid provided most of the scattering is collected by the experimental collection aperture It can be loosely understood by examining the result for an integration over kf of eq (8) This will leave the terms A(k), A(k') unchanged and generate a specimen response Pq(k, k', to) specified in terms of the incident probe spatial frequencies The lntegrand in eq (8) is then just a simple multiplication of Fourier coefficients In real space, therefore, this produces a convolution operation involving the probe mtenslty and the
×e-'k'°fq, dZq±
e - lq ±
P,,o(q, w.0)
(10)
where the first Integral sums over final wavevector, kf up to some k m hmlted by the collection aperture, the second integral sums over the probe size which has a lateral extent, A, and the last integral sums over the system response, expressed in terms of the matrix element, Pn0, weighted by the Coulomb interaction, Vq .-~ 1/q 2 =- 1/(q 2 +ql~), and hmlted roughly by a cutoff wavevector, qc, controlled by electron-electron interactions within the target As usual, the components of q, (qll, q±), are resolved parallel and perpendicular to the swift electron trajectory, and qll = W,,o/Vo The last integral also may be viewed as the contrlbutlon by the nth excited state to the real space amplitude response of the system Thus there are four limiting distances in this problem 1/k.,, A, 1/q c and the bare Coulomb interaction width, VO/tonO In the limit that
l / k , . << mln(A, vo/wno, l/qc),
(11)
then interference terms under the second and third integrals are averaged A summation over final states, n, then obtains eq (9) Eqs (10) and (11) summarize the observation made with eq (3) above, that the spatial resolution is dominated not by the Coulomb interaction term (width vo/w,, o) but by the integral over q of the Coulomb term weighted by the system response This gives the Bessel function, which is much narrower than the bare Coulomb shape
P E Batson /Spattal resolutton m electron energy loss spectroscopy
Secondly, we see that the scattering hmlt on experimental resolution for Pc(w, b), will be given by the probe size, A, provided k m is made large It is also apparently possible to optimize a resolution by adjusting the ranges of kf which are used to in the experiment However, the ultimate limit is given by electron-electron interactions within the material ( ~ 1/q c) In the context of the discussion on high-angle dark field imaging [3,4], eq (9) is a generahzatlon to energy loss scattering of the results for elastic scattering Thus, for a coherent probe propagating down a zone-axis orientation, 1 4 ~ ( p - b ) 12 takes on the form of thin filaments, coincident with the atom columns and weighted by the incident probe intensity, as described with fig 2 Examination of the third integral in eq (10) shows that it is related to the Fourier transform of the solid response to an applied Coulomb potential Thus, it describes the real space disturbance within the solid - the probability amplitude for electron density fluctuations, related to the excited state n, as a function of distance away from a particular position within the swift electron probe The probability of scattering for a classical electron impinging at that position, Pc(P, °)no), is just the square of the third integral It has been suggested in the past that the spatial extent of an inelastic excitation process might be related to the reciprocal of the width of the scattering distribution - r ~ Vo/t%o Thus, we would write o __1
Pc(P, W.o) =__facd2q j-
e-lq±
-'~flno( q, (1))
2, (12)
so that
Pc(O, to) = ~f
1(1)
d2q± e - l q ± ° -q2 Im
141
ously required to be preserved in eqs (6)-(8) above This is analogous to trying to determine a crystalline structure by Fourier analysis of intensities in a diffraction pattern Instead of the true real space structure, we recover the Patterson function, a density-density correlation function In addition, a projection problem exists if eq (13) is applied The Fourier transform is specified with respect to a plane in reciprocal space defined by q± But the scattering distribution ~s really a function of q---(qu, q - ) Thus, the scatterlng intensity observed at small q . does not result from spatial excitations having a wavelength 27r/q. but 2rr/q A reasonable evaluation of eq (13) therefore would require some appropriate co-ordinate transformation prior to the Fourier transform This would then obtain the correct Patterson function for the change density fluctuations related to the system excitation If we want to obtain the true spatial extent of the excitation, we must retain both the magnitude and phase of the probability amplitudes in eq (10) for each particular excitation process, and then Fourier analyze with respect to the total transferred wavevector, q Unfortunately, as in the diffraction example, we do not usually measure the phase of the scattering directly There have been some attempts to derive real space information solely from the scattering probability by making assumptions about the phase For instance, the C h a n g - R a m a n transform which has been used in high-energy physics makes the assumption that the appropriate scattering amplitude IS just the square root of the probablhty intensity [18] Thus, eq (12) becomes
Pc(O, o,)
~(q, to)
'
aqfc
1
2
(13)
(14)
where q 2 I m ( - l / e ) replaces the summation over final states n of the matrix elements [17] We can see that this leads to an incorrect treatment because it ignores the phases of the various components of the dielectric response, which are obvi-
This equation is not rigorous, but conveys the essential point that the real space scattering probability will be much narrower than that predicted by eqs (1) and (13) above, due to taking of the square root prior to the Fourier analysis
P E Batson / Spattal resolutmn m electron energy loss spectroscopy
142
Rltchle et al have recently taken this reasoning further, to derive an energy-transfer transform instead [18,19]
Pc(p, 0)) ~ fq d2q ± fq d2q~ e '°(q±-q'~)
qq'[ 1]
× - -
q2q,2
Im
e( q, ~o)
(15)
Comparing with eqs (8) and (10) above, we can see that this approach apparently includes products of terms involving different spatial frequencies, but makes the assumption that the specimen-dependent density-density correlation function, expressed here in terms of I r a ( - l / e ) , depends only on correlations among identical spatlal frequencies (q = q ' ) The results summarized by eq (15) specify the scattering probability in terms of the impact parameter, a measure of the lateral distance away from the trajectory of the probe electron Evaluatlon of this expression for the process of generatlon of an electron-hole pair gives an excitation probablhty which has a full width of less than 0 5 nm [18] Thus, the very high resolution obtainable in images obtained with secondary electrons can be explained [20]
7. Dielectric screening
In retrospect, this value for the excitation probablhty width is very reasonable because It is of the order of the T h o m a s - F e r m i screening length in typical materials Thus, beyond this length, the swift electron should be completely screened, so that no specimen excitations are expected This would be precisely the case if the probe electron were at rest or moving slowly However, the probe electron in cases of interest here is moving very swiftly Therefore, the screenlng charge - or correlation hole - lags behind the swift probe electron as it traverses the specimen Some distance behind the probe electron, the screening charge collapses, initiating the bulk plasmon excitation It is important to reahze, however, that in the lateral direction away from
the probe electron, the screened Coulomb potential decays with the characteristic T h o m a s - F e r m i screening length Thus the result from eq (15), that the impact p a r a m e t e r scattering probability has a very narrow width, is entirely reasonable An interesting process which can be easily VlSUahzed in this context is the excitation of a locahzed state embedded in a bulk material having a dielectric constant, e(q, to) The Coulomb interaction is screened, becoming e / ( r - r,)e, so that the third Integral given in eq (10) above becomes 1
-Pc(P, ~°.o) "~ fq d2q ± e - l q ± P q2 1
2
×P~o(q, W.o) e(q, W,,o) = I p.o(q .... OJnO) }2 q, × f
1 1 2, d2q± e-lq± p _ q~ e ( q , wm~)
(16) where Pn0 describes the matrix element for excitation of the impurity state by the bare Coulomb interaction and e describes the energy-dependent and spatially dependent screening If the matrix element varies slowly with q, then it may be pulled out of the integral, as shown The result is very similar to eq (12) above, but in this case the phase of 1/e(q, wn0) is preserved The experiment summarized in eq (16) has been done for single misfit dislocations at the G a A s / Ga0ssln015As interface [21] It becomes clear now why it was possible to obtain a useable signal Since the impact p a r a m e t e r dependence is dominated by the screening length rather than the Coulomb scattering kinematics, the spatial resolution is expected to be in the sub-nm range rather than greater than 500 nm, as would be the case if p ~ tY0/O)n0 More recent work involving silicon 2p absorption near the A I / S i ( l l l ) interface also shows a variation in scattering on a length scale of 0 5 nm [22]
P E Batson / Spatial resolution m electron energy loss spectroscopy
8. T h e p l a s m o n
wake
If eq (16), within the squared magnitude, ts Fourier analyzed over an0, then we obtain a real space representation In the z-direction as well as in the lateral, impact parameter, directions The result is precisely the expression which has been derived in the past for the charge density wake which trails a swift charged particle as it traverses the dielectric solid [23] z)=
2O E
V
-~vofq±Jo(q.p) dq_~f do xe ''z/`°
30
t--
e
4'(0,
143
1 q2e(q, o9)
,
N
(17)
where the o) subscript has been dropped for convenience and the two-dimensional integral in q j_ has been partially evaluated by mtegratlon of the azimuthal angle about the z-axas to produce the Bessel function Jo(q ±P) The form of eq (17) IS that of the Fourler-Bessel transform, which IS appropriate for azlmuthally symmetrtc functions [24] The wake has been investigated m detail for swift, heavy tons, but has not been considered extensively in the context of electron energy loss scattering As mentioned above, a probe electron at rest within a solid is screened by a correlation hole having a radius corresponding to the T h o m a s - F e r m i screening d t s t a n c e - of order 0 2 - 0 5 nm If the probe electron moves with a finite veloctty, this correlation hole will move along with it If the probe electron moves very swiftly relative to the material Fermi velocity, the correlation hole lags behind the probe electron, producing an elongated depletion of charge behind the swift electron However, its lateral extent remains much the same as for the static case - 0 2 - 0 5 nm After the swift electron has passed, the correlation hole collapses, initiating plasmon and single-particle excitations A real space representation of the wake, which can be obtained by evaluation of eq (17), is shown in fig 7 Thts calculation used the Llndhard dielectric function appropriate to aluminum, including phenomenological damping followmg Mermln [24] Notice that the bulk plasmon fluctuations become large
10
0
4
2
0 2 p (nm)
4
Fig 7 Representation ot the plasmon wake which trails the swift probe electron a it traverses the sample Notice that the applied potential is largely screened beyond about 0 5 n m laterally away from the swift electron The plasmon wake propagates outwards well behind the probe electron, but does not affect the response close to it
only well behind the swift electron Close to the probe electron, the variation of the potential ts dominated by the short screening length A defect state, located beyond the screening length, will not be perturbed until well after passage of the probe electron, when it is hit by the expanding plasmon wake It may undergo excitation at that point, but this will represent a plasmon decay process, not a swift electron energy loss Thus, direct excitation of a locahzed defect state can occur only at very small tmpact parameters, regardless of the excitation energy Another result of screenmg by the wake has been shown recently using a detailed analysis of the core absorption spectra of silicon, dmmond, aluminum and SIO 2 [25] Apparently, the wake
144
P E Batson / S p a t t a l resolutton m electron energy loss" spectroscopy
strongly affects the spatial extent of the final crystal e l e c t r o n w a v e f u n c t l o n This i n f l u e n c e s the s a m p h n g v o l u m e for n e a r e d g e fine structure, a ll ow m g a s a m p h n g o f the excited crystal c o n d u c h o n b a n d s t r u c t u r e with a lateral spatial resolution consistent with the s c r e e n i n g length A n app h c a t l o n o f this to alloys of GeS1 has shown the d e t a t l e d m o v e m e n t o f h~gh symmetry p o m t s m the B r l l l o u m z o n e as a functton of G e c o n t e n t [26] In summary, then, the spatml d l s t n b u t l o n of exc~tattons within a sohd is highly c l u s t e r e d a b o u t t he trajectory o f t h e swift e l e c t r o n A t very small tmpact pa~'ameters, the T h o m a s - F e r m i s c r e e n m g l e n g t h gtves a g o o d e s t i m a t e of the lateral spatml r e s o l u t i o n o b t a m a b l e within a sohd U s i n g very small probes, a p r o p e r q u a n t u m m e c h a n t c a l t r e a t m e n t suggests that a semi-classical evaluation o f the system r e s p o n s e p o t e n t m l , c o n v o l u t e d with the p r o b e shape, m c e l y explains t h e spatial r e s o l u h o n o b t a i n e d by e x p e r i m e n t s to date This u n d e r s t a n d i n g parallels w o r k c u r r e n tl y u n d e r way tn htgh-angle d a r k field lm a g m g , w h e r e sxmtlar conclusions are r e a c h e d
References [1] E L Hall, J Phys 43 (1982) 239 [2] D B Wllhams, J R Michael, J I Goldsteln and A D Romlg, Jr, Ultramlcroscopy 47 (1992) 121 [3] R F Loane, E J Karkland and J Silcox, Acta Cryst A 44 (1988) 912 [4] S J Pennycook and D E Jesson, Phys Rev Lett 64 (1990) 938
[5] J Sflcox, P Xu and R F Loane, Ultramlcroscopy 47 (1992) 173 [6] P E Batson, Surf Scl 156 (1985) 720 [7] P E Batson, Phys Rev Lett 49 (1982) 936, 49 (1982) 1682 [8] R H Rltchle, Phys Rev 106 (1957) 874 [9] P E Batson, Ultramlcroscopy 11 (1983)299 [10] P M Echemque and J B Pendry, J Phys C 8 (1976) 2936, and in the present context, R RJtchle, 1982, unpubhshed [11] M Schemfem, A Muray and M lsaacson, Ultramlcroscopy 16 (1985) 233 [12] N Zalaba and P M Echemque, Ultramicroscopy 32 (1990) 327 [13] D Pines and P Nozleres, The Theory of Quantum Llqmds (BenJamin, New York, 1966) p 86 [14] H Kohl, Ultramlcroscopy 11 (1983) 53 [15] H Rose, Optlk 45 (1976) 139, 187 [16] R H RJtchle and A Howle, Phil Mag A 58 (1988) 753 [17] D Pines and P Nozleres, The Theory of Quantum Llqmds (Benjamin, New York, 1966) p 204 [18] R H Ritchle, A Howle, P M Echemque, G J Basbas, T L Ferrell and J C Ashley, Scanning Mlcrosc Suppl 4 (1990) 45 [19] R H Rltchle, R N Hamm, J E Turner, H A Wright, J C Ashley and G J Basbas, Nucl Tracks Rad Meas 16 (1989) 141 [20] A L Bleloch, A Howle, R H Mllne and MG Walls, UltramJcroscopy 29 (1989) 175 [21] P E Batson, KL Kavanagh, J M Woodall and J W Mayer, Phys Rev Lett 57 (1986) 2729 [22] P E Batson, Phys Rev B 44 (1991) 5556 [23] P M Echemque, R H Rltchle and W Brandt, Phys Rev B 20 (1979) 2567 [24] P E Batson and J Sdcox, Phys Rev B 27 (1983) 5224 [25] P E Batson and J Bruley, Phys Rev Lett 67 (1991) 350 [26] P E Batson and J F Morar, Appl Phys Lett 59 (1991) 3285
Ultramlcroscopy 47 (1992) 133-144 North-Holland
Spatial resolution in electron energy loss spectroscopy P E Batson IBM Thomas J Watson Research Center, Yorktown Helghts, N Y 10598, USA Recewed at Edltorml Office 8 January 1992
Within the semi-classical theory, energy loss scattering may be calculated in an electrostatic hmlt The range of interaction is then dominated by the response of the probed system and not by the Coulomb potentml of the swift electron General quantum mechamcal considerations lead to the conclusion that the semi-classical approach Is very good for high-energy electron scattering, and prowde a good framework for evaluation of scattering using small probes Excitation of a locahzed electron state, or an electron-hole pair, is highly modified by the presence of dielectric screening within a sohd Th~s screening may be wsuahzed as a response field within the bulk, and is closely related to the charge density wake whLch trads the swift electron The spatml resolution for th~s sttuahon ~s less than 0 5 nm
1. Introduction In the past, inelastic scattering has been treated as an incoherent process for purposes of image formation This view assumes that a swift electron loses phase information during an inelastic scatterlng event, and so cannot subsequently participate in interference effects This approach forbids an analysis of image formation similar to that which is commonly used for elastic scatterlng, and has led to much confusion over the years as to the ultimate limits on spatial resolution in electron energy loss spectroscopy (EELS) This discussion summarizes our present understanding of the Inelastic scattering process, attempting to place it within an intuitively reasonable physical basis I start by briefly discussing probe spreading within the specimen This has been treated incoherently in the past, but recent work on field emission systems has shown the need for coherent treatments, especially in zone-axis orientations I next discuss energy-filtered imaging in general, using plasmon scattering as an example A discussion of the semi-classical treatment of finite impact parameter scattering follows I contmue with the perhaps surprising result that general quantum mechanical considerations provide
a justification for the semi-classical approach This justification also suggests a close connection between energy-filtered Imaging and high-resolution annular dark field STEM imaging Finally, I will discuss the on-going efforts to deduce impact parameter information from inelastic scattering, and point out similarities to the problem of generation of the plasmon wake by a swift charged particle
2. Probe spreading When a narrow electron beam traverses bulk or thin specimens, the diameter of the illuminated region becomes larger as electrons within the probe are scattered through relatively large angles This process has been treated largely in an incoherent manner, because the probe ts assumed to be made up of many, Independently propagating, electrons In fig 1 I show a typical illustration of this analysis [1] done for energy dispersive X-ray (EDS) production A result of this analysts is that the specimen thickness must be closely controlled to minimize analytical signal from regions that are far removed from the initial probe position at the top of the foil Since this
0304-3991/92/$05 00 © 1992 - Elsevier Science Pubhshers B V All rights reserved
P E Batson /Spattal re~olution m electron energ) lo~s"~pectroscopy
134
tOpic IS covered In detail elsewhere in this proceedings [2], I will not go into great depth here However, it is useful to point out one difference in the EELS case Usually, EELS is obtained from a limited angular range near the central beam, suggested by the small aperture in the figure Since the worst probe spreading is the result of large-angle scattering, most of the EELS information originates from a confined region near the center of the probed volume As EELS instrumentation improves with time, and wider scattering angles are utilized, this situation will be less true For the probe sizes m the sub-nm range that are now routinely available, the general consideratlons which lead to flg 1 are less vahd In order to form a sub-nm spot s~ze, a high-brightness source, usually a field emission gun, is required The high-brightness source produces a probe that is best characterized as a coherent superposlt~on of waves, forming a wavepacket When we consider the propagation of this wavepacket within a crystalline solid, we find that the simple probe spreading described above is no longer valid As shown in fig 2, a probe which is incident on a crystal in a low-order zone ax~s rather quickly resolves into distinct filaments which propagate without spreading down the atom columns [3,4] The intensity of each filament is directly proportIonal to the intensity at that locahon of the probe Fig 2 summarizes the results for a 1 nm
i incldent I ] Probe
Spec,men:///////~/~'~//////~^,7 .///]l'/////////r///X//~ "x-Rays //
", EELS ~ Aperture
FJg 1 Representahon of probe spread,ng within a thick sohd assuming incoherent scattering X-ray generation occurs throughout the dlummated volume Electron energy loss scattermg, which is collected, occurs within a more confined region near the center of the dlumlnated volume
probe incident along the (111) zone axis in sihcon Within 10 nm, the probe has broken up into thin filaments As the probe propagates through the crystal, these filaments vary m intensity, but do not significantly spread Elsewhere in this volume, this behavior is discussed in detail [5] In addition to the behavior In fig 2 of the probe intensity, the probe probability amplitude has a well defined value at particular lateral positions Thus, the phase of the swift electron is a well defined quantity As a result, the symmetry of the swift electron wavefunction may be defined relative to the position of an atom column This is trivially assumed for traditional image formation using plane waves, but, perhaps, needs to be re-emphasized within the context of the propagation of a small probe It is emerging that this coherent view is also appropriate when we wish to obtain EELS information and energy-filtered crystal lattice images at the ultimate hmlts that the present instrumentation and physical processes will allow
3. Energy-filtered imaging In the early 1980's, we found that the STEM was very good at obtaining images using electrons which have lost a particular amount of energy within the sample [6] These are obtained by positioning energy-selecting detector Silts on a peak in the energy loss scattering, and then scanning the probe across the sample, point-by-point, to generate a map of the scattering probability as a function of poSltlOn An example is reproduced from ref [6] in fig 3 There, I show an image of an oxide-coated aluminum sphere using slits centered at 23 eV, on the aluminum oxide bulk plasmon Clearly, very local spaUal Information is present m this image The 4 nm oxide layer on the sphere surface is very well defined We interpret this image by realizing that the projected thickness of the oxide layer is largest near the edges, and so the oxide plasmon scattering probablhty is also largest there The localized nature of the scattering, demonstrated in fig 3, is puzzling, when we realize that the wavelength, Ap, of the bulk plasmon for small scattering angles is con-
P E Batson / Spaual resolution m electron energy loss spectroscopy
trolled by the energy loss, hoJp, a n d the swift electron velocity, v 0 Ap - 27rVo/OJ p
(1)
This is 30 n m for a 23 eV loss using 100 keY electrons Thus, it Is n o t clear (1) w h e t h e r a p l a s m o n of that w a v e l e n g t h can exist wtthm the 4 n m v o l u m e that ts available, a n d (2) if it does exJst, how the swift electron scatters to preserve the very htgh spattal r e s o l u t t o n reformation whtch ts e v i d e n t in fig 3
135
Fig 4 shows results which lead to different conclusions T h e r e , I r e p r o d u c e results from ref [6] for a two-sphere system where the lme j o m l n g the two spheres xs p e r p e n d i c u l a r to the swift electron trajectory I show two tmages, o b t a m e d with 4 eV a n d 7 eV energy loss peaks I also show energy loss spectra o b t a i n e d from two different positrons relative to the two-sphere system First, tt ts k n o w n that the two spheres are both alum m u m s u r r o u n d e d with oxide layers T h e r e f o r e , the changes m signal, moving from fig 4a to fig
Fig 2 Intensity distribution for a Gausslan-shaped probe incident m a zone-axis orientation on a single crystal at a depth of about 9 nm The probe current rather quickly resolves into free filaments which propagate down the atom columns The intensity of each filament ~s proportional to the intensity at that lateral posmon within the modent probe Photo from ref [3] prowded courtesy of J Sdcox
136
P E Bat~on /Spattal resolutton m electron energy loss spectroscopy
4b, are not due to changes m local compos~hon In fact, it can be verified that the spectra (c) and (d) are obtained without hitting any sphere at all It turns out that the signals m fig 4 are a result of surface plasmon scattering [7] The 7 eV peak results from the normal A I / A I 2 0 ~ interface The 4 eV peak is due to a coupled surface plasmon mode having bl-spherxcal symmetry involving both spheres This collective mode has predominantly d~polar symmetry, oriented parallel to the hne which joins the centers of the two spheres Thts experiment, therefore, illustrates energy-filtered imaging which is sensitive to the global structure of the object Thus, we can give examples of energy-filtered imaging which show eaher local or global information Explanation of this behavior reqmres a detailed theory which takes strong account of the particular morphology of the specimen, and the impact parameter of the probe
4. Semi-classical treatment
A simple way to investigate the spatial variation of EELS is to consider a semi-classical model, summarized in fig 5 The swift electron is assumed to be a point charge, and to travel with a large constant velocity, v 0, passing some finite distance, b, away from a bounded object The problem can be considered in an adiabatic, electrostatic hmlt provided some conditions are met First, the electron is assumed to be swift, to allow the assumption of constant velocity, but not relativistic, so that the transverse part of the electric field due to the swlft electron may be neglected Second, b should not be too large, otherwise propagation times of response electric fields become large and retardation must be considered Th~s cond~ton is eqmvalent to assuming that a large angular range of scattering is utlhzed As the swift electron passes, the small object will be polarized by the Coulomb electric field of the electron At any instant of time, therefore, a s~mple boundary value problem in electrostatics may be solved to obtain the the response field due to the polarized object This response field produces a retarding force on the swift electron, leading to energy loss The energy loss probability can then be written as an Integral along the swift electron path of the force parallel to the electron velocity This is summarized [8] Pc(w, A, b) -
e ~ hoJ f_" ~¢~xx{4,o(x, A, b) -¢hex,(X, A, b)}~=,,,, dx,
Fig 3 Energy-filtered Image of an oxide-coated AI sphere using the oxide bulk plasmon scattering at 23 eV The spatial resolution is better than 1 nm in sp~te of the expected 30 nm wavelength of the bulk plasmon From ref [6]
(2)
where h w 1s the energy loss, qS0(x, b) the solution for the total electric potential in the system formed by the swift electron and the object, ¢hext(x, b) the potential due to the swift electron alone, h defines a momentum transfer given up to the object during the energy loss process, and b IS the impact parameter The differential is evaluated at the position of the swift electron as it moves past the object The electrostatics problem xs usually solved within a co-ordinate system which slmphfles specification of the boundary conditions at the object
137
P E Batson / S p a t t a l resolutton m electron energy loss spectroscopy
0
5 Energy
10 Loss
0
5
10
(eV)
Fig 4 (a) Fdtered image of a two-sphere system using the single-sphere surface plasmon at 7 eV (b) Image of the same system using the 4 eV coupled bl-sphencal surface plasmon (c) Energy spectrum obtained from position C, on the plane of symmetry which bisects the hne joining the centers of the two spheres (d) Energy spectrum obtained from position D From ref [6]
Homogenous solutions of Laplace's equahon then prowde sets of elgenfunct~ons approprmte to the object Expansion of the swift electron external field m the same co-ordinate system facdltates solutions of Po~sson's equation for the total field Then the energy loss may be computed as above
Appl)ed Field / "
A simple example of this procedure has been gwen for surface plasmons on a flat surface [9] In typical instruments, a collection aperture rotegrates over the m o m e n t u m transfer, A, producing a scattering probabdlty which ~s dependent only on position and energy loss For the case of the flat surface, th~s aperture integration produces an impact p a r a m e t e r dependence [10]
l Resp°nse 'Field
1--E P¢(oJ, b) = --7--~Ko\
I
Fig 50uasl-electrostahc boundary value problem used to evaluate the ~mpact-parameter-dependent scattering m the semi-classical hmlt
(3)
where K 0 is a Bessel function and E describes the dielectric constant of the object material The Bessel function describes the flat surface response to the potential apphed by the swift electron evaluated at the posmon of the swift electron It ~s ~mportant to reahze that ~t does not describe the spatial behavior of the surface plasmon field for a single p o s m o n of the swift elec-
1 ~8
P E Batson / Spatial resolution m electron energy loss spectroscopy
•
I
I
I
--
I
, Aluminum Surface Scattenn J21
a~ -4 0)
}
L I
-6-~ c~
-10 0
~ >~
i 100
b
". Ko(2
b)
..
i _ t i 200 300 400 Energy Loss (eV)
i00
Fig 6 C o m p a r i s o n o f r e s u l t s f r o m e q (3) w i t h e x p e r i m e n t T h e i m p a c t - p a r a m e t e r v a r m t ~ o n c o n f o r m s to t h e B e s s e l f u n c -
hon K0 From ret [6]
tron Instead, it describes the variation of the field, evaluated at the swift electron, as the impact parameter is changed The function Ko(2wb/z' o) falls off rapidly as b is increased Fig 6 reproduces the results from ref [6] for a flat surface of aluminum covered with a than oxide layer The dastance from the surface where the scattering intensity has reduced to about l / e is about 3 nm This is a surprising result considering that the surface plasmon wavelength for small-angle scattering is of order 100 nm The reason that it is so short IS that the surface plasmon response field, which dominates the energy loss process, decays exponentmlly away from the surface Integration over the fairly large-angle collection conditions then produces the Bessel function behavior The singularity at the surface predicted by eq (3) is not observed in the experiment [11] Recently, an analysas has shown that inclusion of the non-local (q-dependent) nature of the dielectric constant removes this singularity [12] The results ot fig 4 may be understood within the framework of the semi-classical theory by considering the symmetry of the response field Notice that spectrum (c), obtained from a pos1taon close to the plane of symmetry between the two spheres m fig 4a, does not show any scatterlng peak at 4 eV Results of the electrostatic
boundary value problem in bl-spherlcal coordinates show that the 4 eV peak is produced by a collective mode which has strong dipolar symmetry oriented along the hne joining the centers of the two spheres Thus, the response electric field is oriented perpendicular to the plane which bisects the line joining the sphere centers Spectrum (c) was obtained by posmonlng the probe so that ~ts trajectory hes within this central plane Thus the response field for the blspherlcal mode was required to be perpendicular to the trajectory of the swift electron Therefore, the work done on the swift electron, defined above in eq (2), must be zero On the other hand, at the position in fig 4b, the response field has a strong component which is parallel to the electron trajectory Energy transfer is therefore possible - a force is available to slow the electron down - and the 4 eV peak appears
5. Wave mechanical treatment
In angle-resolved scattering experiments, electron energy loss scattering is treated within the framework of plane wave scattering Thus, we obtain a scattering probability as a function of energy loss and deflection angle, using Ferml's Golden Rule [13]
Pq(q, w)=
2~-I V,q I
2
~,l(P,i+ ),,o126(~° -oJ,,o) II
(4) Thls expresses the quantum mechanical scattering probability as a sum over final states of a squared matrix element times a densaty of states, weighted by the swift electron Coulomb interaction potential, Vq We refer to a dynamical form factor, S(q, w), which is the extension to energy loss of the well known, static form factor for elastic scattering
Pq(q, oJ) =
2~-1V,,
12S(q, w)
(5)
The problem with this formulation is that there ~s no spatial information preserved within it The initial and final states of the swift electron are assumed to be plane wave states, which extend to
139
P E Batson /Spattal resolution m electron energy loss spectroscopy
mfimty m all directions In 1983, Kohl [14] used an idea lmtlally introduced by Rose [15] as a way of recovering some spatml reformation from the standard formulation They used a mixed dynamical form factor, S(q, q', to), to descnbe the scattermg when the incident electron ~s represented not by a single plane wave, but by two plane waves Impinging in shghtly different directions, and having a well defined phase relationship to each other The mixed factor results from products of matrix elements for transitions involving different values of q Thus, for an lnodent probe which consists of two plane waves having wavevectors (k, k'), inelastic scattering into a small, axially located aperture may be described + 2 Pq(q, to)=2rrE{[Vq] 2I(Pq),,o[ n
+]Vq, 12l(p~).o[: + 2VqVq,(p+) .o(Pq') .o} 6 ( to -
W.o)
(6) Now we can see that, w~th two initial plane waves, we obtain two squared terms plus a cross term The cross term expresses interferences among spatial f r e q u e n o e s w~thln the sample Thus S(q, q', to) may be wewed as a simple product of matrix element amphtudes for two different spatlal frequencies The scattering probaNhty for the cross terms in eq (6) then may be written
Pq(q, q', to) = 27rVqVq,S(q, q', to) 87r 2 -- q e q , e S ( q ,
q ' , to),
(7)
where I have inserted the appropriate Fourier transform for the Coulomb interaction potential for the last equality Thin expression summarizes the response m reciprocal space of the two correlated spatml frequencies, (q, q ' ) The result may be evaluated in real space (p) by weighting the (q, q ' ) component above with the appropriate probe spatial frequency coefficients, A(k), A(k'), for the particular probe specimen geometry which
IS relevant, and then Fourier transforming to obtain
Pq(p, to)= f fA+(k) e 'k" XPq(kf-k,
kf-k',
w) d2k dZk ',
(8) where we have now replaced (q, q ' ) with the appropriate scattering from the lmtlal swift electron states (k, k ' ) to the single final state, k) Kohl showed how the A ( k ) ' s can include the microscope aperture and aberration factors, and evaluated scattering for dipolar surface plasmons using a small probe initial state, and a single plane wave (k 0 final state In particular, referring to fig 4 above, we can use eq (8) to make an interesting observation Kohl pointed out that because there exist two degenerate &polar surface plasmon modes on a single sphere, there will be two contributions to the scattering, (All, A ± ) As the notation indicates these are just the collective dipole resonance oriented parallel and perpendicular to the incident beam direction If the probe is located at the center of the sphere, and kf is confined to the optic axis, then excitation of A .~ is zero This experimental situation is very close to the blspherical e x p e n m e n t above when the probe is positioned on the central plane of symmetry of the two spheres At that position, no scattering to the coupled surface plasmon at 4 eV was obtained (fig 4c) In semi-classical terms, we observed that the dipolar response field was always perpendicular to the swift electron trajectory, and so no work could be done on the swift electron In the language of eq (8), transitions involving A ± are dipole forbidden Thus, the scattering integral ~s precisely zero because its lntegrand is antl-symmetnc upon reflection through the central plane The lntegrand is anti-symmetrlc because the initial and final states of the swift electron are both symmetric about the central plane, and the dipole surface plasmon is obviously antlsymmetrlc If the small probe is moved away from the central plane, then the initial swift electron state is no longer symmetric about the
140
P E Batson / S p a t t a l resolutton m electron energy loss spectroscopy
particle center and the scattering involving A ± is no longer zero This describes the case for the spectrum in fig 4d Thus, although it was not generally reahzed at the time, the experiment described above [7] was the first to show conclusively that energy loss scattering with a small probe must be treated as a coherent process which can therefore be sensitive to the relative symmetries of the lmtlal and final swift electron states
specimen response, which supports the statement made in eq (9) The need for collection of all scattered waves may be explored by examining the basis for eq (9) in more detail Following the approach of Rltchle and Howle [16], we may evaluate the quantum mechanical scattering probabIhty, for excitation to an excited state n
Pq( b, to.o)
fkmd2kf
fa d2p¢(
p -b)
2,
6. Quantum mechanical treatment in real space In the wave mechanical treatment above, the final state of the swift electron was confined to the optic axis for simplicity and because it accurately reflected the scattering geometry which was then attainable Today, electron spectrometers routinely collect wide angular ranges of scatterlng intensity Therefore, it is appropriate to integrate eq (8) over many possible final states, kf, to obtain the total scattering This has been done for the general case by Rltchle and Howle [16] They found, for a swift electron characterlzed by a wavepacket having a lateral extent described by the probablhty [ ¢ ( p - b)[ 2, Pq(to, b) = f d2plg)(p-b)12 p¢(p),
(9)
where the quantum result, Pq, is related to the classical result of eq (2) above, Pc, by a convolution over all possible classical trajectories within an extended probe, weighted by the quantum mechanical probability of finding an electron at that particular position within the probe This result is valid provided most of the scattering is collected by the experimental collection aperture It can be loosely understood by examining the result for an integration over kf of eq (8) This will leave the terms A(k), A(k') unchanged and generate a specimen response Pq(k, k', to) specified in terms of the incident probe spatial frequencies The lntegrand in eq (8) is then just a simple multiplication of Fourier coefficients In real space, therefore, this produces a convolution operation involving the probe mtenslty and the
×e-'k'°fq, dZq±
e - lq ±
P,,o(q, w.0)
(10)
where the first Integral sums over final wavevector, kf up to some k m hmlted by the collection aperture, the second integral sums over the probe size which has a lateral extent, A, and the last integral sums over the system response, expressed in terms of the matrix element, Pn0, weighted by the Coulomb interaction, Vq .-~ 1/q 2 =- 1/(q 2 +ql~), and hmlted roughly by a cutoff wavevector, qc, controlled by electron-electron interactions within the target As usual, the components of q, (qll, q±), are resolved parallel and perpendicular to the swift electron trajectory, and qll = W,,o/Vo The last integral also may be viewed as the contrlbutlon by the nth excited state to the real space amplitude response of the system Thus there are four limiting distances in this problem 1/k.,, A, 1/q c and the bare Coulomb interaction width, VO/tonO In the limit that
l / k , . << mln(A, vo/wno, l/qc),
(11)
then interference terms under the second and third integrals are averaged A summation over final states, n, then obtains eq (9) Eqs (10) and (11) summarize the observation made with eq (3) above, that the spatial resolution is dominated not by the Coulomb interaction term (width vo/w,, o) but by the integral over q of the Coulomb term weighted by the system response This gives the Bessel function, which is much narrower than the bare Coulomb shape
P E Batson /Spattal resolutton m electron energy loss spectroscopy
Secondly, we see that the scattering hmlt on experimental resolution for Pc(w, b), will be given by the probe size, A, provided k m is made large It is also apparently possible to optimize a resolution by adjusting the ranges of kf which are used to in the experiment However, the ultimate limit is given by electron-electron interactions within the material ( ~ 1/q c) In the context of the discussion on high-angle dark field imaging [3,4], eq (9) is a generahzatlon to energy loss scattering of the results for elastic scattering Thus, for a coherent probe propagating down a zone-axis orientation, 1 4 ~ ( p - b ) 12 takes on the form of thin filaments, coincident with the atom columns and weighted by the incident probe intensity, as described with fig 2 Examination of the third integral in eq (10) shows that it is related to the Fourier transform of the solid response to an applied Coulomb potential Thus, it describes the real space disturbance within the solid - the probability amplitude for electron density fluctuations, related to the excited state n, as a function of distance away from a particular position within the swift electron probe The probability of scattering for a classical electron impinging at that position, Pc(P, °)no), is just the square of the third integral It has been suggested in the past that the spatial extent of an inelastic excitation process might be related to the reciprocal of the width of the scattering distribution - r ~ Vo/t%o Thus, we would write o __1
Pc(P, W.o) =__facd2q j-
e-lq±
-'~flno( q, (1))
2, (12)
so that
Pc(O, to) = ~f
1(1)
d2q± e - l q ± ° -q2 Im
141
ously required to be preserved in eqs (6)-(8) above This is analogous to trying to determine a crystalline structure by Fourier analysis of intensities in a diffraction pattern Instead of the true real space structure, we recover the Patterson function, a density-density correlation function In addition, a projection problem exists if eq (13) is applied The Fourier transform is specified with respect to a plane in reciprocal space defined by q± But the scattering distribution ~s really a function of q---(qu, q - ) Thus, the scatterlng intensity observed at small q . does not result from spatial excitations having a wavelength 27r/q. but 2rr/q A reasonable evaluation of eq (13) therefore would require some appropriate co-ordinate transformation prior to the Fourier transform This would then obtain the correct Patterson function for the change density fluctuations related to the system excitation If we want to obtain the true spatial extent of the excitation, we must retain both the magnitude and phase of the probability amplitudes in eq (10) for each particular excitation process, and then Fourier analyze with respect to the total transferred wavevector, q Unfortunately, as in the diffraction example, we do not usually measure the phase of the scattering directly There have been some attempts to derive real space information solely from the scattering probability by making assumptions about the phase For instance, the C h a n g - R a m a n transform which has been used in high-energy physics makes the assumption that the appropriate scattering amplitude IS just the square root of the probablhty intensity [18] Thus, eq (12) becomes
Pc(O, o,)
~(q, to)
'
aqfc
1
2
(13)
(14)
where q 2 I m ( - l / e ) replaces the summation over final states n of the matrix elements [17] We can see that this leads to an incorrect treatment because it ignores the phases of the various components of the dielectric response, which are obvi-
This equation is not rigorous, but conveys the essential point that the real space scattering probability will be much narrower than that predicted by eqs (1) and (13) above, due to taking of the square root prior to the Fourier analysis
P E Batson / Spattal resolutmn m electron energy loss spectroscopy
142
Rltchle et al have recently taken this reasoning further, to derive an energy-transfer transform instead [18,19]
Pc(p, 0)) ~ fq d2q ± fq d2q~ e '°(q±-q'~)
qq'[ 1]
× - -
q2q,2
Im
e( q, ~o)
(15)
Comparing with eqs (8) and (10) above, we can see that this approach apparently includes products of terms involving different spatial frequencies, but makes the assumption that the specimen-dependent density-density correlation function, expressed here in terms of I r a ( - l / e ) , depends only on correlations among identical spatlal frequencies (q = q ' ) The results summarized by eq (15) specify the scattering probability in terms of the impact parameter, a measure of the lateral distance away from the trajectory of the probe electron Evaluatlon of this expression for the process of generatlon of an electron-hole pair gives an excitation probablhty which has a full width of less than 0 5 nm [18] Thus, the very high resolution obtainable in images obtained with secondary electrons can be explained [20]
7. Dielectric screening
In retrospect, this value for the excitation probablhty width is very reasonable because It is of the order of the T h o m a s - F e r m i screening length in typical materials Thus, beyond this length, the swift electron should be completely screened, so that no specimen excitations are expected This would be precisely the case if the probe electron were at rest or moving slowly However, the probe electron in cases of interest here is moving very swiftly Therefore, the screenlng charge - or correlation hole - lags behind the swift probe electron as it traverses the specimen Some distance behind the probe electron, the screening charge collapses, initiating the bulk plasmon excitation It is important to reahze, however, that in the lateral direction away from
the probe electron, the screened Coulomb potential decays with the characteristic T h o m a s - F e r m i screening length Thus the result from eq (15), that the impact p a r a m e t e r scattering probability has a very narrow width, is entirely reasonable An interesting process which can be easily VlSUahzed in this context is the excitation of a locahzed state embedded in a bulk material having a dielectric constant, e(q, to) The Coulomb interaction is screened, becoming e / ( r - r,)e, so that the third Integral given in eq (10) above becomes 1
-Pc(P, ~°.o) "~ fq d2q ± e - l q ± P q2 1
2
×P~o(q, W.o) e(q, W,,o) = I p.o(q .... OJnO) }2 q, × f
1 1 2, d2q± e-lq± p _ q~ e ( q , wm~)
(16) where Pn0 describes the matrix element for excitation of the impurity state by the bare Coulomb interaction and e describes the energy-dependent and spatially dependent screening If the matrix element varies slowly with q, then it may be pulled out of the integral, as shown The result is very similar to eq (12) above, but in this case the phase of 1/e(q, wn0) is preserved The experiment summarized in eq (16) has been done for single misfit dislocations at the G a A s / Ga0ssln015As interface [21] It becomes clear now why it was possible to obtain a useable signal Since the impact p a r a m e t e r dependence is dominated by the screening length rather than the Coulomb scattering kinematics, the spatial resolution is expected to be in the sub-nm range rather than greater than 500 nm, as would be the case if p ~ tY0/O)n0 More recent work involving silicon 2p absorption near the A I / S i ( l l l ) interface also shows a variation in scattering on a length scale of 0 5 nm [22]
P E Batson / Spatial resolution m electron energy loss spectroscopy
8. T h e p l a s m o n
wake
If eq (16), within the squared magnitude, ts Fourier analyzed over an0, then we obtain a real space representation In the z-direction as well as in the lateral, impact parameter, directions The result is precisely the expression which has been derived in the past for the charge density wake which trails a swift charged particle as it traverses the dielectric solid [23] z)=
2O E
V
-~vofq±Jo(q.p) dq_~f do xe ''z/`°
30
t--
e
4'(0,
143
1 q2e(q, o9)
,
N
(17)
where the o) subscript has been dropped for convenience and the two-dimensional integral in q j_ has been partially evaluated by mtegratlon of the azimuthal angle about the z-axas to produce the Bessel function Jo(q ±P) The form of eq (17) IS that of the Fourler-Bessel transform, which IS appropriate for azlmuthally symmetrtc functions [24] The wake has been investigated m detail for swift, heavy tons, but has not been considered extensively in the context of electron energy loss scattering As mentioned above, a probe electron at rest within a solid is screened by a correlation hole having a radius corresponding to the T h o m a s - F e r m i screening d t s t a n c e - of order 0 2 - 0 5 nm If the probe electron moves with a finite veloctty, this correlation hole will move along with it If the probe electron moves very swiftly relative to the material Fermi velocity, the correlation hole lags behind the probe electron, producing an elongated depletion of charge behind the swift electron However, its lateral extent remains much the same as for the static case - 0 2 - 0 5 nm After the swift electron has passed, the correlation hole collapses, initiating plasmon and single-particle excitations A real space representation of the wake, which can be obtained by evaluation of eq (17), is shown in fig 7 Thts calculation used the Llndhard dielectric function appropriate to aluminum, including phenomenological damping followmg Mermln [24] Notice that the bulk plasmon fluctuations become large
10
0
4
2
0 2 p (nm)
4
Fig 7 Representation ot the plasmon wake which trails the swift probe electron a it traverses the sample Notice that the applied potential is largely screened beyond about 0 5 n m laterally away from the swift electron The plasmon wake propagates outwards well behind the probe electron, but does not affect the response close to it
only well behind the swift electron Close to the probe electron, the variation of the potential ts dominated by the short screening length A defect state, located beyond the screening length, will not be perturbed until well after passage of the probe electron, when it is hit by the expanding plasmon wake It may undergo excitation at that point, but this will represent a plasmon decay process, not a swift electron energy loss Thus, direct excitation of a locahzed defect state can occur only at very small tmpact parameters, regardless of the excitation energy Another result of screenmg by the wake has been shown recently using a detailed analysis of the core absorption spectra of silicon, dmmond, aluminum and SIO 2 [25] Apparently, the wake
144
P E Batson / S p a t t a l resolutton m electron energy loss" spectroscopy
strongly affects the spatial extent of the final crystal e l e c t r o n w a v e f u n c t l o n This i n f l u e n c e s the s a m p h n g v o l u m e for n e a r e d g e fine structure, a ll ow m g a s a m p h n g o f the excited crystal c o n d u c h o n b a n d s t r u c t u r e with a lateral spatial resolution consistent with the s c r e e n i n g length A n app h c a t l o n o f this to alloys of GeS1 has shown the d e t a t l e d m o v e m e n t o f h~gh symmetry p o m t s m the B r l l l o u m z o n e as a functton of G e c o n t e n t [26] In summary, then, the spatml d l s t n b u t l o n of exc~tattons within a sohd is highly c l u s t e r e d a b o u t t he trajectory o f t h e swift e l e c t r o n A t very small tmpact pa~'ameters, the T h o m a s - F e r m i s c r e e n m g l e n g t h gtves a g o o d e s t i m a t e of the lateral spatml r e s o l u t i o n o b t a m a b l e within a sohd U s i n g very small probes, a p r o p e r q u a n t u m m e c h a n t c a l t r e a t m e n t suggests that a semi-classical evaluation o f the system r e s p o n s e p o t e n t m l , c o n v o l u t e d with the p r o b e shape, m c e l y explains t h e spatial r e s o l u h o n o b t a i n e d by e x p e r i m e n t s to date This u n d e r s t a n d i n g parallels w o r k c u r r e n tl y u n d e r way tn htgh-angle d a r k field lm a g m g , w h e r e sxmtlar conclusions are r e a c h e d
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