Localization of high-energy electron scattering from atomic vibrations

2.9 MB·

by  Christian Dwyer

Localization of high-energy electron scattering from atomic vibrations
C. Dwyer∗
Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons,
and Peter Gru¨nberg Institute, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany
Electrons with kinetic energies ∼100 keV are capable of exciting atomic vibrational states from
adistanceofmicrons. Despitesuchalargeinteractiondistance,ourdetailedcalculationsshowthat
the scattering physics permits a high-energy electron beam to locate vibrational excitations with
atomic-scale spatial resolution. Pursuits to realize this capability experimentally could potentially
benefit numerous fields across the physical sciences.
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1 Introduction.—Substantial interest surrounds the image. In the case of core-level excitations, this tech-
0 question of whether it is possible, from the perspec- nique enables mapping of a material’s chemical com-
2 tiveofscatteringphysics,tousehigh-energyelectrons position and electronic bonding at the atomic scale
n to locate atomic vibrations with a spatial resolution [3], which is ideal for studying interfaces and nano-
a at,ornear,theatomicscale. Thisinterestistriggered materials, for example. In terms of the scattering
J by recent reports that it may soon be possible to use physics, this spatial resolution is permitted firstly by

4 anatomic-sizedbeaminascanningtransmissionelec- the atomic-sized beam, and secondly by the inelastic
2
tron microscope (STEM) to perform spectroscopy at delocalization length v/ω (v is the electron’s velocity

energy resolutions of the order 10 meV [1], enabling and (cid:126)ω is the energy loss), which is typically a few
]
i access to vibrational excitations in a transmission ge- ˚Angstroms or less for core-level excitations.
c
s ometry. The ability to detect atomic vibrations with On the other hand, using incident electrons to ac-
-
high spatial resolution would offer substantial advan-
l cess vibrational excitations presently requires tech-
r tages in a number of technologically-important fields,
t niques such as high-resolution EELS (HREELS),
m such as catalysis. However, its feasibility has been
whichuseslowincidentenergies(afeweV)inareflec-
. questioned due to the large degree of “inelastic delo-
t tion geometry [4, 5]. The superior energy resolution
a calization”, which measures the distance from which
of this technique can reveal vibrational excitations
m
a passing electron can induce an excitation [2]. For
as sharp peaks in the spectra, at energy losses cor-
- a 100 keV electron (typical for a STEM) and an en-
d responding to vibrational transitions. HREELS has
ergy loss of 10-100 meV, the delocalization length is
n proven to be extremely powerful in studies of adsor-
o estimatedtobe1-10µm, i.e.far greaterthanthesize bate molecules on surfaces, for example, where it has
c of an atom or molecule, which would imply that high
enabled fundamental advances in our understanding
[ spatial resolution is impossible.
of molecule-surface interactions. However, the spatial

1 In this Letter, we demonstrate that, in fact, atomic resolution of HREELS is limited, falling well short of
v
spatialresolutionofvibrationalexcitationsis permit- the atomic scale.
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ted by the scattering physics. We demonstrate this
0 The benefits of combining atomic-spatial and
3 via explicit calculations of vibrational-loss STEM im- vibrational-energy resolutions could potentially en-
6 agesofselectedmoleculesbasedonaquantumtheory
able fundamental advances in numerous fields. To
. of inelastic electron scattering. We show that, while
1 assess the spatial resolution afforded by the physics,
0 delocalization effects can be significant, they do not we have used a quantum theory of inelastic electron
4 necessarily preclude atomic spatial resolution. The
scatteringtocalculatevibrationalEELSimagesofse-
1 interpretation of the image contrast, however, can be
lected molecules. We assume a STEM-EELS geome-
:
v non-trivial. These results will be of central impor- try, whereby the incident electrons form a focused co-
i tance to the development of high spatial resolution
X herent beam, and the images are assumed to be gen-
vibrationalspectroscopyasananalyticaltechniquein
erated by extracting the vibrational signals in anal-
r
a the physical sciences. ogy with the description above. A 100 keV beam
Background.—Over the last decade, high spatial withaconvergencesemi-angleof30mradisassumed,
resolution electron energy-loss spectroscopy (EELS) as appropriate for a state-of-the-art STEM equipped
in the STEM has become an extremely powerful tool with aberration-corrected beam-forming optics. Such
for the analysis of materials. In this technique, an a beam is capable of 0.7 ˚A spatial resolution. How-
energy-loss spectrum is acquired for each position of ever, in addition to the scattering mechanism itself,
the electron beam. The signal from inelastic scatter- theactualspatialresolutionachievedisalsoinfluenced
ing processes of interest is extracted from each spec- by the spectrometer’s collection angle [6], and so we
trum(bysubtractinganyother“background”signals) have considered three collection angles in what fol-
andplottedasafunctionofbeampositiontoforman lows. Our consideration of the collection angle also
2
(a) (b) (c) (d)
mrad) 70)− 60)− 50)−
( 1× 1× 1×
hift y( y( y(
S t t t
hase ntensi ntensi ntensi
P I I I
Distance(˚A) BeamPosition(˚A) BeamPosition(˚A) BeamPosition(˚A)
FIG.1. CalculatedvibrationalEELSimagesofanH moleculewithitsmolecularaxislyingperpendiculartotheelectron
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beam: (a) Møller potential for excitation of the H stretching mode; (b–d) vibrational EELS images for detector semi-
2
angles 5, 30 and 80 mrad, respectively. Each subfigure shows a square area of size indicated by abscissa values with H
2
molecule at centre, while graphs show line traces along the molecular axis. Units in (a) correspond to the phase shift
that would be experienced by a 100 keV incident plane wave, and the square of this phase is a measure of scattering
probability. Images(b–d)assumeanaberation-free100keVbeamwitha0.7˚Acrossover,andtheintensityisnormalized
with respect to the incident beam intensity.
haspracticalimplications,inthatexperimentallythis “sees” when it excites the vibrational mode ν. Below
angle influences the spectrometer’s energy resolution. we refer to V (x,y) simply as “the Møller potential”.
ν
As targets we consider molecular H and CO. The Thewavefunctionofaninelasticallyscatteredelec-
2
simplicity and small size of these molecules allows us tron which excites mode ν is given by
to discuss spatial resolution in a straightforward con-
text, and they are exemplary of molecules with and ψν(x,y;x0,y0)=−iσVν(x,y)ψ0(x−x0,y−y0), (2)
without an electric dipole moment, which has impli-
cations that will be made clear below. Moreover, the where ψ0(x,y) is the wave function of the incident
conclusions we draw will have relevance to potential electron beam, (x0,y0) is the beam position, and σ is
applicationsofvibrationalEELSimagingincatalysis. an interaction constant [9]. The image intensity ob-
tainedforabeamposition(x ,y )isgivenbyintegrat-
We employ a theory of molecular vibrations based 0 0
ingtheinelasticintensityfallingwithinspectrometer’s
on quantum mechanics and the harmonic approxima-
entrance aperture in the far field:
tion. Weusethemolecularpropertiesaspredictedby
density functional theory under the pseudopotential
(cid:90) ∞ (cid:90) ∞
and generalized-gradient approximations [7]. Molec- I(x ,y )= du dvD(u,v)|ψ˜ (u,v;x ,y )|2,
0 0 ν 0 0
ular vibrational properties were computed using the −∞ −∞
(3)
finite-displacement method [8], assuming a tempera-
where ψ˜ (u,v;x ,y ) is the Fourier transform of
ture of 300◦K. ν 0 0
ψ (x,y;x ,y ), and D(u,v) is the detector function
A key quantity in our discussion is the two- ν 0 0
which equals unity for positions in the far-field inside
dimensional Møller potential for creating one addi-
the entrance aperture and equals zero otherwise.
tional quantum in vibrational mode ν, given by
Example 1: H molecule.—We consider an H
2 2
(cid:90) ∞ molecule, which we fix fictitiously in free space.
Vν(x,y)= dz(cid:104)nν +1|Vˆ(r)|nν(cid:105)e−i(ων/v)z This molecule has a bond length calculated to be
−∞ (1) 0.749 ˚A (experimental value is 0.750 ˚A), and a vibra-
=(cid:18)(cid:126)(nν +1)(cid:19)1/2(cid:88)eν(κ) ·∇ V(x,y), tional stretching mode of calculated energy (cid:126)ωstr. =
2ων κ m1κ/2 κ 529 meV (experiment: 517 meV [10]).
Fig. 1 shows calculated vibrational EELS images
where|n (cid:105)isavibrationalstatecontainingn quanta of an H molecule lying perpendicular to the electron
ν ν 2
inmodeν,Vˆ(r)istheCoulombinteractionenergyfor beam [11]. The Møller potential for excitation of the
an electron at position r, (cid:126)ω is the energy loss, the H stretchingmode(Fig.1a)isrelatedtoaspatialgra-
ν 2
z axis coincides with the beam direction, κ labels the dientoftheelectrostaticpotential(seeEqn.(1)). Asa
atoms, e (κ) is a polarization vector, m an atomic result, the Møller potential reverses sign at positions
ν κ
mass, and ∇ V(x,y) is the gradient of the projected which are very close to the equilibrium H positions
κ
electrostatic potential with respect to the equilibrium (where there is a peak in the electrostatic potential).
position of the nucleus of atom κ. Loosely speak- Crucialtoourassessmentofspatialresolution, wesee
ing, V (x,y) is the potential that an incident electron that, despiteadelocalizationlengthofv/ω ∼0.2µm,
ν
3
(a) (b) (c) (d)
mrad) 70)− 50)− 50)−
( 1× 1× 1×
hift y( y( y(
S t t t
hase ntensi ntensi ntensi
P I I I
Distance(˚A) BeamPosition(˚A) BeamPosition(˚A) BeamPosition(˚A)
(e) (f) (g) (h)
)
d
a y y y
mr sit sit sit
( n n n
hift Inte Inte Inte
S 0 0 0
ase og1 og1 og1
h l l l
P
Distance(˚A) BeamPosition(˚A) BeamPosition(˚A) BeamPosition(˚A)
FIG. 2. Effect of dipole scattering on vibrational EELS imaging of a CO molecule lying perpendicular to the electron
beam (C on the left): (a) impact potential for excitation of CO stretching mode (solid line, left ordinate) with dipole
potential overlaid (dashed line, right ordinate); (b–d) vibrational EELS images for detector semi-angles of 5, 30 and 80
mrad;(e)dipolepotentialshownoveralargerfieldofview;(f–h)vibrationalEELSimagesshownonalogarithmicscale
and over a larger field of view. Beam parameters match Fig. 1.
the Møller potential is contained within an area ap- sitioned immediately after the O-K onset.
proximately2˚A×2˚A.Thereforeweimmediatelyan- Example 2: CO molecule—Theinversionsymmetry
ticipate that the vibrational EELS images (Figs. 1b– of the H molecule considered in the previous exam-
2
d) should be capable of exhibiting atomic spatial res- ple precludes a very important consideration: A free
olution, as indeed they do: For a detector semi-angle H molecule has no electric dipole moment [14]. On
2
β = 5 mrad, the vibrational image contains a maxi- theotherhand,manymoleculespossesselectricdipole
mum between the H atoms which, in principle, could momentsarisingfromtheredistributionofchargethat
beusedtolocatethepositionoftheH2 moleculewith takesplaceduringbondformation. Excitationofavi-
a spatial precision of at least 1 ˚A. However, the im- brationalmodewillcausethemolecule’sdipolefieldto
age intensity is very weak and would be difficult to oscillate with period ω , giving rise to inelastic scat-
ν
observe in practice (reducing the convergence angle tering. Here, the most important point is that the
would increase the normalized intensity at the cost dipolefieldsarelong-ranged,extendingoverdistances
of spatial resolution). Moreover, this vibrational im- much larger than the molecules we are considering.
agealsocontainstwostrongsecondarymaxima(three Hence dipole scattering potentially precludes atomic
maximaintotal),aswellasnon-intuitiveminimaclose spatial resolution.
to the equilibrium atomic positions which result from We split the Møller potential into two parts: The
thezerosoftheMøllerpotential. Whenβ isincreased first part, called the “impact potential”, arises from
to 30 mrad, these minima weaken. For β = 80 mrad changes, due to vibrations, of the molecular charge
the H2 molecule appears as a more-intuitive “dumb- distribution that would result when electronic bond-
bell” shape with a shallow minimum at the centre. ing is “switched off”. The impact potential is derived
Such behaviour, where atomic-resolution STEM im- from the atomic potentials, and it decays exponen-
ages are easier to interpret if the detector angle is tially at large distances so that the scattering from it
larger than the convergence angle (here 30 mrad), is is inherently localized. The second part of the Møller
encountered in other STEM imaging modalities too, potential, called the “dipole potential”, arises from
for example, bright-field imaging [12] and core-level changes, due to vibrations, of the so-called “bonding
EELS imaging [13]. For β =80 mrad the peak inten- charge” [15]. The names of these potentials connect
sityiscomparabletothatobtainedincore-levelEELS with the existing literature on HREELS [4, 16]. How-
fromasingleoxygenatomusinga100eVwindowpo- ever,whileinconventionalHREELSisunnecessaryto
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considertheatomic-scalestructureofthetargetwhen (a) + Oscillatingelectron charge (b) +
calculating dipole scattering, here it is crucial.
WeconsideraCOmoleculefixedinfreespace. This − −
molecule has a calculated bond length of 1.142 ˚A (ex- + z + z
2πv
periment: 1.128 ˚A) and a single stretching mode of ω − x,y − x,y
calculated energy (cid:126)ω =265 meV (experiment: 267 + +
str.
meV [17]). The permanent electric dipole moment Target Target
− −
was calculated to be 0.143 Debye (experiment: 0.112
+ +
Debye[18]). WhilethepermanentdipoleofCOisrel-
atively small compared to other diatomic molecules, − d! 2πωv − d! 2ωπv
we find that its dynamic dipole (due to vibrations) is + +
comparabletothatofdiatomicmoleculeswithperma-
nent dipoles that are one order of magnitude larger. FIG. 3. The physical origin of spatial localization in in-
elastic electron scattering. (a) If the distance d of closest
Fig. 2 shows the effects of dipole scattering on the
approach to the target well exceeds the oscillation period
vibrational EELS images of a CO molecule lying per-
2πv/ω of the incident electron’s charge distribution then
pendicular to the electron beam. As in the H ex-
2 theMøllerinteractioncancels;(b)Ifthedistanceofclosest
ample above, the impact potential for excitation of approachiscomparabletoorsmallerthan2πv/ω thenan
the CO stretching mode (Fig. 2a solid line) has zeros interaction is possible.
at the equilibrium C and O positions. In contrast,
the dipole potential (Fig. 2a dashed line, and Fig. 2e)
has a large antisymmetric component. It is evident also diminish the desirable impact signal).
from Fig. 2e that the dipole potential extends far be- Note that this situation is complementary to
yond the molecule. However, its amplitude is much HREELS, which uses low energy electrons in a reflec-
smaller than the impact potential. In the vibrational tion geometry, and where dipole scattering is often
EELS images (Figs. 2b–d), the dipole scattering can selected by employing a small acceptance angle [4].
introduce considerable asymmetry. For β = 5 mrad, In this context, it is interesting to recall a statement
wherethedipolescatteringmakesasignificantoverall made over 30 years ago by Ibach and Mills: “...in
contribution, the effect is such that the CO molecule the impact scattering regime, the total excitation ef-
appears as a “dumbbell” displaced from its true po- ficiency (dS/dΩ) increases as the electron [incident]
sition. This effect can be interpreted as arising from energy increases. It thus would be most favourable to
the quantum interference of impact and dipole scat- study large-angle inelastic electron scattering at [in-
tering. The apparent displacement of the molecule cident] energies substantially larger than that used in
persists for β = 30 mrad. For β = 80 mrad, dipole present generation [low energy] experiments, if suit-
scattering produces only a minor effect, and now the ablespectrometerscouldbeconstructed.”[4]. Indeed,
“dumbbell” closely coincides with the molecule’s true our detailed calculations show that this idea is very
position. Crucially, the vibrational EELS images in favourable for achieving high spatial resolution.
Figs.2b–dexhibitatomicresolutiondespiteadelocal- It was noted earlier that only for a detector angle
izationlengthofv/ω ∼0.4µm. Figs.2f–gshowthevi- larger than the convergence angle is the image con-
brationalimagesonlogarithmicscaleandoveralarger trastinFigs.1and2intuitivelyinterpretableinterms
field of view in order to exhibit the “dipole tails”. of the molecule’s structure. We have also confirmed
Whenthebeamismoved20˚Afromthemolecule,the this behaviour for other atomic structures, including
image intensity is seen to drop by about 4 orders of solidsandadsorbatemoleculesonsurfaces(tobepub-
magnitude. lished elsewhere). In addition to better interpretabil-
Discussion.—The CO example demonstrates that, ity, a larger detector has the benefit of a stronger sig-
at high spatial resolution, the long-ranged nature of nal. Experimentally, on the other hand, larger de-
dipole scattering is counterbalanced by its small am- tector angles do place greater demands on the spec-
plitude: For small detector angles its contribution is trometer optics to achieve a given energy resolution,
comparable to impact scattering, whereas for larger sothatacompromisebetweeninterpretabilityanden-
detector angles impact scattering dominates. While ergy resolution is likely to be necessary in practice.
we do not claim that this must hold for absolutely Returning to the general question of delocaliza-
all targets, the strength of the CO dynamic dipole tion versus spatial resolution in the inelastic scat-
is, however, representative of a fairly large class of tering of high-energy electrons, considerable insight
molecules. Incaseswherethedipolescatteringiseven can be gained by considering the form of Møller po-
stronger, its effect could be circumvented by employ- tential. This potential represents the interaction be-
ing, for example, an annular detector that excludes tween the charge distributions of the scattering elec-
thedipolescatteringatlowangles(thoughthiswould tron and the transitioning target. In the case of the
5
electron, thechargedistributionincorporates themo-
mentum change along the direction of motion (the
consequences of the change in transverse momentum
[email protected]
are negligible for high-energy electrons). Hence the ∗
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[11] WeuseFFT-basedmultislicecalculationswithamax-
calling that in quantum field theory the Yukawa force imum scattering angle of 316 mrad. The impact po-
is “carried” by particles with mass, here, by analogy, tentials (defined later) are derived from the atomic
the carrier particles move in two-dimensions with a electronscatteringfactorsofRef.[? ].Elasticscatter-
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2 2
wherethereisanabsenceofadipolefieldatlargedis- included in Fig. 1, but its affect on spatial resolution
tances,isacaseinpoint. Evenwhenthetransitioning is minor so we omit its discussion.
[15] The dipole potentials were calculated by applying
targetdoesproduceadipolefieldatlargedistances,if
the Born-Oppenheimer approximation to all-electron
this field is sufficiently weak then the interaction can
bondingchargedensitypredictedbyDFT.Byourdef-
still be effectively more localized than v/ω, as in the
inition, the dipole potentials also contain the effects
CO example above. of quadrupole and higher-order moments.
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