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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: RMS - Royal Microscopical Society
- Sprache: Englisch
A detailed presentation of the physics of electron beam-specimen interactions
Electron microscopy is one of the most widely used characterisation techniques in materials science, physics, chemistry, and the life sciences. This book examines the interactions between the electron beam and the specimen, the fundamental starting point for all electron microscopy. Detailed explanations are provided to help reinforce understanding, and new topics at the forefront of current research are presented. It provides readers with a deeper knowledge of the subject, particularly if they intend to simulate electron beam-specimen interactions as part of their research projects. The book covers the vast majority of commonly used electron microscopy techniques. Some of the more advanced topics (annular bright field and dopant atom imaging, atomic resolution chemical analysis, band gap measurements) provide additional value, especially for readers who have access to advanced instrumentation, such as aberration-corrected and monochromated microscopes.
Electron Beam-Specimen Interactions and Simulation Methods in Microscopy offers enlightening coverage of: the Monte-Carlo Method; Multislice Simulations; Bloch Waves in Conventional and Analytical Transmission Electron Microscopy; Bloch Waves in Scanning Transmission Electron Microscopy; Low Energy Loss and Core Loss EELS. It also supplements each chapter with clear diagrams and provides appendices at the end of the book to assist with the pre-requisites.
- A detailed presentation of the physics of electron beam-specimen interactions
- Each chapter first discusses the background physics before moving onto simulation methods
- Uses computer programs to simulate electron beam-specimen interactions (presented in the form of case studies)
- Includes hot topics brought to light due to advances in instrumentation (particularly aberration-corrected and monochromated microscopes)
Electron Beam-Specimen Interactions and Simulation Methods in Microscopy benefits students undertaking higher education degrees, practicing electron microscopists who wish to learn more about their subject, and researchers who wish to obtain a deeper understanding of the subject matter for their own work.
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Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Series page
Title Page
Copyright
Preface
Chapter 1: Introduction
1.1 ORGANISATION AND SCOPE OF THE BOOK
REFERENCES
Chapter 2: The Monte Carlo Method
2.1 PHYSICAL BACKGROUND AND IMPLEMENTATION
2.2 SOME APPLICATIONS OF THE MONTE CARLO METHOD
2.3 FURTHER TOPICS IN MONTE CARLO SIMULATIONS
2.4 SUMMARY
REFERENCES
Chapter 3: Multislice Method
3.1 MATHEMATICAL TREATMENT OF THE MULTISLICE METHOD
3.2 APPLICATIONS OF MULTISLICE SIMULATIONS
3.3 FURTHER TOPICS IN MULTISLICE SIMULATION
3.4 SUMMARY
REFERENCES
Chapter 4: Bloch Waves
4.1 BASIC PRINCIPLES
4.2 APPLICATIONS OF BLOCH WAVE THEORY
4.3 FURTHER TOPICS IN BLOCH WAVES
4.4 SUMMARY
REFERENCES
Chapter 5: Single Electron Inelastic Scattering
5.1 FUNDAMENTALS OF INELASTIC SCATTERING
5.2 FINE STRUCTURE OF THE ELECTRON ENERGY LOSS SIGNAL
5.3 SUMMARY
REFERENCES
Chapter 6: Electrodynamic Theory of Inelastic Scattering
6.1 BULK AND SURFACE ENERGY LOSS
6.2 RADIATIVE PHENOMENA
6.3 SIMULATING LOW ENERGY LOSS EELS SPECTRA
6.4 SUMMARY
REFERENCES
Appendix A: The First Born Approximation and Atom Scattering Factor
References
Appendix B: Potential for an 'Infinite' Perfect Crystal
Appendix C: The Transition Matrix Element in the One Electron Approximation
Reference
Appendix D: Bulk Energy Loss in the Retarded Regime
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 (a) HAADF STEM image of a [110]-oriented, AlAs–GaAs superlattice. The background subtracted column intensity ratio values for all dumbbells in (a) are shown in (b) as a histogram. (c) plots experimental and multislice simulated values for the AlAs–GaAs interface width, as defined from the column intensity ratio profile, as a function of specimen thickness. Supercells for the multislice simulation are constructed assuming a linear diffusion model and a more accurate diffusion model (Moison
et al.,
1989) valid for the AlAs–GaAs system. Results are shown for superlattices grown on GaAs (labelled ‘AlAs-on-GaAs’) and AlAs (labelled ‘GaAs-on-AlAs’) substrates respectively. (a) and (b)
Figure 1.2 (a) Supercells used for simulating the EELS edge shape for fourfold and threefold coordinated silicon dopant atoms in graphene. For the latter, a planar structure and a distorted structure, where the silicon atom is displaced out of the graphene sheet, are assumed. (b) shows the corresponding EELS spectra, with the experimental measurement represented as a solid line and the simulated result superimposed as a filled spectrum.
Chapter 2: The Monte Carlo Method
Figure 2.1 Rutherford elastic scattering of an electron of speed
v
by a nucleus of charge +
Ze
. The impact parameter is
b
and the electron trajectory is described using (
r
,
ϕ
) polar coordinates.
θ
is the scattering angle.
Figure 2.2 Schematic illustrating the differential scattering cross-section. Electrons uniformly illuminate a target of unit area and thickness
t
. Solid circles represent individual atoms in the target. An electron incident within the thin annular ring centred about a given atom is scattered between the angles
θ
and (
θ
+
dθ
). The differential scattering cross-section
dσ
(
θ
) is the area of an annular ring and is proportional to the fraction of electron intensity scattered within the angular range
θ
to (
θ
+
dθ
).
Figure 2.3 The solid angle Ω is defined as the area subtended on a sphere of unit radius by a given angle. Consequently,
d
Ω is the shaded area in the figure, that is, the difference in subtended areas for the angles
θ
and (
θ
+
dθ
). The shaded area can be unfolded into a rectangular strip with length 2
π
sin
θ
and width
dθ
to give
d
Ω = (2
π
sin
θ
)
dθ
.
Figure 2.4 Collision parameters for inelastic electron–electron scattering. The incident electron is denoted by the black circle and the atomic electron is the grey circle.
Figure 2.5 Bethe continuous energy loss stopping power for electrons in a copper target, calculated using Eqs. (2.30) and (2.43), plotted as a function of incident electron energy.
Figure 2.6 Schematic illustrating the collision sequence assumed in a Monte Carlo simulation. The incident electron is first scattered at
P
N
with scattering angle
θ
and azimuthal angle
φ
, the angular deviations being defined with respect to the electron incident direction (downward vertical arrow). The scattered electron travels a distance
s
before being re-scattered at the point
P
N+1
.
Figure 2.7 Monte Carlo simulated trajectories for (a) 100 keV, (b) 400 keV electrons in Si and (c) 100 keV, (d) 400 keV electrons in Au. The target is a thin film of 100 nm thickness. For visual clarity only 500 electron trajectories are shown.
Figure 2.8 Monte Carlo simulated trajectories for (a) 5 keV, (b) 20 keV electrons in Si and (c) 5 keV, (d) 20 keV electrons in Au. The target is a bulk solid. For visual clarity, only 500 electron trajectories are shown. Note the change in scale between Figure (a), (c) and (b), (d).
Figure 2.9 (a) Variation of the backscattering coefficient with target atomic number for 20 keV electrons at normal incidence and (b) backscattering coefficient as a function of beam tilt angle for 20 keV electrons incident on Au (‘×’ symbols are Monte Carlo calculated values, while the other symbols represent experimental measurements).
Figure 2.10 Spatial variation of emitted backscattered electron intensity for 20 keV electrons incident on Au. In (a) the electron beam is normal to the target while in (b) the beam tilt angle is 45
°
. Figure (a) is symmetrical with respect to each quadrant, while Figure (b) is symmetrical only about the
xz
-plane.
Figure 2.11 (a) Ionisation cross-section for Cu K X-rays as a function of overpotential, calculated using Eq. (2.47). (b) and (c) are respectively the Monte Carlo simulated and experimental
φ
(
ρz
) curves for characteristic X-rays from tracer elements in a Ag matrix. The electron beam energy is 20 keV.
Figure 2.12 (a) Monte Carlo simulated electron–hole pair generation volume for a 15 keV electron beam in CdTe. The electron beam current is 1.6 nA and the carrier density is plotted on a logarithmic scale. (b) is the ‘quasi-steady state’ carrier distribution after numerically integrating for three lifetimes. The surface recombination velocity is 5 × 10
5
cm/s and the minority carrier diffusion length is 500 nm.
Figure 2.13 Schematic of a thought experiment used to define classical versus quantum mechanical scattering regimes. An opaque screen with a narrow slit is used to collimate a beam of electrons. If the electron is strictly a particle only a single ‘ray’ will be selected with a well-defined impact parameter of
b
. However, owing to the wave nature of the electron diffraction from the narrow slit can take place. The spreading of the Hüygen wavelets gives rise to an uncertainty Δ
b
in the impact parameter.
Figure 2.14 Effect of spin–orbit coupling on electron scattering by an atomic nucleus of charge +
Ze
. The electron is incident into the plane of the paper at a distance
r
from the atom nucleus. The direction of the angular momentum
L
for an electron passing the nucleus on the left- and right-hand sides is indicated. The potential energy due to spin–orbit coupling (dashed lines) either adds to or subtracts from the intrinsic Coulomb potential energy (solid line) depending on the relative orientation of the electron spin
S
with respect to
L
. This in turn modifies scattering. See text for further details.
Chapter 3: Multislice Method
Figure 3.1 Schematic illustration of the multislice technique for an incident plane wave. The specimen is divided into thin slices, such as, slices AB, BC, etc., which correspond to a single atom row. The electron beam is scattered by the atoms in a given slice before propagating through free space to the next slice. The electron exit wavefunction is calculated by repeated application of scattering and propagation for all slices within the specimen.
Figure 3.2 Unscattered (
0
) and ±
g
diffracted beam wavefunctions predicted by the multislice method. The incident wavefunction is a plane wave of unit amplitude and only the first two slices are shown. See text for details.
Figure 3.3 Schematic of the shortening of the de Broglie wavelength for an incident electron in a region of high potential, such as a single atom (depicted as a filled circle). The electron undergoes a phase shift compared to free space propagation. This is evident by comparing the relative positions of the first maximum in the electron wavefunction beyond the atom for the two cases (vertical dashed lines).
Figure 3.4 Elastic and thermal diffuse scattered intensity (TDS) of 200 kV electrons by a single silicon atom plotted as a function of scattering vector magnitude (
q
). The elastic scattered intensity increases rapidly at small
q
and extends beyond the intensity scale of the Figure The phase contrast transfer function for an objective lens of 1 mm spherical aberration coefficient at Scherzer defocus (−61 nm) is also superimposed (beam semi-convergence angle and defocus spread are 0.5 mrad and 6 nm respectively). The TDS intensity at spatial frequencies between the point resolution and information limit of the microscope is delocalised due to oscillations in the transfer function. The spatial frequency of the 111 zero order Laue zone reflection in silicon is also indicated.
Figure 3.5 Special case of the Fresnel–Kirchhoff integral formula used for deriving the propagator function. A (fictitious) circular aperture is placed at the specimen plane; the source
S
is symmetrically located about the aperture and at the origin of the spherical cap.
n
is the surface normal to the spherical cap at the general position (
x
′,
y
′,
z
′). Each position along the spherical cap can potentially contribute to the wavefunction beyond the aperture, such as, at the arbitrary point
O
(
x
,
y
,
z
), which is related to (
x
′,
y
′,
z
′) by the position vector
r
.
Figure 3.6 (a) Spreading of a Hüygen wavelet (solid arc) from the source
S′
in free space. The observation plane
PP′
is at a distance Δ
z
from
S′
. The phase of the wave at
O′
is different to that at
O
due to the additional optical path length
t
along
S′O′
compared to
S′O
. (b) Geometry of the deviation parameter (
s
) for the reciprocal vector
u
of scattering angle 2
θ
. For simplicity the specimen is assumed to be untilted, so that its surface normal, and hence
s
, is parallel to the incident wave vector (i.e. the line joining the Ewald sphere centre to the origin of the reciprocal lattice
O
). The equation of the Ewald sphere in the
xy
-plane is
x
2
+ [
y
− (1/
λ
)]
2
= (1/
λ
)
2
. This gives , which simplifies to
s
= −½
λu
2
for
λu
≪ 1 (by convention
s
is negative if
u
lies outside the Ewald sphere).
Figure 3.7 Distortion of the electron wavefunction due to lens aberrations, in this case spherical aberration. ‘Rays’ emitted at an angle
α
to the optic axis and transverse wave vector component
u
by a point
O
in the specimen converge to the point
I
for a perfect lens. In the ideal case, the electron wavefunction is a converging spherical wavefront (dashed arcs) and the ray paths are along the dashed lines. With spherical aberration the rays (solid lines) are brought to a premature focus. The image of
O
is therefore no longer a point, but blurred in the image plane. The wavefront (solid arcs) is distorted by a distance
W
compared to the ideal case.
Figure 3.8 Illustration of the column approximation used for calculating the kinematic electron diffraction intensity for a specimen of thickness
t
. Owing to small scattering angles the diffracted intensity from a high energy incident plane wave is confined to a narrow column centred about the diffracted beam observed at the exit surface point
O
. Scattering at different depths along the column, such as at the point
P
, will contribute to the net diffracted beam intensity observed at
O
.
Figure 3.9 (a) Projected potential of [110]-Si. A pair of dumbbell atom columns is circled for reference. The inset shows the multislice simulated diffraction pattern for a 50 nm thick specimen; the circle is the outline of the objective aperture used in HREM image simulations. (b) Simulated HREM image for a 3.8 Å thick specimen (i.e. a single AB stacking sequence) at −61 nm Scherzer defocus. (c) and (d) are HREM images for the same specimen but at −100 and −140 nm defocus respectively. (e) Variation of and 000 beam intensities as a function of depth. In (f) the variation in phase for the 000, and beams are plotted with respect to depth. The Scherzer defocus HREM images for 12.6, 5 and 20 nm thick specimens are shown in (g), (h) and (i) respectively. Traces of the () and () planes have been superimposed (solid lines) on the left-hand corner of each image. For all simulations the following parameters were assumed: 200 kV accelerating voltage, 1 mm spherical aberration coefficient, 0.5 mrad beam semi-convergence angle, 6 nm defocus spread and 8.5 mrad objective aperture radius. Absorption is not included. The supercell structure and scale bar in (a) are common to all HREM images.
Figure 3.10 (a) Phase image of the [112]-Si exit wave reconstructed from a tilt-focus series. The dumbbell atom columns, separated by only 0.78 Å, are clearly resolved.
Figure 3.11 (a) Multislice simulated CBED pattern for a 50 nm thick, [112]-Si specimen at 120 kV and 4 mrad probe semi-convergence angle. Atomic vibrations are not included. (b) The CBED pattern obtained from a single frozen phonon configuration and (c) after averaging over 20 frozen phonon configurations. The Kikuchi pattern is clearly visible in (c), including the [223] minor zone-axis to the left of the ‘image’. The sharp cut-off in intensity at large spatial frequencies is due to bandwidth limiting in the simulation to avoid aliasing artefacts. All CBED patterns are plotted on a logarithmic intensity scale.
Figure 3.12 (a) Pendellösung plots for a STEM probe incident on an atom column of [110]-Si determined from multislice simulations with and without atomic vibrations (i.e. with and without TDS). The pendellösung intensity was calculated along the atom column on which the probe is incident. The probe parameters were 200 keV energy, 20 mrad semi-convergence angle and zero electron-optic aberrations. Twenty frozen phonon configurations were averaged for the pendellösung plot with atomic vibrations. (b) and (c) are the HAADF intensity profiles across the dumbbell atom columns, calculated with and without TDS, for 10 and 50 nm thick specimens respectively. Note the change in intensity scale between the two figures.
Chapter 4: Bloch Waves
Figure 4.1 Intensity (
I
) of Bloch states (a) 1 and (b) 2 as a function of position
x
for two-beam conditions. In the former the electrons channel along the diffracting crystal planes, while for the latter the channelling is between the planes.
Figure 4.2 Dispersion surface diagram for two-beam diffracting conditions (not drawn to scale).
k
mean
is the incident beam wave vector after correcting for the mean inner potential.
k
(1)
and
k
(2)
are the wave vectors for the two Bloch waves; note that the wave vector component along the specimen surface normal
n
is altered by
γ
(1)
and
γ
(2)
respectively. The Bloch wave dispersion surfaces are highlighted by the solid lines, while the two circles of radius
k
mean
centred about the origin
O
and reciprocal lattice point
G
are indicated by the dashed lines. The diffracted beam wave vector is along
k
D
and has deviation parameter
s
g
.
Figure 4.3
g
= ±220 (a) bright-field and (b) dark-field images of silicon showing thickness pendellösung fringes. (c) and (d) are
g
= 220 two-beam, convergent beam electron diffraction patterns from approximately 90 and 215 nm thick (estimated using electron energy loss spectroscopy) regions of silicon respectively. Positive and negative deviation parameters (
s
g
) are indicated. The electron beam energy is 200 keV.
Figure 4.4
g
= 220, bright-field and dark-field rocking beam patterns in 200 nm thick silicon, calculated using Eqs. (4.34) and (4.35). A ratio was assumed. In (a) the intensity of the transmitted beam (
I
0
) is plotted as a function of the dimensionless deviation parameter
w
(=
s
g
ξ
g
), while in (b) the intensity is that of the diffracted beam (
I
g
). The intensities are normalised, so that the total intensity incident on the sample is unity. The electron beam energy is 200 keV.
Figure 4.5 (a) Scattering geometry used for calculating the TDS intensity distribution.
k
I
and
k
D
are the incident and Bragg diffracted beam wave vectors superimposed on the Ewald sphere. The TDS wave with wave vector
k
S
is due to TDS scattering of the incident beam along the vector
q
. The diffracted beam can also contribute via scattering along (
q
−
g
). (b) shows the scattering geometry used for calculating the total TDS intensity. Integration is carried out over all scattering vectors
q
in three-dimensional reciprocal space.
Figure 4.6 (a)–(g) show the intensity distribution of the first seven Bloch states in [111]-Mo at 300 kV normal incidence. The atom column positions are indicated by the open circles. The inset in each Figure lists the real and imaginary parts of
γ
for the Bloch state.
Figure 4.7 (a)–(f) show the intensity distribution of the first six Bloch states in [110]-Si at 300 kV normal incidence. The crosses in each Figure represent the atom column positions of the central dumbbell. The symmetries of the Bloch states are 1s, 1s*, 2p
x
, 2p
y
, 2s and 2p
y
* respectively. The insets in (c), (d) and (f) indicate the orientation and polarity of the overlapping 2p atom column orbitals. (g) is a schematic of the energy (
E
) levels of two atomic orbitals (AO1 and AO2) and the molecular orbitals (MO) created by their overlap.
Figure 4.8 (a) and (b) show the As 1s and Ga 1s Bloch states for [110]-GaAs at 300 kV normal incidence. The corresponding Bloch states for [112]-GaAs are shown in (c) and (d) respectively. The atom column positions in the central dumbbell are indicated by the crosses in each Figure (As is the atom column to the right).
Figure 4.9 Scattering geometry used for calculating the hollow cone dark-field intensity and coherence volume. The incident wave vector
k
, with polar angle
α
and azimuthal angle
φ
, is scattered by an atom along the wave vector
k′
. The scattering lies within the objective aperture. See text for further details.
Figure 4.10 Coherence volume plots for HAADF detector inner angles of 50 mrad (a and b) and 100 mrad (c and d) in silicon. The detector outer angle is fixed at 200 mrad. Thermal vibrations are included in (b) and (d), but not (a) and (c). In (a) the atomic structure of silicon along [001], with [110] parallel to the (vertical) optic axis, is superimposed as a guide to the size of the coherence volume. The electron beam energy is 100 keV.
Figure 4.11 Pendellösung plots for a 300 kV, aberration-free STEM probe in [110]-Si with (a) 10 mrad and (b) 30 mrad probe semi-convergence angle. The probe is incident on a dumbbell atom column and the intensity is calculated along the same column as a function of depth. The dashed and solid lines represent results with and without TDS. Note the difference in scale (intensity and depth) for the two Figure (c) and (d) respectively show the excitation at 300 kV of the 1s Bloch state and Bloch state 50 in [110]-Si as a function of the incident plane wave transverse wave vector component (expressed as an angle in mrad units). The horizontal and vertical axes are along the 002 and reciprocal vectors respectively. (e) and (f) are two-dimensional cross-sections of STEM probe propagation in [110]-Si. The contrast is inverted for visual clarity, so that black represents regions of high electron intensity. The horizontal axis measures distance along [001], with the dumbbell atom columns at ±0.7 Å, while the vertical depth axis is along [110]. In (e) the probe is incident on the dumbbell atom column at −0.7 Å, while in (f) it is incident at the dumbbell centre (0 Å). The probe parameters are 300 kV, 10 mrad semi-convergence angle and zero electron optic aberrations.
Figure 4.12 (a) Schematic of the true geometry for which the ‘column’ approximation holds under two-beam conditions. The bright-field, dark-field intensity at the specimen exit surface point P is governed by scattering within the triangle APB, which has sides parallel to the incident (
k
I
) and diffracted (
k
D
) beam wave vectors. (b) Schematic of an inclined stacking fault within a crystal.
n
1
and
n
2
are the surface normals for the specimen and stacking fault respectively. (c) Illustration of Bloch wave scattering by the stacking fault in (b) under two-beam conditions. Solid lines represent the Bloch wave dispersion surfaces, while the dashed lines are circles of radii
k
mean
(compare with Figure 4.2). The Bloch waves excited at the specimen entrance surface are given by the points A and B. At the stacking fault the two Bloch waves are scattered to points C and D respectively.
Figure 4.13 Change in STEM probe intensity (with respect to a perfect crystal) as a function of depth for [111]-Fe containing a single substitutional atom of W. Curves for the W atom at depths of 18, 46 and 66 Å are shown. The STEM probe is incident on the atom column containing the W atom, and the intensity is calculated along the same column. Simulations are for a 300 kV aberration-free probe with 20 mrad semi-convergence angle.
Figure 4.14 (a) Schematic of the detection geometry used in ABF imaging. An annular detector is placed within the bright-field disc in STEM mode. (b) An ABF image of [010]-oriented YH
2
. The insets show the projected crystal structure (purple and green atoms represent yttrium and hydrogen respectively) as well as the multislice simulated ABF image. The image was acquired using an aberration corrected STEM operating at 200 kV with 22 mrad probe semi-convergence angle. The ABF detector inner and outer angles are 11 and 22 mrad respectively.
Figure 4.15 SEM backscattered images of (a) a polycrystalline CdTe thin film.
Figure 4.16 (a) Schematic of the scattering geometry assumed for calculating the ECCI signal intensity. See text for further details. (b) illustrates the backscattered electron (BSE) intensity expected under two-beam conditions for an edge dislocation lying parallel, and in close proximity, to the specimen entrance surface. The electron beam is incident at the Bragg angle within the undistorted regions of the crystal.
Chapter 5: Single Electron Inelastic Scattering
Figure 5.1 Schematic diagram illustrating the inelastic scattering vector
q
and its relation to the incident (
k
0
n
0
) and scattered wave vectors.
θ
E
is the characteristic scattering angle. See text for further details.
Figure 5.2 Two-dimensional electron density plots for the (1s + 2p
0
) and (1s + 2p
1
) coherent states at quarter period (
T
) time intervals are shown in (a) and (b) respectively. The axis of quantisation for the atom is parallel to the vertical axis in (a) and along the viewing direction in (b). The (1s + 2p
0
) state is an oscillating linear dipole, while (1s + 2p
1
) is a rotating dipole.
Figure 5.3 (a) Generalised oscillator strength (
f
nl
) for a hydrogen atom plotted as a function of momentum transfer for transitions to the second Bohr orbit. The momentum change is plotted as ln(
qa
o
)
2
, where
q
is the magnitude of the inelastic scattering vector and
a
o
is the Bohr radius. The total and individual final state oscillator strengths are indicated along with the ‘optical’ (i.e.
q
→ 0) value denoted by ‘opt’. The oscillator strength
df
/
dE
for ionisation is shown in (b) for a range of energy losses
E
expressed in Rydberg energy ‘
R
’ units; (c) is the Bethe surface for hydrogen.
Figure 5.4 EELS spectra for graphite measured for different values of the scattering angle
θ
. The graphite basal planes are oriented perpendicular to the electron beam.
Figure 5.5 (a) Inelastic scattering from an atom (grey circle) due to two incident plane waves with wave vectors
k
1
n
1
and
k
2
n
2
. The two waves give rise to an interference intensity pattern at the atom, so that the inelastic scattering consists of two direct Lorentzian contributions and an interference contribution. (b) Inelastic scattering in a STEM geometry; see text for details.
Figure 5.6 200 kV, aberration-free STEM EELS intensity profiles calculated for an individual Si atom. (a) and (b) correspond to the Si L
2,3
and K-edges for a 20 mrad STEM probe semi-convergence angle and point detector. The Si atom is at the origin and the parallel and perpendicular contributions to the total intensity (i.e.
A
‖
and
A
⊥
terms in Eq. (5.44)) are indicated. In (a) the curve for the total intensity largely overlaps with the perpendicular contribution. (c) shows the intensity profile for the Si L
2,3
-edge with the detector collection semi-angle increased to 40 mrad, but keeping all other parameters unchanged. (d) is the intensity profile for the Si K-edge with 10 mrad probe convergence semi-angle and point detector.
Figure 5.7 Projected potential 1 eV above the Si L
2,3
-edge onset for transition to the (a)
l
= 0,
m
= 0 and (b)
l
= 3,
m
= 0 final states, where
l
,
m
are the orbital and magnetic quantum numbers respectively. The former is a dipole transition, while the latter is a quadrupole transition. The incident electron energy is 100 keV. The potential is plotted over a 16 Å × 16 Å area with the silicon atom in the centre.
Figure 5.8 (a) HAADF image of [0001]-Si
3
N
4
. The summed EELS spectrum extracted from the spectrum imaging area in (a) is shown in (b). The background subtracted Si L
2,3
intensity maps in Figure (c) to (f) are extracted from the corresponding energy windows labelled in (b). Figure (g) is the background subtracted N K-edge intensity.
Figure 5.9 (a) HAADF image of a 91 nm thick, [110]-Si specimen acquired at 100 kV beam energy. The background subtracted Si L
2,3
intensity maps extracted from a 20 eV wide energy window centred about 153 and 290 eV are shown in (b) and (c) respectively. The HAADF and EELS data were acquired simultaneously from the same spectrum image. For each ‘image’ the atomic structure is overlaid along with the simulation result.
Figure 5.10 (a) Ti L
2,3
EFTEM map acquired from a 30 nm thick, [110]-SrTiO
3
specimen using a
C
s-
and
C
c
-corrected microscope operating at 200 kV. The underlying atomic structure and simulation result are overlaid on the experimental data. Intensity profiles for the Ti–O and Sr/O row of atoms are shown in (b) and (c) respectively. The solid black line and red dashed line are extracted from the experimental and simulated images respectively.
Figure 5.11 (a) Si L
2,3
-edge photoabsorption spectrum (solid line) and simulated result for a silicon atom (dashed line). (b) Generalised oscillator strength for different values of the Si L
2,3
energy loss plotted as a function of the magnitude of the inelastic scattering vector
q
. The energy loss at the Si L
2,3
-edge onset is arbitrarily assigned a zero value.
Figure 5.12 (a) Potential scattering pathways for an outgoing photoelectron that is redirected towards the emitting atom (grey circle). The neighbouring atom labelled ‘A’ is involved in single scattering, while atoms ‘B’ and ‘C’ give rise to multiple scattering. (b) The inelastic mean free path for a photoelectron plotted as a function of its kinetic energy; note the log scale for both axes. The graph is a ‘universal curve’ based on measurements for several different elements.
Figure 5.13 C K-edge in TiC simulated with and without a core hole, using the FEFF multiple scattering code. The results are plotted on an arbitrary energy scale. The core hole was modelled using the
Z
* approximation, such that the electronic configuration of the excited C atom is 1s
1
2s
2
2p
3
. The core hole increases the relative intensity of the peak labelled ‘1’ at the edge onset, as well as shifts it to lower energy loss.
Figure 5.14 (a) Experimental set-up for electron magnetic circular dichroism (EMCD) measurements. The unscattered and Bragg diffracted beams in the diffraction plane are denoted by
O
and
G
respectively. An EELS detector at the ‘+’ or ‘−’ positions has an inelastic scattering vector that is equivalent to circular polarised X-rays of opposite helicity. (b) EMCD data for the Fe L
2,3
-edge measured from a pure iron specimen. The EELS spectra recorded at the ‘+’ and ‘−’ detector positions are shown along with the difference spectrum. The 002 reflection is used for the Bragg diffracted beam.
Chapter 6: Electrodynamic Theory of Inelastic Scattering
Figure 6.1 (a) Calculated energy loss function Im[−1/
ϵ
(
ω
)] for a solid containing bound electrons with 10 eV resonant energy. The plasmon energy of the free electrons (0.9 oscillator strength) is 15 eV, while the damping energy for all electrons is 1 eV. The dielectric function
ϵ
(
ω
) is modelled using Eq. (6.25). (b) Plots of the real (
ϵ
1
) and imaginary (
ϵ
2
) parts of the dielectric function as a function of energy for the solid in (a).
Figure 6.2 (a) Ion displacements in acoustic (in-phase) and optical (out-of-phase) transverse phonon modes. The positive and negative ions are depicted by black and grey circles. (b)
ω–k
dispersion plots for the acoustic and optical phonon modes. The wave number
k
is expressed as a fraction of the first Brillouin zone. (c) Real (
ϵ
1
) and imaginary (
ϵ
2
) parts of the dielectric function
ϵ
(
ω
) given by Eq. (6.31), with
ħω
T
= 41.25 meV,
ħω
L
= 45.40 meV and
γ
= 10
11
s
−1
. The energy loss function Im[−1/
ϵ
(
ω
)] is shown in (d).
Figure 6.3 (a) Schematic of an electron travelling parallel to an interface between two materials ‘A’ and ‘B’. The interface is arbitrarily positioned at
x
= 0 and the impact parameter of the electron, which is travelling in material ‘B’ with speed
v
, is
x
0
. An image charge is ‘induced’ on the opposite side of the interface. (b) Plot of the modified Bessel function
K
0
(
u
) with respect to the dimensionless variable
u
. The inset shows ln[
K
0
(
u
)] versus
u
. (c) Shows the measured EELS signal for different energy losses of a 60 kV electron probe as a function of distance from a graphene sheet edge. The EELS signal has been normalised for direct comparison. The normalised annular dark field (ADF) signal is also shown superimposed.
Figure 6.4 (a) Charge density and electric field lines for the surface plasmon at the interface between a semi-infinite solid and vacuum. Regions of negative charge are displayed as filled strips while positive charge regions are displayed as open strips. Only the electric field lines within the solid are shown. (b) and (c) are the equivalent diagrams for the symmetric and anti-symmetric surface plasmon modes in a thin-foil respectively. (d) Intensity of 1.75, 2.70 and 3.20 eV surface plasmon modes as excited by a 100 kV STEM electron probe scanned over a triangular shaped silver nano-prism. The specimen boundary is indicated in the figure.
Figure 6.5 Schematic of Cerenkov radiation emitted by an electron travelling faster than the speed of light in a given medium. From the origin
O
the electron moves a distance
vt
to the new point
P
, where
v
is the electron speed and
t
is the time elapsed. Cerenkov radiation is emitted at an angle
θ
c
. Radiation emitted from the origin
O
travels a distance
ct
/
n
(
ω
), where
c
is the speed of light in vacuum and
n
(
ω
) is the refractive index.
Figure 6.6 (a) Experimentally measured
ω
–
q
⊥
diagram for GaN, where
ħω
is the EELS energy loss and
q
⊥
is the transverse momentum change of the incident electron expressed as a scattering angle. The feature at 19.6 eV corresponds to the plasmon energy (labelled
E
p
). The three arrows at the bottom indicate the locations where individual EELS spectra were extracted; these correspond to 0, 20 and 60 μrad scattering angle respectively and are shown superimposed in (b).
Figure 6.7 (a) Polar diagram representation of the emission pattern for transition radiation at a vacuum–metal boundary. The distance of the butterfly wing-shaped contour from the origin is proportional to the intensity emitted along that particular direction. (b) Radiation generated by an incident electron traversing a thin-foil specimen. ‘P1’ and ‘P3’ represent transition radiation generated at the electron entrance and exit surfaces respectively, while ‘P2’ represents any Cerenkov radiation generated from within the specimen. It is assumed that the radiation can undergo both reflection and transmission at the vacuum–specimen boundaries.
Figure 6.8 TEM-cathodoluminescence (CL) spectra measured in the backward emission direction (i.e. anti-parallel to the incident electron direction) for mica of thickness 290, 600, 1030 and 1150 nm respectively at 200 keV beam energy.
Figure 6.9 (a) TEM-cathodoluminescence (CL) spectrum measured at 200 kV from a ∼280 nm thick CdTe specimen. The peak at ∼820 nm is due to electron–hole pair recombination and the broad peak centred at ∼600 nm is due to transition radiation. (b) and (c) show intensity maps extracted from a CL spectrum image over the wavelength windows 760–850 nm and 400–700 nm respectively. Grain boundaries are visible in the former, as indicated by the arrows. The anomalous intensity in some of the pixels in (b) and (c) are due to light collection artefacts.
Figure 6.10 Excitation amplitudes for three surface plasmon modes at 1.9, 2.9 and 3.4 eV in a triangular silver nano-prism as simulated by the boundary element method. The specimen boundary is indicated in the Figure The results should be compared with the experimental measurements shown in Figure 6.4d (the nano-prism dimensions are identical for both simulation and experiment).
Appendix A: The First Born Approximation and Atom Scattering Factor
Figure A.1 Elastic scattering of an incident electron of wave number
k
by an atom (filled circle).
n
0
and
n
1
are unit vectors in the direction of incidence and scattering (i.e.
r
) respectively. The scattering angle is 2
θ
, and
q
is the scattering vector.
List of Tables
Chapter 1: Introduction
Table 1.1 The simulation methods discussed in this book and some of their advantages and disadvantages
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Preface
In writing this book, I have attempted to introduce electron beam scattering in the context of simulation methods, since the practicing microscopist is usually concerned with the latter. Despite the availability of user-friendly software, some understanding of the underlying physics is essential, both to ‘optimise’ the simulation and recognise its limitations and to correctly interpret and/or generalise the results. In highlighting applications, I have selected examples that have been made possible by the tremendous advances in instrumentation, such as aberration correction and monochromation. The examples are limited to those of general interest, that is, mainly imaging and spectroscopy. The field is, however, progressing rapidly and it is only a matter of time before electron beam simulation methods proliferate into new areas, such as time-resolved microscopy, in situ microscopy in gaseous and liquid environments and unconventional probes geometries in the form of vortex beams.
The book is intended to be as self-contained as possible, with derivations provided for the main results. The level of mathematics and physics assumed is largely limited to graduate level calculus and quantum mechanics. Certain topics, such as Maxwell's equations, may need refreshing, and in such cases I have referenced some of the many excellent textbooks that are widely available. It is also assumed that the reader has some familiarity with electron microscopes and basic techniques, such as HREM, HAADF imaging and EDX, EELS spectroscopy.
The interactions I have had over the years with colleagues and students have helped shape my own understanding of the subject. I am also grateful to Ian Jones, Mervyn Shannon, Rik Brydson, Alan Craven and Tom Lancaster for providing helpful comments on individual chapters. Finally, I would like to dedicate this effort to my parents, who have unconditionally supported me in my formative years.
Budhika G. Mendis Durham
Introduction
Electron beam scattering has had a long and distinguished history. Some of the essential physics was investigated even before the first electron microscope was built. The unsuspecting reader may find it surprising to come across familiar names such as Bethe, Bohr, Rutherford, Fermi and Mott in this book. While electron beam scattering is a mature theory its widespread use in electron microscopy measurements is arguably a more recent phenomenon. This is primarily due to two reasons. The first is the processing speed of modern computers; even a standard desktop computer can now produce useful results within a reasonable time and thankfully there are many software packages that take advantage of this. The second reason is the emergence of a new generation of electron microscopes that can resolve atom columns that are less than an angstrom apart, that have ∼10 meV energy resolution or less for measuring vibronic modes and that can record events separated in time by femtoseconds. With such a wealth of new information there is a strong emphasis on extracting quantitative information about the sample. Electron beam scattering calculations are often indispensable for correct data interpretation.
Two examples help illustrate the advantages of combining experimental results with simulation. The first is using high angle annular dark field (HAADF) imaging in a scanning transmission electron microscope (STEM) to characterise the interface in an AlAs–GaAs superlattice (Robb and Craven, 2008; Robb et al., 2012). Figure 1.1a shows the HAADF image of a [110]-oriented, epitaxial AlAs–GaAs superlattice acquired using an aberration corrected STEM. In this orientation the Group III–V elements are distributed as closely spaced (i.e. ∼1.4 Å) atom column pairs or ‘dumbbells’. The HAADF signal increases monotonically with the atomic number of the scattering element, so that using AlAs dumbbells as an example, the intensity of an As column is larger than that of Al. Figure 1.1b is a histogram of the background subtracted column intensity ratio values for all dumbbells in Figure 1.1a. The two prominent peaks are due to dumbbells in ‘bulk’ AlAs and ‘bulk’ GaAs respectively. However, there are also intermediate values for the column intensity ratio (arrowed region in Figure 1.1b) and further analysis reveals these to be due to dumbbells located at the AlAs–GaAs interface region (Robb and Craven, 2008). An interface ‘width’ can be defined based on the 5–95% variation in column intensity ratio across the interface. The interface width is found to be independent of the specimen thickness for superlattices grown on an AlAs substrate, but not on GaAs substrate.
Figure 1.1 (a) HAADF STEM image of a [110]-oriented, AlAs–GaAs superlattice. The background subtracted column intensity ratio values for all dumbbells in (a) are shown in (b) as a histogram. (c) plots experimental and multislice simulated values for the AlAs–GaAs interface width, as defined from the column intensity ratio profile, as a function of specimen thickness. Supercells for the multislice simulation are constructed assuming a linear diffusion model and a more accurate diffusion model (Moison et al., 1989) valid for the AlAs–GaAs system. Results are shown for superlattices grown on GaAs (labelled ‘AlAs-on-GaAs’) and AlAs (labelled ‘GaAs-on-AlAs’) substrates respectively. (a) and (b)
From Robb and Craven (2008). Reproduced with permission; copyright Elsevier. (c) From Robb et al. (2012). Reproduced with permission; copyright Elsevier.
It is not clear if the interface width is due to chemical inter-diffusion, electron beam spreading within the sample or interfacial roughness. This can, however, be tested by constructing supercells representing the different scenarios and performing multislice simulations (Chapter 3). Figure 1.1c shows the simulated results for chemical diffusion. The interface width, as deduced from the column intensity ratio values, is plotted as a function of specimen thickness for a linear composition profile and a more realistic diffusion model valid for the AlAs–GaAs system (Moison et al., 1989). The latter accurately reproduces the experimental results, suggesting diffusion as a likely candidate. In fact, simulations for a saw tooth-shaped and smooth interface did not agree with experiment, so that interfacial roughness and beam spreading have only a secondary effect on the measurement (Robb et al., 2012).
The second example is the use of electron energy loss spectroscopy (EELS) to extract the local electronic density of states for a silicon dopant atom in graphene (Ramasse et al., 2013). As illustrated in Figure 1.2a, the silicon atom can be incorporated either through direct substitution (i.e. threefold coordination) or as a fourfold coordinated atom in defect regions of the graphene sheet. The dopant atom can be readily identified using HAADF imaging in an aberration corrected STEM, taking advantage of the higher atomic number of silicon compared to carbon. The solid lines in Figure 1.2b are the Si L2,3-EELS edges measured from the two different dopant atom configurations. Owing to the nature of inelastic scattering (Chapter 5) the shape of the EELS spectrum is governed by the angular momentum resolved unoccupied density of electronic states. The filled spectra in Figure 1.2b are the results obtained from density functional theory simulation. There is excellent agreement between theory and experiment for the fourfold coordinated atom. For the threefold coordinated atom, however, accurate results are only obtained if it is assumed that the silicon dopant atom is displaced out of the graphene sheet (Figure 1.2a). This can be justified by the slightly longer Si–C bond length compared to graphene (note that the structure was relaxed to its lowest energy configuration prior to EELS simulation; Ramasse et al., 2013). The out-of-plane displacement of the silicon atom is not evident in the HAADF image and was only revealed through a careful quantitative analysis of the EELS result with the aid of simulation.
Figure 1.2 (a) Supercells used for simulating the EELS edge shape for fourfold and threefold coordinated silicon dopant atoms in graphene. For the latter, a planar structure and a distorted structure, where the silicon atom is displaced out of the graphene sheet, are assumed. (b) shows the corresponding EELS spectra, with the experimental measurement represented as a solid line and the simulated result superimposed as a filled spectrum.
From Ramasse et al. (2013). Reproduced with permission; copyright American Chemical Society.
1.1 ORGANISATION AND SCOPE OF THE BOOK
There are many ways to simulate electron beam scattering. Although the fundamental physics is unchanged, there are differences in the manner in which it is implemented and consequently the information that can be extracted. For example, if the interest is in images formed from high energy electrons passing through a thin foil, such as in transmission electron microscopy (TEM), then the strongest signal will be due to elastically scattered electrons. The less probable inelastic scattering events can be treated phenomenologically or in certain cases (e.g. a single graphene sheet) ignored altogether. This approach considerably simplifies and speeds up the calculation while still providing the required information.
Four different simulation methods are discussed in this book, namely Monte Carlo (Chapter 2), multislice (Chapter 3), Bloch waves (Chapter 4) and electrodynamic theory (Chapter 6). Chapter 5 deals with inelastic scattering of core atomic electrons and extends the multislice, Bloch wave methods to include simulation of inelastic images in the form of chemical maps. Together these form a core body of techniques for analysing a large range of electron microscopy data. Electronic structure calculations, based on either density functional theory or multiple scattering, are also widely used for simulating the fine structure of EELS spectra, but are not discussed here in any great detail. This is a vast area separate from the main topic of this book and the interested reader should consult textbooks such as Martin (2004) for further details. Table 1.1 lists some of the advantages and disadvantages of each of the simulation techniques. It should give some indication of which technique to use for a given problem and which techniques to avoid.
Table 1.1 The simulation methods discussed in this book and some of their advantages and disadvantages
Simulation method
Advantages
Disadvantages
Monte Carlo (probabilistic scattering of particles, e.g. incident electrons)
Both elastic and inelastic scattering are readily incorporated
Applicable for a wide range of specimen geometries (SEM
a
and TEM)
Large range of signals can be simulated (e.g. images, electron beam induced current, X-ray generation, etc.)
Cannot reproduce channelling or diffraction in a crystal
Inelastic scattering is often modelled as a continuous energy loss (i.e. stopping power). Hence ‘straggling’ is not observed
Multislice(physical optics approach based on transmission and propagation of the incident electron wave)
Used in both TEM and STEM image simulations
Channelling and dynamic diffraction in a crystal are reproduced
Supercells can be constructed, such as defects, amorphous materials, etc.
Most software packages only simulate elastic scattering, with thermal diffuse scattering modelled as a pseudo-elastic scattering event (i.e. frozen phonon). Core level inelastic scattering can be included, but at the expense of computing time
Computing time increases with thickness of the specimen
Can be difficult to interpret the underlying scattering mechanisms
Bloch wave(based on Schrödinger's equation for an electron in a periodic potential)
Used in both TEM and STEM image simulations
Channelling and dynamic diffraction in a crystal are reproduced
Intuitive description of dynamic diffraction
Computing time does not increase with specimen thickness (unless information at different depths is required)
Most software packages only simulate elastic scattering, with thermal diffuse scattering modelled phenomenologically via an ‘optical potential’. Core level inelastic scattering can be included, but at the expense of computing time
Only useful as a computational technique for periodic crystals with small unit cells. Column approximation and/or perturbation methods can nevertheless be used to obtain useful information about defect crystals
Electrodynamic theory(based on Maxwell's equations)
Vastly simplifies inelastic scattering events involving many electrons, for example, plasmons
Only the specimen dielectric function is required to calculate the energy loss
Analytical solutions are available for relatively simple specimen geometries (e.g. thin films, spheres, etc.)
Radiative phenomena, such as Cerenkov and transition radiation, can also be analysed
Only valid when the energy loss is a negligible fraction of the incident electron energy
Simulation methods exist for arbitrary specimen geometries, but can be computationally intensive
a Scanning electron microscope.
Finally, it should be noted that it is impossible to give an exhaustive treatment of electron beam scattering in a book of this size. Instead, the emphasis is on describing the essential physics, so that the reader is able to comprehend a large part of the literature and develop an understanding of simulation packages beyond a mere ‘black box’. As for the vast literature on the subject, a word of caution is appropriate: unfortunately, there is no universally accepted notation in scattering theory. In this book, I have tried to be as consistent as possible, following the procedure outlined below:
i.
The relativistic mass of the high energy electron is distinguished from its rest mass by using
m
for the former and
m
o
for the latter. There are, however, several examples in the text where the kinetic energy is expressed as ½
mv
2
, rather than the correct relativistic formula (
γ
− 1)
m
o
c
2
, where
v
is the speed of the electron,
c
the speed of light and
γ
= [1 − (
v
/
c
)
2
]
−½
. This is standard practice in most of the literature, although for a 300 keV electron beam the fractional error is as large as 18%. The magnitude of the momentum
p
=
mv
