detecting strain in scanning transmission electron...
TRANSCRIPT
Detecting strain in scanning transmission electron
microscope images
Part II Thesis
Ina M. Sørensen
Mansfield College, Oxford
10th of June, 2015
1
Abstract
Scanning Transmission Electron Microscopes (STEM) can produce directly interpretable images
with atomic resolution, and atomic number, Z, contrast which allows elements to be distinguished
based on their relative brightness. In a strained sample, superimposed strain contrast prohibits
Z-contrast based elemental determinations. This project aimed to explain the origins of strain
contrast using two different simulation approaches. Multislice simulations were used to explore
dependency of strain contrast on sample thickness, detector angles and defect position, then
Bloch wave simulations examined interband transitions between Bloch waves due to strain. It
was found that strain contrast arises from increased elastic scattering to high angles, due to
electrons transitioning from s-state Bloch waves to waves with higher angle elastic scattering.
These results challenge the existing hypotheses on the origin of strain contrast, and the interband
code developed paves the way to quantitatively removing strain effects from STEM images.
2
Acknowledgements
I would like to thank Prof Peter Nellist, Dr Hao Yang and everyone in my research group for
the many hours of instructions and discussion that were invaluable to this project. I would also
like to thank Peter Clark for all his help in writing this report and my family and friends for
continuous support throughout.
3
Contents
Abstract 2
Acknowledgements 3
Table of Contents 4
1 Introduction and engineering context 8
2 The Scanning Transmission Electron Microscope 11
2.1 The design of the STEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Thermal Diffuse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Incoherent imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Simulating STEM images 16
3.1 Electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The multislice approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 The incorporation of TDS and the frozen phonon model . . . . . . . . . . 18
4
3.3 The Bloch wave approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Dispersion surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 Probe illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.3 TDS in the Bloch wave approach . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.4 From electron wave function to STEM image . . . . . . . . . . . . . . . . 24
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Strain contrast in ADF STEM 26
4.1 Dependence of strain contrast on experimental variables . . . . . . . . . . . . . . 26
4.2 Modelling strain contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Interband transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Mathematical definition of interband transitions . . . . . . . . . . . . . . 28
4.4 Explaining strain contrast with interband transitions . . . . . . . . . . . . . . . . 30
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Multislice simulation setup 32
5.1 Creating the model for multislice simulations . . . . . . . . . . . . . . . . . . . . 32
5.1.1 Silicon crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1.2 Single boron dopant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1.3 1atomic%B-Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.4 Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Exploring the parameter space for strain contrast . . . . . . . . . . . . . . . . . . 35
5
5.4 Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Multislice simulation results 38
6.1 Accuracy of multislice simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 The parameter space of strain contrast . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 Explaining strain contrast using multislice . . . . . . . . . . . . . . . . . . . . . . 45
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7 Creating the Bloch wave code 48
7.1 Calculating eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 48
7.1.1 Dispersion surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 Bloch waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 Electron wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3.1 Electron wave with planewave illumination . . . . . . . . . . . . . . . . . 51
7.3.2 Electron wave with probe illumination . . . . . . . . . . . . . . . . . . . . 52
7.3.3 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.4 Interband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.4.1 The strain field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8 Results of the Bloch wave approach 58
8.1 Scattering angles of the Bloch waves . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 Changes in electron wave intensity and diffraction pattern . . . . . . . . . . . . . 60
6
8.2.1 Unstrained crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.2.2 Simple shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.2.3 Spherical strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.3 Development of a new interband code . . . . . . . . . . . . . . . . . . . . . . . . 63
8.3.1 Spherical strain employing new interband code . . . . . . . . . . . . . . . 64
8.3.2 Interband transitions in the spherical strain field . . . . . . . . . . . . . . 65
8.4 Importance of Bloch wave results . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9 Conclusion and future work 68
10 Project management 70
Appendices 79
A Multislice simulation setup parameters 80
B The interband transition code 82
Bibliography 89
7
Chapter 1
Introduction and engineering context
The Scanning Transmission Electron Microscope, or STEM, can produce atomic resolution im-
ages. It is a very useful characterisation technique, providing images that are robust to changes
of variables such as sample thickness. With the use of an Annular Dark Field (ADF) detector,
directly interpretable images with contrast that is related to the atomic number of the elements,
Z-contrast, are obtained. These can be directly compared to a model of the crystal lattice (fig.
1.1[1]).
Z-contrast is one of the major reasons for the wide use of ADF STEM. It allows the exact position
of a dopant in a sample to be determined, which is useful in the semiconductor industry.[2, 3, 4]
It can also be employed to look at the structure and size of nanoparticles,[2, 4] as well as
their composition when coupled with other characterisation techniques like Electron Energy
Loss Spectroscopy or Energy Dispersive X-ray Spectroscopy.[5, 6] STEM can also be applied to
biological samples.[7]
However, problems arise when the sample is strained, that is, a sample in which the atoms are
displaced from their ideal positions. This can occur when a sample contains an unintentional
defect, like a dislocation, but also appears around intentional artefacts, like dopants and the
interfaces in a layered structure. Strain leads to a change in the image contrast which masks
the Z-contrast. In some instances it can make regions of lower atomic number appear brighter
than regions of higher atomic number,[8, 9] preventing qualitative composition information to
be drawn from the images (fig. 1.2). This strain contrast also depends on parameters that
8
(a) GaAs lattice 〈110〉: Ga (blue), As (red).
(b) ADF STEM image of GaAs 〈110〉. Intensity plot gives the intensity variations
inside the white rectangle. Reproduced from Nellist et al.[1]
Figure 1.1: As atoms have higher intensities due to Z-contrast since they have a higher atomic
number than Ga.
ADF STEM images are normally independent of.[8, 10] Attempts have been made at modelling
this strain contrast to gain knowledge of how it can be interpreted.[11, 10, 12] Even though the
contrast can be modelled empirically, there have been few attempts to explain the origin of the
strain contrast.[11, 8]
The main aim of this project has been to find a physical explanation for strain contrast in ADF
STEM. By gaining an understanding of the underlying mechanisms it may be possible to develop
a method that removes strain contrast from images, allowing Z-contrast to be visible again. This
study was conducted by performing multislice simulations to empirically model strain to give a
9
point of reference for the subsequent Bloch wave simulations. The Bloch wave approach to ADF
STEM image simulation was employed because it allows the scattering events inside the sample
to be studied in detail, and could therefore be a step towards explaining the strain contrast.
Although many defects lead to strain contrast, this project was limited to the study of a single
substitutional boron dopant in a silicon sample. This was chosen because B doped Si is brighter
than pure Si[11, 8] even though B has a lower atomic number than Si (fig. 1.2). Hence, this
system is ideal for distinguishing the effect of strain on the image contrast because the contrast
increase arises purely from strain.
Figure 1.2: Si sample with B doped layers and dislocations. B doped layers are brighter than
pure Si. The contrast of both defects depends on depth in sample. Reproduced from Perovic et
al.[8]
10
Chapter 2
The Scanning Transmission Electron
Microscope
2.1 The design of the STEM
In STEM, electrons are transmitted through the sample to a detector (fig. 2.1). As the electrons
travel through the sample, they are scattered to different angles by the atoms. Before entering
the sample, the electrons are focused into an illuminating probe which is scanned across the
sample surface, as each probe position represents a pixel in the final image.
All the lenses in a STEM sit above the sample and their main task is to focus the illuminating
electrons into a small point or a probe. Although the resolution of the microscope is limited
by the aberrations in all the lenses, the main limitation comes from the objective lens since it
provides the final and largest focussing step.[2] The aberrations are more significant at higher
angles, hence an objective aperture is utilised to set a maximum angle of the electron beam.[13]
In microscopes with aberration correctors, sub-angstrom resolution can be achieved.[14]
2.2 The detectors
There are different methods of detection in electron microscopy, generating different types of
images. In STEM, the detectors collect the diffraction pattern. For each position of the probe,
11
Figure 2.1: A simplistic layout of the STEM. Some elements, such as the condenser lenses, are
not shown.
the electrons are collected and integrated over the detector to give the intensity at that particular
point, representing one image pixel.[2, 15, 1]
Due to the fact that the sample is illuminated by a probe rather than a plane wave, as used in a
transmission electron microscope (TEM), the diffracted electrons form discs instead of spots (fig.
2.2), where electrons diffracted from reciprocal lattice point g make up disc g, reciprocal lattice
point -g makes up disc -g, and so on. The final diffraction pattern is quite different from the
plane wave diffraction patterns that are seen in a TEM. Interference between electrons scattered
at different positions in the sample, like 0 and g, takes place in the disc overlap regions and it
is this interference that gives contrast to the STEM images.[13]
There are two principal detectors in a STEM: a bright field (BF) detector and an annular dark
field (ADF) detector (fig. 2.2). The BF detector is centred at the unscattered (0th order) disc,
detecting interference between the 0, +g and -g discs. As such the main signal to the BF detector
comes from elastic, or Bragg, scattering.[13] The BF detector produces phase contrast images
which are similar to those in TEM, as stated by the principle of reciprocity.[16, 13] However,
because the BF detector is smaller than the 0th order disc, signal is lost due to electrons having
angles greater than the detector. Thus, the BF STEM image is more affected by noise than
TEM images. The BF image also has an intrinsic resolution limit as the 0, +g and -g discs must
overlap to produce contrast.[13]
12
Figure 2.2: Formation of the convergent beam diffraction pattern on the STEM detectors.
The ADF detector is very different from the BF detector. Firstly, it collects electrons over
a larger angle, which means that more overlap regions are detected. However, the type of
scattering is no longer purely elastic scattering. At these high angles, thermal diffuse scatter-
ing (TDS) becomes important (section 2.2.1).[17, 18] Secondly, it produces incoherent images
(section 2.2.2).[17] Unlike coherent images produced by the BF detector and most TEMs, the
incoherent images can be directly interpreted, making the ADF technique more popular than
the BF technique.
2.2.1 Thermal Diffuse Scattering
It is common to assume that all scattering events are elastic Bragg scattering. However, it
has been found that at the higher angles of the ADF detector this assumption is no longer
valid.[19] In this case, TDS must also be considered. This type of scattering arises from the fact
that the atoms in a crystal are vibrating, so at any given time, the atoms are shifted slightly
out of their ideal positions. When an electron scatters off a displaced atom, the symmetry of
Bragg scattering is lost, leading to a decrease in intensity and a blurring of the Bragg diffraction
spots.[20] This decrease in intensity is given by the Debye-Waller factor (DWF):
exp(−16π2u2ssin2θB
λ2) (2.1)
13
where u2s is the mean square displacements of the atoms due to vibrations, θB is the Bragg
scattering angle and λ is the wavelength of the incident electrons.[20] The decrease in intensity
of the elastic scattering is due to electrons thermally scattering to high angles. In a TEM, where
the objective apertue is below the sample, the high angles means that the electrons would hit the
aperture and not contribute to the image signal. Thus, TDS is often referred to as absorption.
In STEM, however, the aperture is above the sample and the ADF detector measures high-angle
scattering, meaning that thermally scattered electrons make up most of the signal in the ADF
detector. The exponential decrease in eq. 2.1 therefore relates to the decrease in electrons that
have been elastically scattered in the STEM, since they are being thermally scattered instead.[15]
Due to the different origin of the signal, the contrast in ADF imaging is different to the contrast
obtained in BF imaging. TDS is dependent on atomic number, Z, and the proximity of the
incident electrons to the atomic nuclei: an incident electron is more likely to thermally scatter
when it travels close to the atomic nuclei and when the atomic elements have a high atomic
number.[15, 18] This is the origin of Z-contrast in ADF STEM (fig. 1.1). The dependency of
absorption on the position of the incident electrons gives rise to anomalous absorption, which
will be discussed later (section 3.3.1).[18]
2.2.2 Incoherent imaging
Incoherent imaging was described by Lord Rayleigh as an image where interference between
waves scattered from spatially separated atoms does not occur.[21] As there is less interference
in an incoherent image than a coherent one, the former is easier to interpret. Whereas TEM
images, which are usually coherent, have to be compared to simulated images, the incoherent
images of the ADF STEM are related to what would be seen in ”real life” (fig. 1.1). Coherence
in ADF STEM is destroyed partly by the geometry of the ADF detector and partly by TDS.
The removal of coherence gives the characteristic Z-contrast, and an intensity dependence on
the number of atoms in a column (fig. 2.3)[15, 13].
14
Figure 2.3: GaAs[110] lattice: Ga (red), As (blue). The box indicates a column and the arrow
indicates the direction of the incident electrons.
2.3 Summary
In STEM the incident electrons are focused into a probe which is scanned over the sample to
create an incoherent image. The electrons are transmitted through the sample and detected by
an ADF detector, which mainly measures TDS because thermal scattering gives much higher
scattering angles than elastic scattering.
15
Chapter 3
Simulating STEM images
Obtaining a mathematical description of electron scattering has been a pursuit since 1928 when
Bethe first used the Schrodinger equation to describe scattering.[22] The result has been two
different approaches to describe the process: multislice and Bloch wave. The multislice approach
is based on an optics point of view, dividing the sample into slices, whilst the Bloch wave
approach describes the crystal as a set of periodic potentials.[23] Both methods produce the same
image, so the aim of the simulation determines which is used. The main advantage of multislice
is that it is able to simulate accurate images very quickly.[23] The Bloch wave approach is slower
and requires more computing memory, however, it gives greater insight into the scattering of
electrons inside the sample.[23]
In this project, the multislice approach was employed to simulate strain contrast before using
the Bloch wave approach to investigate the origins of strain contrast. This chapter will describe
two fundamental models in electron scattering before discussing the two simulation approaches.
3.1 Electron scattering
As electrons travel through a sample, they scatter off the atoms repeatedly, providing them
with different scattering angles as they move towards the detector. In attempts to describe
electron scattering, several assumptions have been made for simplification. One commonly
used approximation is the kinematical model which assumes that electrons only scatter once on
16
their way through the sample.[24] The approximation is likely to be valid in thin samples, but
commonly fails in thicker samples where the amplitudes of the scattered waves usually become
greater than that of the unscattered wave.[25] As the main signal to the ADF detector arises
from TDS, the amplitude of the scattered waves is consistently larger than the unscattered wave
in the measured signal. Thus kinematical theory was not applicable, and instead the dynamical
model was used which incorporates multiple scattering of the electrons.[24] Multiple scattering
is also important when analysing strain contrast, as will be seen later (section 4.3).
3.2 The multislice approach
The multislice approach involves treating a sample as a stack of thin slices, usually only one
atomic layer thick (fig. 3.1).
Figure 3.1: Schematic of multislice approach treatment of sample.
The electron wave function is calculated as the electrons move through the crystal by transmit-
ting through a slice and then propagating to the next slice. This process is described by eq.
3.1
ψn+1(x, y) = pn(x, y,∆zn) ∗ [tn(x, y)ψn(x, y)] (3.1)
where ψn and ψn+1 are the electron wave functions before and after slice n respectively, pn is
the propagation function, tn is the transmission function, ∆zn is the thickness of slice n and
∗ indicates a convolution.[23] The transmission function, tn(x, y) describes the interactions of
the electron wave with the atoms as it moves through a slice, while the propagation function,
17
pn, describes the movement of electrons between each slice. This is approximated to movement
through a vacuum, as no forces are acting on the electrons since the atomic potentials do not
reach the edge of a slice.[23]
The convolution in eq. 3.1 can be very time consuming so instead the Fast Fourier Transform
(FFT) algorithm has been introduced to reduce the computational time. The incorporation of
the FFT is one of the reasons why multislice is much faster than the Bloch wave method.[23]
3.2.1 The incorporation of TDS and the frozen phonon model
As the majority of the signal measured by the ADF detector comes from TDS, it is important
to incorporate this type of scattering in the simulations. In the multislice approach, this can be
done in two ways: absorptive potential (AP) or frozen phonon.[23, 26]
The AP calculations incorporate absorption by calculating the decrease in Bragg scattering
due to thermal scattering events, analogous to incorporating the Debye-Waller factor (DWF)
(section 2.2.1) into eq. 3.1. However this model is inaccurate because it incorrectly combines
elastic scattering and TDS.[27]
The frozen phonon model has proven more accurate.[23, 27, 26] The concept of this model is
that the incident electrons see snapshots of the vibrating lattice.[28] In each slice, the atoms are
slightly displaced from their ideal positions, but they do not move from these positions as the
electrons are transmitted through the slice. This model can be justified by the fact that the
electrons move much faster than the atoms vibrate.[29] The simulation is then run several times
and the results averaged to give as many different vibrational positions as necessary to obtain
a converging result. By incorporating the frozen phonon model, the multislice approach is able
to simulate very accurate images very quickly.
3.3 The Bloch wave approach
The Bloch wave approach is better able to express the physical origins of contrast in STEM
because it can give a detailed view of electron scattering in a strain field. The name comes
from the use of Bloch waves to build the electron wave that describes the electrons and their
18
movements.[23]
As shown by Bethe[22], the electron scattering processes can be described by the time-independent
Schrodinger equation (eq. 3.2)[1]
∇2Ψ(r) +8π2me
h2[E + ϕ(r)]Ψ(r) = 0 (3.2)
where Ψ(r) is the electron wave, E is the total energy, ϕ(r) is the atomic potential of the crystal
lattice, m is the relativistic electron mass, e is the electron charge and h is Planck’s constant.
The solutions to this equation can be given by Bloch waves (eq. 3.3)[1]
ψ(j)(r) = Σgφ(j)g exp(−2πi(k(j) + g) · r) (3.3)
where ψ(j) is the jth Bloch wave, φ(j)g is the amplitude of the wave at reciprocal lattice point
g, and k(j) is the wave vector of the jth Bloch wave. It can be seen that this equation satisfies
Bloch’s Theorem[30] as it has a periodically repeating component, φ(j)g , and a travelling wave
component, exp(−2πi(k(j) + g) · r).
The wave vector, k, and the position vector r can be separated into parts that are perpendicular
(transverse) and parallel (z-direction) to the electron beam (eq. 3.4)
k = k(j)t + k(j)z (3.4a)
r = R + z (3.4b)
where kt and R are the transverse components and kz and z the z-direction components. Equa-
tion 3.3 then becomes:
ψ(j)(r) = Σgφ(j)g exp(−2πi(k
(j)t + g) ·R)exp(−2πik(j)z z) (3.5)
19
In order to calculate the Bloch waves, φ(j)g and k
(j)z must be found. This can be done by re-
inserting eq. 3.5 into Schrodinger’s equation (eq. 3.2). After some manipulation, the result is
an eigenvalue problem[24]:
Aφ(j) = k(j)z φ(j) (3.6)
where φ(j) are the eigenvectors:
φ(j) =
φ(j)0
φ(j)g
φ(j)h
(3.7)
and k(j)z are the eigenvalues. A is the dynamical matrix:
−k2
t U−g U−h
Ug −(kt + g)2 Ug−h
Uh Uh−g −(kt + h)2
(3.8)
where Ug is the Fourier coefficient of one of the atomic potentials. The above equations are
presented for a 3 beam case, meaning that only 3 diffraction spots or reciprocal lattice points:
0, g and h, are included in the calculation. The dynamical matrix, A, increases in size for each
beam added to the simulation. By solving the eigenvalue problem (eq. 3.6), the eigenvectors
and eigenvalues can be found.
Finally, the total electron wave, Ψ (eq. 3.2) can be found by summing all the Bloch waves
multiplied by their relative amplitudes (eq. 3.9)[1]:
Ψ(r) = Σjα(j)ψ(j) (3.9)
where Ψ(r) is the total electron wave function, α(j) are the relative amplitudes or excitations
and ψ(j) are the Bloch waves (eq. 3.5). The excitations, α(j), can be found by solving for the
20
boundary conditions which state that Ψ and ∇Ψ at the surface (z=0) must be equal to Ψ and
∇Ψ of the incident wave. From this it is found that α(j) = φ∗(j)0 , the complex conjugate of φ
(j)0 .
Furthermore, k(j)t is found to be equal to the incident wave vector, Ki, for all Bloch waves, j, and
k(j)z is dependent on Ki.[24, 18] The dependency of k
(j)z on Ki can be presented in a dispersion
surface plot, which will be discussed later (section 3.3.1).
Finally, inserting equation 3.5 into 3.9 gives the electron wave as follows:
Ψ(R, z,Ki) = Σjφ∗(j)0 (Ki)Σgφ
(j)g (Ki)exp(−2πi(Ki + g) ·R)exp(−2πik(j)z (Ki)z) (3.10)
Probe illumination and TDS still need to be incorporated, and will be discussed later. It will
also be mentioned how the above equation can be used to simulate the STEM image. First,
however, the dispersion surface will be introduced.
3.3.1 Dispersion surface
The dispersion surface is a plot of the allowed kz values for a given Ki (fig. 3.2[31]). It is the
electron diffraction equivalent of the band structure (energy vs wave vector) plots in electron
band theory.[24] In addition, Bloch waves tend to resemble electron orbitals, as in the case of
silicon where the first two Bloch waves resemble the 1s bonding and antibonding states[31].
Figure 3.2: Dispersion surface of Si [110] at 100kV, including the first 9 Bloch waves. The kz
values constitute the y-axis.[31]
21
As the name suggests, dispersion surfaces plot how dispersive each Bloch wave is, that is, the
degree to which the wave spreads out as it moves through the sample. The first two Bloch waves,
the 1s states, are relatively flat (fig. 3.2), indicating that these states are concentrated at the
atomic columns. This is called channelling because the electron wave is confined to a small space.
The other Bloch waves are more dispersive as these have more more curved dispersion surfaces.
The distinction between dispersive and non-dispersive states is important for ADF imaging. As
previously mentioned, electrons that are concentrated at the atomic nuclei are more susceptible
to thermal scattering. As a result, the 1s Bloch states, which are centred at the atomic nuclei,
will be the main contributors to the TDS signal on an ADF detector.[18] The dispersiveness
of each Bloch wave also leads to the anomalous absorption effect, which was mentioned earlier
(section 2.2.1). Some Bloch waves have symmetries that position the electrons at the atomic
nuclei whilst others position the electrons between atoms (fig. 3.3[18]). Since proximity to the
atomic nuclei determines the likelihood of thermal scattering, different waves will be absorbed
to different extents at a given depth. This difference in absorption is what is called anomalous
absorption.[18]
Figure 3.3: A schematic of anomalous absorption. Redrawn from Hirsch et al.[18]
3.3.2 Probe illumination
When a sample is illuminated by a plane wave, all the incident waves have the same wave vector,
Ki, as they are all in phase. In probe illumination, there is a range of wave vectors because
22
the probe is made up of partial plane waves. The range of Ki values is limited by the objective
aperture.
Probe illumination is represented by eq. 3.11
P (R−R0) =
∫A(Ki)exp[−2πiKi · (R−R0)]dK (3.11)
where R is the real space variable, R0 is the probe position, Ki is the wave vector of the
incident electrons and A(Ki) is the aperture function. If the objective lens has no aberrations,
then A(Ki) is 1 inside the objective aperture and 0 outside.[1] This is multiplied with the wave
function in eq. 3.10, to give the electron wave with probe illumination (eq. 3.12)[1]. This
equation calculates the electron wave as a function of R, at a depth z in the crystal and probe
position R0.
Ψ(R, z,R0) =
∫A(Ki)exp(2πiKi ·R0)Σjφ
∗(j)0 (Ki)exp(−2πik(j)z (Ki)z)
x Σgφ(j)g (Ki)exp(−2πi(Ki + g) ·R)dKi
(3.12)
3.3.3 TDS in the Bloch wave approach
Thermal diffuse scattering can be described in the Bloch wave approach by introducing complex
values. In optics, absorption of light in a material is usually described by introducing a com-
plex refractive index.[32] Similarly, absorption in STEM can be simulated by complex atomic
potentials.[24]
ϕ(r) = ϕ′(r) + iϕ′′(r) (3.13a)
Ug = U ′g + iU ′′g (3.13b)
Eq. 3.13 indicates how the atomic potentials, ϕ, and the corresponding Fourier coefficients, Ug,
can be described as complex values. The imaginary part of the potential, ϕ′′(r), is introduced
23
because of the inelastic scattering processes where the incident electrons lose energy to the
crystal by energising the phonons, the particles responsible for the atomic vibrations. From the
complex potentials it follows that the wave vectors of the electrons must be complex.[24] Since
the transverse components of the wave vectors are all equal to the incident wave vector Ki,
which is real, only the longitudinal component, kz, can have complex values (eq. 3.14).
k(j)z = k′(j)z − ik′′(j)
z (3.14)
In eq. 3.12, k(j)z is in an exponential. This exponential can be rewritten, taking into account
that k(j)z is complex, which gives an exponential decrease in the electron waves (eq. 3.15).
exp(−2πik(j)z z) = exp(−2πik′(j)z z)exp(−2πk
′′(j)z z) (3.15)
As wih the DWF, this is not an overall decrease in the ADF detector signal, but a decrease in
the elastic scattering due to electrons being thermally scattered instead, thus contributing to
TDS (section 2.2.1).
The effect of channelling and absorption on the electron wave intensity as a function of depth
can be observed (fig. 3.4[33]). It can be seen that the intensity peaks slightly below the sample
surface and that the intensity oscillates with depth. This occurs due to channelling which causes
the electrons to enter the s-states, thus creating the peak in intensity at the atomic columns,
and oscillations because the electrons are confined to a small space.[33] The overall intensity
decrease with depth arises from the exponential decay due to absorption (eq. 3.15).
3.3.4 From electron wave function to STEM image
So far, the mathematical derivations have only described the electron wave inside the sample.
However, the Bloch wave approach can also simulate ADF STEM images by incorporating an
equation describing the ADF detector.[1] These images are obtained by Fourier Transforming
eq. 3.12 with respect to R to get the diffraction pattern on the detector. The intensity of the
diffraction pattern is multiplied with the detector function and integrated over reciprocal space,
24
Figure 3.4: Intensity of a column as a function of depth in GaAs 〈110〉, at 300kV. Reproduced
from Cosgriff et al.[33]
giving the intensity of one pixel in the ADF STEM image. As the probe position is moved,
the 2D image is created. However, the image would neglect the contribution from TDS, as the
previous equations only calculate the elastic component.
3.4 Summary
The dynamical model must be used when simulating ADF STEM due to the importance of
multiple scattering. The two approaches to ADF STEM simulation have been shown to have
different advantages: the multislice approach produces accurate simulations quickly, whilst the
Bloch wave approach allows the contribution of each Bloch wave to the final image to be in-
spected, which will be useful when investigating the origin of strain contrast.
25
Chapter 4
Strain contrast in ADF STEM
ADF STEM is a powerful characterisation technique in a perfect crystal, due to its atomic
resolution and Z-contrast. However, in imperfect crystals the image becomes more complicated
as Z-contrast is overshadowed by contrast which arises due to the strain induced by a defect.
This strain contrast is a problem because it prevents direct interpretation of the ADF STEM
images.
Strain contrast occurs when defects lead to displacements of the surrounding atoms.[11, 9] It
has often been seen in studies of semiconductor materials where strain can arise from lattice
mismatch across layers of different composition. In most cases the defects cause an increase
in the image intensity which is unrelated to composition variations.[34, 35] Interstingly, boron
doped silicon has a higher intensity than pure silicon even though it has a lower average atomic
number (fig. 1.2).[11, 8, 36] Research has been conducted on strain contrast to gain an insight
into the variables it depends on and simple theories of the physical origin have been proposed.[34,
37, 8, 12] However, few attempts have been made at fully explaining the underlying principles.
In this chapter, studies of the causes and effects of strain contrast will be explored.
4.1 Dependence of strain contrast on experimental variables
Although ADF STEM contrast is robust to changes in most variables, it becomes more sus-
ceptible when a defect is introduced. A number of different defects have been studied and,
26
although most lead to an increase in brightness, they tend to show different dependencies on
the experimental parameters. For example, Wu et al. found that both Si-Ge and Si-C layers
are brighter than pure Si, but the contrast in the Ge-containing layers (ZGe > ZSi) increased
with increasing inner angle of the detector, whilst in the C-containing layers (ZC < ZSi), the
contrast decreased with increasing inner angle. Some of these parameters were explored in this
project to gain an idea about the strain contrast in B doped Si.
The dependence of strain contrast on detector angles has frequently been studied. It has been
shown that a single substitutional B dopant in a 200A thick Si sample gave brighter contrast
compared to pure Si for detector inner angles below 90mrad and darker contrast for angles
above.[36] Similar results have been found in other work.[10, 9]
While intensity in perfect crystals increases monotonically with sample thickness,[13] strain
contrast has a more complicated dependency on sample thickness. The interface between crys-
talline and amorphous silicon in a 150A thick sample was bright in a low-angle ADF (LAADF:
20-64mrad) detector and dark in a high-angle ADF (HAADF: 64-200mrad) detector. For sam-
ples below 100A, however, the interface was dark in both detectors.[10] Further, it has been
reported that contrast change compared to pure Si increased with increasing thickness in a Si-C
layer.[38] The position of the defect within the sample also affects strain contrast as the intensity
of a dislocation has been shown to oscillate with depth.[8]
The effect of sample tilt on strain contrast has also been studied. In a GaAs sample containing
thin layers of InGaAs, there was a dip in intensity at the layer interfaces. As tilt was introduced,
the interface became brighter on one side of the GaInAs layer and darker on the other.[12]
4.2 Modelling strain contrast
There have been several attempts at simulating strain contrast, some of which have been given
better results than others. The simplest model is the static Debye-Waller factor (DWF) model.
The DWF was previously introduced as a description of the decrease in intensity due to dis-
placements of atoms arising from thermal vibrations (section 2.2.1). Since strain contrast comes
from displacements of atoms near the defect[11, 9] a second term describing these displacements
is added to the original DWF. Although this model has held under some conditions [38, 34] it
27
is not universal.[10].
A more accurate method for simulating strain contrast is to use multislice simulations. In this, a
model of the defect crystal is loaded into the multislice software, which then simulates the image
with strain contrast. Although it provides accurate images of strain contrast[10, 12, 35, 38], it
does not give a detailed account of the behaviour of electrons in a strain field. To that end,
interband transitions must be studied.
4.3 Interband transitions
The electron wave inside the crystal is made up of Bloch waves with relative excitations (section
3.3). In a perfect crystal, these excitations are given by the boundary conditions at the top
surface, however, in an imperfect crystal these excitations can change when the electrons scatter
off displaced atoms. This is called interband transitions as the electrons move between the Bloch
waves (i.e. bands in the dispersion surface, fig. 3.2). In this project, the origins of strain contrast
was contemplated by studying these interband transitions.
4.3.1 Mathematical definition of interband transitions
To calculate the change in excitations due to interband transitions the sample was divided into
slices (fig. 4.1).
Figure 4.1: Division of sample into slices of thickness dz.
28
The change in Bloch wave excitation due to interband scattering in each slice is given by eq.
4.1[17, 39, 40]:
dΨ = 2πi{exp(−2πik(j)z z)}Φ−1{β′g}Φ{exp(2πik(j)z z)}Ψdz (4.1)
where dΨ is the excitation at the top of a slice and dΨ is the change in excitation within that
slice. That is, dΨ1 is the change in slice 1 and is related to Ψ1. The excitation at the top of
slice 2, Ψ2, is given by Ψ1 + dΨ1, and so on. k(j)z is the eigenvalue for Bloch wave j and the
curly brackets represent a diagonal matrix (eq. 4.2 for a three-beam case):
{exp(2πik(j)z z)} =
exp(2πik
(1)z z) 0 0
0 exp(2πik(2)z z) 0
0 0 exp(2πik(3)z z)
(4.2)
Φ is a matrix containing the eigenvectors (eq. 4.3 for a three-beam case):
Φ =
φ(1)0 φ
(2)0 φ
(3)0
φ(1)1 φ
(2)1 φ
(3)1
φ(1)2 φ
(2)2 φ
(3)2
(4.3)
where each column represent a Bloch wave and each row a diffraction spot, g. Φ−1 is the inverse
of Φ. Finally β′g relates to the displacements that arise due to a defect[17, 39]:
β′g =d
dzgR(z) = g
dR(z)
dz(4.4)
where g is the diffraction spot vector and R(z) is the atom displacements dependent on depth,
z.
In interband calculations the column approximation is used: all columns except that on which
the probe is focused are ignored, and scattering to the other columns is assumed negligible.[24]
This approximation is commonly used in imperfect crystals, however there are some limitations.
Firstly, the approximation fails if the sample is tilted such that the electrons do not travel
29
parallel to the columns, and secondly if the strain is so large that electrons scatter far from the
Bragg directions. These limitations prevent eq. 4.1 from being applied to dislocation cores.[24]
Displacements of the atoms can be separated into those within a slice, R, and those along
the beam direction, Rz, which have no effect on image contrast since g · dRzdz = 0 for all g.
For most defects, R is dependent on the depth of the each slice relative to the defect, hence
R = R(z). Eq. 4.1 also gives a selection rule for interband transitions: transitions with dΨ = 0
are prohibited.[17]
Interband transitions are important in this research because Bloch waves have different sym-
metries. Thus, excitation changes in the Bloch waves can change the symmetry of the electron
wave inside the sample. This could change the scattering events, providing an explanation to
strain contrast.
4.4 Explaining strain contrast with interband transitions
Some research on strain contrast has suggested that strain contrast is due to atom displace-
ments giving dechannelling.[34, 35, 37] However, there are few detailed explanations of strain
contrast.[8, 12]
One attempt was made by Perovic et al. to explain the intensity oscillation along a dislocation
through interband transitions. Strain was stated to have caused a re-excitation of the s-state
Bloch waves, such that the re-excited Bloch waves interferred with the original s-state electrons,
causing intensity oscillations.[8] This explanation was specific to the case of a dislocation, how-
ever it provided a basis for the initial hypothesis for this project: the brightness increase in B
doped Si relative to pure Si comes from a re-excitation of the s-states where electrons transitions
from higher-order Bloch waves to the s-states, thus increasing TDS since the s-states are the
main contributors to TDS.
30
4.5 Summary
Although the multislice approach provides accurate simulations of strain contrast, interband
transitions gives a detailed account of the scattering events involved. The hypothesis for this
project was that strain leads to a re-excitation of the s-state Bloch waves, giving increased
absorption, and hence, increased image intensity.
31
Chapter 5
Multislice simulation setup
All multislice simulations were performed on a software called µSTEM.[41] The software required
an input model of the imperfect crystal and a file containing the parameters for the simulation,
some of which controlled the accuracy of the simulated images. The simulations produced
images like those obtained in the microscope which were analysed using Absolute Integrator,[42]
a software that provides quantitative analysis of simulated ADF STEM images.
5.1 Creating the model for multislice simulations
In this project, silicon was chosen as the base crystal due to a large body of research on this
material being available for comparison.[8, 10, 36] Different defects were then introduced into the
model. All models were made by codes written in MATLAB[43] and viewed in CrystalMaker[44],
a software used to look at crystal structures. The main model adopted to investigate strain
contrast was a silicon crystal containing a single substitutional boron atom. However, a few
models containing other defects were produced to verify the accuracy of the main model.
5.1.1 Silicon crystal
Throughout this project, the electron probe in the ADF STEM simulations was set to the
[110] direction in Si, which has a diamond cubic lattice. Hence the silicon unit cell was setup
with z-axis in the [110] direction, giving dumbbells on the plane perpendicular to the beam,
32
characteristic of the [110] viewing direction (fig. 5.1).
Figure 5.1: Si down [110]. a=b=5.431A and c=3.84A. The blue atoms represent the 4 atoms
in a unit cell.
In order to employ the FFT (section 3.2), the input model is repeated indefinitely in the di-
rections perpendicular to the beam.[23] This is not an issue for the perfect silicon crystal, but
becomes problematic once a defect is introduced. If a unit cell with a single dopant is repeated
then the close proximity of the dopants will generate overlapping strain fields, causing errors in
the simulated image. To prevent this, a supercell is created. This is a larger cell consisting of
repeats of the unit cell in the x- and y-directions perpendicular to the probe. The number of unit
cells in the z-direction gives the thickness of the sample as µSTEM will not repeat the model in
this direction. The supercell was produced by creating a matrix containing the coordinates of
the atom positions.
5.1.2 Single boron dopant
The single B dopant was introduced by defining a position in the silicon supercell. The position
had to be one of the initial Si positions so that the B atom became substitutional. Since B has
a smaller atomic radius than Si, the surrounding lattice will contract around the dopant. This
spherical strain field was described by the following equation[11]:
d = − c
r3r (5.1)
where c=1.4A3
is the strain constant, r is the vector going from the B dopant to one of the
33
Si atoms, r is the length of r and d is the displacement vector for a given Si atom. The new
positions of the Si atoms were calculated based these displacement vectors (fig. 5.2).
Figure 5.2: Spherical strain around B (red) with amplified strain field of c=5.4A3
Image dopants
It is known that the free surface of a real crystal has to be traction free. This was accounted for by
the introduction of image dopants, analogous to the more commonly known image dislocations.
The image dopant produces a compressive stress field that combines with the tensile stress field
of the real dopant to remove the forces at the free surface. In the model developed for this
project six image dopants were implemented, on either side of the top and bottom surface, to
obtain converging values. The sample was assumed to be sufficiently wide that no electrons
would exit through the sides of the crystal, eliminating the need of image dopants. The overall
effect of the image dopants was found be small, as the largest displacement due to an image
dopant was less than the smallest displacement due to the real dopant at the sample surface.
5.1.3 1atomic%B-Si
A model was developed that included more than one boron dopant, to see if a larger number
of dopants would give more strain contrast. Each dopant was introduced as before, in that the
positions of the Si atoms would change but not the positions of the other dopants. The total
number of dopants was set to 1 at%B based on the effect noted by Hall et al.[11]
34
5.1.4 Dislocation
An edge dislocation was introduced based on a code by H. Yang using isotropic elasticity dis-
placements. The dislocation line was set to lie across the beam direction (fig. 5.3).[45]
Figure 5.3: Schematic of edge dislocation with dislocation line along [110] and Burgers vector,
b, along [110]
5.2 Simulation parameters
With an input model produced, the simulation parameters had to be set. Most of these define
the setup of the microscope, but some relate to how the model is sampled and how the image is
simulated. These last parameters are important as they determine the accuracy of the simulated
image. The parameter setup can be found in appendix A.
5.3 Exploring the parameter space for strain contrast
Not all parameters that affects strain contrast could be covered (section 4.1), so the project was
limited to investigating the effects of detector angle, sample thickness and defect position. The
values tested are as follows:
• Detector angles
– Low Angle ADF (LAADF): 30-90mrad
– High Angle ADF (HAADF): 90-150mrad
35
• Sample thickness
– 73A
– 353A
– 500A
• Dopant depth in sample as fraction of sample thickness
– 1/4
– 1/2
– 3/4
5.4 Image analysis
The images were analysed with Absolute Integrator to gain a quantitative description of the
intensity variations. Absolute Integrator integrates the intensities in a Voronoi cell, centered at
the atomic columns, to produce the scattering cross-section.[42] An intensity map is generated
where the colours represent the cross-sectional intensities in Mb (10−28m2) (fig. 5.4b).
36
(a) µSTEM image. Points define the positions of the atomic columns
in Absolute Integrator.
(b) Absolute Integrator output.
Figure 5.4: Input and output figures of Absolute Integrator.
37
Chapter 6
Multislice simulation results
The aim of the multislice simulations was to inspect the strain contrast around a single, substi-
tutional B dopant in Si, to give a point of reference for when the interband code was used. The
multislice results could also give an indication of the validity of hypothesis of re-excitation of
the s-states, by comparing the strain contrast to the contrast based purely on elastic scattering.
However, it was first important to identify potential errors in the multislice simulations and how
to reduce them.
6.1 Accuracy of multislice simulations
The main errors in µSTEM simulations were believed to come from the limits set for sampling,
but unrealistic strain models could also lead to inaccuracies.
Some of the input variables required in µSTEM control the accuracy of the simulation results
(appendix A). Two input parameters that can give errors due to a lack of convergence are phonon
runs and supercell dimensions. 20 sets of phonon displacements were employed, which has been
shown to give reasonably accurate results.[27] The supercell dimensions were set to 21.7Ax21.7A
which was slightly less than advised in the µSTEM manual,[26] hence the cross-sections near the
edges were excluded from analyses. To verify the accuracy of these variables and assumptions,
simulations of the pure Si crystal were performed as every column in a uniformly thick Si crystal
should have the same intensity. A maximum fluctuation of 1.6% in the column intensities was
38
revealed (table. 6.1), confirming that the multislice simulations were highly accurate.
Table 6.1: Average intensities and errors
Thickness (A) Detector Average intensity Standard deviation Percentage error
73LAADF 6.2 0.1 1.6HAADF 1.08 0.01 1.1
353LAADF 22.9 0.2 0.9HAADF 5.31 0.03 0.5
500LAADF 30.5 0.2 0.6HAADF 7.27 0.03 0.4
To prevent mistakes in the defect models, CrystalMaker[44] was used to inspect the models.
However, an issue emerged when the ADF STEM image of the single B doped Si model was
simulated: the intensity change with respect to the pure Si was very small, making it unclear
how the strain contrast varied with increasing distance from the dopant column (fig. 6.1). The
strain was therefore amplified by increasing the strain constant, c, from 1.4 to 5.4. The results
were compared to more realistic samples, like 1at%B-Si and an edge dislocation, to show that
the new strain field was not unreasonable.
(a) LAADF (b) HAADF
Figure 6.1: Intensity change compared to pure Si for a 73A thick single B doped Si sample, B
at 1/4. The black cross in a) shows the B-containing column in all multislice simulations.
The contrast in the 1at%B-Si sample (fig. 6.2) was larger and matched reported contrasts,[8]
however, it was difficult to detect the effect of each dopant. Hence, the single B doped Si with
amplified strain was chosen to investigate strain contrast.
To show that the amplified strain gave realistic displacements and that the strain contrast
trends did not change due to amplification, the contrast surrounding a pure edge dislocation
39
(a) LAADF (b) HAADF
Figure 6.2: Intensity change relative to Si for a 73A thick 1at%B-Si sample.
was simulated. The displacements near the dislocation were much greater than around the
B dopant, even with the increased strain constant. Both models showed the same trend, with
brighter contrast in the LAADF detector (fig. 6.3a and 6.3b) and darker contrast in the HAADF
detector (fig. 6.3c and 6.3d), validating the results from single B doped Si samples with amplified
strain.
6.2 The parameter space of strain contrast
Several images of B doped Si were simulated to explore the parameter space of strain contrast. In
place of comparing every image to inspect the trends, the intensities were compared in intensity
change vs neighbour number plots. These demonstrated how the intensities in a column varied
with the distance from the B-containing column. Based on this distance the columns were given
neighbour numbers (fig. 6.4). For equidistant columns, the average intensity of these was used.
The data showed a clear trend for the two detector angles: brighter images in the LAADF
detector and darker images in the HAADF detector (fig. 6.5). This trend was more distinguished
in the 73A sample than in the 500A sample, with the percentage change in intensity being greater
the thinner the sample, indicating that strain contrast decreased with increasing thickness. There
were some irregularities in these trends. Firstly, there was an intensity dip at the 9th neighbour
for both the 353A and 500A samples, as well as a brightness increase as the neighbour number
increased. Secondly, the trends in the HAADF for the two thicker samples differed from the 73A
40
(a) LAADF Dislocation (b) LAADF amplified B doped Si
(c) HAADF Dislocation (d) HAADF amplified B doped Si
Figure 6.3: Both samples are 73A thick. The arrow indicates the position and direction of the
dislocation line, the Burgers vector is perpendicular to the page. The B dopant is at 1/4 of the
thickness.
41
Figure 6.4: Neighbour numbers indicated by different colours. The remaining columns are
excluded because of proximity to the sample edge.
sample. In the latter (fig. 6.5b) the contrast tended to zero as the neighbour number increased,
as was expected since strain decreases with distance from the dopant. The discrepancies could
have occured due to proximity to the sample edge, as the edge errors could have a greater effect on
the thicker samples where the strain contrast was smaller. To assess whether the inconsistencies
were strain contrast effects, images were simulated with a greater strain constant, c, of 10.4A3
(fig. 6.6).
With c=10.4A3, the overall trends for the contrast became more like expected: a contrast
increase in the LAADF detector and a decrease in the HAADF detector was seen, both tending
to zero as the distance to the dopant column increased. The unexpected dips were also gone.
The dip at neighbour number 1, which became more pronounced with increasing strain constant
(fig. 6.5a), has not been described in literature. It is likely that this occurs due to the large
displacements in this column, either causing the atoms to leave the column entirely or the
channelling in the column to break down.
Dopant position seemed to have little effect on the HAADF images, but some contrast change
occured in the LAADF detector. In the 73A thick sample (fig. 6.5a and 6.6a), the dopant at
1/2 gave the highest strain contrast whereas, in the 353A sample (fig. 6.5c and 6.6c), it was the
dopant at 1/4. It was concluded that strain contrast depends on the exact depth of the dopant,
which was likely related to the amount of absorption that had occured by this point.
The simulated intensity changes related to detector angle correlated well with literature.[36].
42
(a) LAADF, 73A thickness (b) HAADF, 73A thickness
(c) LAADF, 353A thickness (d) HAADF, 353A thickness
(e) LAADF, 500A thickness (f) HAADF, 500A thickness
Figure 6.5: Percentage contrast changes compared to pure Si for different dopant positions,
sample thicknesses and detector angles.
43
(a) LAADF, 73A thickness (b) HAADF, 73A thickness
(c) LAADF, 353A thickness (d) HAADF, 353A thickness
Figure 6.6: Percentage contrast changes compared to pure Si, with c=10.4A3
44
However, the decrease in intensity change with increasing thickness was inconsistent with the
increase in intensity change with thickness in Si-C.[38]. It was reasonable to assume that the
contrast in B doped Si would resemble the Si-C contrast since both C and B have lower atomic
numbers than Si. However, the strain in the Si-C layers was reported to increase with increasing
thickness whereas in B doped Si fewer electrons would be affected by strain as thickness increased.
This was because the dopants moved further into the sample with increasing thickness, hence
making it more likely that the electrons had undergone thermal scattering before they reached
the dopant.
6.3 Explaining strain contrast using multislice
µSTEM could also be employed to test the hypothesis of re-excitation of the s-states by looking
at the elastic scattering component of the strain contrast. If the hypothesis was correct, the
strained sample would give less elastic scattering since the s-states mainly contribute to TDS.
The strain contrast results from the previous QEP simulations, based on the accurate frozen
phonon model (appendix A), were compared to the change in elastic contrast and absorption
contrast, based on AP simulations (fig. 6.7). Exact contrast change was plotted, rather than
percentage change, to make comparison easier.
The QEP strain contrast in the LAADF detector correlated with the elastic contrast rather
than the absorptive contrast (fig. 6.7a, 6.7c and 6.7e). This indicated that the strain contrast in
the LAADF detector came from an increase in elastic scattering to the angles spanned by that
detector. Similar results had been reported when looking at scattering near different defects.[37]
In the HAADF detector, the QEP strain contrast matched up with the absorption contrast (fig.
6.7b, 6.7d and 6.7f), probably because the HAADF detector primarily measures TDS, whilst
the LAADF detector measures both elastic scattering and TDS. These results disagreed with
the initial hypothesis, so a new hypothesis was formed: strain contrast arises from increased
elastic scattering to high angles, giving an intensity increase in the LAADF detector which also
measures high angle elastic scattering, and a decrease in the HAADF detector, which mainly
measures TDS. The decrease in the HAADF detector comes from electrons being elastically
scattered rather than thermally scattered, thus decreasing the TDS signal.
45
(a) LAADF, dopant at 1/4 (b) HAADF, dopant at 1/4
(c) LAADF, dopant at 1/2 (d) HAADF, dopant at 1/2
(e) LAADF, dopant at 3/4 (f) HAADF, dopant at 3/4
Figure 6.7: Exact contrast changes compared to pure Si in QEP, elastic and absorptive scattering
in a 73A thick sample, c=5.4A3.
46
6.4 Summary
In this chapter, it was found that the contrast increased in the LAADF detector and decreased in
the HAADF detector compared to pure Si. The change in contrast also decreased with thickness
because the dopants were set deeper in the sample, increasing the likelihood of electrons being
absorbed before reaching the defect. Contrary to the hypothesis of re-excitation of the s-states,
comparisons with pure elastic scattering revealed an increase in high angle elastic scattering.
47
Chapter 7
Creating the Bloch wave code
The Bloch wave codes were produced based on the equations that make up the Bloch wave
approach to STEM image simulations (section 3.3). The majority of the codes were written
specifically for this project, but could be used in other projects. The output of each code was
verified by comparison with literature or to hand calculations based on simple systems. The
following sections give a detailed description of the production of each code.
7.1 Calculating eigenvectors and eigenvalues
The eigenvalue problem (eq. 3.6) was solved by the simCBED code written by M. Saunders[46]
and translated to MATLAB by H. Yang. The eigenvalues, k(j)z (Ki), and eigenvectors, φ
(j)g (Ki),
are defined for each beam (giving the number of j and g) and partial planewave wave vectors,
Ki (eq. 7.1), hence the accuracy of the calculation is dependent on the number of beams and
Ki included. A large number of beams is important in ADF STEM[39] as it determines the
maximum scattering angle, but it also drastically increases the computational time.[23] In this
project, 249 beams were included based on literature standards.[1, 33] The number of partial
planewaves determines how well sampled the probe is. 754 Ki values were included based on
literature standards.[1] Throughout the simulations, Higher Order Laue Zones (HOLZ) were
omitted because of their negligible effect on the simulations.[47]
48
Ψ(R, z,R0) =
∫A(Ki)exp(2πiKi ·R0)Σjφ
∗(j)0 (Ki)exp(−2πik(j)z (Ki)z)
x Σgφ(j)g (Ki)exp(−2πi(Ki + g) ·R)dKi
(7.1)
The original code supplied for this work underwent a number of corrections. Initially the code
did not work, and each step had to be understood and the outputs verified to locate the error.
One significant change was including all Ki values allowed by the aperture in the calculations
rather than just those included in the 1st Brillouin zone. Although this increased the computing
time, it meant that the addition of Brillouin zones did not have to be contemplated.
7.1.1 Dispersion surface
In order to validate the corrected simCBED code, a dispersion surface for the first 9 Bloch waves
was plotted (fig. 7.1).
Figure 7.1: Dispersion surface for Si[110], at 100kV.
This was found to match the disperion surface produced by Pennycook and Jesson (fig. 3.2)[31]:
the first two Bloch waves were non-dispersive and the other Bloch waves shared similar curves.
Although the kz values differed, ∼ 0.01A−1
for the s-states was reasonable since this would give
49
channelling oscillations with a period of 100A.[48]
7.2 Bloch waves
Bloch waves were calculated based on eq. 3.3 and plotted by a code written by H. Yang. After
corrections were made, such as providing all components with real units so the units in the
exponential would cancel, the final code was used to visualise the Bloch waves.
(a) 1st Bloch wave. (b) 2nd Bloch wave.
Figure 7.2: Real part of the first two Bloch waves in Si[110], 200kV.
As mentioned earlier (section 3.3.1), Bloch waves resemble electron orbitals, with the first two
Bloch waves in silicon resembling the bonding and antibonding orbitals (fig. 7.2). In the bonding
state, the phase has the same sign over both atoms (fig. 7.2a), while in the antibonding state,
the phase has opposite signs over the two atoms (fig. 7.2b).[31] These are the states that give
channelling.
7.3 Electron wave
The next step was to create a code that would simulate the total electron wave inside the
crystal. This code was written entirely during this project and divided into three stages. First,
the electron wave with planewave illumination was calculated, then probe illumination was
introduced, and finally, absorption was added. The codes calculated for several depths at once,
thus allowing the electron wave along the thickness to be analysed.
50
7.3.1 Electron wave with planewave illumination
To calculate the electron wave with plane wave illumination eq. 3.10 had to be solved for
Ki=[000]. This was done by separating the equation into a component that is dependent on
both summation variables, g and j, (square brackets in eq. 7.2) and a component that is only
dependent on the second summation variable, j, (before square brackets):
Ψ(R, z,Ki) = Σjφ∗(j)0 (Ki)exp(−2πik(j)z (Ki)z)[Σgφ
(j)g (Ki)exp(−2πi(Ki + g) ·R)] (7.2)
The part in square brackets is the equation used to calculate the Bloch waves in the Bloch wave
code (section 7.2). Hence, each Bloch wave from the Bloch wave code was multiplied with the
excitation, φ∗(j)0 (Ki), and the eigenvalue term, exp(−2πik
(j)z (Ki)z), and summed over all Bloch
waves, j, to give the electron wave with plane wave illumination (fig. 7.3).
(a) Electron wave at 200A depth.(b) Electron wave down the crystal thickness. X-axis along the orange line in a)
Figure 7.3: Electron wave intensity with plane wave illumination for Si[110], 200kV.
Although the electron wave intensity looks like an image of the silicon crystal lattice (fig. 7.3a),
it’s important to note that this is not the final ADF STEM image. The intensity peaks around
the atomic columns because of channelling causing electrons to enter the s-state Bloch waves.
Channelling oscillations can be seen along the colum (fig. 7.3b), where the intensity peaks
around 130A depth and then decreases again.
51
7.3.2 Electron wave with probe illumination
To incorporate probe illumination, eq. 7.3 had to be incorporated into eq. 7.2, giving eq. 7.4.
The probe part of the equation does not depend on which diffraction spot, g, or Bloch wave,
j, is being calculated for and could therefore be setup separately. This was done by creating a
2D matrix for the aperture function, A(Ki), where each matrix element represents a certain Ki
and is set to 1 or 0 depending on whether this Ki is inside the objective aperture or not. The
aperture function was multiplied with the results from the planewave code (section 7.3.1) and
the integral over Ki was approximated to a sum over Ki.
P (R−R0) =
∫A(Ki)exp[−2πiKi(R−R0)]dK (7.3)
Ψ(R, z,R0) =
∫A(Ki)exp(2πiKi ·R0)Σjφ
∗(j)0 (Ki)exp(−2πik(j)z (Ki)z)
x Σgφjg(Ki)exp(−2πi(Ki + g) ·R)dKi
(7.4)
The resulting electron wave intensity, with probe illumination, displayed channelling oscillations
down the column the probe was positioned over (fig. 7.4). The probe code was created so that
the probe position could be moved around, as this could be useful in future work.
(a) Electron wave at 200A depth. (b) Electron wave down the crystal thickness.
Figure 7.4: Electron wave intensity with probe illumination for Si[110], 200kV.
52
7.3.3 Absorption
The final step in simulating the electron wave was to incorporate absorption of the elastic
scattering due to TDS. The simCBED code [46] allowed the inclusion of absorption to be switched
on an off. The planewave and probe codes were modified so that the complex values of k(j)z (Ki)
were used correctly. This was done by separating the exponential with k(j)z (Ki) as in eq. 3.15.
To verify the code, a plot of the electron intensity against depth in GaAs (fig. 7.5) was compared
to fig. 3.4. A similar overall trend of exponentially decreasing channelling oscillations was
observed.
Figure 7.5: Fractional probe intensity over Ga in GaAs[110], 300kV.
It is important to note that the code created only incorporates elastic scattering and its reduction
due to TDS. In order to analyse the thermally scattered electrons, the absorbed part would
have to incorporated into the simulations. However, it was assumed that thermally scattered
electrons could not be subsequently elastically scattered, hence excluding TDS would not affect
the accuracy of the elastic result.[15] The effect of interband transitions on TDS could be inferred
from the transitions to and from the s-states, as an increase in the s-states would give an increase
in thermal scattering, since these states are the main contributors to TDS.
53
7.4 Interband Transitions
Changes in the excitations of the Bloch waves due to strain (section 4.3) was calculated based
on eq. 7.5.
dΨ = 2πi{exp(−2πik(j)z z)}Φ−1{β′g}Φ{exp(2πik(j)z z)}Ψdz (7.5)
There have been attempts at writing programs to calculate the change in Bloch wave excitations
due to interband scattering,[12, 49] however these attempts have been limited to only including
the 1s states. The interband code developed in this project is possibly the first attempt at an
interband transition code which includes a large number of Bloch waves in order to give a more
accurate description of strain contrast in ADF STEM.
Initially, the sample was sliced so dΨ could be calculated for each slice (section 4.3). An equation
for R(z) then had to be derived so that β′g = g dR(z)dz could be evaluated. This was one of the
main difficulties in producing the interband code and will be described subsequently (section
7.4.1). Once β′g was found, each of the matrices in eq. 7.5 were populated for a given depth,
z, before they were multiplied to give dΨ for each slice. This was then added to the intial
excitations to give the excitation at the bottom surface, from which the electron wave could
be calculated. Alternatively, the excitations could be used to find the electron wave at a given
depth in the sample.
7.4.1 The strain field
One of the greatest difficulties in producing the interband code was determining the displace-
ments of atoms as a function of depth, z; R(z). The first attempt was made by introducing the
simplest possible strain field, namely a simple shear perpendicular to the beam (fig. 7.6). From
this it was found that R(z) = dt z and hence β′g = d
tg (fig. 7.6a).
This simple system was validated by comparing the results to calculations based on eq. (7) in
a paper by Nellist et al.[17] for a simplified model of 5 beams and no absorption. The code
was also checked by ensuring that the impossible transitions listed by Nellist et al.[17] were
54
(a) Reality (b) Model
Figure 7.6: Schematic of simple shear model.
prohibited.
To analyse the results of the multislice simulations using the Bloch wave approach, the more
complicated spherical strain field surrounding a dopant was modelled. The column approxima-
tion was applied when calculating the interband transitions, which means that, in the case of the
strain field around a dopant, only displacements in the column the probe is over are considered.
The displacements were calculated based on eq. 7.6[11], neglecting displacements parallel to the
beam as these would have no effect on the contrast (fig. 7.7).
d =c
r3r (7.6)
Introducing the spherical strain field in the interband code was complicated because the calcu-
lation of a displacement vector for each atom, as in the multislice model (section 5.1.2), was
insufficient. Instead, an equation for the displacements had to be found as a function of z so
that β′g could be calculated. Once R(z) was derived, both R(z) and dR(z)dz were plotted to ensure
that the values were reasonable (fig. 7.8). It was seen that R(z) has a dip at the dopant position
because the slice at the same depth as the B dopant has the largest displacement in the direction
of the dopant (negative direction). dR(z)dz is negative when R(z) decreases and is positive when
R(z) increases just as the differential of R(z) should be. This indicated that the model of the
spherical strain field is correct.
55
(a) Strain parameters (b) Strained column
Figure 7.7: Schematic of spherical strain model.
Figure 7.8: R(z) and dR(z)dz for 73A thick Si, B dopant at z=36.48A.
7.5 Summary
In this chapter, it was shown how codes were developed so the following could be calculated:
• The eigenvectors and eigenvalues
• The Bloch waves
• The electron wave inside the sample
– With or without absorption
56
– With plane wave or probe illumination
• The change in excitations due to interband transitions
• The electron wave with interband transitions
The codes were written generally such that they could be applied in other studies. The only
limitation was that only simple shear and spherical strain has been incorporated to date. Other
defects would require the R(z) function to be found before the code can be used.
57
Chapter 8
Results of the Bloch wave approach
The aim of the interband code was to assess whether strain causes interband transitions from
s-states to higher-order states. The s-states give channelling, and hence tend to have small
elastic scattering angles, but are the main contributors to TDS. The higher-order Bloch waves
were believed to elastically scatter to higher angles than the s-states, but contribute less to
TDS. If some of these Bloch waves scatter to angles spanned by the LAADF detector, and
strain causes transitions to these, then increased high angle elastic scattering, as found in the
multislice results, could be the cause of strain contrast.
Initially, the main scattering angle of each Bloch wave had to be determined. The electron
wave inside the strained crystals and the resulting diffraction pattern were then compared to
the perfect crystal to detect any changes, before the Bloch wave amplitudes were examined for
evidence of transitions.
8.1 Scattering angles of the Bloch waves
To identify the scattering angle of each Bloch wave, the electron wave was examined in reciprocal
space, where it forms the diffraction pattern measured by the ADF detector. Based on the
Bloch wave approach, it was possible to look at the individual contribution of each Bloch wave
to the final diffraction pattern. It has been found that higher-order Bloch waves, with larger
eigenvalues, kz, scatter to specific angles giving a ring shaped diffraction pattern (fig. 8.1).[33]
58
By determining the position of the peak intensity for each Bloch wave, the scattering angle could
be obtained. An overall increase in the scattering angle with kz was found (fig. 8.2).
Figure 8.1: Contribution of the 23rd Bloch wave in GaAs to the diffraction pattern. Reproduced
from Cosgriff et al.[33]
Figure 8.2: Scattering angles from a 200A Si sample without absorption, against kz values for
Ki = [000].
This data can be tied to the dispersion surfaces, where the low kz Bloch waves were non-
dispersive, whereas the higher kz waves were dispersive (fig. 3.2). The same is seen here where
the low kz Bloch waves give small scattering angles and the high kz Bloch waves scatter to larger
angles.
59
8.2 Changes in electron wave intensity and diffraction pattern
8.2.1 Unstrained crystal
To assess electron wave intensity and diffraction pattern changes in strained samples, the perfect
crystal first had to be simulated. Due to channelling oscillations within the space of the crystal
(a) Electron wave intensity. (b) Log of electron wave intensity.
(c) Diffraction pattern. (d) Log of diffraction pattern.
Figure 8.3: Perfect Si crystal of 73A thickness. Dotted line represents the LAADF detector.
(fig. 7.4b), the intensity peaked at a 20A depth then dispersed (fig. 8.3a, 8.3b). In the diffraction
pattern, an intensity peak at the centre was observed (fig. 8.3c), arising from the majority of
the electrons populating the low-order Bloch waves (fig. 8.2). The degree of elastically-scattered
electrons reaching the LAADF detector was visualised by indicating the position of the detector.
It is important to note that the total signal to the detector is not shown as TDS is omitted. The
logarithmic-scale plots highlighted that even without strain, there was some elastic scattering
to high angles (fig. 8.3b and 8.3d).
60
8.2.2 Simple shear
Simple shear was calculated in a sample with 1.92A slices displaced by 0.0054A, giving an
overall shear of 0.4A in the 73A thick sample. The most notable difference in the electron wave
intensity with respect to the perfect crystal was the deviation of the wave from the column
near the bottom surface (fig. 8.4a and 8.4b). This was manifested in reciprocal space by the
(a) Electron wave intensity. (b) Log of electron wave intensity.
(c) Diffraction pattern. (d) Log of diffraction pattern.
Figure 8.4: Si crystal of 73A thickness with simple shear.
diffraction pattern no longer being centered (fig. 8.4c). Quantitative changes in the diffraction
pattern were assessed by subtracting the pattern of the perfect crystal (fig. 8.5a). This showed
a decrease in intensity at the centre and an off-centre increase in intensity, at a lower angle than
the inner angle of the LAADF detector. To examine changes in the LAADF detector specifically,
a matrix representing the detector was multiplied with the subtracted diffraction pattern. The
result showed an overall decrease in intensity compared to the perfect sample (fig. 8.5b). The
decrease was at odds with the multislice results, potentially indicating an increase in absorption
61
due to re-excitation of the s-states (explained further in the following section).
(a) Overall change. (b) Change in LAADF detector.
Figure 8.5: Change in diffraction pattern intensity between simple shear sample and perfect
crystal.
8.2.3 Spherical strain
Spherical strain was simulated as for multislice, with a B doped Si sample of 73A thickness
and dopant depth of 1/2. Initial interband calculations were performed with the real strain
constant, c=1.4A3,[11] and the probe positioned over the first nearest neighbour colum (fig.
6.4). Although unexpected dips in intensity for this position had been observed under certain
conditions, using c=1.4A3
or 5.4A3
and a dopant at 1/2 thickness would avoid this.
An intensity peak below the dopant was observed which supported the theory of re-excitation
of the s-states (fig. 8.6a). However, an increase in the elastic scattering to the LAADF detector
relative to the perfect crystal was also observed (fig. 8.6d and 8.3d). To verify these contradicting
results, total intensity over the field of view was calculated. This was found to increase with
depth, highlighted by an order of magnitude increase in intensity compared to the perfect crystal
(fig. 8.6a and 8.3a). The result was unphysical, indicating an error in the interband code and
likely invalidating the results obtained for the two strain models.
62
(a) Electron wave intensity. (b) Log of electron wave intensity.
(c) Diffraction pattern. (d) Log of diffraction pattern.
Figure 8.6: Si crystal of 73A thickness, B at 1/2 of thickness.
8.3 Development of a new interband code
Upon examination of the interband code, it was found that one of the assumptions made in the
derivation of the initial equation (eq. 8.1) was invalid in the case of ADF STEM. This was an
important realisation as this equation has been used to describe electron scattering in imperfect
crystals since the publication of ”Electron Microscopy of Thin Crystals”[39] in 1965.
dΨ = 2πi{exp(−2πik(j)z z)}Φ−1{β′g}Φ{exp(2πik(j)z z)}Ψdz (8.1)
To obtain β′g in eq. 8.1, a first order expansion related to g ·R(z) was used (eq. 8.2). This first
order approximation is reasonable in calculations involving few beams, but fails in ADF STEM
simulations where a large number of beams must be used. This is because at high scattering
angles, g is large, making the higher order terms significant.
63
exp(2πig ·R(z)) = 1 + 2πig ·R(z) (8.2)
To avoid the invalid approximation, a new equation for interband transitions was derived by P.
Nellist (eq. 8.3), and a new version of the interband code was developed (appendix B). The new
equation was arrived at by excluding the first order approximation and then integrating eq. 8.1
to find the exact change in excitations between slice n and n+1.
Ψn+1 = {exp(−2πik(j)z z)}Φ−1{exp(2πig ·∆Rn(z))}Φ{exp(2πik(j)z z)}Ψn (8.3)
Ψn+1 and Ψn are the excitations at the top of slice n+1 and n respectively, and ∆Rn(z) is the
change in displacement between slice n and n+1.
8.3.1 Spherical strain employing new interband code
The spherical strain model was simulated with an amplified strain field of c=5.4A3. The total
intensity was found to be slightly less than for the perfect crystal, suggesting an increase in
absorption. Although this could potentially be due to increased re-excitation of s-states, wave
intensity data did not indicate this (fig. 8.7a, 8.4a), thus the drop in intensity was likely a
sampling inaccuracy. Even though further work must be done before the interband code can be
used quantitatively, the qualitative trends in the spherical strain model seemed reasonable.
The wave intensity resembled that of the perfect crystal, with a channelling peak around 20A
(fig. 8.7a, 8.7b), but spread out after passing the dopant. At the depth of the dopant, the
wave appeared to have pixel-like shifts (fig. 8.7b). This artefact arose because of the way the
interband equation calculates strain: instead of displacing the slice, the illuminating wave at
the top of each slice is shifted. As such, the movement of the wave is seen, not the movements
of the slices. This could be removed by shifting the wave-pixels back by the same displacement
seen by the slice at this depth.
The logarithmic plot of the diffraction pattern showed an increase in scattering to the LAADF
detector (fig. 8.7d). To highlight changes, the diffraction pattern of the perfect crystal was
64
again subtracted. As seen for simple shear, the peak intensity was shifted off-centre (fig. 8.8a).
However, unlike the simple shear sample, the intensity on the LAADF detector increased (fig.
8.8b).
(a) Electron wave intensity. (b) Log of electron wave intensity.
(c) Diffraction pattern. (d) Log of diffraction pattern.
Figure 8.7: Si crystal of 73A thickness, B at 1/2 of thickness, calculated with eq. 8.3.
8.3.2 Interband transitions in the spherical strain field
To assess interband transitions, the contribution of each Bloch wave to the peak intensity change
on the LAADF detector was determined by calculating the moduli at this point (fig. 8.9).
Although the s-state contribution was considerable in both samples, the strained sample had
a large contribution from some of the higher eigenvalue Bloch waves (fig. 8.9b). As higher
eigenvalue Bloch waves have larger elastic scattering angles, this signified an increase in high
angle elastic scattering. Further, the decrease of s-state moduli could imply transitioning from
these to higher eigenvalue states.
65
(a) Overall change. (b) Change in LAADF detector.
Figure 8.8: Change in diffraction pattern intensity between the spherically strained crystal and
the perfect crystal.
(a) Unstrained crystal. (b) Spherically strained crystal.
Figure 8.9: Moduli of the Bloch waves at the LAADF peak intensity point.
8.4 Importance of Bloch wave results
The conclusion that strain contrast arises from increased elastic scattering to high angles due
to interband transitions from s-states to higher-order Bloch waves explains the multislice ob-
servations. Increased contrast in the LAADF detector arises from increased elastic scattering
to those angles, whilst the decrease in the HAADF results from electrons transitioning out of
the s-states and decreasing TDS. Further, the thickness and dopant position dependencies are
explained as s-state electrons are more likely to have thermally scattered with increasing depth,
preventing transitions to higher-order Bloch waves.
66
Elastic scattering to higher angles agrees with recorded experimental results,[37] and supports
previous suggestions of dechannelling as the cause of strain contrast.[34, 35, 37] However, the
results appeared to disprove the s-state re-excitation theory[8] as no re-excitation was observed.
Still, interband transitions around a dislocation, for which this theory was proposed, would have
to be simulated to confirm this.
The elastic scattering basis for strain contrast draws parallels with beam tilt. The strain causes
the columns to bend such that they are ”tilted” relative to the beam, giving contrast trends
similar to beam tilt.[50] This was further supported by the effect of sample tilt on strain contrast
(section 4.1).[12] The relation to beam tilt suggested that the Ewald sphere construction could
give an indication of the processes at play in strain contrast: by bending the columns due to
strain, the Ewald sphere is tilted so higher-order diffraction spots intersect the sphere, giving
higher angle elastic scattering. However, the Ewald sphere can not descirbe multiple scattering
leading to channelling, thus the interband code is required to accurately simulate strain effects.
8.5 Summary
The initial interband code proved to be based on an invalid assumption. Hence a new equation
and code omitting this was developed that gave a qualitative description of strain contrast and
interband transitions. Strain contrast was concluded to result from increased elastic scattering
to high angles due to transitions from the s-states to higher-ordrer Bloch waves.
67
Chapter 9
Conclusion and future work
The aim of this project was to explain strain contrast in ADF STEM images by using two
different simulation approaches: the multislice approach and the Bloch wave approach. The
multislice simulations showed that B doped Si gives an intensity increase in the LAADF detector
and a decrease in the HAADF detector compared to pure Si. Strain contrast was also found to
depend on sample thickness and dopant position within the sample. The first indication that
strain contrast comes from increased elastic scattering to high angles was shown by comparisons
of the multislice results with pure elastic scattering and pure absorption. This was confirmed
through interband transition calculations which showed that electrons transition from the s-state
Bloch waves to Bloch waves with larger elastic scattering angles. This explained the increase
in the intensity on the LAADF detector since this detector can measure high angle elastic
scattering. The decrease in intensity on the HAADF detector was due to the transitions out of
the s-states which are the primary contributors to TDS. As part of this work, a new equation for
excitation changes in an imperfect crystal was derived and an interband code developed which
calculates interband transitions including a large number of Bloch waves.
In future the interband code could be used to calculate the intensity contribution from strain
on an ADF detector, which would allow Z-contrast to be decoupled from strain contrast. Before
the interband code could be employed in quantitative analysis, further verification of the code is
needed. This could be done by comparison of results to hand calculations of simple systems, with
or without absorption, and to literature results, in addition to a section by section inspection
of the code. Development of a code which simulates the final ADF STEM image would also be
68
beneficial, producing images that could be compared to real microscope images. Production of
this final code was begun but could not be completed due to computational demands and time
restraints. Finally, it would be useful to test applicability and versatility of the code written for
this project, and the new equation it was based on, by testing it on large strain fields, such as
a dislocation.
Only a few defects were investigated in this project, however, there are a vast number of defects
which induce strain contrast. Future work that simulates other defects would be beneficial,
especially studying dislocations along the beam direction to see if the explanation put forth
by Perovic et al.[8] is valid. The results in this project showed no re-excitation of the s-state,
however this might change in the case of a dislocation. It could also be interesting to incorporate
TDS into the Bloch wave code as a step towards a model involving all possible scattering events.
Finally, investigation of more variables involved in strain contrast could provide a more detailed
model, and explanation, of strain contrast.
69
Chapter 10
Project management
The original objectives of this project, as stated in the Project Management Form 1 (PMF1),
were to explain the origin of strain contrast in ADF STEM and to find a way to decouple strain
contrast from Z-contrast. This was to be done entirely by computational simulations of the
strain contrast using two different approaches: multislice and Bloch wave. The two methods
had quite different requirements in terms of work to be done. The multislice simulations involved
making a model of the sample and learning to use the existing simulation software. The Bloch
wave simulations, however, required writing the simulation codes from start to finish.
Initial plan
Producing an initial plan for this project was problematic. Due to limited programming ex-
perience and limited knowledge regrading the mathematical theory of electron microscopy, it
was difficult to estimate the time required to understand the theoretical aspect of the project
and the time involved in writing the various codes. Planning was also complicated due to the
interdependency of the codes. Until one aspect of the code was working, I could not progress
to the next. At the beginning I therefore relied on the milestones and timeline set down by my
supervisor (see PMF1). Later on, the PMF1 was used as my long term plan, with short term
plans resulting from weekly meetings with my supervior.
The initial plan was to complete all multislice simulations before the end of Michaelmas. We
also wished to have a working Bloch wave code which simulated the electron wave including
70
probe illumination and absorption by this point. Hilary term would then be devoted to writing
the interband code. Work on this code was initially given a whole term because it had never
been attempted before and the timescale of this task was therfore unpredictable. The final term
was intended for write up and collecting results.
Project review
The first two months of the project was spent understanding the theory of electron microscopy
and learning to use MATLAB. Most of my knowledge about MATLAB was gained through trial
and error but I also attended a course at the IT Services. Early in the project I started modelling
the strained sample in MATLAB. This proved to be one of the easier tasks of this project and
provided a great opportunity to familiarise myself with MATLAB and CrystalMaker. In the
early stages I received a large amount of guidance from my research group in the use of MATLAB
and the multislice simulation softwares µSTEM and QSTEM. However, with time, my coding
became self-sufficient. The interband code especially was entirely my work since no one in my
group works with this.
The aim to finish the multislice simulations before the end of Michaelmas was not realised.
This was due to several setbacks. Firstly, QSTEM turned out to be computationally slow when
running simulations on samples as thick as 500A. As a result, I decided to only use µSTEM
since it was a great deal faster and provided more accurate results.[27] Secondly, to create the
input file for µSTEM, CrystalMaker had to be used, which resulted in errors in the model due
to CrystalMaker repeating the crystal structure. Thirdly, I also had some issues detecting strain
contrast in my models and it took a few weeks before it was realised that the strain in the sample
had to be amplified. These issues delayed the simulations which meant that useful results were
not obtained until the beginning of Hilary. Once I had obtained reasonable results however,
collecting results for different parameters was quite quick as the simulations could be run in
parallel to other work.
Due to the delays in the multislice simulations, the work on the Bloch wave code had to be
done simultaneously. The first task in building the code was to debug the translated simCBED
code since it was not giving reasonable results when I first got it. The original code was written
in the 1970’s without many comments, presenting a substantial task and risk. After nearly
71
two months spent understanding every line of the code, the mistake was identified as a type
error mistaking a single j for an i in a 600 lines code. Once the error was found, the work on
writing the codes for the electron waves could be commenced. The work on the Bloch wave
code progressed quickly and by the end of Michaelmas both planewave and probe illumination
had been included. Absorption still needed to be incorporated, but work on this had already
commenced and was not a great concern. The first version of the interband code was completed
during Hilary. However, some difficulties were encountered when trying to introduce spherical
strain field into the code. This resulted in a slight delay which meant that the results from the
interband code with a spherical strain field was not obtained until Trinity. Due to this delay
and the fact that collecting results with a higher number of beams would take time, the second
main objective of decoupling strain contrast from Z-contrast, could not be realised in the time
available. Finally, about a two weeks before the submission deadline, it was realised that the
results from the interband code were incorrect. Thus within a week, a new interband transition
equation had been derived and implemented into the code, so that a final set of results could be
obtained. Due to the computational time involved in running the code, it was not possible to
try more than a single system in the updated code.
Throughout the project, I had meetings with my supervisor to discuss my progress. In prepa-
ration for these meetings I made power point presentations of the results I had gained since
our last meeting. This was a way for me to continuously analyse my results and use them to
determine what my next steps should be. The meetings were also helpful because I often did
not know what to expect from the code output. My supervisor was able to provide useful advice
on what the results should be and why, as well as how to break down my codes to find potential
errors.
The write up was begun towards the end of Michaelmas so that I had a first draft of the literary
review at the beginning of Hilary. From then on I was continuously writing sections whilst codes
were running. The result sections were not formally started until Trinity, however they were
planned as results were obtained.
72
Difficulties and learning outcomes
This project has been a good learning experience, especially in software development. My lack
of coding experience meant that my initial plan for the project was inconsistent with reality
but weekly planning helped me keep on track for the main goal. One of the risks I had not
considered was that due to the dependency of the codes, an error in the initial code would impact
all subsequent codes. Another issue was that the changes made to a code could introduce errors.
For this reason I tried to document each iteration of the code such that I could return to these
if I needed. As it turned out, I needed better vigilance in this task, though I was hampered
by not knowing what information would be useful in the long run. In hindsight, I realise that
what would have been most beneficial would have been to keep a detailed log of the changes
made to each code, saving them with new names each time and using the lab book to note down
the changes made. Another idea would have been to save the results from each simulation so
that I could have used previous results, instead of having to run codes again. With respect to
the write up, I found it difficult to know when to stop pursuing results and start writing which
meant that the writing got slightly neglected at certain stages and I felt fairly rushed towards
the end trying to produce the final code and write up. However, this did mean that I was able
to include some very good results at the end. Ultimately, I think my project management was
sufficient to keep my project on track, although I have learned a great deal that I would take
with me if I were to go through a similar process again.
73
74
75
76
77
78
Appendices
79
Appendix A
Multislice simulation setup
parameters
Microscope setup
• Accelerating voltage: 200kV.
• Aberrations: Set to zero for simplicity.
• Defocus: Probe focused at sample surface.
Sampling setup
• Supercell dimensions
– All supercells were set to 21.7A x 21.7A in the dimensions perpendicular to the beam,
except for the dislocation supercell which was set to 32.6A x 32.6A.
• Phonon sampling
– The DWF was calculated at 300K based on constants found in literature[51].
– 20 phonon runs were included, meaning that the results depict an average over 20
different vibration displacements.
• Reciprocal and real space samplings
80
– Reciprocal space sampling was set to 512x512 pixels (max. scattering angle: 195mrad)
for all samples, except the dislocation sample, which had a 1024x1024 pixels sampling
(max. scattering angle: 257mrad).
– The probe sampling was set to 0.31A/pixels for all simulations.
• Higher Order Laue Zone (HOLZ) scattering
– Included for accuracy although their effect was believed to be negligible[47].
• Slicing
– Manual slicing was used, to confirm that each slice contained a single atom layer. For
the dislocation, however, manual slicing would be difficult and time consuming, so
the slices were set by µSTEM.
Thermal diffuse scattering (TDS)
It was mentioned that incorporating TDS into multislice simulations can be done using two
different approaches; AP or frozen phonon, called quantum excitation of phonons (QEP) in
µSTEM (section 3.2.1). QEP has been shown to give more accurate results[27] and was therefore
used in most of the µSTEM simulations, except for when the intention was to only look at elastic
Bragg scattering.
81
Appendix B
The interband transition code
The interband transition code based on the new interband transition eqauation, derived by P.
Nellist (eq. 8.3).
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Interband t r a n s i t i o n s wi th probe i l l u m i n a t i o n% Based on new in te rba nd t r a n s i t i o n equat ion d e r i v e d by P. N e l l i s t%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Clear memory%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++clear a l lload Bloch var . mat
s t r a i n f i e l d =2;
po in t s=s ize ( or1 , 1 ) ;Nobloch = s ize ( g rea l , 2 ) ;
g0Index=f ix ( ( Nobloch +1)/2) ;
dz=t s t ep ;
%Index f o r Ki =[0 ,0 ,0 ]Ki0Index=f ix ( ( po in t s +1)/2) ;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Parameter s e t up%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
gdRz = zeros ( Nobloch , Nobloch , nth ick +1);DeltaRz = zeros ( nth ick +1 ,3) ;
82
exp gDeltaRz = zeros ( Nobloch , Nobloch , nth ick +1);
arg = zeros ( Nobloch , 1 ) ;kzz = zeros ( Nobloch , 1 ) ;exp arg = zeros ( Nobloch , 1 ) ;abs fac = ones ( Nobloch , 1 ) ;abs fac neg = ones ( Nobloch , 1 ) ;
d i ag kz = zeros ( Nobloch , Nobloch ) ;d iag negkz = zeros ( Nobloch , Nobloch ) ;
s tep4 = zeros ( Nobloch , Nobloch ) ;
M = zeros ( Nobloch , Nobloch ) ;dI = zeros ( Nobloch , 1 ) ;r e c o r d d I = zeros ( Nobloch , nth ick +1, po ints , po in t s ) ;
r e c o r d e x c i t e = zeros ( Nobloch , nth ick +2, po ints , po in t s ) ;
ewave state = zeros ( Rsampling , Rsampling , Nobloch ) ;Bloch ewave = zeros ( Rsampling , Rsampling , Nobloch , nth ick +1);Bewave = zeros ( Rsampling , Rsampling , Nobloch , nth ick +1);ewave = zeros ( Rsampling , Rsampling , nth ick +1);
modulus e = zeros ( Rsampling , Rsampling , nth ick +1);i n t e n s e e = zeros ( Rsampling , Rsampling , nth ick +1);modulus state = zeros ( Rsampling , Rsampling , Nobloch , nth ick +1);i n t e n s e s t a t e = zeros ( Rsampling , Rsampling , Nobloch , nth ick +1);
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% C a l c u l a t e e i g e n v e c t o r s%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
c s t o r ( : , : , : , : ) = n c s t o r r ( : , : , : , : ) + ( sqrt (−1).∗ n c s t o r i ( : , : , : , : ) ) ;d s to r ( : , : , : , : ) = d s t o r r ( : , : , : , : ) + ( sqrt (−1).∗ d s t o r i ( : , : , : , : ) ) ;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Set up probe%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Ki ( : , : , 1 ) = or1 ( : , : ) . / a ;Ki ( : , : , 2 ) = or2 ( : , : ) . / b ;Ki ( : , : , 3 ) = or3 ( : , : ) . / c ;
Ki dot Ro = Ro ( 1 ) . ∗Ki ( : , : , 1 ) + Ro ( 2 ) . ∗Ki ( : , : , 2 ) + Ro ( 3 ) . ∗Ki ( : , : , 3 ) ;probe ( : , : ) = exp ( (2∗ pi∗sqrt (−1)) .∗Ki dot Ro ) . ∗ Aperture ( : , : ) ;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Give r e a l v a l u e s to g
83
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
grea lu (1 , : )= g r e a l ( 1 , : ) . / a ;g r ea lu (2 , : )= g r e a l ( 2 , : ) . / b ;g r ea lu (3 , : )= g r e a l ( 3 , : ) . / c ;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Set up d i sp lacement f i e l d%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%Decide which type o f d i sp lacement f i e l d
i f s t r a i n f i e l d==1
dRz = 0 . 0 0 5 4∗ [ 0 , 0 , 1 ] ;
tcount =0;for z=t s t a r t : t s t ep : ( ( nth ick ∗ t s t ep )+ t s t a r t )
tcount=tcount +1;for i gg =1:Nobloch
gdRz ( igg , igg , tcount )=( grea lu (1 , i gg )∗dRz ( 1 , 1 ) ) . . .+( g r ea lu (2 , i gg )∗dRz(1 ,2))+( g rea lu (3 , i gg )∗dRz ( 1 , 3 ) ) ;
endend
e l s e i f s t r a i n f i e l d==2%Choosing p o s i t i o n o f dopant atom% Give d i s t a n c e in angstroms from i n i t i a l probe p o s i t i o n f o r x and y% and g i v e depth o f dopant in angstroms ( zero i s a t the entrance% s u r f a c e )Dopant x=0;Dopant y =0.68;Dopant z =36.48;
%S t r a i n cons tantconst =5.4 ;
%Find dRz[ gdRz , dRz , Rz]= s t r a i n 5 (Ro , grea lu , Dopant x , Dopant y , Dopant z , . . .
t s t a r t , t s tep , nthick , Nobloch , const ) ;
endtcount =0;
for z=t s t a r t : t s t ep : ( t s t a r t +(nth ick ∗ t s t ep ) )tcount=tcount +1;i f z==t s t a r t
DeltaRz ( tcount ,3)=Rz( tcount ,3)−0;else
DeltaRz ( tcount ,3)=Rz( tcount ,3)−Rz( tcount −1 ,3) ;end
84
for i gg =1:NoblochgDeltaRz=( grea lu (1 , i gg )∗DeltaRz ( tcount , 1 ) ) . . .
+( g r ea lu (2 , i gg )∗DeltaRz ( tcount , 2 ) ) . . .+( g r ea lu (3 , i gg )∗DeltaRz ( tcount , 3 ) ) ;
exp gDeltaRz ( igg , igg , tcount)=exp ( (2∗ pi ∗( sqrt (−1)))∗ gDeltaRz ) ;end
end
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Sum over d i f f e r e n t Ki to g e t probe i l l u m i n a t i o n%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
for sweep1=1: po in t sfor sweep2=1: po in t s
d i s p l a y ( [ num2str( sweep1 ) , ’ / ’ ,num2str( sweep2 ) ] ) ;
tot ewave = zeros ( Rsampling , Rsampling , nth ick +1);%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% C a l c u l a t i n g new e x c i t a t i o n s and e l e c t r o n wave%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
tcount =0;
for z=t s t a r t : t s t ep : ( ( nth ick ∗ t s t ep )+ t s t a r t )
d i s p l a y ( z ) ;
tcount=tcount +1;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% C a l c u l a t e Bloch waves f o r p e r f e c t c r y s t a l% Returns Cgexp(−2∗ p i ∗ i ∗gR) exp(−2∗ p i ∗ i ∗KiR)%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
i f z==t s t a r t % So only c a l c u l a t e t h i s once s i n c e doesn ’ t depend on z[ Bloch wave ]= b l o c h c a l c ( c s tor , or1 , or2 , or3 , g rea l , a , b , c , Rstart , . . .
Rend , Rsampling , sweep1 , sweep2 , Nobloch , vx , vy ) ;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Generate exp ( kz ) f o r e l e c t r o n waves and exp ( kz ) d i a g o n a l s%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
for j =1: Noblocharg ( j )=((−( sqrt (−1))∗( z ) )/ (2∗ kk ) )∗ ( e v a l r ( sweep1 , sweep2 , j ) ) ;exp arg ( j )=exp( arg ( j ) ) ;
85
i f im f fkzz ( j )=(( z )/(2∗ kk ) ) . ∗ ( squeeze ( e v a l i ( sweep1 , sweep2 , evkey ( j ) ) ) ) ;
i f kzz ( j )<180abs fac ( j )=exp( kzz ( j ) ) ;abs f ac neg ( j ) = exp(−kzz ( j ) ) ;
endi f abs fac ( j )<1.0e−20
abs fac ( j )=0;abs fac neg ( j )=0;
endendend
for j =1: Nobloch
d iag kz ( j , j )=exp arg ( j )∗ abs fac ( j ) ;
d iag negkz ( j , j )=exp(−arg ( j ) )∗ abs fac neg ( j ) ;
endend
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Set e x c i t a t i o n f o r curren t depth%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
i f z==t s t a r te x c i t e o l d = squeeze ( ds to r ( sweep1 , sweep2 , g0Index , : ) ) ;r e c o r d e x c i t e ( : , tcount , sweep1 , sweep2)= e x c i t e o l d ;
% d i s p l a y ( e x c i t e ) ;else
e x c i t e o l d = exc i t e new ;r e c o r d e x c i t e ( : , tcount , sweep1 , sweep2)= e x c i t e o l d ;
% d i s p l a y ( e x c i t e ) ;
end
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Matrix m u l t i p l i c a t i o n s e c t i o n%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
ctemp = squeeze ( c s t o r ( sweep1 , sweep2 , : , : ) ) ;
i f det ( ctemp)˜=0
86
s tep1 ( : , : ) = ctemp∗ d iag kz ;
s tep2 ( : , : ) = exp gDeltaRz ( : , : , tcount )∗ s tep1 ;
s tep3 ( : , : ) = ( inv ( ctemp ))∗ s tep2 ;
s tep4 ( : , : ) = diag negkz ∗ s tep3 ;
end
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Find e x c i t a t i o n s at the bottom of a s l i c e%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
exc i t e new=step4 ∗ e x c i t e o l d ;
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Find e l e c t r o n wave at t h i c k n e s s z+dz%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Find e x p k z and a b s f a c f o r z+dz%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
for j =1: Noblocharg ( j )=((−( sqrt (−1))∗( z+dz ) )/ (2∗ kk ) )∗ ( e v a l r ( sweep1 , sweep2 , j ) ) ;exp arg ( j )=exp( arg ( j ) ) ;
i f im f fkzz ( j )=(( z+dz )/(2∗ kk ) ) . ∗ ( squeeze ( e v a l i ( sweep1 , sweep2 , evkey ( j ) ) ) ) ;
i f kzz ( j )<180abs fac ( j )=exp( kzz ( j ) ) ;abs f ac neg ( j ) = exp(−kzz ( j ) ) ;
endi f abs fac ( j )<1.0e−20
abs fac ( j )=0;abs fac neg ( j )=0;
endendend
for j =1: Nobloch
d iag kz ( j , j )=exp arg ( j )∗ abs fac ( j ) ;
d iag negkz ( j , j )=exp(−arg ( j ) )∗ abs fac neg ( j ) ;end
87
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% C a l c u l a t e wave f o r z+dz%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
for j =1: Nobloch
ewave state ( : , : , j ) = ( exc i t e new ( j )∗ ( exp arg ( j )∗ abs fac ( j ) ) ) . . ..∗ Bloch wave ( : , : , j ) ;
Bloch ewave ( : , : , j , tcount ) = ewave state ( : , : , j ) ;
tot ewave ( : , : , tcount ) = tot ewave ( : , : , tcount ) + ewave state ( : , : , j ) ;
end
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% I n t e g r a t e over Ki%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Bewave ( : , : , : , tcount)= Bewave ( : , : , : , tcount)+probe ( sweep1 , sweep2 ) . . ..∗ Bloch ewave ( : , : , : , tcount ) ;
ewave ( : , : , tcount)=ewave ( : , : , tcount)+probe ( sweep1 , sweep2 ) . . ..∗ tot ewave ( : , : , tcount ) ;
end
r e c o r d e x c i t e ( : , tcount +1,sweep1 , sweep2)=exc i t e new ( : ) ;
endend
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% Find modulus and i n t e n s i t y%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
for T=1:( nth ick +1)
modulus e ( : , : ,T)=ewave ( : , : ,T) . ∗ ( conj ( ewave ( : , : ,T) ) ) ;i n t e n s e e ( : , : ,T)=modulus e ( : , : ,T) . ∗ modulus e ( : , : ,T) ;
for j =1: Nobloch
modulus state ( : , : , j ,T)=Bewave ( : , : , j ,T) . ∗ ( conj ( Bewave ( : , : , j ,T) ) ) ;i n t e n s e s t a t e ( : , : , j ,T)=modulus state ( : , : , j ,T) . ∗ modulus state ( : , : , j ,T) ;
endend
88
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