Detecting strain in scanning transmission electron microscope images. Part II Thesis

Transcription

1 Detecting strain in scanning transmission electron microscope images Part II Thesis Ina M. Sørensen Mansfield College, Oxford 10th of June, 2015

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3 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

4 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

5 Contents Abstract 2 Acknowledgements 3 Table of Contents 4 1 Introduction and engineering context 8 2 The Scanning Transmission Electron Microscope The design of the STEM The detectors Thermal Diffuse Scattering Incoherent imaging Summary Simulating STEM images Electron scattering The multislice approach The incorporation of TDS and the frozen phonon model

6 3.3 The Bloch wave approach Dispersion surface Probe illumination TDS in the Bloch wave approach From electron wave function to STEM image Summary Strain contrast in ADF STEM Dependence of strain contrast on experimental variables Modelling strain contrast Interband transitions Mathematical definition of interband transitions Explaining strain contrast with interband transitions Summary Multislice simulation setup Creating the model for multislice simulations Silicon crystal Single boron dopant atomic%B-Si Dislocation Simulation parameters Exploring the parameter space for strain contrast

7 5.4 Image analysis Multislice simulation results Accuracy of multislice simulations The parameter space of strain contrast Explaining strain contrast using multislice Summary Creating the Bloch wave code Calculating eigenvectors and eigenvalues Dispersion surface Bloch waves Electron wave Electron wave with planewave illumination Electron wave with probe illumination Absorption Interband Transitions The strain field Summary Results of the Bloch wave approach Scattering angles of the Bloch waves Changes in electron wave intensity and diffraction pattern

8 8.2.1 Unstrained crystal Simple shear Spherical strain Development of a new interband code Spherical strain employing new interband code Interband transitions in the spherical strain field Importance of Bloch wave results Summary Conclusion and future work Project management 70 Appendices 79 A Multislice simulation setup parameters 80 B The interband transition code 82 Bibliography 89 7

9 Chapter 1 Introduction and engineering context The Scanning Transmission Electron Microscope, or STEM, can produce atomic resolution images. 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

10 (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

11 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

12 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

13 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

14 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 scattering (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 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( 16π2 u 2 ssin 2 θ B λ 2 ) (2.1) 13

15 where u 2 s is the mean square displacements of the atoms due to vibrations, θ B 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] 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

16 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

17 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

18 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 transmitting through a slice and then propagating to the next slice. This process is described by eq. 3.1 ψ n+1 (x, y) = p n (x, y, z n ) [t n (x, y)ψ n (x, y)] (3.1) where ψ n and ψ n+1 are the electron wave functions before and after slice n respectively, p n is the propagation function, t n is the transmission function, z n is the thickness of slice n and indicates a convolution.[23] The transmission function, t n (x, y) describes the interactions of the electron wave with the atoms as it moves through a slice, while the propagation function, 17

19 p n, 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] 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 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

20 movements.[23] As shown by Bethe[22], the electron scattering processes can be described by the time-independent Schrodinger equation (eq. 3.2)[1] 2 Ψ(r) + 8π2 me h 2 [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) = Σ g φ (j) g exp( 2πi(k (j) + g) r) (3.3) where ψ (j) is the j th 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 j th 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 r = R + z (3.4a) (3.4b) where k t and R are the transverse components and k z and z the z-direction components. Equation 3.3 then becomes: ψ (j) (r) = Σ g φ (j) g exp( 2πi(k (j) t + g) R)exp( 2πik z (j) z) (3.5) 19

21 In order to calculate the Bloch waves, φ (j) g and k (j) z must be found. This can be done by reinserting 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: k 2 t U g U h U g (k t + g) 2 U g h U h U h g (k t + h) 2 (3.8) where U g 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) = Σ j α (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

22 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 k (j) z is found to be equal to the incident wave vector, K i, for all Bloch waves, j, and is dependent on K i.[24, 18] The dependency of k (j) z surface plot, which will be discussed later (section 3.3.1). on K i can be presented in a dispersion Finally, inserting equation 3.5 into 3.9 gives the electron wave as follows: Ψ(R, z, K i ) = Σ j φ (j) 0 (K i )Σ g φ (j) g (K i )exp( 2πi(K i + g) R)exp( 2πik z (j) (K i )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 Dispersion surface The dispersion surface is a plot of the allowed k z values for a given K i (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 k z values constitute the y-axis.[31] 21

23 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] Probe illumination When a sample is illuminated by a plane wave, all the incident waves have the same wave vector, K i, as they are all in phase. In probe illumination, there is a range of wave vectors because 22

24 the probe is made up of partial plane waves. The range of K i values is limited by the objective aperture. Probe illumination is represented by eq P (R R 0 ) = A(K i )exp[ 2πiK i (R R 0 )]dk (3.11) where R is the real space variable, R 0 is the probe position, K i is the wave vector of the incident electrons and A(K i ) is the aperture function. If the objective lens has no aberrations, then A(K i ) 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 R 0. Ψ(R, z, R 0 ) = A(K i )exp(2πik i R 0 )Σ j φ (j) 0 (K i )exp( 2πik z (j) (K i )z) (3.12) x Σ g φ (j) g (K i )exp( 2πi(K i + g) R)dK i 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 complex refractive index.[32] Similarly, absorption in STEM can be simulated by complex atomic potentials.[24] ϕ(r) = ϕ (r) + iϕ (r) (3.13a) U g = U g + iu g (3.13b) Eq indicates how the atomic potentials, ϕ, and the corresponding Fourier coefficients, U g, can be described as complex values. The imaginary part of the potential, ϕ (r), is introduced 23

25 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 K i, which is real, only the longitudinal component, k z, 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) 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 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

26 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 inspected, which will be useful when investigating the origin of strain contrast. 25

27 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 susceptible when a defect is introduced. A number of different defects have been studied and, 26

28 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 (Z Ge > Z Si ) increased with increasing inner angle of the detector, whilst in the C-containing layers (Z C < Z Si ), 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 200Å 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 crystalline and amorphous silicon in a 150Å thick sample was bright in a low-angle ADF (LAADF: 20-64mrad) detector and dark in a high-angle ADF (HAADF: mrad) detector. For samples below 100Å, 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 displacements 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

29 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 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

30 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 z (j) z)}φ 1 {β g}φ{exp(2πik z (j) 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, dψ 1 is the change in slice 1 and is related to Ψ 1. The excitation at the top of slice 2, Ψ 2, is given by Ψ 1 + dψ 1, 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 z (1) z) exp(2πik z (2) z) exp(2πik z (3) z) (4.2) Φ is a matrix containing the eigenvectors (eq. 4.3 for a three-beam case): φ (1) 0 φ (2) 0 φ (3) 0 Φ = 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 dr(z) gr(z) = g dz 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

31 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, R z, which have no effect on image contrast since g drz dz = 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 symmetries. 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 displacements 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, however 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

32 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

33 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 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

34 characteristic of the [110] viewing direction (fig. 5.1). Figure 5.1: Si down [110]. a=b=5.431å and c=3.84å. 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 directions 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 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 r 3 r (5.1) where c=1.4å 3 is the strain constant, r is the vector going from the B dopant to one of the 33

35 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=5.4å 3 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 atomic%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

36 5.1.4 Dislocation An edge dislocation was introduced based on a code by H. Yang using isotropic elasticity displacements. 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 [1 10] 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): mrad 35

37 Sample thickness 73Å 353Å 500Å 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 28 m 2 ) (fig. 5.4b). 36

38 (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

39 Chapter 6 Multislice simulation results The aim of the multislice simulations was to inspect the strain contrast around a single, substitutional 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 21.7Åx21.7Å 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

40 revealed (table. 6.1), confirming that the multislice simulations were highly accurate. Table 6.1: Average intensities and errors Thickness (Å) Detector Average intensity Standard deviation Percentage error 73 LAADF HAADF LAADF HAADF LAADF HAADF 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 73Å 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

41 (a) LAADF (b) HAADF Figure 6.2: Intensity change relative to Si for a 73Å 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 73Å sample than in the 500Å 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 353Å and 500Å 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 73Å 40

42 (a) LAADF Dislocation (b) LAADF amplified B doped Si (c) HAADF Dislocation (d) HAADF amplified B doped Si Figure 6.3: Both samples are 73Å 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

43 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 10.4Å 3 (fig. 6.6). With c=10.4å 3, 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 73Å thick sample (fig. 6.5a and 6.6a), the dopant at 1/2 gave the highest strain contrast whereas, in the 353Å 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

44 (a) LAADF, 73Å thickness (b) HAADF, 73Å thickness (c) LAADF, 353Å thickness (d) HAADF, 353Å thickness (e) LAADF, 500Å thickness (f) HAADF, 500Å thickness Figure 6.5: Percentage contrast changes compared to pure Si for different dopant positions, sample thicknesses and detector angles. 43

45 (a) LAADF, 73Å thickness (b) HAADF, 73Å thickness (c) LAADF, 353Å thickness (d) HAADF, 353Å thickness Figure 6.6: Percentage contrast changes compared to pure Si, with c=10.4å 3 44

46 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

47 (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 73Å thick sample, c=5.4å 3. 46

48 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

49 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 z (j) (K i ), and eigenvectors, φ (j) g (K i ), are defined for each beam (giving the number of j and g) and partial planewave wave vectors, K i (eq. 7.1), hence the accuracy of the calculation is dependent on the number of beams and K i 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 K i 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

50 Ψ(R, z, R 0 ) = A(K i )exp(2πik i R 0 )Σ j φ (j) 0 (K i )exp( 2πik z (j) (K i )z) (7.1) x Σ g φ (j) g (K i )exp( 2πi(K i + g) R)dK i 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 K i 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 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 k z values differed, 0.01Å 1 for the s-states was reasonable since this would give 49

51 channelling oscillations with a period of 100Å.[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

52 7.3.1 Electron wave with planewave illumination To calculate the electron wave with plane wave illumination eq had to be solved for K i =[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, K i ) = Σ j φ (j) 0 (K i )exp( 2πik z (j) (K i )z)[σ g φ (j) g (K i )exp( 2πi(K i + 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 (K i ), and the eigenvalue term, exp( 2πik z (j) (K i )z), and summed over all Bloch waves, j, to give the electron wave with plane wave illumination (fig. 7.3). (a) Electron wave at 200Å 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 130Å depth and then decreases again. 51

53 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 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(K i ), where each matrix element represents a certain K i and is set to 1 or 0 depending on whether this K i 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 K i was approximated to a sum over K i. P (R R 0 ) = A(K i )exp[ 2πiK i (R R 0 )]dk (7.3) Ψ(R, z, R 0 ) = A(K i )exp(2πik i R 0 )Σ j φ (j) 0 (K i )exp( 2πik z (j) (K i )z) (7.4) x Σ g φ j g(k i )exp( 2πi(K i + g) R)dK i 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 200Å depth. (b) Electron wave down the crystal thickness. Figure 7.4: Electron wave intensity with probe illumination for Si[110], 200kV. 52

54 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 (K i ) were used correctly. This was done by separating the exponential with k (j) z (K i ) as in eq To verify the code, a plot of the electron intensity against depth in GaAs (fig. 7.5) was compared to fig 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

55 7.4 Interband Transitions Changes in the excitations of the Bloch waves due to strain (section 4.3) was calculated based on eq dψ = 2πi{exp( 2πik z (j) z)}φ 1 {β g}φ{exp(2πik z (j) 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 The strain field One of the greatest difficulties in producing the interband code was determining the displacements 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) = d t z and hence β g = d t g (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

56 (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 approximation 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 r 3 r (7.6) Introducing the spherical strain field in the interband code was complicated because the calculation 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

57 (a) Strain parameters (b) Strained column Figure 7.7: Schematic of spherical strain model. Figure 7.8: R(z) and dr(z) dz for 73Å thick Si, B dopant at z=36.48å. 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

58 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

59 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, k z, scatter to specific angles giving a ring shaped diffraction pattern (fig. 8.1).[33] 58

60 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 k z was found (fig. 8.2). Figure 8.1: Contribution of the 23 rd Bloch wave in GaAs to the diffraction pattern. Reproduced from Cosgriff et al.[33] Figure 8.2: Scattering angles from a 200Å Si sample without absorption, against k z values for K i = [000]. This data can be tied to the dispersion surfaces, where the low k z Bloch waves were nondispersive, whereas the higher k z waves were dispersive (fig. 3.2). The same is seen here where the low k z Bloch waves give small scattering angles and the high k z Bloch waves scatter to larger angles. 59

61 8.2 Changes in electron wave intensity and diffraction pattern 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 73Å thickness. Dotted line represents the LAADF detector. (fig. 7.4b), the intensity peaked at a 20Å 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

62 8.2.2 Simple shear Simple shear was calculated in a sample with 1.92Å slices displaced by Å, giving an overall shear of 0.4Å in the 73Å 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 73Å 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

63 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 Spherical strain Spherical strain was simulated as for multislice, with a B doped Si sample of 73Å thickness and dopant depth of 1/2. Initial interband calculations were performed with the real strain constant, c=1.4å 3,[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=1.4å 3 or 5.4Å 3 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

64 (a) Electron wave intensity. (b) Log of electron wave intensity. (c) Diffraction pattern. (d) Log of diffraction pattern. Figure 8.6: Si crystal of 73Å 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 dψ = 2πi{exp( 2πik z (j) z)}φ 1 {β g}φ{exp(2πik z (j) 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

65 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. Ψ n+1 = {exp( 2πik z (j) z)}φ 1 {exp(2πig R n (z))}φ{exp(2πik z (j) z)}ψ n (8.3) Ψ n+1 and Ψ n are the excitations at the top of slice n+1 and n respectively, and R n (z) is the change in displacement between slice n and n Spherical strain employing new interband code The spherical strain model was simulated with an amplified strain field of c=5.4å 3. 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 20Å (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

66 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 73Å thickness, B at 1/2 of thickness, calculated with eq 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

67 (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 observations. 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