MAT-VET-F 20001
Examensarbete 15 hp Juni 2020
Reducing the dynamical diffraction effects in EMCD by electron beam precession
Arvid Forsberg
Teknisk- naturvetenskaplig fakultet UTH-enheten
Besöksadress:
Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0
Postadress:
Box 536 751 21 Uppsala
Telefon:
018 – 471 30 03
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
Abstract
Reducing the dynamical diffraction effects in EMCD by electron beam precession
Arvid Forsberg
Dynamical effects are known to reduce the signal to noise ratio in EMCD
measurements making them highly dependent on sample thickness. Precession of the electron beam has been shown to reduce these effects in ordinary crystallography.
This work investigates precession of the electron beam as a method of reducing the dynamical effects in EMCD using simulations. Simulations are run on BCC Fe in two and three beam conditions. The results show significant effects on the EMCD signal.
However, whether these improve the signal quality seems dependent on sample orientation and thickness range. The initial findings reported here are promising and motivate further research.
Examinator: Martin Sjödin Ämnesgranskare: Ocean Cheung Handledare: Ján Rusz
Contents
1 Introduction 3
1.1 Project goal . . . 3
2 Theory 3 2.1 Transmission electron microscopy (TEM) . . . 3
2.2 Crystal structures and atomic planes . . . 4
2.3 Bragg’s law . . . 5
2.4 Three and two beam conditions . . . 6
2.5 X-ray magnetic chiral dichroism (XMCD) . . . 8
2.6 Electron energy loss magnetic chiral dichroism (EMCD) . . . 9
2.7 Dynamical effects in TEM and EMCD . . . 11
2.8 Precession electron diffraction (PED). . . 12
2.9 Quantifying the quality of an EMCD signal . . . 12
2.10 Simulations . . . 13
3 Method 14 4 Results 16 4.1 Three beam condition . . . 16
4.2 Two beam condition - (110) plane. . . 18
4.3 Two beam condition - (200) plane. . . 20
5 Discussion 22
6 Conclusions 22
7 Populärvetenskaplig sammanfattning av projektet 23
1 Introduction
Mapping magnetic properties of materials is crucial for the development of new lighter, cheaper and more functional composites used in electronics and related areas. Electron energy-loss magnetic chiral dichroism (EMCD) is a relatively new and under development method of measuring magnetic properties of materi- als using the transmission electron microscope (TEM).[1] The precursor, X-ray magnetic chiral dichroism (XMCD), on which the ideas of EMCD build utilizes X-rays instead of electrons and is limited in the exper- imentally achievable spatial resolution. An analogy to XMCD was for a long time thought of as impossible with the available technology, but the proposed method could eventually be verified experimentally.[2] Stud- ies on EMCD are ongoing and one of its greatest challenges is its generic low signal-to-noise ratio. This combined with its dependence on sample thickness due to so-called dynamical effects makes it somewhat unreliable in its present state as magnetic materials could appear non-magnetic.[3,4, 5] Efforts have been made to theoretically investigate the impact of detector position, electron beam type as well as material sample orientation.[6, 7] While these results are welcomed by the experimentalists, the optimal setup will vary with sample thickness. It would be of benefit to be able to use somewhat similar setups regardless of sample thickness. Precession of the electron beam has been shown to reduce dynamical effects in ordinary crystallographic measurements in the TEM and is an established method feasible in many modern TEMs.[8]
Research as of today has not been conducted on beam precession in EMCD measurements, which is the aim of this project.
1.1 Project goal
The goal of this project is to, using simulations, investigate how electron beam precession affects the thickness dependency of the EMCD signal for BCC iron using three different sample orientations.
2 Theory
2.1 Transmission electron microscopy (TEM)
Transmission electron microscopy (TEM) is an effective way of investigating material properties. A high energy beam of electrons (often 100+ kV) is directed at a thin material sample often between 10 and 100 nanometers thin. The electrons interact strongly with the sample, affecting their directions of propagation and velocities. Exactly to which extent the electrons interact with the sample is a probability distribution, just as the wave-like quantum behaviour of the electrons. Most electrons just go directly through the sample, barely interacting with the material due to their high kinetic energy. Many electrons have their directions altered just a bit, while a few have their directions altered more significantly. This is mainly due to the fact that the atoms in the material really are not very dense. Their electron "cloud" allows for incident electrons to have a high chance of passing through.
Measuring the characteristics of the outgoing electrons after the interaction can tell much about the structure and general properties of the material used. The distribution of the outgoing electron directions will form a so called diffraction pattern, where some directions occur more frequently while other form "dark areas".
The brighter areas are generally associated with elastic scattering while darker areas are associated with in- elastic scattering.[9] Electrons elastically scattered do not lose any kinetic energy in the event and therefore their de Broglie wavelength is unchanged. Often, the elastically scattered electrons are scattered in regular directions, forming structured patterns in the diffraction image. This is an effect of the structured order of the atoms in the sample, and large groups of atoms generally together determine the nature of the elastic scattering. Inelastically scattered electrons lose some kinetic energy which is transferred to the sample. The
scattering directions of such electrons are a bit harder to predict since it generally is due to interaction with individual atoms, and not the overall structure of the material. In the diffraction image these can be found between and inside the bright spots that are caused primarily by the elastically scattered electrons.
Elastically scattered electrons generally hold information about distances between atoms in the sample, which is crucial for mapping material structures. On the other hand, inelastically scattered electrons carry more detailed information about the material. This is generally extracted with a technique called electron energy loss spectroscopy (EELS). By measuring the difference in energies between incident and outgoing electrons after sample interaction as well as scattering directions, element specific information such as elec- tronic structure can be understood. Incident electrons which succesfully excite some atom in the sample have their kinetic energy lowered by a corresponding amount. Different elements have characteristic energy levels of the electron shells, so this can be used to for example identify such.
Figure 1: Example of a diffraction pattern image off of a crystalline structure. The brighter areas indicate where electrons are more frequently scattered. Image from Wikimedia Commons.
2.2 Crystal structures and atomic planes
To properly utilize the TEM technique and be able to interpret the resulting diffraction patterns, it is crucial to have an understanding of the structure of the material and how it translates to the diffraction patterns.
In this project iron (Fe) is investigated. Iron is a crystalline material. Such materials are recognized as an ordered structure of repeated smaller groups of atoms. The smallest group of atoms identifiable withinin the crystal is called the unit cell. The structure of the unit cell defines the overall structure of the crystal.[10]
The unit cell of iron is a so called body-centered cubic. It is a cube of side length a = 286.65 pm with one atom in each corner as well as one in the middle, as illustrated in figure2. Since the crystal is constructed entirely unit cells stacked in large numbers, it is often sufficient to define a direction or group of atoms within the crystal only in terms of the unit cell.
In this project, atomic planes are discussed reccurently. Within a crystal, it is possible to identify sev- eral different sets of parllell planes formed by atoms systematically arranged in rows. To index different sets of atomic planes, the unit cell is used. By defining x, y and z directions in the unit cell, different plane orientations are defined by their repective intersections with the axes. Contrary to ordinary coordinates, they are inverted after they have been identified. In crystallography this is common practice and is called
Miller indices. In this project, diffraction off of the sets of atomic planes with Miller indices (110) and (200) are investigated. The first one indicates that in the unit cell, the plane intersects the x and y axises at 1 while it never intersects the z axis. The second index implies that the plane intersects the x-axis at 12 while it does not intersect any other axis. These plane orientations are illustrated in figure3. It is important to remember that to fully visualize the sets of atomic planes in the crystal, many unit cells have to be put together. Also, a Miller index refers to the whole set of planes within the crystal fulfilling that orientation, not single plane.
In TEM, the atomic plane Miller indices are commonly used to refer to different bright spots in the diffrac- tion pattern. That is because different sets of atomic planes can cause scattering diffraction effects, which is further discussed in the next section.
Figure 2: The unit cell of iron, a body centered cubic.
Figure 3: The orientations of the sets of planes (a) (110) and (b) (200) in the BCC unit cell.
2.3 Bragg’s law
An important diffraction phenomenon in TEM is diffraction from parallell planes of atoms. The idea can be thought of much like reflection off of thin foils in optics, although using electrons, it is not exactly reflection taking place, but rather scattering off of atoms. When two coherent plane waves of electrons are incident on two different parallell planes of atoms, the electrons will have probabilities to scatter in many different
directions off of both of the planes. This scattering event is elastic, so the de Broglie wavelength of the electrons does not change. Much like light waves, different electron waves can experience constructive or destructive interference between each other. Bragg’s law (1) describes in what direction this interference will be constructive.[11] It is derived from the path difference between the two electron waves as one of them has to travel a longer distance to hit the second plane of atoms, as illustrated in figure4.
When constructive interference occurs, the incident angle of the electron waves is called the Bragg angle for that particular set of planes and the resulting beam in that direction is often referred to as the Bragg diffracted beam. All in all, the situation is referred to as to fulfill the Bragg condition. The d in Bragg’s law denotes the distance between the planes and λ the de Broglie wavelength of the electrons. n is an integer starting from 1. Higher integers than 1 are referred to as higher order diffraction, but in this project only n = 1 is regarded of. The distance between adjacent planes is easy to derive geometrically from the unit cell. In iron, the (110) planes are a distance √a
2 apart and the (200) planes are a distance a2 apart.
nλ
2d = sin θB (1)
In TEM, the Bragg angle often is not larger than 1◦ and is often conveniently expressed in milliradians (mrad). Although the Bragg condition may be fulfilled for a certain situation, there still will be electrons in the electron beam which do not scatter away from their incoming direction since they have such high kinetic energy. Such electrons form what is referred to as the direct beam and usually the direct and Bragg diffracted beam simultaneously travel through the sample.
Figure 4: Illustration of Bragg’s law which describes when two beams scattered off of two separate parallell planes of atoms experience constructive interference. In reality, there are many more planes stacked together, but the superposition of all planes only form one Bragg diffracted beam. Image from Wikimedia Commons.
2.4 Three and two beam conditions
Generally, diffraction patterns consist of many distinct spots as in figure 1. This is due to both Bragg diffraction off of different crystal planes and also the so called grating effect of the atoms, comparable to the classical double slit experiment. Often, material scientists are interested in some special feature of the material. To investigate such, there are ways to tilt the sample such that some very specific diffraction dominates the diffraction pattern.
The first such orientation handled in this project is a so called three beam orientation. In such an ori- entation, three distinct spots can be distinguished in the diffraction image. One of the spots correspond to ordinary Bragg diffraction from some set of planes and is often referred to as G. The other spot corresponds to Bragg diffraction but from the other side of the planes and is referred to as −G as this beam is scattered in the “opposite” direction. This condition is practically achieved by first identifying which atomic planes should be included in the diffraction image. Then, an axis called the zone axis which is perpendicular to the plane normals has to be identified. Lastly, the sample is tilted such that the incoming beam has a 5◦− 15◦ tilt from the zone axis. In this orientation, the Bragg condition is not perfectly fulfilled for any atomic planes and the grating effect from the atoms is minimized. Although, it is possible to have diffraction spots even though the Bragg condition is not perfectly fulfilled, which is explained with a quantity called the excitation error, s. It can be seen as a measure of the deviation from a perfect Bragg condition.[11] When s = 0 for some set of planes, the Bragg condition is perfectly fulfilled and the intensity of the diffraction spot from these planes is maximized. The intensity of the spot weakens when the excitation error increases. If the excitation error is too big, the intensity approaches zero. To increase or decrease the excitation error for some particular set of planes, the sample needs to be tilted accordingly. In the three beam condition, the exciation error of two diffraction spots is sufficiently small to make their intensities non-negligible together with the direct beam in the middle. A simulated example of a three beam diffraction pattern can be seen in figure5below.
The second case investigated is a so called two beam condition. Contrary to the previous case, the sample is oriented such that the exciation error is exactly zero for a single set of planes in the crystal and greater than zero for all other sets of planes, making their diffraction intensities negligible. Generally, when a two beam condition is set, the sample is first tilted to a three beam condition. From there, the sample is further tilted by the Bragg angle of the set of planes of interest. By doing this, the excitation error of one of the spots in the three beam condition increases while the other one is strongly suppressed. When this condition is fulfilled, two distinct spots can be identified in the diffraction image which is the direct beam and one perfectly Bragg diffracted beam, as shown in figure5.
Figure 5: Example simulated inelastic scattering energy loss diffraction patterns for a (a) three beam and (b) two beam diffraction condition in which diffraction off of some specific set of planes is isolated. Note that the intensities between the bright spots are non-zero. This is due to the inelastically scattered electrons, as explained in section2.1.
2.5 X-ray magnetic chiral dichroism (XMCD)
Before the electron microscopes were invented, mainly X-rays (0.03-3 nm) were used in material science ex- periments aiming to map properties on the atomic level. Just like electrons, as described previously, X-rays can interact with electrons and atoms as well. X-ray transmission can be used to give insight to the inner structure of materials and in X-ray spectroscopy the rays are used to knock out electrons in the material causing radiation to be emitted. This is an established method used for element identification.
There is however a method of investigating magnetic properties of materials using X-rays. This method is called X-ray magnetic chiral dichroism. Dichroism is the phenomenon of how different polarizations of light interact differently with certain materials. The usual setup is to shine a monochromatic circularly polarized X-ray beam through a material sample and measure the intensity of the outgoing beam. Measur- ing the difference in intensity using right and left circularly polarized X-rays allows for calculations on the magnetic moments of the atoms in the material.
Magnetic materials have electron shells which in themselves are split due to electron spin. A circularly polarized X-ray beam is more probable to excite electrons with one specific spin, depending on if the beam is left or right circularly polarized. When atoms are excited, photons get absorbed, decreasing the overall intensity of the outgoing photon beam at the incident wavelength. Often, when transition metals are inves- tigated, these measurements are carried out at the L electron shell of the material. This layer is the first energy level which provides energy level splitting due to electron spin. Also, higher levels are less likely to be excited. To target this shell, the incident X-ray photons need to have sufficient energy to excite these electrons. The X-ray beam can be made very thin, making it possible to pinpoint small areas of a sample.
Practical limitations in the experimental setups of X-ray spectroscopy limit what wavelengths effectively can be utilized. To focus the X-ray beam, so-called Fresnel lenses are used which need to be very precise.
When the beam wavelength becomes very small, these lenses need to be extremely precise, which today is a limiting factor of the spatial resolution achievable with XMCD.
Figure 6: Schematic of a XMCD measurement where M+ denotes right circular polarized X-rays and M− denotes left circular polarized X-rays. The two distinct peaks are due to the spin-orbital interaction splitting of the L energy level in this sample, giving information about the magnetical properties of the atoms. Image from Wikimedia Commons.
2.6 Electron energy loss magnetic chiral dichroism (EMCD)
Electrons have smaller de Broglie wavelengths than the wavelength of photons of same energy. This makes it possible to achieve higher spatial resolution using electrons. TEMs use complicated combinations of magnetic fields to focus the electron beam, which are easier to fine-tune than Fresnel lenses used in X-ray spectroscopy and XMCD. To overcome the spatial resolution limits of XMCD, scientists wanted to figure out ways of using TEMs to map magnetic properties of materials. A method was proposed by Schattschneider et al. in 2006 which later could be experimentally verified.[1, 2] The main idea was to find a reliable analogy to circular polarization using electrons instead of photons. Spin-polarized electron beams already was an established phenomenon, but the available technology could not produce such with sufficient intensity. For that cause, it was long thought to be impossible to design a TEM experiment mimicing the XMCD method.
However, it was found that momentum transfer due to inelastic scattering of the incident and Bragg diffracted electron beams mathematically could be regarded an analogy to the effect of polarized X-rays. For absorption of X-rays on a group of atoms, such as in XMCD, the expression
σ =X
i,f
4π2~αω| hf | · R|ii|2δ(Ei− Ef+ E) (2) over tangibly describes the probability for photons of energy ~ω to be absorbed where is the polarization vector of the photon wave and R is the position of atom i. hf | and |ii are the final and initial states of the atom, Ei and Ef are the initial and excited energies of the atom and E is the energy loss in the event.
Similarly, looking at the expression
∂2σ
∂E∂Ω =X
i,f
4γ2kf
a20q4ki
| hf |q · R|ii|2δ(Ei− Ef+ E) (3)
describes the probability for an incident electron with wave vector kito be inelastically scattered with energy loss E within the small differential solid angle ∂Ω off of a group of atoms with single electrons in their shells and using a dipole approximation for the Coloumb potentials eiq·R ≈ 1 + iq · R. Higher intensities in the diffraction pattern imply that many electrons whose kinetic energy has been lowered by E is scattered into that direction. kf is the wave number of the scattered electron wave and q = kf − ki is the momentum change in the scattering event. This expression is called a double differential scattering cross section.
When an X-ray beam is circularly polarized, the polarization vector takes the form = + i0 which represents two orthogonal E-field vector components with π2 phase shift in between with equal amplitude which gives the circular polarization. To find an analogy to the circular polarization of photons for electrons, we consider the two beam condition in TEM. In this case, we have two strong electron beams which travel through the sample, one the Bragg diffracted beam and the other one the direct beam. The electrons of these two beams share the same wavelength, since the Bragg diffraction is an elastic scattering event. If we consider inelastic scattering taking place after the Bragg diffraction has happened, electrons may deviate from either one of the two beams as they lose kinetic energy due to some interactions in the material. Focus- ing on some arbitrary point away from any bright spots corresponding to elastic scattering in the diffraction pattern, electrons scattered there have either undergone a momentum change of q or q’ each corresponding to inelastic scattering away from the direct beam and the Bragg diffracted beam respectively.
The complete derivation is not made here, but it can be shown for the described situation that if it is assumed that |q| = |q’|, q ⊥ q0and that there is a phase shift of π2 between q and q’, the q in equation (3) can be replaced with q + iq’. This expression roughly tells us how big the probability is to find electrons inelastically scattered with the energy loss E into the semiangle ∂Ω with either the momentum change q or q’. It might seem like we introduced a complex momentum transfer, but it really just is a convenient
mathematical form arising from the dipole approximation and that there are two coherent waves propagating through the sample. The phase shift of π2 comes for free from the Bragg diffracted beam, much like phase shifts during reflections in optics.
The result is convenient, showing a clear similarity to the polarization vector in XMCD. This means that using an electron beam instead of photons and manipulating q and q’ accordingly, the same effects can be achieved as those from manipulating the polarization of X-rays in XMCD. To manipulate q and q’, nothing is done with the electron beam before it interacts with the sample. Instead, the detector needs to be placed at positions where the momentum transfers fulfill the proposed conditions. This is quite contrary to XMCD, where the photon beam is manipulated before it interacts with the material. To mimic the change from right to left circularly polarized light, the detector needs to change position. The detector positions can be seen in figure 7 for a two beam condition. In a three beam condition there will be four areas in the diffraction pattern from which EMCD signal can be extracted. Generally, because of the symmetry of the crystal structure, to get an idea of the behaviour of the total EMCD signal, it is sufficient to study the behaviour of the signal in one of these positions. This is a practice applied in the execution of this project.
The signal which holds information about magnetic properties in the material can be further derived from equation (3) for a two beam approximation. The magnetic information is woven into the imaginary part of a term called the mixed dynamic form factor which emerges.[3] In electron energy loss simulations and experiments, (3) is what is measured. From this, the magnetic information sequentually can be extracted.
In principle, the TEM-EMCD measurement is setup to detect electrons which have had an energy loss cor- responding to excitation of the L-electron shell of the material. Because of relativistic effects the 2p electron shell will be split into two energy levels. When the detector placement is changed to mimic the change of polarization in XMCD, due to magnetism, the likelihood of energy loss processes for one of these levels will be enhanced at one of the two detector positions and suppressed at the other. A reversed situation is observed for the other of the two levels. Essentially, this is what provides magnetic information. If there is no difference between the measurements in the two detector positions, the material is non-magnetic.
Figure 7: Detector placements (red) for an EMCD measurement in a two beam condition with the direct beam in the middle and the Bragg diffracted beam to the right. The designated positions fulfill the required conditions - |q| = |q’| and q ⊥ q0, to achieve the same effects as those from photon polarization in XMCD.
To mimic the change of polarization direction, the detector has to change between the two positions, and a difference of the signals is taken.
2.7 Dynamical effects in TEM and EMCD
Transmission electron microscopy utilizes how electrons interact with a material as they travel through it.
As previously described, an important process is the Bragg scattering off of crystal planes within the mate- rial. However, electrons can interact with the sample more than one time on their way through the depth of the sample. A Bragg diffracted beam actually perfectly fulfills the Bragg condition again and could be rediffracted into the direct beam direction. This rediffracted beam also fulfills the Bragg condition, and could be diffracted once more into the diffracted beam direction.[11] The concept is, somewhat simplicated, illustrated for a two beam condition in figure8. The effect this has in TEM is that the intensity of diffraction spots vary with thickness. In the two beam condition, the two beams are said to be coupled, which means that when one increases in intensity the other has its intensity reduced. The normalized intensities of the two progapating waves fulfill the condition I0+ Ig= 1 where subscript 0 denotes the direct beam and subscript g denotes the Bragg diffracted beam. The oscillations of the intensities as functions of sample thickness are called Pendellösungs and are proportional to the squared cosine and sine functions, which can be shown by solving the so called Howie-Whelan equations.[11]
Recent studies on EMCD, both experimental and theoretical, show that the EMCD signal inherits a consid- erable sample thickness dependency as well.[3,6] The experimental EMCD signal generally is relatively weak and often comparable in strength to signal noise. Since the EMCD signal is dependent on the diffracted beam intensities, the dynamical effects present in TEM reduce the reliability of the EMCD method. Small deviations in sample thickness could completely wipe the measureable EMCD signal out.
Figure 8: Illustration of the dynamical diffraction effects within the crystal in a two beam condition. The black lines represent parallell planes in the crystal and the blue lines represent the incoming beam which is split up into two directions as it propagates through the crystal. In reality, the distance between planes is much shorter than displayed, the angle of incidence of the incoming beam is much smaller and the incoming beam does not only hit one single plane.
2.8 Precession electron diffraction (PED)
Precession electron diffraction (PED) is a TEM technique widely used in crystallography for structural anal- ysis experiments.[12] The method amounts to introducing a small tilt between the sample zone axis and the electron beam and then precessing the beam essentially forming a conical shape in space as illustrated in figure9. The technique was developed as materiologists wanted to make the TEM more similar to X-ray crystallography. Electrons interact much more strongly with a material than X-rays, and therefore allows for more unwanted disturbances in the diffraction pattern.[9] This is somewhat of a trade-off since electrons also can provide more information about the material and with a better spatial resolution. Precession of the electron beam has been shown both experimentally and in simulation to reduce the dynamical effects in crystallography.[12,8] When the technique was introduced, materials whose atomic structures previously were unknown could be determined.[13] The theory is quite complicated but in principle the precession re- duces the impact of each individual unwanted dynamical interaction of the electron beam within the sample and as an average is taken, the overall effect is a less disturbed diffraction image. Research as of today has not been done combining the EMCD method and PED. It is still to be investigated if this technique could bring some advantages to EMCD as well.
Figure 9: Illustration of electron beam precession in a simplified two beam situation. The sample is tilted to fulfill the wanted diffraction condition. The incoming beam is then further tilted and precessed around the full lap maintaining the tilt angle θ, forming a cone in space.
2.9 Quantifying the quality of an EMCD signal
To draw any conclusions from the simulations, some quantitative measures should be defined to use as ref- erence to more clearly assess the results. In general, the EMCD signal has an oscillative behaviour as a function of sample thickness. The ideal EMCD signal would be a strong signal indifferent to the sample thickness.
To quantify how close a signal is to the ideal, three measures are used in this project. Firstly - the mean intensity of the signal, secondly - the mean squared deviation from the mean intensity and thirdly - the minimum intensity of the signal within a chosen interval of sample thicknesses. Ideally, the mean intensity would be high, meaning that for whatever sample thickness, a sufficiently strong signal could be expected.
However, a signal could have a high mean but still display an oscillative pattern. To pinpoint this, the mean
squared deviation from the mean intensity should provide good indications. This measure would be low if oscillations in the signal were not as prevalent. The minimum intensity of the signal is of interest since a low minimum intensity imply that experimental measurements would have trouble extracting anything useful for certain thicknesses as the noise could dominate the signal. Together, these three measures help quantify the quality of a EMCD signal.
2.10 Simulations
Simulating the transmission electron microscope, a sturdy mathematical model is required to describe the propagation of the electron wave function through the sample and eventually how it looks when it exits the sample, as that is where measurements are made. Since iron has a crystal structure, it’s electric potential can be described as a periodic function. For such periodic potentials, the electron wave function which solves the Schrödinger equation within the crystal is modulated by a periodic function determined by the crystal structure and orientation. Such wave functions are called Bloch waves. Bloch’s theorem states that the elec- tron energy eigenstates within a crystal is a superposition of Bloch wave basis states.[11] The mathematics are rather involved, but the main idea is that within the crystal, the incoming wave can be elastically scat- tered in many different directions, and the Bloch wave method relates the periodic crystal structure to the intensities of the different scattered waves. In principle, this amounts to solving large systems of equations with boundary conditions. An important parameter in the simulations for the precision of the calculations is how many of the Bloch waves should be taken into consideration in the final diffraction pattern. This has to be done by running convergence tests for different thresholds.
3 Method
Simulations were performed utilizing the MATS v.2 software which applied the Bloch wave method to a given crystal potential, sample orientation and electron beam energy.[14] An electron beam of 200 kV was used. The simulation calculated the distribution of electrons scattered with the energy loss corresponding to the L3 edge of iron. Also, it calculated the EMCD signal for all points in the diffraction pattern. Electron beam precession was simulated for a few different sample orientations. This means that the electron beam was tilted and then systematically aimed from different directions to cover a whole lap, in essence forming a cone in space. The precession could of course not be simulated as a continous motion, so it was tested how few steps the full lap could be divided into while maintaining sufficient accuracy since simulations generally were time consuming. From figure 10, a satisfactory amount was found to be 20 steps, meaning that two succeeding simulations in the precession were 36020◦ = 18◦apart. Convergence tests were also run to find how small coefficients in the Bloch wave calculation needed to be taken into consideration. From figure11it was concluded that waves with coefficients of size smaller than 2 · 10−4could be disregarded of while maintaining sufficient accuracy. In the figure, four different images are presented each showing the relative difference between the magnetic signal using a certain coefficient c0 threshold and the threshold c0= 1 · 10−4, which was known to be a rather small threshold. Simulations were run for sample thicknesses in the range 10-60 nanometers since that is the most common range of thicknesses used in experiments. The final result of a simulation consisting of many steps over the lap was constructed by averaging over the results from the individual steps. Generally, the cone angles of the precession simulated were chosen arbitrarily based on previous simulation results but the aim was to cover some range of cone angles to form a perception of how it affected the EMCD signal.
Figure 10: Convergence tests comparing how the signal is affected by how many steps the precession lap is divided into for two different precession beam tilt angles (a) 1 mrad and (b) 5 mrad. When the number of steps is more than 10, the signal is more or less indifferent to further increases in the number of steps.
Figure 11: Relative difference in magnetic signal diffraction patterns between using Bloch wave coefficient threshold c0 = 0.00010 and (a) c0 = 0.00050, (b) c0 = 0.00030, (c) c0 = 0.00020 and (d) c0 = 0.000050.
From this, it was concluded that c0= 0.00020 provided sufficient accuracy.
Simulations were done using BCC iron. The diffraction setups were the three beam (¯1¯10) and (110) orien- tation as well as the two beam (110) orientation set up by a tilt of around 10◦ from the [001] axis. These configurations were chosen since they were known to exhibit rather frequent dynamical effects, meaning many oscillations could be observed for a given thickness range. Also, these configurations are often used in experiments. Some simulations were finally run on the two beam (200) spot which was set up by a tilt of around 18 degrees from the [001] axis. This configuration was known to exhibit less frequent dynamical effects, allowing for some comparison with the previous two configurations. For the two first situations, the simulated diffraction pattern was an area of 21 × 21 pixels, covering a total of 40 milliradians in x- and y-direction. The detector was implemented as a single pixel in the diffraction pattern. In the last situation, the simulated area was 32 × 32 pixels, covering a total of 62 milliradians in x- and y-directions, and a square 2 × 2 detector was implemented by averaging over their intensities. This was because the ideal detector placement was situated in the middle of these four pixels.
4 Results
4.1 Three beam condition
The following figure shows the EMCD signal as a function of sample thickness at one of the two proposed detector positions in the three beam condition as the overall behaviour of the total EMCD difference signal is analogous. The different colours represent different precession cone angles in a range from 0.5 to 3.5 mrad.
The intensity is in arbitrary units.
Figure 12: Simulated three beam EMCD signal sample thickness dependency for different precession cone tilt angles.
The signal without precession exhibits an oscillatory behaviour, with rather constant period and amplitude.
This compares well to previous studies where the same simulation has been run.[4] As precession is introduced, the amplitude varies more over the thickness range and the wavelength is not completely constant. Generally, the maxima and minima using precession are not as high/low and some of the signals never reach zero or even sub-20 intensity for higher thicknesses. A tendency of two thickness ranges with slightly different behaviour can be identified. One lower range between 10 nm to around 28 nm and one upper range from around 28 nm to 60 nm. In the upper range, the wavelength of most of the signals increases as well as the amplitudes vary more. The three quantifying measures were calculated for each of the colored lines in the figure above.
These are presented for the two thickness ranges in the following two figures. One colored line is therefore represented as a single dot in the graphs.
Figure 13: (a) Mean intensity, (b) mean squared deviation from mean intensity and (c) minimum intensity in the lower thickness range of the three beam condition precession simulations as a function of precession cone tilt angle.
Figure 14: (a) Mean intensity, (b) mean squared deviation from mean intensity and (c) minimum intensity in the upper thickness range of the three beam condition precession simulations as a function of precession cone tilt angle.
Both figures show that all the three quantifying measures are sensitively dependent on the precession cone angle. All three measures are of similar parabola shapes. Although, in the lower thickness range the shapes are smoother with less edges. There are maxima or minima for all measures, which indicates that the used range of cone angles should be sufficient to draw some conclusions on the cone angle dependency. The position of the maxima and minima seem to coincide between all the three measures in both of the thickness ranges - which is very convenient. However, these are not situated at the same cone angles for both thickness ranges. The lower range has a higher cone angle as optimum (approximately 3 mrad) while the higher range has a smaller cone angle as optimum (approximately 1.5 mrad).
4.2 Two beam condition - (110) plane
Figure 15: Simulated (110)-two beam EMCD signal sample thickness dependency for different precession cone tilt angles.
The behaviour of the signals is similar to the previous case and the same two thickness ranges (10-28 nm and 28-60 nmm) with slightly different appearences can be identified. The quantifying measures are again presented separetely in the following two figures.
Figure 16: (a) Mean intensity, (b) mean squared deviation from mean intensity and (c) minimum intensity in the lower thickness range of the (110)-two beam condition precession simulations as a function of precession cone tilt angle.
Figure 17: (a) Mean intensity, (b) mean squared deviation from mean intensity and (c) minimum intensity in the upper thickness range of the (110)-two beam condition precession simulations as a function of precession cone tilt angle.
The higher thickness range is a bit more smooth here compared to in the previous section. Maybe it could be explained by the fact that in a two beam condition, there are less dynamical diffraction events taking place in the sample compared to in a three beam condition. As previously, the three measures have maximas and minimas coinciding at the same cone angles and the lower thickness range advocates a higher cone angle (approximately 2.75 mrad) while for the higher thickness range the opposite is true (optimum cone angle approximately 1.5 mrad).
4.3 Two beam condition - (200) plane
Figure 18: Simulated (200)-two beam EMCD signal sample thickness dependency for different precession cone tilt angles.
The signal exhibits much less frequent oscillations than in the previous two cases, which was expected.[5] The wavelength of the signal without precession is however around 5-8 nm longer than what has been observed in previous studies but the overall shape is similar. This could be due to the exact orientation of the sample not being the same and the electron beam settings. As precession is introduced, the maximum intensity of the signal decreases significantly at the same time as the minimum intensity increases. Two distinct thickness ranges cannot really be identified here, so the following figure shows the three quantifying measures over the full thickness range.
Figure 19: (a) Mean intensity, (b) mean squared deviation from mean intensity and (c) minimum intensity in full thickness range of the (200)-two beam condition precession simulations as a function of precession cone tilt angle.
The shapes of the quantifying measures are also a bit different compared to the two previous cases since there are not any extremas covered by the simulated cone angles. Higher angles were not simulated since the mean intensity of the signal started getting very low as well as the fact that the two beam condition would start to get dissolved with a too big of a tilt. However, there is a clear dependence on the cone angle here as well and most significantly the mean squared deviation from the mean was around 10 times smaller with the highest cone angle of 4 mrad. This is also clear in figure 18where the signal almost approaches a constant as the cone angle increases. Although it is almost constant, which is what the ideal signal would be, it comes at a cost of mean intensity reduced by approximately a factor of two, compared to the situation without precession.
5 Discussion
The results are promising, indicating that electron beam precession has a considerable effect on the EMCD signal. For the first two setups the results were similar. This may be due the fact that they include the same atomic planes (110) and that they are relatively close to each other - a tilt of about 6 mrad apart. For these setups, the resulting signals were analyzed in two different thickness ranges. The three quantifying measures were consistent in which precession cone angle resulted in the best signal. Although, in the lower thickness range a higher precession angle was most advantageous while in the higher thickness range the contrary was true. While the mean intensity of the signals was not very precession angle sensitive, the mean squared deviation from the mean intensity was highly precession dependent. At best, it was decreased with a factor of more than 6, as can be seen in figures17&14. This is also apparent in figures15&18where the amplitude of oscillations decreased considerably for some precession angles around 1-2 mrad. This means that the minimum intensity of the signal also increased from zero or sub zero intensity to over 20 which can be seen in figures12&15. This is crucial for increasing the signal-to-noise ratio in experiments. In the lower thickness range, the minimum was not increased as much but still a considerable amount from around 4 to around 12, which is three times stronger.
The third setup with the (200) diffraction planes show somewhat different results. The overall behaviour of the signal when beam precession is introduced is similar to the previous cases, but the intensity peak is very sensitive to the precession angle. Looking at the quantifying measures in figure19, the mean intensity of the signal significantly decreased with an increasing precession angle while the oscillatory behaviour was reduced drastically. This makes the result a bit ambigous. While the mean and maximum intensity decreased signif- icantly, the minimum intensity noticeably increased. Looking at the full thickness range, the interval where the precession improved the signal is rather short - 47 to 60 nm. If a sample is within this range it seems advantegous to use precession, but for lower thicknesses the advantages of using precession are reduced.
All three setups seem to have in common that for higher sample thicknesses, precession results in a less oscillative and more stable EMCD signal. This is especially useful since, in experiments, the EMCD signal relative to the non-magnetic signal gets weaker when the sample thickness increases since the non-magnetic signal increases with sample thickness.
An important detail all the results have in common is that precession never increased the maximum signal strength but rather decreased it. However, it always increased the minimum signal strength. Therefore, it seems like precession should not be used in regions where the signal already has a maximum while it appears advantageous for minimum intensity regions. It could also be concluded that if the sample thick- ness for some reason would be unknown for an experimentalist, it would probably be good practice to use precession since it would minimize the risk of hitting a zero-intensity thickness region. This also means that fluctuations in the sample thickness would not have as great of an impact as when precession is not used.
6 Conclusions
In this stage, the results can be interpreted such that electron beam precession should not be seen as a guarantee for a stronger EMCD signal regardless of the sample thickness and diffraction condition setup but that it certainly can improve the quality for certain diffraction condition setups and ranges of sample thicknesses. Further research on the subject should aim to try and closer map how precession affects the EMCD signal for different diffraction setups to clarify when precession is advantageous and not. This should also be done using different materials.
7 Populärvetenskaplig sammanfattning av projektet
Nya material bidrar i stor utsträckning till den teknologiska utvecklingen av samhället. Med fokus på egen- skaper såsom styrka, vikt och formbarhet tillsammans med elektriska och magnetiska egenskaper utforskas möjligheter att konstruera material som kan förbättra dagens teknik eller möjliggöra helt ny sådan. Transmis- sionelektronmikroskopet har historiskt sett varit ett viktigt instrument i forskningen kring materialstrukturer.
En relativt ny metod som kallas electron magnetic chiral dichroism (EMCD) utnyttjar elektronmikroskopet för att utforska även de magnetiska egenskaperna hos material. Tidigare har röntgenstrålning använts för samma syfte, men den metoden är begränsad i den uppnåeliga rumsliga upplösningen i mätningarna. EMCD möjliggör för en högre upplösning i mätningarna men resulterar däremot i en relativt svag signal som lätt hamnar under brus i mätningarna. Detta projekt undersöker i simuleringar hur signalkvalitén i EMCD påverkas av att precessera elektronstrålen i elektronmikroskopet, det vill säga luta elektronstrålen och rotera den längs med en konformad yta i rummet. Detta görs på järn i tre olika orienteringar. Resultaten pekar på att metoden kan ha en fördelaktig effekt på signalen, men inte helt oberoende av materialets orienter- ing och tjocklek. Vidare forskning bör fastställa mer precist när precession av elektronstrålen är fördelaktigt.
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