Image Series Reconstruction for Transmission Electron Microscopy

Martin Ek Division of Polymer & Materials Chemistry Lund University

Supervisor: Professor Reine W Examiner: Senior Lecturer Lars S

February 2, 2009

i Abstract The transmission electron microscope (abbreviated as TEM, which also refers to the method of transmission electron microscopy), has since its creation in the 1930’s become a very powerful instrument for materials analysis at an atomic level. Although able to image individual atomic columns in many materials, the inherent aberrations in the round magnetic lenses used in conventional TEMs severely limits the interpretable resolution in the images. For semiconductor nanowires made of group III and V elements this has meant that full atomic resolution, meaning the ability to separate and differentiate between the different atom columns, hasn’t been possible on conventional intermediate voltage TEMs, which in turn has made it difficult to determine the structure of the twin planes that are very common in these crystals. Using convergent beam electron diffraction on bulk gallium phosphide and cross sectional scanning tunneling micrscopy on gallium arsenide nanowires these twins have previously been determined to be of the ortho type, meaning that they don’t disrupt the crystal polarity. Several techniques have been developed to correct or work around the lens aberrations which would make it possible to analyze these twins by high resolution TEM imaging instead. Aberration correctors consisting of multipole lenses make it possible to directly compensate for the distortions of the other lenses, but require extensive modification of the instrument. Image series reconstruction is on the other hand an indirect method that relies on post processing the images, but has the advantage of being applicable to any instrument and would still bring benefits to microscopes already equipped with hardware correctors. For each image in the series the aberrations are changed (by changing the lens or illumination settings), measured and finally corrected. To reduce the computing times the two last steps are usually made with the assumption that the sample is thin, something that isn’t strictly true for the nanowires which can be several tens of nanometers thick. Methods that measure the aberrations using the surrounding thin amorphous material and changes of lens focus rather than illumination tilt are less sensitive to the thickness, but should still be validated with simulations. Two focus series of indium phosphide nanowires have been captured with sufficiently low specimen drift and sample misorientation to allow reconstructions. Comparisons with multislice simulated image series indicate that the twin bound-

ii aries were of the ortho type and In–P dumbbells could be seen for some of the twins which indicates that the resolution has been improved from the 0.17 nm point to point resolution to about 0.14 nm. Unfortunately in both cases every second twin appeared smeared in the images, most likely due to misorientation, that made the exact structure hard to see and no definite conclusion could be made. Since the thickness varies due to the geometry of the wires even in a best case scenario only a few of the outermost atomic layers will give useful reconstructions with this method and other reconstruction methods that don’t rely on the thin sample assumption should be tested.

iii Sammanfattning Transmissionselektronmikroskopet (förkortat TEM, vilket används både för instrumentet och metoden) skapades på 30-talet och har sedan dess utvecklats till ett mycket kraftfullt verktyg för materialanalys på atomnivå. Fastän det går att urskilja kolumner av atomer i många material så är upplösningen kraftigt begränsad av aberrationer som oundvikligen finns i de runda magnetiska linser som används i konventionella TEM. För halvledande nanotrådar bestående av grupp III och V atomer betyder detta att full atomär upplösning inte har varit möjlig med konventionella mikroskop med intermediär accelerationsspänning, vilket i sin tur innebär att det inte har varit möjligt att karakterisera de tvillingplan som är vanliga i dessa material. Dessa tvillingplan har tidigare undersökts med hjälp av konvergent stråle elektrondiffraktion och tvärsnittssveptunnelmikroskopi för galliumfosfid respektive galliumarsenid och har visats sig vara av orto typen som, till skillnad från para, innebär att kristallens polaritet är oförändrad över tvillingplanet. Flera tekniker har utvecklats för att korrigera eller på andra sätt komma runt linsaberrationerna vilket skulle göra det möjligt att analysera tvillingplanen utifrån högupplösande TEM bilder istället. Linspaket bestående av till exempel oktapoler har gjort det möjligt att direkt korrigera förvrängningarna från de andra linserna men kräver stora modifikationer av mikroskopet. Bildserierekonstruktioner är å andra sidan en indirekt metod som använder sig av efterbehandling av bilderna, vilket innebär att metoden kan användas på vilket mikroskop som helst. För varje bild i serien måste aberrationerna ändras (genom att ändra fokus eller strålningsriktningen), mätas och slutligen korrigeras. För att minska beräkningstiderna görs ofta de två sista stegen med antagandet att provet är mycket tunt, något som egentligen inte stämmer på nanotrådarna som kan vara flera tiotals nanometer tjocka. Metoder som mäter aberrationerna utifrån omkringliggande tunna amorfa områden samt ändrar fokus istället för strålningsriktningen är mindre känsliga för tjockleken, men bör ändå valideras med simuleringar. Två fokusserier på nanotrådar av indiumfosfid togs där drift och orientering av proverna var tillräckligt bra för att rekonstruktion skulle vara möjlig. Jämförelse med multislice-simulerade bilder antyder att trådarnas tvillingar är av orto typ och på ett par ställen är det möjligt att skilja de närliggande indium och fosforatomerna åt, vilket innebär att upplösningen har förbättrats från 0.17 nm till ungefär 0.14 nm.

iv Tyvärr är hälften av tvillingarna mycket otydligare i bilderna, antagligen på grund av att trådarna inte är helt rätt orienterade, vilket gör att inga definitiva slutsatser kan dras. Eftersom trådarnas tjocklek varierar mycket på grund av sin geometri så kan den rekonstruktionsmetod som använts även i bästa fall bara ge atomär upplösning för de allra yttersta atomlagren och andra metoder som inte förutsätter tunna prov bör testas.

v Acknowledgements I would like to thank Reine Wallenberg for the opportunity to do this, in my opinion, very interesting and fun project as well as Jan–Olle Malm for first suggesting it. The visit to Oxford was much appreciated and really helped me getting started. All the practical microscopy work in this report was made possible by Lisa Karlsson, who gave me great crash-courses in all of the techniques and theories involved in the focus series reconstructions and also took time to answer my many questions and look over my results with a critical eye after my week in Oxford. I would also like to thank Kimberly Dick for lending me the nanowire samples, Jacob Wagner for writing the atoms2jems-2 program which made the image simulations possible, and everyone at the department. I have felt very welcome from the very start of this project and the tutoring work from Lars Stenberg was very rewarding and needed. Finally I would like to thank my family for all the support and encouragement they have given me over the years.

Contents 1 Introduction 1.1

1

The basics of image series reconstruction . . . . . . . . . . . . . . . . .

2 Methods and materials 2.1

2 4

Transmission electron microscopy . . . . . . . . . . . . . . . . . . . .

4

2.1.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1.2

Image formation, aberrations and resolution . . . . . . . . . . .

5

2.1.3

Methods for improving the resolution . . . . . . . . . . . . . .

13

Image series reconstruction . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.1

Focus or tilt series? . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.2

Measuring the aberrations . . . . . . . . . . . . . . . . . . . .

15

2.2.3

Reconstruction methods . . . . . . . . . . . . . . . . . . . . .

17

2.3

Multislice simulations . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.4

InP nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2

3 Experimental section

24

3.1

Method overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2.1

Camera characterisation . . . . . . . . . . . . . . . . . . . . .

24

3.2.2

Multislice simulations . . . . . . . . . . . . . . . . . . . . . .

25

3.2.3

Image series . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

4 Discussion 4.1

35

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

35

CONTENTS

viii

4.2

4.1.1

Camera characterisation . . . . . . . . . . . . . . . . . . . . .

35

4.1.2

Multislice simulations . . . . . . . . . . . . . . . . . . . . . .

36

4.1.3

Image series . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2.1

43

Bibliography

Suggestions for further work . . . . . . . . . . . . . . . . . . .

44

1 Introduction The transmission electron microscope (TEM) is unique in its ability to detect and measure the structure of crystals and their defects on an atomic scale. Since its creation in 1931 the performance of the TEM has steadily improved with better lens design, emitters, stable power supplies and so on. In 1956 the first image showing a crystal lattice was taken, requiring a resolution of 1.2 nm, which in a sense started the field of high resolution transmission electron microscopy (HRTEM). Today the resolution of an intermediate voltage microscope (operating with an accelerating voltage of around 300 kV) with a conventional lens design is around 0.15 nm and is mainly limited by the aberrations in the objective lens. One type of aberration, spherical aberration (which causes electrons with different angles to the optical axis to be focused at different distances), is actually unavoidable in the round magnetic lenses that are conventionally used in the TEM and has been seen as setting a limit to the obtainable resolution, beyond which the images will not be interpretable. A resolution of 0.17 nm (corresponding to the JEOL 3000F at the National Centre for High Resolution Electron Microscopy, nCHREM) is enough to resolve individual columns of atoms in many materials. There are, however, still many materials and applications that require even better resolution than can be directly obtained and methods have been invented that can “work around” the normal limits of a TEM. One such method is based on acquiring series of images with varying microscope settings that can later be combined to form a single image with improved resolution. [1] To understand how image series can improve the resolution one needs to have some knowledge of how images are formed in the TEM and what its limits are. This is discussed in the first sections of the next chapter together with an overview of the different methods

1

CHAPTER 1. INTRODUCTION

2

for improving the resolution beyond the classical limits. The image series reconstruction methods will then be discussed more in depth and this will be used to motivate the particular method used in the experimental section of this thesis. In this thesis the image series reconstruction method is applied to indium phosphide (InP) nanowires to determine the type of the twin planes that are formed when they are grown with the zincblende structure. In general III-V semiconductors, such as InP, can form either coherent twins with III–V bonds or incoherent twins with either III–III or V–V bonds which will have large effects on the properties of the wires. To separate the different twin types one needs to be able to separate the indium and the phosphorous atom columns in the HRTEM image, which requires a resolution of about 0.14 nm in the case if InP. The structure of these nanowires and the problem of determining the twin type is discussed in section 2.4. [2]

1.1

The basics of image series reconstruction

In the next chapter the theory behind image formation and image series reconstruction will be given in detail but it might be useful to have a general idea of how they work. Imagine having only two columns of atoms separated by a certain distance. Under certain circumstances that will be discussed later they will show up as two separate black spots in the image. If the atom columns were moved closer to each other the two spots in the image would ideally also move closer to each other until the distance between the atoms becomes too small for the microscope to resolve and only one larger spot would be visible. This is known as the information limit of the microscope. Due to the aberrations in the objective lens this simple description doesn’t hold true for the TEM. Just as before the two atom columns will show up as two black spots if the distance between them is sufficiently large, but below a certain distance called the point resolution, which is roughly in the order of twice the information limit, this will change. Instead of being shown as the two easily interpretable black spots from before the appearance of the two atom columns will now vary rapidly depending on their separation, sometimes showing up as two bright spots on a dark background, sometimes as two dark spots and sometimes as a single spot. This behavior continues until the

1.1. THE BASICS OF IMAGE SERIES RECONSTRUCTION

3

separation between the two atom columns reaches the information limit where they again will be seen as a single black spot. The situation in a real sample where there are many atom columns with many different separations is even more complicated, but generally the same limits apply which can make TEM images very hard to interpret. In practice one usually uses an aperture that cuts off all distances smaller than the point resolution, making them appear as single black spots whick makes the images interpretable, but means that a lot of information about the sample is lost. The purpose of image series reconstructions is to instead compensate for the aberrations, giving interpretable images without loosing any information. To illustrate how this is done it is easiest to describe the electrons as a wave which is modulated as it passes through the sample (which gives us information about the sample) and the objective lens (which distorts this information but is necessary to form an image). Image series reconstruction can now be seen as solving the following problem: given a set of images where we know the aberrations and how they affect the electron wave, what was the electron wave just after the sample, before it was destorted by the objective lens? To find the answer we need to know how to measure the aberrations from the images (section 2.2.2) and how to solve this system of equations (section 2.2.3), neither of which is very straight forward!

2 Methods and materials 2.1 2.1.1

Transmission electron microscopy Overview

Figure 2.1 shows a very simplified illustration of how the electrons travel through the TEM column. Although it leaves out a lot of components that are present in a real microscope it is still useful when discussing how the images are formed in an idealised way.

Figure 2.1: Simple illustration of how an image is formed in a TEM, where r is the realspace coordinates in the sample or image, u is the reciprocal space coordinates and θ is the scattering angle. These notations and the names shown for the wave functions are used throughout this thesis. The illumination system consists of the electron gun, accelerator and condenser lenses. The gun emits electrons either by heating a filament to overcome the work function of 4

2.1. TRANSMISSION ELECTRON MICROSCOPY

5

the electrons or by field emission where the electrons are tunnelled from a fine tip due to a very high electric field. The latter type of emitter, called a field emission gun (FEG), is superior as it gives a brighter and most importantly a more coherent illumination (the benefits of this will be shown in the next section). In conventional transmission electron microscopy the condenser lenses then form a broad parallel beam on the sample. The condenser aperture is used to exclude electrons emitted at high angles from the gun, which will decrease the brightness but improve the quality (as these peripheral electrons are less coherent) of the illumination. Electrons interact strongly with all materials and as they pass through the sample a multitude of processes takes place. The electrons that have passed through the sample (which should be the vast majority) are used to form an image by the objective lens which is then magnified by the intermediate lenses and projector lens in an analogous way to an ordinary light microscope. In high resolution TEM only electrons scattered at small angles and without loosing any energy will contribute to the information in the image and while all the others (electrons that have lost energy, been scattered multiple times or at high angles) also contains important information about the sample, they will at best give a diffuse background in the images. The objective aperture can be inserted into the back focal plane to cut off peripheral electrons, the effects of which will be discussed in a later section. The selected area diffraction (SAD) aperture is used to exclude parts of the area being viewed so as to make sure the diffraction pattern originates in a small, specific area. To actually see the electrons they are either projected onto a viewing screen coated with a scintillating material (a material that generates light when impinged by electrons) or onto a scintillator coupled with a CCD camera for capturing images. [3]

2.1.2

Image formation, aberrations and resolution

The specimen In high resolution TEM the formation of images are best described by the interaction of electron waves with first the sample and then with the objective lens (which, being the strongest lens, will have the largest effect). The specimen is usually described by a

CHAPTER 2. METHODS AND MATERIALS

6

projected potential model where the atomic potentials will change the phase (and only the phase) of the electron wave as it passes through the material, which gives the phase object approximation (POA). This is then further simplified when the specimen is thin and the phase change is small by making a linear approximation which is called the weak phase object approximation (WPOA) shown in equation 1.

q(r) = exp (−iσVt (r)) ≈ 1 + iσVt (r)

(1)

In this expression σ is an interaction constant and Vt is the projected potential. It is important to know that if the sample is thick (more than a few nm) it’s not only the assumption that only the phase is changed that breaks down but also the whole notion of a projected potential as the electrons will be scattered more than once within the sample. Assuming that the electron wave is perfectly even and coherent (the illumination electron wave ψillum = 1) before it hits the specimen and that the WPOA is valid, the electron wave after the sample is described by equation 2, where ψo is called the object wave function or the exit plane wave function. To emphasize the physical meaning of this expression, it is sometimes divided into one part, 1, corresponding to the direct beam and one part related to the scattered electrons, ψo,s . [4]

ψo = ψillum q(r) ≈ 1 + iσVt (r) = 1 + ψo,s

(2)

The objective lens An ideal lens would have no effect on the electron wave except focusing the spherical wave fronts coming from a point in the object plane to a point in the image plane. The magnetic lenses in a TEM are, however, far from ideal and will change the phase and amplitude of the electron wave as it passes through. This is especially true for the objective lens which, being the strongest lens in the TEM, will have the largest effects. Figure 2.2 below shows the effects of a lens on the wave fronts coming from a point in the object plane. This particular lens has a spherical aberration that causes electron rays with a high incident angle to be focused to a point closer to the lens than rays closer to the optical

2.1. TRANSMISSION ELECTRON MICROSCOPY

7

Figure 2.2: The effects of the objective lens on the spherical wavefronts from a point source in the object plane. The aberration displaces the wave a distance W and the point object will no longer be focused to a point in the image but to a disc (shown as red). This distance, called the wave aberration function (WAF), depends on the scattering angle θ and where the point source is located in the object plane, r. Only the scattering angle is shown in the figure since r and its effect on the WAF is usually very small in a TEM.

axis. In the wave description the aberration causes the ideal spherical wave front to be displaced by a distance, W , (which is called the wave aberration function) that depends on the position, r, where the wave originated from and the angle, θ, to the optical axis. Since the area on the sample that is viewed at any time in a TEM is very small the wave aberration function’s dependence on r is usually ignored, which is known as the isoplanatic approximation. In table 2.1 below the wave aberration function is expanded in polar notation to fourth order (with φ being the direction and k the magnitude of the angle to the optical axis) and the different aberrations are described. In this list only the spherical aberration, C3 , is completely unavoidable but defocus, C1 , is also almost always present since it is used to reduce the effects of the spherical aberration by slightly underfocusing the objective lens (which corresponds to a negative C1 ). Usually the other aberrations are left out since they are much smaller then the two main aberrations and also since it is possible, at least in theory, to correct them. If the scattering angle is substituted for the position in the back focal plane, u, according to k ∼ = λu (the exact expression is sin k = λu, but for the very small scattering angles involved sin k ≈ k) and the difference in distance W is converted to a phase difference χ, the expression in equation 3 is derived.

CHAPTER 2. METHODS AND MATERIALS

8

Table 2.1: The wave aberration function and the different aberration constants in polar coordinates with φ being the direction and k the magnitude of the angle to the optical axis. Astigmatism can be thought of as smearing the image. For defocus and spherical aberration an alternative notation is often used. W (k, φ) = |A0 | k cos (φ − α0 ) + 12 |A1 | k 2 + 12 C1 k 2 + 13 |A2 | k 3

cos 2 (φ − α1 )

Image shift Twofold astigmatism Defocus (alt: ∆f )

cos 3 (φ − α2 )

Threefold astigmatism

+ 13 |B2 | k 3 cos (φ − β2 )

Axial coma

+ 14 |A3 | k 4 cos 4 (φ − α3 )

Fourfold astigmatism

+ 14 |B3 | k 4 + 14 C3 k 4

Axial star

cos 2 (φ − β3 )

χ (u) =

Spherical aberration (alt: Cs )

2π 1 W (λu) = π Cs λ3 u4 + πC1 λu λ 2

(3)

Equation 3 is a useful simplification when discussing image formation in the TEM, but for image series reconstructions one needs to consider some of the other aberrations as well, depending on how high resolution that is required. Twofold astigmatism can, as an example, be clearly seen in normal HRTEM images if the operator doesn’t compensate for its effects and at resolutions approaching 0.1 nm (which is possible with image series reconstruction) the low order astigmatisms are very important. The effect of the objective lens on the object wave function is described by equation 4 below where Ψi is the Fourier transform of the image wave function.

B(u) = exp (iχ(u)) = cos (χ(u)) + i sin (χ(u)) Ψi (u) = F T [ψo (r)] B(u)

(4)

The actual image, I(r), that is recorded is then the square amplitude of the inverse transform of Ψi (u) but the effects of the aberrations are more apparent when their effects directly on Ψi (u) is considered. [5]

2.1. TRANSMISSION ELECTRON MICROSCOPY

9

The phase contrast transfer function Equation 4 together with the WPOA implies that only the imaginary part of B(u) contributes to the image contrast, as in equation 5, where the last part (sin χ) is sometimes called the phase contrast transfer function (pCTF).

Ψi (u) = δ + Ψo,s (u)A(u)E(u)2 sin χ(u)

(5)

In the above expression A is an aperture function (1 for all u to a certain value and then 0) describing the effects of the objective aperture and E is the envelope functions which describe, among other things, the effects of limited coherence which will be discussed in the next section. The physical interpretation is as follows: when the electron wave interacts with a sample the scattered parts will be −π/2 out of phase with the direct beam. These scattered beams will then pass through the objective lens at an angle which will cause an additional phase change before being combined with the direct beam in the image plane. If the objective lens also gives a −π/2 phase change (which means that the phase contrast transfer function is -1) the scattered beams will be −π out of phase and interfere destructively with the direct beam. Since the atoms in the sample are the scattering centers, they will appear darker than the background in the image. This is known as linear imaging conditions. To transfer as much information about the sample as possible in a single image it is possible to set the defocus C1 to a suitable negative number (called the optimum defocus) to compensate for the effects of the spherical aberration, C3 , and have a large interval from 0 to some spatial frequency, u1 , where sin χ ≈ −1. This is illustrated in figure 2.3 where the aperture function has been set to cut of the pCTF after the first crossing of the u-axis where it starts to oscillate rapidly. Although information about the specimen is transferred after this point had it not been for the aperture in this case, it will not contribute to the image contrast in an easily interpretable way since it will be transferred out of phase with the rest of the image and will therefore be smeared out over a large area and will actually make it harder to interpret the image. The spatial frequency, u1 , where the phase contrast transfer function crosses the u-axis the first time at the optimum defocus setting defines the point to point resolution of the microscope and is the maximum resolution where the

10

CHAPTER 2. METHODS AND MATERIALS

Figure 2.3: The phase contrast transfer function at optimum defocus (C1 = −40 nm on a 300kV microscope with a spherical aberration of C3 = 0.6 mm) giving a resolution of 0.17 nm which corresponds to about 5.8 nm−1 in reciprocal space. In this case the pCTF is reduced to zero by the aperture function A after it crosses the x-axis the first time.

images are directly interpretable. [3]

Envelope functions The discussion in the previous section assumed a perfectly coherent illumination and the resulting phase contrast transfer function continued to oscillate between +1 and -1 even at very large spatial frequencies, which is not the case with a real microscope. Variations in electrical currents in lenses, accelerating voltage and gun emission together with specimen drift and vibration, a slightly non-parallel illumination and attenuation in the CCD camera all pose limits on the phase contrast transfer function. These limits are described by envelope functions, E, which will gradually dampen the phase contrast transfer function and eventually reduce it below the noise at some spatial frequency. Again, it’s important to remember that the notion of envelope functions as described in this section is only applicable under linear imaging conditions. For all modern microscopes this happens at higher spatial frequencies than the point to point resolution and defines the informa-

2.1. TRANSMISSION ELECTRON MICROSCOPY

11

tion limit of the microscope beyond which no information is transferred at all. Three of these envelope functions depend on factors that are inherent to the microscope (unlike vibration) and will be described more in detail in this section. Temporal coherency effects comes from the small instabilities over time in accelerating voltage and gun emission, which will give the illumination a small energy spread, and from variations in the lens currents, which means that the focus will vary with time. The total effect of these variations (since electrons with different energy are focused at different planes) is to create a spread of focus (usually of a few nm), δ, which will mean that the phase contrast transfer function is averaged over these defoci. At high spatial frequencies where the phase contrast transfer function varies rapidly this averaging will cause large reductions. The exact form of the temporal coherency envelope is given in equation 6 below. 

1 Et (u) = exp − (πλδ)2 u4 2

 (6)

Spatial coherency arises from the fact that the illumination will never be perfectly parallel and is actually slightly convergent at best, which can described as having a distribution of different illumination tilts. When the illumination is tilted the effect on the phase contrast transfer function is to shift it about the y-axis and, in the same way as for the temporal coherency, the phase contrast transfer function is averaged over this tilt distribution (which is described by the convergence semi–angle of the illumination α). Equation 7 below describes the effect of this. It should be noted that as the envelope function for spatial coherency contain a defocus term, C1 , it can be affected by the operator unlike the temporal coherency.    
2 πα 2 3 3 Es (u) = exp − C3 λ u + C1 λu λ

(7)

The combined effect of these two envelope functions on the phase contrast transfer function is shown in figure 2.4 below for the optimum defocus condition. [6] The detector envelope function can be divided into two parts: one due to the scattering processes taking place in the scintillator and fibre optical coupling and one which

12

CHAPTER 2. METHODS AND MATERIALS

Figure 2.4: The effect of the temporal (Et ) and spatial (Es ) coherence envelopes on the phase contrast transfer function. The spread of focus has been set to δ = 4 nm and the convergence semi-angle to α = 0.2 mrad, which corresponds to a JEOL 3000F microscope. No aperture function has been applied.

arises when the continuous wave function is turned into a discrete image having a finite pixel size. The situation is further complicated by the fact that the detector as a whole will have different transfer characteristics for the signal and the noise which are described by the modulation transfer function (MTF) and the noise power spectrum (NPS) respectively, both of which will need to be characterised experimentally. The shape of these two functions vary slightly from camera to camera depending on the type and thickness of the scintillator, pixel size etc. and can be determined by capturing and processing images of a sharp edge, flat images with even illumination and dark images. Ideally a sharp edge should result in an image with an instantaneous jump in the number of electrons detected between the dark and the illuminated area but due to scattering in the detector the result will instead be a continuous, if very steep, transition. The difference between the ideal and the measured edge profile is related to the MTF. [7, 8] To avoid having the contrast transfer function dampened too much by these effects it is important to choose the magnification carefully since it is impossible in a pixel image to detect spatial frequencies less than the reciprocal pixel size (which is known as the

2.1. TRANSMISSION ELECTRON MICROSCOPY

13

Nyquist limit). To put it in another way, it is impossible to separate two objects in an image if they are closer than one pixel away from each other. Choosing a magnification where the pixel size is at least half, and preferably one fourth, of the desired minimum resolvable distance reduces this problem. [9]

2.1.3

Methods for improving the resolution

There are many different ways to improve the resolution or increase the amount of information that can be extracted from an image (which in a sense increases the useful resolution). By increasing the accelerating voltage it is possible to have the transfer function extend to higher spatial frequencies, but at the same time the microscope increases in size and the samples will be damaged faster. Image simulations can be used to compare experimental images with model structures even at spatial frequencies above the point to point resolution, but this requires one to have an accurate idea about the sample structure from the beginning. [1] There are three main methods that truly extend the resolution of high resolution TEM: aberration correction, image series reconstruction and electron holography. The latter requires specially designed microscopes and computer processing of the images and except for special circumstances the direct method of aberration correction or the easier image series reconstruction is preferred. In an aberration corrected microscope a combination of hexapole or octapole lenses are used in a corrector which lacks circular symmetry and therefore doesn’t have to have a positive spherical aberration like normal, round magnetic lenses. The total spherical aberration (and the other low order aberrations) of the objective lens and this corrector can then be adjusted by the operator, which can greatly increase the point to point resolution by choosing a small spherical aberration. This has also made it possible to use negative spherical aberrations giving bright atoms on a dark background, which actually increases the contrast in the images compared to images taken with an equally large, positive spherical aberration. Image series reconstructions compensate for the oscillations in the phase contrast transfer function by processing a series of images taken at either different defocus or illumination tilts. By having images with different aberrations in the series (due to the

CHAPTER 2. METHODS AND MATERIALS

14

different illumination tilts or defocus) one makes sure that all spatial frequencies are included in the reconstruction as each single image will have have frequencies where the phase contrast transfer function is zero. Although this method is indirect and requires extensive post processing of the image series (unlike images from aberration corrected microscopes) it can be used on any TEM and results in the complex object wave function which, unlike a normal image, gives detailed information about the specimen’s effect on both the amplitude and the phase of the electron wave. [10] Even if all microscopes in the future will have aberration correctors the image series method will still be useful as the two methods actually benefit from being used together. By reducing the main aberrations the corrector enables the image series reconstruction to deal with the higher order aberrations instead as well as making larger beam tilts possible for tilt series. On the other hand the image series reconstruction can compensate for any residual aberrations from the corrector, improve signal to noise ratio, improve transfer of low spatial frequencies and makes it possible to do quantitative comparisons with the object wave function. [9] The combination of image series reconstruction and aberration corrected microscopy would also make it possible to correct local variations in the aberrations over the field of view. [11]

2.2 2.2.1

Image series reconstruction Focus or tilt series?

There are several factors that need to be taken into account when choosing whether to use focus or tilt series when doing an image series reconstruction, the most obvious of which is the desired resolution. The phase contrast transfer function in equation 5 shows that the temporal coherence envelope will set the limit for the maximum resolution obtainable with focus series since it is not affected as the focus is changed. When the illumination is tilted instead the whole pCTF is shifted, including the envelope functions, and the resolution will depend on the maximum tilt angle. There are, however, several reasons to prefer focus series over tilt series when the resolution needed is less than the axial information limit (the limit set by the temporal coherence under axial illumination): there are fewer microscope setting that needs to be adjusted, it is easier to correct for

2.2. IMAGE SERIES RECONSTRUCTION

15

specimen drift as there is an added image shift when tilting the illumination and parallax effects pose a limit to the specimen thickness for tilt series. The parallax effect is illustrated in figure 2.5 below. [9]

Figure 2.5: The parallax effect which places a limit on the maximum beam tilt. For a certain tilt the effective beam displacement is larger for a thicker sample and one is no longer looking at the same projected structure. This also means that if a sample is tilted the displacement of the atoms in a thicker sample will be larger. In the case of III-V nanowires focus series are preferable since the resolution required to separate the two different atom columns (0.14 nm in the case of InP) in a [110] projection is well within the axial information limit. In this particular projection the nanowires are also quite thick as will be shown in figure 2.9 in the nanowire section, which will limit the maximum tilt due to parallax effects. If one would be interested in the [112] direction instead tilt series would be more useful as the projected III–V distance is shorter and the wires are thinner along the edges. Since the focus series method is less demanding and provide high enough resolution it will be used through out this thesis.

2.2.2

Measuring the aberrations

The methods for measuring the aberrations can be broadly divided into two groups based either on measuring image shifts when the illumination is tilted or on diffractograms of amorphous materials. When the illumination is tilted the images will appear to have different aberrations compared to the axial images. These new aberrations will depend on the original, axial, aberrations and the tilt. For this discussion it is sufficient to know that the new image shift will depend on all the other aberrations which can therefore be measured from the

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16

distance the image is shifted compared to an axial image for a certain tilt. Measuring image shifts doesn’t require any amorphous material in the vicinity of the sample which makes it applicable to a large variety of materials, but image shifts are hard to separate from specimen drift which reduces the accuracy of this method. [12] The diffractogram based methods utilizes the fact that for a thin amorphous sample the diffractogram will be directly related to the phase contrast transfer function in the sense that all the zero crossings in the pCTF will show up as dark rings. This is used in the method where pre-calculated and experimental diffractograms are matched, which is insensitive to sample drift. Unfortunately it is not possible to do these comparisons when there is crystalline material present as its strong reflections will mask the amorphous signal. Although the crystalline reflections have higher amplitude than the amorphous signal they are limited to a smaller area in the diffractogram and hence the phase (but not the amplitude) of the entire diffractogram is approximately described by a weak phase amorphous object. This is true even for many completely crystalline materials due to a thin, disordered layer on the surface or the carbon support film on the TEM grid and this forms the basis of a method that is both insensitive to drift and applicable to almost all materials. In a first step two images are compared using a phase correlation function (PCF) with a trial function, t, containing an estimated defocus difference, ∆C1 , between the two images. When this estimation corresponds to the actual defocus difference the PCF collapses to a single peak at a position which corresponds to how the specimen has moved due to drift.

P CF (r) = F T

−1



t(∆C1 , u)c∗1 (u)c2 (u) F (u) |t(∆C1 , u)c∗1 (u)c2 (u) + h|

 (8)

In equation 8 F is used to reduce the effects of high frequency noise and h simply prevents a zero denominator. The result from the PCF is then used in a second trial function, the phase contrast index (PCI), which contains estimations of the absolute values of the defocus and twofold astigmatism together with the known spherical aberration. The new trial function works roughly by compensating for the phase variations in the experimental diffractogram, making the PCI one for all spatial frequencies when the estimated defocus and astigmatism are correct. [5]

2.2. IMAGE SERIES RECONSTRUCTION

17

Under axial illumination conditions it is only possible to measure the defocus and the twofold astigmatism with the methods described in this section. The spherical aberration is given by the microscope manufacturer or needs to be measured separately. Sometimes this is not enough since the other aberrations cannot be ignored at resolutions approaching 0.1 nm and it might be necessary to measure also the axial coma and especially the threefold astigmatism. In the above method this is done by tilting the illumination, which will change the apparent defocus and astigmatism by some amount depending on the tilt and the original aberrations. The difference in the measured astigmatism and defocus between the axial and the tilted images is then directly related to the previously unknown aberrations. Needless to say this requires much more work and for lower resolutions it can be enough to use a previously measured threefold astigmatism or assume that it is negligible. [13, 14]

2.2.3

Reconstruction methods

The objective of all reconstruction methods is to find an object wave function ψo,s given a set of measured image contrasts , c, and their aberrations , B (from equation 4). The reconstruction methods can be broadly divided into two groups depending on if they are based on the weak phase object approximation or the more general phase object approximation. Although they give a better estimation of the object wave function, the methods based on the phase object approximation don’t have closed form solutions and have to be computed iteratively which is very time consuming. [5] The WPOA based methods on the other hand are fast but require much thinner samples to work well; even with a sample only a few nm thick there will be a measurable difference compared to the POA methods. [15] There are many different weak phase object methods available, such as the parabaloid method, the Wiener filter method [5] and the iterative wave function reconstruction [16], that have been developed into commercial applications. In many aspects these different approaches are equivalent but the Wiener filter method has some advantages over the two others: it is not an iterative method (unlike the iterative wave function reconstruction) and it is more flexible and better at low spatial frequencies than the parabaloid method, in the sense that it can be used on tilt series and unevenly spaced focus series. There

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18

might be special circumstances where the other methods are better (such as low dose imaging [17]), but for most applications the Wiener filter has more advantages and will therefore be used in the experimental section. [18] In the Wiener filter method the Fourier transform of the image contrast for each image in a series is weighted by a filter function, r, to give the object wave function 0 . estimate, ψo,s

0 ψo,s =

X

ri ci

(9)

i

The filters depend on the aberrations of all the images in the series according to equation 10. For a Fourier component that is present in all images the filter simply averages them (and makes sure that they have the same phase), if the component is present only in one image it is retained but with the correct phase. For areas where the phase contrast transfer function is low or zero for all images the filter will tend to zero due to the noise term, ν. [5] W (−u)Bi∗ (k) − C ∗ (u)Bi (−u) ri (u) = W (u)W (−u) − |C(u)|2 + ν(u) X W (u) = |Bi (u)|2

(10)

i

C(u) =

X

Bi (u)Bi (−u)

i

To see the effect of such a reconstruction one can apply the filters directly onto Bi to get a phase contrast transfer function for the whole, reconstructed series. Figure 2.6 shows the result from a 21 member focus series which shows a remarkable improvement over the optimum focus pCTF and also how the temporal coherence limits the maximum resolution possible in a focus series.

2.3

Multislice simulations

The multislice simulation (MS) method is one of many ways to simulate TEM images from a crystal model. By cutting the model of a thick crystal into many thin slices,

2.3. MULTISLICE SIMULATIONS

19

Figure 2.6: Wiener filter reconstruction applied toPthe phase contrast transfer functions of a focus series and a tilt series as < i ri Bi . The tilt series only contains one tilt azimuth and 11 images which doesn’t give a true picture of the reconstruction, but illustrates how it can exceed the axial information limit set by the temporal coherence. The optimum defocus pCTF is included to illustrate the improvement.

typically around 0.2 nm thick, perpendicular to the illumination it is possible to use the weak phase object approximation to describe how an electron wave interacts with each slice. After the electron wave has passed through one slice a free space propagator is applied before it reaches the next slice and so on until it has passed through all the slices that make up the crystal. At this point the simulated electron wave function is the object wave function of the crystal model and it is then possible to add the effects of aberrations and turn the wave function into an image. [3] Even though each individual slice is limited by the WPOA the simulation as a whole will include non linear effects due to using many slices and allowing the electron wave to propagate between them. Images simulated by this method won’t be identical to the real TEM images but will show a significantly larger contrast. In most simulations a multitude of factors, such as inelastic scattering and beam damage, could contribute and ad up to this discrepancy. [19] Even if the imaging conditions are matched very precisely to the simulations and a sample with a well known geometry is used a the contrast will still be approximately twice as large for the simulations. [20] Using off-axis holography to exclude essentially all non–elastically scattered electrons from the images the most likely candidate for this remaining difference, known as the Stobbs factor, has been determined to be thermally diffuse scattering. [21]

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20

The nanowires that are used in this thesis are, as will be shown in the next section, too thick for the WPOA to be a good approximation, contrary to the assumptions in the Wiener filter. It will therefore be important to compare the experimental object wave reconstructions with results from simulated image series. Such simulations can help determine what contrasts the different atomic species will have in the images as well as the effects of an imperfectly aligned sample.

2.4

InP nanowires

The nanowires used in the experimental section are all indium phosphide, which was chosen due to having a larger unit cell and higher potential difference between its two elements than most other III-V compounds. This means that compared to many other possible nanowire materials, such as GaAs, a smaller resolution improvement will be needed and the atom columns should show a larger contrast difference making them easier to differentiate. Nanowires are grown by metal organinc vapor phase epitaxy (MOVPE) catalysed by gold particles. In the case of indium phosphide, phosphine (PH3 ) and trimethylindium

Figure 2.7: The structure of a zincblende (ZB) nanowire with four ortho twin planes. The atom dimensions have been exaggerated to make the illustration clearer: they wires are typically at least 5 times as wide and each segment twice as high.

2.4. INP NANOWIRES

21

(In(CH3 )3 ) is used. The exact growth mechanism differs for the different III-V combination and isn’t known in detail, but a general and very simplified description is as follows: the two gaseous compunds decompose at the surface of the substrate where they move around due to diffusion at the high temperatures used during the growth, until they hit the gold interface where they assemble into epitaxial layers. [22] When grown with a diameter, which is determined by the size of the gold particle, larger than about ten nm these wires tend to have the zincblende stucture (which is the same as the bulk structure) limited by {111} type surfaces which gives them the characteristic shape shown in figure 2.7. [23] Twins in zincblende structured III-V compounds have been studied for a long time, but mostly for bulk crystals. [24] The projected structure of a wire in a low order zone axis, [110], is shown in figure 2.8 together with the indium to phosphorous distances that needs to be resolved in order to determine the twin type. Since the focus series restoration method is limitied by the axial information limit (about 0.10 nm for the JEOL 3000F used in this work) it will not be possible to separate the indium from the phosphorous atom columns in the images of the [112] direction. The atomic structure of twins in III-V materials have previously been investigated in the bulk materials by convergent beam electron diffraction (CBED) where scattering from higher order Laue zone reflections into the 002 type disks will give constructive or destructive interference depending on the polarity (direction of the III–V bonds) of the material. These measurements have been confirmed with chemical etching methods and have shown the ortho type boundary to be the most common, at least for the bulk materials. This method could also be used for nanowires, especially for the thicker ones where multiple scattering will dominate, which makes HRTEM difficult but is actually beneficial for CBED. [26] Another method which is designed specifically for the nanowires is cross-sectional scanning tunneling microscopy (XSTM). In this method a scanning tunneling microscope is used to probe the electronic states of a cross–section surface with atomic resolution, showing the exact positions of the atoms with highest atomic number. In order to create the cross–sectional slices the wires are usually grown at an angle to the substrate and then encased, which does not reflect the growing conditions of a typical nanowire. Using

CHAPTER 2. METHODS AND MATERIALS

22

Figure 2.8: The atomic structure of the three twin types in the [110] and [112] projections. The In–In and P–P distances have been estimated from the sum of their tetrahedral covalent radii. [25] this method only orto twins have been found in III-V nanowires, which is reasonable since they should be energetically favoured over the para twins. [27] The method used in this work, focus series restoration, has been used previously for the determination of defects in thin film gallium arsenide using an aberration corrected microscope. Indium phosphide has the advantage of having a larger difference in atomic potential, which should make it easier to determine the polarity of the crystals from the images. [28] The geometry of the nanowires is, however, complicated and the thickness will vary greatly with the projection and location on the wire as seen in figure 2.9, which will make the interpretation of the HRTEM images and the reconstructions more complicated.

2.4. INP NANOWIRES

23

Figure 2.9: The projected thickness for a wire with three segments of similar dimensions to those used in the experimental section. Each level represents an increased thickness of about 2.9 nm. In the [110] projection a wire 30 nm wide with 5 nm twin planes will be just over 14 nm at its thinnest.

3 Experimental section 3.1

Method overview

Indium phosphide nanowires doped with zinc and grown by gold particle catalysis had previously been deposited on a copper TEM grid covered by a solid thin carbon film by simply pressing the grid against the substrate. The nanowires were all around 50 nm at the base and tapered to less than 30 nm closest to the gold particle with the smallest being around 20 nm thick. No other sample preparation, such as plasma cleaning, was done. The microscopy work was performed on two JEOL 3000F microscopes (one at the Department of Materials, Oxford and one at nCHREM, Lund) both with an accelerating voltage of 300 kV, a spherical aberration of 0.6 mm and an assumed spread of focus of 4 nm and a beam convergence of 0.15 mrad. The Oxford microscope was equipped with a 1x1 k CCD and the Lund one with a 2x2 k CCD. All images were captured using DigitalMicrograph (DM), a program developed by Gatan Inc. for both microscope control and image processing. The camera characterisation and the capturing and processing focus series was done using the FTSR v 1.0 script package, which is a plug-in for DM which uses the phase correlation method for determining the aberrations and a Wiener filter approach for the reconstructions. [29]

3.2 3.2.1

Results Camera characterisation

The camera characterisation was done, using the same microscope settings as was used for the image series, with the beam stopper as the sharp edge and the viewing screen to block 24

3.2. RESULTS

25

Figure 3.1: Modulation transfer functions and noise power spectra as determined for the two microscopes by the FTSR plug-in for DM. Both axis have arbitrary units, although the x-axis is related to reciprocal distance measured in reciprocal pixels. Oxford ref was included in the FTSR package for an example image series. out the illumination for the dark current images. [8] In Oxford the first condensor lens was set to spot 1 and two characterisations were carried out with two different settings of the second condensor lens (different brightness) resulting in an average of about 3k and 6k counts per pixel. In Lund the brightness was set to illuminate approximately twice the CCD area and two different spot settings were used: spot 3 giving about 1k counts per pixel and spot 2 giving about 3k counts per pixel. The reason for choosing a smaller spot size and a less intense illumination for the Lund image series was to minimize the risk of beam damage. The resulting MTF and NPS, as calculated by the FTSR script, are shown in figure 3.1 together with a reference from the Oxford microscope that was included in the FTSR script package as an exercise. [30]

3.2.2

Multislice simulations

A model nanowire was created with the ATOMS software based on four segments 6 nm high and around 23 nm wide when seen in the [110] direction. The segments are enclosed by {111} type surfaces only and all bond lengths have been set to the values suggested in section 2.4. Two models were used: one containing ortho twins only and one having a mixture of the two para types. The models were tilted to a [110] zone axis and were cut using atoms2jems-2 into 0.2 nm thick slices which were imported into JEMS, which is a

CHAPTER 3. EXPERIMENTAL SECTION

26

java based electron microscopy simulation program, for the multislice simulations. The simulations were carried out using a real space multislice method with PRDW atomic form factors with an amorphous carbon layer below the crystal. The amorphous carbon layer deteriorates the quality of the images but is necessary for the reconstruction as it is used for the aberration measurement. For each model twenty 1024x1024 pixel images were simulated to make a focus series with a defocus range of -190 nm to 0 nm with a 10 nm defocus step. These were then processed using the FTSR scripts in the same way as for the experimental images, except using the camera characteristics from equation 11 where ustep is the reciprocal distance measured in reciprocal pixels. Equation 12 represents the effect of an ideal detector where the pixel size is the only factor affecting the MTF. [7] The validity of the reconstructions were checked by comparing the estimated defocus and astigmatism from the reconstruction with their known values from the simulation. The phase of the reconstructed object wave function for the ortho model is shown in figure 3.2. In figure 3.3 both the phase and the amplitude for a small area of the wire model is shown for all three twin types. It is very unlikely that the real nanowires were imaged exactly on axis. Even using bulk GaAs and automated sample adjustments misorientations of 2 mrad have been measured. [28] Nanowires are more likely to change position during imaging and only using manual tilt adjustment means that sample misorientation in this case is most likely even larger. To see what effects this would have on the reconstructions the ortho model was tilted almost 7 mrad (0.4◦ = 6.98 mrad) off the [110] zone in three different azimuths; parallel to the wire growth direction, perpendicular to the wire and in the direction of the indium to phosphorous bonds in one of the twins. The phase of the reconstructed object wave function of these simulations are shown in figure 3.4.

 mtf = nps = 1

sin πustep πustep

2 (11)

3.2. RESULTS

27

Figure 3.2: The phase of the object wave function for a model with orto type twin planes estimated from a focus series simulated using the multislice method. The large square shows the area displayed in figure 3.3 for the different models and the atom positions from the model are shown for four In-P pairs across the twin boundary in the smaller square. The reconstruction shows a contrast reversal in the central area with white atoms instead of black. In the two experimental series it is not possible to see the atomic possitions in this area but a similar reversal of contrast is seen there as well.

28

CHAPTER 3. EXPERIMENTAL SECTION

Figure 3.3: Object wave function for the three twin types from focus series simulated using the multislice method. The images have been Fourier filtered to reduce the contrast from the amorphous carbon background. The amorphous carbon layer was necessary for the reconstruction method, but makes the details in the images harder to see afterwards.

3.2. RESULTS

29

Figure 3.4: The effects on the reconstructed object wave function when tilting the ortho model slightly off the [110] zone axis. The arrows indicate the tilt direction from the top left model, which is exactly on the [110] zone. The images have been Fourier filtered to reduce the contrast from the amorphous carbon background. A is tilted parallel to the wire, B perpendicular and C in the In–P bond direction.

CHAPTER 3. EXPERIMENTAL SECTION

30

Figure 3.5: Full view of the two InP wires that were used for the focus series. Both are taken close to the optimum defocus but without any objective aperture. The blue boxes show where the areas in figure 3.7 and 3.8 are located and also the area used as a reference during the aberration measurement.

3.2.3

Image series

Suitable wires were identified at low magnification by looking for wires that were thin, not surrounded by other wires, showed a zig-zag pattern and high contrast (which suggests that they were close to a [110] zone) and were unbent. They were then tilted to roughly the [110] zone by using successively smaller SAD apertures to select the area of interest and making sure that the intensity in the diffraction pattern (DP) was evenly distributed around the direct beam. This was repeated at larger camera lengths to make any differences more apparent. For the final adjustment, images were recorded of the DP and the intensity measured for the various spots, which was then used to find the direction needed for the tilt. Figure 3.5 shows the two wires that were used for the focus series. The image series, consisting of 20 images, were captured with no binning (meaning that the images had the same number of pixels as the CCD), a 1 s exposure and aiming for a 10 nm defocus step between the images giving a defocus range of -190 nm to 0 nm. The actual defocus for the two series are shown in figure 3.6. Images from the best series are shown in figure 3.7 and 3.8. Since each image is very large only a small section of a few of the images in each series is shown. The drift between the images was determined as a step in the measurement of the aberrations and the images have been shifted by this

3.2. RESULTS

31

amount to show the same area. The reconstructions were carried out with the FTSR scripts using the microscope parameters from the method overview section, a 0.10 nm information limit, 0.03 nm vibration (which will act as an extra envelope function [6]) and zero threefold astigmatism (since this isn’t measured in a focus series). The sampling interval, s, was determined for each series according to equation 12 from the number of reciprocal pixels, n, between two crystalline reflection from a known spacing, d (usually the 0.0565 nm distance between the two 333 type spots) and the total number of pixels, ntot . [30]

s=

d×n ntot

(12)

A small (256x256 and 512x512 for the Oxford and Lund series respectively) area in the centre of the images which contained both the nanowire and some of the surrounding carbon film was was used as the reference during the aberration measurement since the registration algorithm benefits from having a clearly defined edge in this area. The reconstruction was carried out on the whole image and a small section of the resulting object wave function is shown in figure 3.9 and 3.10.

Figure 3.6: Estimated defocus from the FTSR script for the two image series showing the average defocus step between the images. In both cases the aim was for a defocus range of -190 nm to 0 nm with a 10 nm step.

32

CHAPTER 3. EXPERIMENTAL SECTION

Figure 3.7: Oxford focus series. Images were taken using the spot 1 setting for the first condensor lens aiming for around 3000 counts per pixel. Only every other image is shown except around the optimum defocus of -42 nm. The sampling interval was determined to 0.0420 nm.

3.2. RESULTS

33

Figure 3.8: Lund focus series. Images were taken using the spot 2 setting for the first condensor lens aiming for around 1000 counts per pixel. The tilt wasn’t adjusted using diffraction pattern images. Only every other image is shown except around the optimum defocus of -42 nm. The sampling interval was determined to 0.0248 nm.

34

CHAPTER 3. EXPERIMENTAL SECTION

Figure 3.9: Object wave function as estimated from the Oxford image series in figure 3.7 by the FTSR script package.

Figure 3.10: Object wave function as estimated from the Lund image series in figure 3.8 by the FTSR script package.

4 Discussion 4.1 4.1.1

Results Camera characterisation

By comparing the three modulation transfer function results from the Oxford instrument in figure 3.1 one can see that the reference characterization is slightly higher, especially at the mid range spatial frequencies and, although it probably can’t bee seen very clearly in the figure, smoother. One reason for this is most likely that the reference has been averaged over several characterizations whereas the two others represent results from just one each. The exact conditions used during the acquisition for the reference were not included in the manual, but from the two characterizations done just before the focus series in this report it seems that the microscope settings don’t influence the result much, which is reasonable as it is a measurement of properties related to the scintillator, fibre optics and CCD chip and should be independent of the illumination as long as the CCD chip is evenly illuminated. This small discrepancy between the reference and the two others is therefore hard to explain, but should not make a large difference for the reconstructions as long as one doesn’t intend to use the estimated object wave functions for quantitative comparisons. The MTF of the Lund instrument is lower for essentially all spatial frequencies. This can be explained by the fact that the higher resolution CCD chip necessarily has a smaller pixel size which means that the scattering of the incoming electrons and the generated photons in the scintillator will affect more neighboring pixels. At first glance this might seem to be a disadvantage for the higher resolution CCD, but keep in mind that the x-axis is related to the number of reciprocal pixels for a given spatial frequency. Since the 35

CHAPTER 4. DISCUSSION

36

two image series were taken at a similar magnification, 400kx, the Fourier components of the images will be further to the right for the Oxford instrument. Apart from the pixel size there are two more factors that probably have had a negative effect on the results from the Lund microscope: a round beamstopper and a faulty cable to the camera. The beamstopper is used to form images of a sharp edge, which are then used to calculate the MTF, but the beamstopper on the JEOL 3000F instrument in Lund is round and is therefore not the ideal object to use. This effect is actually noticeable when the edge images are compared for the Lund and Oxford instruments where the latter has a slightly sharper profile. Although it is possible to instead use another object, such as a partially inserted HAADF detector, to form the edge this is not supported in the FTSR scripts. The 2x2 k CCD chip on the Lund microscope consists of four individual 1x1 k quadrants. During the first characterisation, at an exposure level of 1000 counts per pixel (cpp), a problem with the signal cables resulted in a significantly different gain between the quadrants. Although the underlying problem with the cables was corrected before the 3k cpp characterisation, there is still a noticeable difference between the quadrants in the non-illuminated images. This can possibly explain the oscillations in the NPS from the 3k cpp images but not why the results from the 1k cpp images look so much better. The role of the MTF is that of an envelope function and as such it is not critical in a qualitative reconstruction (compared to factors such as defocus and spherical aberration) since its only effect is to attenuate high frequency Fourier components. In a quantitative restoration, on the other hand, it is important to know the exact form of these envelopes to be able to rescale all the Fourier components accordingly. With the exception of the Lund NPS with very large ripples these results are very much adequate for the intended task and the samples are too thick anyway for quantitative measurements to be meaningful.

4.1.2

Multislice simulations

In the simulated object wave function reconstructions in figure 3.3 one can see that when viewing the phase the atoms are dark on a brighter background and that in the amplitude

4.1. RESULTS

37

image it is the other way around. Since the actual focus series have been treated the same way this should hold true for them as well. Furthermore the phase images are slightly better in the sense that the indium and phosphorous atom columns are visible in the entire image, whereas in the amplitude images one of them seems to disappear in certain areas. This effect is even more noticeable in the results from the real image series and I will therefore focus on the phase images for this discussion. The most encouraging result from these simulations is that despite the large changes in sample thickness over the whole field of view the reconstruction is interpretable for fairly large areas, although definitely not over the entire twin interface. The reconstructed phase is best, in the sense that it shows little change and is interpretable, close to the “tips” of the wire where it’s at its thinnest, which is what was expected. In these areas it is possible to differentiate the indium and the phosphorous atom columns as the latter appear larger and darker, which means that it is possible to determine the twin type of the model from the reconstruction. The contrast difference is contrary to what one would expect from the WPOA as indium has a much larger projected potential, but even at these thin areas the wires are much too thick for this approximation to be applicable and non-linear effects will change the contrast difference between the two atom species. Even when the model is tilted away from the [110] zone as in figure 3.4 it is still possible to resolve the In–P dumbbells, although it is not as easy to differentiate the two for some of these tilts. The reason for choosing a 7 mrad tilt is that due to the thickness of the wire model the atoms are shifted a relatively large distance and already at these tilts some of the atom columns starts overlapping. It is also much more difficult to do reconstructions from the tilted models compared to the on zone ones as the phase correlation routine wrongly estimates the astigmatism to be large, when there in fact is none at all and subsequently gets all the defocus wrong as well. Even for the best reconstructions the FTSR script recorded a twofold astigmatism in the order of 0.2 nm for the tilted wires, which is smaller than the typical residual astigmatism in the real focus series of about 1 nm but not zero, as it should be. Of these three tilted models the one where the tilt direction is parallel to the In-P bonds is the most interesting. It represents a worst case scenario as the In–P dumbbells for half the twin planes are smeared, making it very hard to see the individual atom columns

CHAPTER 4. DISCUSSION

38

and even harder to differentiate between them. Even in this case there still remain a few small areas at the very thinnest parts of the wire where it is possible to determine the twin type, but now one has to draw conclusions about the entire twin interface from just seeing a couple of the dumbbells closest to the very edge of the wire.

4.1.3

Image series

The reconstructed object wave functions from the two focus series have a similar appearance as both show In–P dumbbells in one of the twins but not the other where the image looks smeared in this direction. At a few places, shown in figure 4.1 and 4.2, it is possible to see an arrangement of a big black spot followed by a smaller lighter one and then the reverse on the other side of the twin boundary, which is consistent with an ortho type interface since it has a similar pattern in the simulations. Since the model doesn’t exactly match the sample in terms of thickness it is not possible to tell which type of spot belongs to the indium or phosphorous atoms. Furthermore these dumbbells are only visible at a few locations in each twin boundary and to draw a general conclusion about the twin type is not possible. The two image series are discussed separately below as they in part have different faults, but one reason they have in common for not resulting in better restorations is that both nanowires appear damaged. Compared to even just slightly wider wires the sides looks jagged instead of showing a clear zigzag pattern. The smaller wires might be more susceptible to electron beam damage and tilting them is usually more time consuming which means that they are exposed to the electron beam longer before the series are recorded. The threefold astigmatism has been assumed to be zero during these reconstructions. Since the focus series method can’t measure this astigmatism to see if it really is negligible its effect will not be corrected, should the assumption be wrong. This might actually affect the measurement of the other aberrations as well since the phase correlation method would try to compensate any threefold astigmatism with defocus and twofold astigmatism instead. Axial coma would have a similar effect and it is known through multislice simulations of focus series on GaAs that if these aberrations are present the dumbbells in the images can be reversed, making the gallium atoms appear darker than the arsenic. [28]

4.1. RESULTS

39

To determine if any significant threefold astigmatism was present in the Oxford series a previously measured value was used during the reconstruction and very little, if any, change in the images could be seen which indicates that it is negligible, at least for the resolution needed. Contrast reversal due to axial coma would make it impossible to determine which contrast features in the images belong to indium and phosphorous, but as long as the two appear different it is possible to determine the twin type. Oxford The first thing to notice about this series is that the images have a pixel size of 0.0420 nm, which is larger than one fourth of the In-P distance which is 0.0367 nm and should be seen as an upper limit to the pixel size for good reconstructions. One clearly seen effect of this is that the peak separation in the dumbbells is significantly larger than the correct value as shown in figure 4.1. Although this could be explained by the dumbbells being an artifact not related to the In-P distance it but is more likely due to the difficulty of accurately determining sub pixel positions.

Figure 4.1: Close up of the object wave function phase for the Oxford image series. An ortho model has been added on top which shows the similarities. A linescan in the direction of the In-P bond shows their separation to be larger than expected, but this could be due to the large pixel size making it hard specify the peak positions. On the lower half of the image the dumbbells can be seen not just at the very edge, but

CHAPTER 4. DISCUSSION

40

also further into the structure where their appearance and direction seem to change. The former can be explained by the wire getting thicker which means that non-linear effects will be much stronger further in from the edge but the apparent shift in In-P direction is harder to explain. A similar but much weaker effect can be seen in the simulations of tilted wires which seems to indicate, together with the fact that the half of the twins show a much worse reconstruction, that the wire is not exactly on the [110] axis. The reason that this has affected the real wire much more than the simulated one can, at least in part, be related to the former being thicker and any tilt will therefore lead to a larger shift of the atom positions. Lund The 0.0248 nm pixel size has greatly improved the peak positions in the linescan in figure 4.2 compared to figure 4.1 shows a separation of 0.1464 nm which is very close to the correct value of 0.1467 nm. Compared to the Oxford series fewer In-P pairs are clearly resolved, although many show up as a gray area adjacent to the darker spots and in the “correct” direction as determined from the ones that do show dumbbells. The change in In-P direction is also smaller than in the Oxford reconstruction, although the lower half has a bond direction that is consistently different from the model. Again this is best explained by the sample being tilted off the [110] axis, either from the start or due to shifting position as the image series was recorded. Despite being better than the Oxford series in some regards, the Lund series has a few additional problems. It is taken with a less intense illumination (spot 2 instead of spot 1 was chosen to minimize beam damage) and shows a less steady drift (changing direction and speed) which has resulted in a worse registration during the aberration measurement. Especially the drift might be a problem as the Fourier transforms of the affected images show a clear dampening in one direction for some of the images, indicating a large drift during the exposure. The envelope function due to drift is described by equation 13 [6]. 

1 Ed (u) = exp − (πud)2 6

 (13)

The drift distance during the exposure, d, can be calculated from the pixel displace-

4.2. CONCLUSIONS

41

Figure 4.2: Close up of the object wave function phase for the Lund image series. An ortho model has been added on top which shows the similarities. Only a few of the In–P pairs show up as two separate peaks, although most have a gray “tail” instead.

ment between two images (as measured during the reconstruction) and the times these two images were recorded. For the image showing the most drift (0.54 nm over 7 seconds) this envelope only reduces the pCTF by half at 8 nm−1 , which is not enough to explain the appearance of the diffractograms. The uneven drift indicates that the acutal drift speed might have been larger than was measured during some of the exposures. Leaving the images showing obvious effects from drift out of the reconstruction prevents them from having a negative effect on the reconstructed object wave function but as this creates a very uneven series in terms of defocus difference between the images no improvement could be seen.

4.2

Conclusions

Compared to the two other methods discussed briefly for enhancing the resolution, aberration correctors and holography, the main advantage of the image series method is that it can be implemented on any microscope, although it benefits greatly from FEG electron sources due to the increased coherency of the illumination. For both the aberration corrected microscopes and the most common image series methods nanowires are much

CHAPTER 4. DISCUSSION

42

thicker than the ideal samples. The effect of the thickness on the latter are small enough to give interpretable results for some areas of the wires as can be seen from the image series presented in this thesis, but it is unclear exactly how it would affect the aberration corrected images. The chosen image series reconstruction method of phase correlation aberration measurement and a Wiener filtering was able to successfully resolve the In–P dumbbells in a nanowire sample. This means that the resolution has been improved from the 0.17 nm point to point resolution as determined by the spherical aberration of the objective lens to less than 0.147 nm corresponding to the 004 spacing of the dumbbells. Although the Wiener filter is based on a weak phase object model it was still possible to get a consistent reconstruction of the outermost atomic layers that was directly interpretable. Both image series indicate that the twins are of the ortho type, which is consistent with all previous studies on both bulk and nanowire III–V materials, but due to the unknown (both in magnitude and direction) sample misorientation no definite conclusions should be made. In the reconstructions from the simulated tilted nanowires there seems to be an upper limit of about 7 mrad away from the [110] zone after which the aberration measurements often fail and are highly dependent on the reference area used. Since the models differ from the real wires in at least two aspects (slightly different dimensions and the model amorphous carbon is always perfectly perpendicular to the illumination) these limits can’t be directly transferred to the real image series. However the simulations do indicate that the reconstructions will give useful results, even if the sample is slightly off the zone axis. Even if the thickness of the wires didn’t directly hinder the reconstructions it is still a large problem when orienting the sample to the [110] zone. Small tilts away from the zone axis that probably would have been acceptable for conventional HRTEM imaging of thin crystal edges result in very large displacement of the atoms due to parallax shift. The result of this can be seen clearly in the object wave phase where some of the In–P dumbbells are smeared or show a different direction from what one would expect. Just trying to spread the intensity evenly around the direct beam in the diffraction pattern on the viewing screen is not sufficient for orienting the samples.

4.2. CONCLUSIONS

4.2.1

43

Suggestions for further work

The most necessary improvement is to get the sample closer to the [110] zone axis in a more consistent way. A more detailed analysis of how the intensity is divided among the 220 type spots for different misorientations would be a good start as they should be of equal intensity for each twin when the sample is exactly on the zone axis, independent of which III-V is used. This could in turn either be measured from images of the diffraction pattern or by doing reconstructions of small image series. The former involves adjusting diffraction focus and brightness to make the spots larger and less intense to avoid saturating or damaging the CCD. The latter is problematic since the image series reconstructions still take time, both during the set up and during the calculations. Even though the threefold astigmatism didn’t seem to affect the reconstructions it would still be useful to measure it separately. This would require making tilt series or tilt tableaus on a separate sample, but the results can then be used for all reconstructions on that instrument. If better image series are acquired it would be useful to also run them through a full, non-linear restoration. These would be very time consuming but for an object as thick as the nanowires it would be very interesting to see how the result would differ from the Wiener filter approach, especially when one moves away from the very outermost atomic layers. Another option would be to use a lacey carbon film instead of the solid one used in this thesis. The aberration measurements would then have to be done on a separate area close to the vacuum edge, but the actual reconstruction could use the part of the wire just outside the carbon film. This should improve the quality of the restored object wave function as it would be without an amorphous background, but it would be much harder find suitable wires as they would have to have to a good area just on the edge of the carbon film.

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