Strain mapping in semiconductor nanowires using X-ray diffraction / eingereicht von: Mario Keplinger

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UNIVERSIT¨AT LINZ

JOHANNES KEPLER

_JKU

Technisch-Naturwissenschaftliche Fakult¨at

Strain mapping in semiconductor nanowires

using X-ray diffraction

DISSERTATION

zur Erlangung des akademischen Grades

Doktor

im Doktoratsstudium der

Technischen Wissenschaften

Eingereicht von:

Mario Keplinger

Angefertigt am:

Institute of Semiconductor and Solid State Physics

Beurteilung:

a.Univ.Prof. Dr. Julian Stangl (Betreuung)

a.Univ.Prof. Dr. Alois Lugstein

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Eidesstattliche Erkl¨arung

Eidesstattliche Erklärung Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Dissertation ist mit dem elektronisch ¨

ubermittelten Textdokument identisch.

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Kontrollierte mechanische Spannungen in Halbleitermaterialien erlauben die Herstellung neuester elektronischer Bauteile. Die elektronischen Eigenschaften von Halbleitern sind bestimmt durch ihre chemischen Bauelemente, also die Atome, und wie diese Bauelemente räumlich angeordnet sind, was sich bei Halbleitern in ihrer Kristallstruktur manifestiert. Wenn ein Halbleiter einer mechanischen Spannung ausgesetzt ist, verändern sich die Positionen der Atome im Gitter und das Material wird, entsprechend seiner mechanischen Festigkeit, gedehnt. Elastische Dehnung verändert die Bandstruktur und die elektronischen Eigenschaften des Halbleitermaterials. Die Bandstruktur eines Halbleiters kann als eine Karte seiner elektronischen Eigenschaften interpretiert werden.

Röntgenbeugung ist ein mächtiges Werkzeug zur Charakterisierung von kristallinen Materialien und eine ausgezeichnete Methode zur Bestimmung von Veränderungen im Kristallgitter. Mit diesem Werkzeug wurden während dieser Arbeit Verspannungszustände verschiedenster Nanodrähte untersucht. Diese Nanodrähte sind stabförmige Objekte mit Durchmessern von normalerweise unter 100 nm und Längen im Bereich von mehreren Mikrometern. Diese Größenordnung birgt für die elektronischen Eigenschaften signifikante Änderungen, und das Material wird im quantenmechanischen Sinne eindimensional, was sich in einer Änderung der Bandstruktur niederschlägt. Diese Eigenschaft von Nanodrähten ermöglicht die Herstellung von Halbleiterbauteilen mit überlegenen elektronischen und optoelektronischen Eigenschaften im Vergleich zum aktullen Stand der Technik.

Mit Hilfe von R¨ontgenbeugungsexperimenten in Verbindung mit Ergebnissen aus elektronenmikroskopischen Untersuchungen wurden InAs/InAs1−xPx

Nanodraht-Heterostrukturen untersucht. Durch unterschiedliche Gitterkonstanten der verwendeten Materialien weisen diese Heterostrukturen sich räumlich verändernde Spannungszustände auf. Für diese Untersuchungen war die Entwicklung neuer experimenteller Methoden vonnöten und eine umfangreiche Datenauswertung in Verbindung mit Finite-Elemente-Simulationen. Dies erlaubte eine quantitative Aussage über die mittleren Eigenschaften, wie die chemische Zusammensetzung und die Dimensionen in den untersuchten Nanodraht-Ensembles. Unter systematisch variierten Wachstumsbedingungen hergestellte Proben wurden charakterisiert, um Rückschlüsse auf den Wachstumsprozess ziehen zu können. Es werden aber auch, zum Beispiel für die Entwicklung von Bauelementen mit einzelnen Nanodrähten mit genau bestimmten physikalischen Eigenschaften, Verfahren benötigt, die über das Messen von statistischen Mittelwerten über mehrere Nanodrähte hinausgehen. Auch könnte der Verspannungszustand eines einzelnen Nanodrahts durch das ihn einbettende Bauelement verändert werden. Dadurch wird es nötig, ein Verfahren mit hoher räumlicher Auflösung zu verwenden, wie Röntgenstrahlen mit Strahlgrößen im Submikrometer-Bereich. Glücklicherweise wurden solche Untersuchungen durch die Entwicklung von Synchrotronen der dritten Generation in Verbindung mit neuartigen Fokussiermethoden von Röntgenstrahlen ermöglicht. Mit solch einem Verfahren konnten wir ein einzelnes axiales InAs1−xPx Segment in einem InAs

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Nanodraht mit nano-fokussierter Synchrotronstrahlung beleuchten und durch die Auswertung des aufgenommenen Beugungsbildes die genaue räumliche Verteilung der Verspannungszustände in und um das Segment bestimmen. Dies erfolgte mittels Finite-Elemente-Simulationen, aber es gibt auch die theoretische Möglichkeit, ohne den Umweg über Simulationen die Verspannungswerte direkt aus der gemessenen Intensitätsdistribution zu bekommen. Dieses Verfahren braucht aber gewisse a-priori Informationen über das untersuchte Objekt. Algorithmen wurden implementiert mit der Randbedingung, dass so wenig wie möglich a-priori Information vorausgesetzt wird, was in Verbindung mit verspannten Objekten auch nach dieser Arbeit eine Herausforderung bleibt.

Für ein anderes Projekt, nämlich die Untersuchung von Nanodrähten in Bauelementen welche den Draht dehnen, verwendeten wir einen einfacheren Ansatz. Nur die Position von Bragg-Reflexionen im reziproken Raum wurde in Abhängigkeit der Position des nano-fokussierten Synchrotronstrahls entlang des Nanodrahts bestimmt. Das Dehnen eines Drahtes ist motiviert durch den Versuch, aus dem Halbleitermaterial Germanium mittels Dehnung einen direkten Halbleiter zu machen. Dies wurde theoretisch vorhergesagt für uniaxial verspanntes Germanium. Mit nano-fokussierter Synchrotronstrahlung wurden elongierte Germanium- aber auch Silizium-Nanodrähte auf ihren Verspannungszustand untersucht. Bauelemente, entwickelt an der TU Wien, lieferten einzelne vorgespannte Germanium-Nanodrähte sowie Silizium-Nanodrähte mit während des Experiments verstellbarer Verspannung. An den vorgespannten Germanium-Drähten wurde die Verspannung in radialer Ebene durch die Bestimmung der Position eines bestimmten Bragg-Reflexes gemessen, während der nano-fokussierte Synchrotronstrahl über die Probe rasterte. Die resultierende Spannung entlang des Drahtes wurde mit Finite-Elemente-Simulationen verglichen, um die axiale Spannung des Nanodrahtes zu bekommen. Auch konnte mit dem simulierten Spannungstensor die theoretische Ramanverschiebung berechnet und mit den Resultaten von Mikro-Raman-Messungen verglichen werden. Durch die Bestimmung der Position mehrerer Bragg-Reflexe, war es möglich, die axiale und radiale Spannung eines verspannten Silizium-Nanodrahtes zu messen. Dies resultiert in der Bestimmung eines Poisson-Verhältnisses für einen uniaxial und tensil verspannten Silizium-Nanodraht. Die Verspannung des Drahtes wurde danach sukzessiv erhöht und gemessen, bis hin zur plastischen Verformung, also dem Reißen, des Nanodrahtes. Damit wurde auch gezeigt, dass Röntgenbeugung in Verbindung mit nano-fokussierter Synchrotronstrahlung eine Analyse komplizierter Verspannungszustände von Objekten mit Dimensionen im Nanometerbereich zulässt.

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Strain engineering in semiconductor materials facilitates the fabrication of novel electronic and optoelectronic devices. The electronic properties of semiconductors are determined by their building blocks, i.e. their atoms, and by the way how these building blocks are arranged, i.e. their crystal structure. If stress is applied, the spatial arrangement of the atoms is altered and the material is strained according to its mechanical response property, the stiffness. Within the elastic regime, strain results in a changed band structure of the material. The band structure of a semiconductor can be seen as a map of its electronic and optoelectronic properties.

X-ray diffraction, being a versatile structural characterization method, probes the atom’s electron clouds in crystalline materials, and is widely used to deduce the strain in semiconductor materials. In this work, X-ray diffraction is used to probe the strain distribution in semiconductor nanowires. Nanowires are several µm long rod shaped semiconductor structures with diameters below 100 nm. Due to quantum confinement effects the nanowires are one-dimensional in terms of their electronic properties. This allows for the fabrication of various devices, being superior compared to similar state of the art devices.

Standard ensemble X-ray diffraction experiments in connection with results from electron microscopy investigations were performed to investigate InAs/InAs1−xPx

nanowire heterostructures. These nanowire structures exhibit a strain state according to the material’s lattice mismatch and the nanowire geometry. Novel characterization methods were developed based on extensive data evaluation and Finite Element simulations, to deduce the nanowires’ crystal structure, as well as their average chemical composition and dimension. The characterization experiments of several samples fabricated under systematically varied environmental conditions provide conclusions on the process of nanowire formation.

The information on single nanowires beyond statistical averages becomes interesting when, for the fabrication of devices, the exact strain distribution in one particular nanowire is crucial. Moreover, the strain state of one single nanowire could be altered in or by some device, pushing the need for its precise strain monitoring. For example, the knowledge on the complex strain distribution of a short hetero-segment in a nanowire will allow the fabrication of strain engineered devices. This requires X-ray beams with sub micrometer spot sizes. Fortunately, 3rd _{generation synchrotron facilities fulfill the}

preconditions to perform micro- as well as nano-focused synchrotron radiation diffraction experiments. Therefore, strain investigation of a single InAs1−xPxnanowire segment and

its vicinity were performed, by illuminating it with a nano-focused synchrotron radiation beam. Its three dimensional strain distribution was found by comparing the measured scattered intensity distribution with simulations. However, there is also the theoretic possibility of deducing the strain distribution directly from the measured scattering density, i.e. phase retrieval. Such phase retrieval algorithms were implemented and investigated in this work. This however, still needs some a-priory information of the sample, and the attempt to find phase retrieval algorithms which only need to “know” a rough approximation of the samples dimension remains challenging.

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For the investigation of nanowires inside straining devices, we employed a simpler approach, evaluation the Bragg reflection’s position in reciprocal space as a function of the position of the nano-focused synchrotron radiation spot along the nanowire. These studies are motivated by an attempt to obtain direct band-gap germanium by applying tensile strain. It has been theoretically predicted that uniaxial strain alters the band structure of germanium, converting it into a direct semiconductor. Therefore, a quantitative strain analysis of strained germanium as well as silicon nanowires was performed using nano-focused synchrotron radiation diffraction. Straining devices as developed at the TU vienna provided pre-strained germanium nanowires, as well as silicon nanowires which can be gradually elongated during the experiment. For the germanium nanowires, the strain perpendicular to its growth axis was deduced by probing the position of a proper Bragg reflection while sampling the region of the nanowire with the focused synchrotron radiation beam. Finite Element simulations were compared with the measured strain values resulting in the nanowire’s axial strain. Moreover, with the simulated strain tensor the theoretic Raman shift was calculated and compared to µ-Raman experiments. By recording several Bragg reflections the strain in different directions, e.g. in axial direction and in the radial plane, is accessible. With this method a silicon nanowire at evaluated strain state was investigated, and a value of the Poisson ratio for an uniaxially strained silicon nanowire was found. Its strain was then increased and monitored up to a plastic deformation of the nanowire. Nano-focused synchrotron diffraction proved to be a powerful tool for strain analysis allowing for a detailed investigation of objects in the nanometer scale.

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Chapter 1 Introduction

In semiconductor research, the characterization of nanometer-scaled objects and devices is crucial for a full understanding of their physical properties, as well as of their fabrication process. Self assembled crystallization, i.e. self assembled growth, of semiconductor nano-structures has proven to be a versatile tool, resulting in the fabrication of crystallites of different shapes, dimensions, and material combinations, see Refs [1, 2, 3, 4]. The process follows basically the scheme of providing a crystalline substrate and a constant supply of atoms, being the building blocks of the desired structures. Under proper environmental conditions, for instance temperature and pressure, the atoms form crystalline structures on the substrate’s surface with particular size and shape. A connection between the growth conditions and the resulting crystallite’s properties can only be made by extensive and systematic investigations. Therefore, precise methods to deduce the sample’s physical properties are needed. X-ray diffraction has been successfully used to deduce the crystal structure and quality as well as average structural parameters of a high number of different samples and sample structures. It is in fact a very powerful characterization tool in semiconductor industry and research. One big advantage of X-ray diffraction is that the average strain within the investigated structures is detectable, and quantitative results can be determined. With the fabrication of more and more complicated nano-scaled objects and sample structures, characterization using only one method, however, might become not sufficient to provide all desired information. Therefore, results from electron microscopy as well as Raman scattering investigations were used in this work, to connect them with the performed X-ray diffraction experiments. This results then in a full picture of the investigated samples. Moreover, with the advent of 3rd generation synchrotron facilities, which provide very intensive and coherent synchrotron radiation, spatially resolved scattering experiments became possible in the last years. The synchrotron radiation is focused down to focal spot sizes of several 100 nm allowing for probing single nano-scaled objects, or also locally resolved diffraction experiments on several micrometer sized objects. In this work, nano-focused synchrotron diffraction experiments were performed on different semiconductor nano-structures, resulting in a detailed picture of their strain states.

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By using such nano-structures as building blocks, new designs for future electronic and photoelectronic devices have been developed, since low-dimensional semiconductor architectures result in enhanced device properties compared to the current state of the art. Reducing the dimensions of structures to few 10 to 100 nm, their physical properties become different to three-dimensional bulk due to quantum mechanical effects, like an increase in the density of states or changes in the band structure as well as band alignment due to quantum confinement [5]. While two-dimensional heterolayer structures are already widely used, one-dimensional nanowires (NWs) offer even greater flexibility. Being 1D in the quantum mechanical sense, by including heterosegments of different materials the formation of 0D quantum dots or tunnel barriers becomes possible. Due to elastic relaxation, a larger mismatch between materials can be tolerated during fabrication of NWs without introduction of defects, as compared to 2D systems, which further enhances the design freedom [6, 7, 8, 9, 10, 11]. Therefore, materials with high electron mobilities like InAs and InAs1−xPxcan be combined to form

heterostructured NWs still showing good crystalline properties. Moreover, due to the small contact area of the NWs with their substrate’s surface, they can be combined with state of the art silicon technology, even tough InAs/InP and Si have a lattice mismatch of ≈ 8 − 11 %. Additionally, wurtzite/zinc-blende crystal phase transitions were observed in NWs, which are reported to be controllable with proper growth conditions, and might serve as an additional tweaking parameter of the material’s properties. Kriegner et al. [12, 13] found different lattice parameter for the wurtzite phase which introduces additional strain into a wurtzite/zinc-blende stacked NW. Inhomogeneous strain fields are intimately connected to heterostructures of either different materials or also different crystal phases, which need to be controlled, but can also be exploited. Strain alters the band-structure of semiconductor materials dramatically, which was for example used by M. Chu et al. [14] to enhance the electron mobility in nano-scale MOSFETs. The need for high-efficiency light emitting or detecting devices based on Si-technology, also drives the efforts of altering the physical properties of germanium, see Refs [15, 16, 17]. The high mobility values, especially for holes, along with the compatibility with complementary metal-oxide-semiconductor processing, would make it a promising material for novel on-chip light sources and detectors, see Refs [18, 19, 20, 21]. However, intrinsic Ge is is an indirect semiconductor, i.e. the lowest energy of the valence band valley at the Γ point is 140 eV higher than the valence band valley at the L point. Electron-hole recombination from the lowest energy valleys are therefore only possible in connection with Coulomb and phonon scattering, rendering this recombination more inefficient than for a direct band-gap material. The goal is, by altering the material’s properties, to make Ge a direct semiconductor. There are various suggestions and theoretical works providing road maps to do so, such as band structure modifications by quantum confinement, alloying, or zone folding and quantum confinement. The latter can be achieved by fabricating alternating layers of Si/Ge, which unfortunately shows only a weak transition strength [22]. Alloying Ge with Sn was reported to show direct band-gap behaviour [23], but this material combination is difficult to fabricate due to the low

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CHAPTER 1. INTRODUCTION 13

solubility of Sn in Ge. Another attempt is strain engineering, which is a powerful tool to alter the band structure of semiconductor materials in beneficial ways. By applying uniaxial tensile strain of around 4 % on Ge, studies predict a a direct band-gap at the Γ point [24]. This would give the possibility for optical emission in this technologically important group-IV element. An experimental verification of this prediction is highly desirable. A good ansatz for a device structure, which allows to strain Ge without defects, is using NWs, where it has been shown that very high tensile strain values up to 10 % are achievable before mechanical fracture [25]. When going into this research direction, it is as important as it is non-trivial to fabricate devices which provide strained NWs. To develop suitable processes, the accurate measurement of the strain state is of high importance. A common way is using Raman scattering, which proved to be an accurate tool for bulk samples and samples with sizes in the µm regime, i.e. micro structures. However, there are are some difficulties like a certain penetration depth, or at dimensions in the nm range confinement effects, and a (possible) dependence of the phonon deformation potentials on the lattice structure.

Therefore, several NW structures and devices were investigated in this work, using standard ensemble XRD techniques as well as nano-focused synchrotron diffraction. Growth series of InAs/InAs1−xPxheterostructured NWs where the object of an extensive

study, with NWs fabricated by Bernhard Mandl using a self seeded particle assisted growth scheme in a metal oxide chemical vapor phase epitaxy system. Moreover, in order to find the chemical composition of an InAs1−xPx segment in a single NW as

well as the heterosegment dimensions and hence the strain distribution in this region, a nano-focused synchrotron radiation diffraction experiment was performed. During more nano-focus experiments, sustained Si and Ge NWs were subject of extensive strain characterizations. To achieve germanium NWs with uniaxial tensile strain, Lugstein et al. developed according sample layouts based on silicon on insulator (SOI) chips. Devices with single pre-strained NWs as well as devices allowing for the variation of the strain in single NWs were fabricated. With these samples strain experiments on elongated Si and Ge NWs were possible, and already performed by Johannes Greil using µ-Raman scattering.

A detailed X-ray diffraction investigation of InAs/InAs1−xPx heterostructured NWs

with a controlled variation of the P concentration as well as the heterostructure segment length is presented. Two growth series of InAs/InAs1−xPx heterostructured NWs

were extensively characterized using XRD in connection with results from scanning electron microscopy (SEM), high resolution transmission electron microscopy (HRTEM), and energy dispersive X-ray spectroscopy (XEDS). The results from X-ray diffraction provide, along with inputs from complementary electron microscopy, a full picture of the average chemical compositions in the NW’s hetero-part, the NW dimensions, and the strain distribution in the NWs. Note, the ensemble average over a high number of illuminated nanowires gives certain ranges in the deduced values, according to different properties from nanowire to nanowire. The results of two selected samples were additionally compared with investigations of single NWs using XEDS in TEM,

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as an independent confirmation of the resulting chemical composition. A stacking of wurtzite/zinc-blende crystal polytypes was confirmed both by TEM and XRD. This polytypism had to be taken into account when deducing the strain state and the chemical composition of the NW ensemble. New methods for X-ray diffraction ensemble characterization were developed, since not only NWs but also crystallites formed on the sample surfaces. Additionally, the complicated strain distribution inside the NWs did not allow a straight forward data evaluation, i.e. Finite Element simulations and from that calculations of the diffusely scattered intensity were necessary. Therefore, a simple statistic distribution of the NW properties in the ensemble was assumed, and a linear dependency of the chemical composition on the III/V materials supply ratio, and a dependency of the NWs aspect-ratio on the growth temperature was found. Furthermore, the resulting strain fields due to the wurtzite/zinc-blende stacking make the average polytype segment length and the wurtzite content of the wires accessible in the X-ray experiment, and these parameters have been obtained from ensemble XRD measurements as well.

However, the traditional XRD experiment setups are not feasible for the investigation of single nano-structures. Therefore, nano-focused synchrotron radiation diffraction was used in this work to get the desired real space resolution. The possibility of focusing synchrotron radiation down to 300 × 300nm2 _{full width half maxima (FWHM) spot}

sizes, results in the opportunity to perform diffraction experiments on segments (with a length according to the focal spot size) along strained Si and Ge NWs. This eventually gives the lattice plane spacings along the probed Bragg reflection, which can then be, by comparing with lattice parameters from literature, transformed into the strain along the according direction. Moreover, by measuring the position of a second, properly chosen Bragg reflection, the axial as well as radial strain, and thus the Poisson ratio of the uniaxially strained NW can be deduced. Using nano-focused synchrotron radiation diffraction, the radial strain along the axial direction of several pre-strained germanium NWs was measured. The results were then compared with strain values obtained by Finite Element simulations of elongated NWs using their shape and dimension deduced with scanning electron measurements (SEM). This rendered an axial strain distribution of the investigated NW, which compared reasonable with strain values from µ-Raman scattering experiments. However, simulations of the Raman shift with the strain tensor from Finite Element simulations gave no proper agreement with the measured Raman shift. The reason might be an incorrect description of the NW’s material parameters, since literature parameters of bulk germanium were used for all simulations. As a second system silicon NWs were investigated, with the possibility to vary the strain, i.e. their elongation, during the experiment. These NWs were strained until plastic deformation, i.e. until they broke. Additionally, as already mentioned, the Poisson ratio of one silicon NW at an elevated strain state was determined, which gave a value of 0.079 ± 0.004. The experiment showed the Poisson ratio being a factor two smaller than the theoretically expected Poisson ratio (for germanium strained uniaxially in [111] direction). The highest detected radial strain, before the investigated NW actually

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CHAPTER 1. INTRODUCTION 15

broke, was −0.38 ± 0.01 %. With the deduced Poisson ratio, the highest axial strain in the investigated uniaxially strained silicon nanowire is around 4.8 %.

With nano-focused synchrotron diffraction experiments not only the positions of the Bragg reflections can be evaluated. When a single nano-scaled object like a heterostructured NW is coherently illuminated, its scattering density as well as the displacement of the atoms in the crystal direction of the probed Bragg reflection are accessible. Basic research requires insight into single NWs, or even parts of them as the NW-substrate interface, to complement information from other methods like transmission electron microscopy, where sample preparation is difficult and often destructive. Therefore, a single InAs1−xPx segment in an InAs NW was coherently

illuminated and the diffusely scattered intensity was recorded, resulting in a three dimensional reciprocal space map (RSM). With simulations of the scattered intensity the shape of the measured intensity distribution was described, giving a good picture of the strain distribution in the InAs1−xPx segment and its vicinity. This, however, is

an approach of the evaluation of coherent diffraction imaging with the need of a lot of a-priori information on the investigated object.

Another Ansatz of dealing with intensity distributions of Bragg reflections recorded at coherent diffraction imaging experiments, is performing phase retrieval on the measured data. But, let’s start from the beginning. The incoming monochromatic and coherent electromagnetic waves are scattered at the investigated object, and the scattered waves would actually carry almost all information on the electron density distribution inside the object. However, only the numbers and directions of scattered photons are probed. Therefore, one loses some information which is, due to the fact that an electromagnetic wave field is complex valued and the measured intensity is its squared Fourier modulus, the phase of the scattered wave. Fortunately, there is the theoretical possibility to recover the phase of properly measured scattered intensity distributions by using phase retrieval algorithms. The most common and promising ones are iterative algorithms needing a certain a-priory knowledge of the investigated object. However, this should basically be just a rough knowledge of the objects dimensions, which is a reasonable restriction since this needs to be known anyways, to be able to choose the proper size of the focused beam. Coherent diffraction experiments on heterostructured core-shell nanowires were performed. Due to different material constants and the core-shell structure, these NWs exhibit complicated strain distributions. This corresponds to a displacement of the crystal’s atoms, and thus the electron cloud around the atoms. Coherent diffraction imaging can probe this displacements, if one succeeds to retrieve the phase from such a scattering experiment. However, the reconstruction of the complex scattering density in strained core-shell NWs without using a priori information of the investigated object was not successful so far. In order to develop better algorithms with improved convergence, studies of promising algorithms on simulated data from model structures with the same geometry were performed. The convergence for strained NWs is very poor when solely the support constraint is used in real space, since the amount of unknown information increases

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tremendously when the phase of the measured electromagnetic field is varying in a wide range. Using a variation of the guided hybrid input-output [26] algorithm along with the shrink- wrap algorithm [27], and a proper value for the feedback parameter β outside the usual range (see Ref. [28]), the convergence for strained test objects was investigated: By applying internal constraints like the positivity of the real part of the real space scattering density (corresponding to limiting the phase in real space to the range of −π/2 to π/2 ), weakly strained core-shell NW were retrieved providing an error value withing the machine precision. When probing the (111) Bragg reflection, the threshold of the P concentration for having a phase shift within the mentioned range is at about 2.5 %. For phase retrieval without using such internal constraints, convergence is already an issue for non-strained objects, and becomes poorer and poorer for increasing strain values, and hence range of phase variation. The question where the limits of current algorithms lie, and whether it is in practise possible to retrieve highly strained objects without a priori information, remains open. Hence, theoretical limitations and other restrictions are discussed.

1.1 Structure of the thesis

The thesis is divided into three main parts, reflecting the three main projects I was working on. They are embedded in an introduction to the applied methods as well as some additional material for further information in the end. At the very end i want to briefly introduce the reader to a small side project we had during my PhD, see AppendixE.

• Theory part covering the basic theoretic background for this work.

• Methods and their technical aspects of focusing of X-rays.

• Phase Retrieval.

• InAs1−xPx Nanowires.

• Suspended germanium and silicon Nanowires.

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Chapter 2 Theory

2.1 Diffraction for Dummies

2.1.1 X-rays/Synchrotron Radiation

X-rays as well as synchrotron radiation are basically electromagnetic waves. X-rays were first discovered in 1895 by W. C. R¨ontgen. He was studying cathode radiation in various types of vacuum tubes, when he found what he called X (the mathematical symbol for an unknown variable) rays, which were emitted from an aluminum target foil illuminated by cathode radiation. For details I want to refer to R¨ontgen’s publication “Eine neue Art von Strahlen” [29] and vonLenard’s Nobel lecture [30]. This new radiation became a hot topic, eventually, it was found that the spectrum of X-rays consists of Bremsstrahlung due to deceleration of the electrons when approaching the target, and of characteristic radiation. The anode material is subject to multiple electronic excitations leading to characteristic radiation, i.e. radiative recombinations when electrons fall back to their initial energy state. X-rays quickly became a tool for clinical diagnostics, and also for material science and crystallography (for more information please have a look at Laue’s Nobel lecture [31]). On the other hand, synchrotron radiation was discovered and announced in 1947 by F. Elder, A. Gurewitsch, R. Langmuir, and H. Pollock in a General Electric [32] synchrotron accelerator, as the electromagnetic radiation emitted when charged particles are accelerated radially (a ⊥ v). Synchrotron radiation covers a broad spectral range and also includes radiation in the X-ray energy regime. These two types of radiation are of course basically the same, it is just sometimes named differently due to the different kinds of generation. In synchrotrons the radial acceleration of the charged particles is normally done using one of three different devices, resulting in radiation with different intensities and coherence properties. A bending magnet simply alters the trajectory of the particle beam, using a homogeneous magnetic field created by one or several dipole magnets, into a certain segment of circle, which then results in the emission of a continuous radiation spectra. On the other hand, insertion devices like a Wiggler alter the beam direction several times into a wiggly trajectory by using a periodic structure of dipole magnets with diametrically opposed pole orientations.

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The amplitude of this movement is higher in a Wiggler than it is in an Undulator. The radiation from every oscillation, which is actually emitted into a relativistic cone with tangential direction, simply sums up in a Wiggler, and becomes a comparable continuous emission spectra as from the bending magnet, but with a significantly higher intensity. The Undulator’s magnets are not so strong and their strength as well as their periodicity are chosen in a way that the generated radiation sums up coherently and displays interference patterns leading to narrow energy bands in the radiation spectra. To define a measure of the beam “quality” we introduce the brilliance

brilliance = photons

seconds · mrad2_{· mm}2_{, ·0.1 % BW} (2.1)

which is the produced number of photons per second, referred to the angular divergence and cross-sectional area of the beam as well as to a fraction of the bandwith of the produced spectra. From X-rays and synchrotron radiation one extracts the radiation of the desired energy using X-ray optics, but to describe such optics we need some theoretical background. First, an electromagnetic plane wave can be expressed as:

E(r, t) = ePE0e−i(ωt−kr) (2.2)

This equation describes the spatial and temporal variation of an electromagnetic field, with eP being the unit vector in polarization direction and k being the wave-vector

pointing in propagation direction. Electromagnetic waves are transversal, i.e. ePk = 0.

The energy of a wave is given by E = ~ω (~ is the Dirac constant and ω = 2πλc,

with c being the speed of light in vacuum), and defines the absolute value of the wave-vector: |k| = k = 2π_λ . However, the phase velocity of an electromagnetic wave depends on the medium it is travelling in. This deviation is given by the complex valued refractive index n = _v c

phase (with vphase =

ω

k), which is depending on the energy of

the electromagnetic wave according the the medium’s dispersion relation. The real part of the refractive index describes the refraction and the imaginary part the absorption, and both are connected via the Kramers-Kroning relations. Refraction, as we commonly know it, is a surface effect, and would not be observable by assuming a monochromatic electromagnetic wave travelling inside just one medium. But when the wave reaches a surface with a transition from one medium to another the commonly known change of propagation direction appears (Snell’s Law). X-rays are in an energy range where the index of refraction is almost 1 for any dielectric material, moreover it is actually even slightly smaller than 1. This means, in terms of geometrical optics, that X-rays are refracted away from the surface normal of the refracting surface, whereas for visible light, the waves are refracted towards the surface normal. This makes our beloved X-rays quite special, and they penetrate almost without deviation through a very broad range of materials. Now let us write the refractive index as:

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CHAPTER 2. THEORY 19

Fig. 2.1: Calculation of the penetration depth depending on the incidence angle αi, for various exit

angles. The calculation was done for CuKα1radiation being scattered on the surface of a Si waver. The

left and the right graph show the resulting values on a linear and a half-logarithmic scale, respectively.

One can calculate from Snell’s law the critical angle of total external reflection for X-rays hitting a surface under very small angles. For simplicity we set β = 0, since in most materials β is one order of magnitude smaller than δ, as well as δ2 _{= 0.}

cos(α) = n cos(α0) with α0 = 0 and α = αC =⇒ αC ≈

√

2δ (2.4)

This total external reflection occurs when the incidence angle, i.e. the angle between the incoming X-ray beam and the sample surface, is below or equal to the critical angle, and plays a crucial role for X-ray diffraction experiments. The refracted beam normal to the sample surface is exponentially decaying with a magnitude according to the incidence angle. Therefore, the damping till a 1_e intensity fraction of the evanescent wave inside the material can be qualified as the penetration depth and calculated, as elaborated by Dosch et al. [33], according to:

Λpen = 1 Λi + 1 Λf , with (2.5) Λi,f = 1 k√2 q

sin2(αi,f) − 2 Re(δ)]2+ 4 Im2(δ) − [sin2(αi,f) − 2 Re(δ)]

−1₂

. (2.6)

The value for δ depends on the material, as well as on the energy of the X-rays. Therefore, the penetration depth is tunable by changing αi and αf. Figure2.1 shows

how the penetration depth behaves with the change of this two angles. Please note that the penetration depth is no sharp edge, and the accessible depth is very sensitive on αi. In practice, especially with rough samples, this value can only be determined

to a certain precision, hence there are restrictions of the penetration depth accuracy in the experiment. The amount of how much of the incoming X-ray intensity is reflected or transmitted when hitting the surface of some dielectric material can be calculated

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with the Fresnel equations, see for example in [34]. Electromagnetic waves exposing matter can be refracted, adsorbed, and also diffracted when getting in contact with matter and/or when travelling from one medium to another one. The real physical picture is however a mixture of these interactions. For this work we assume all materials being dielectric and we solely discuss elastic scattering of X-rays on single crystalline semiconductor samples. Additionally, we assume monochromatic and coherent plane waves, rendering e−i(ωt) a prefactor which is not depending on the position of the wave. Therefore we will omit it subsequently until we reach the coherence discussion.

2.1.2 Scattering

Scattering of a monochromatic and plane electromagnetic wave, according to a scattering potential given by the operator

ˆ

V (r) = graddiv − k2χ(r) (2.7)

(with χ(r) = r(r) − 1 being the polarizability and r(r) the relative permittivity of the

material), is described by the following scalar wave equation

(∆ + k2)E(r) = ˆV (r)E(r). (2.8)

This equation is, for the case of non-magnetic materials, generally derived from Maxwell’s equations, and can be written in integral form as

E(r) = Ei(r) +

Z

d3r0G0(r − r0) ˆV (r0)E(r0), (2.9)

where Ei is the vacuum wave equation’s solution:

(∆ + k2)Ei(r) = 0. (2.10)

Ei is physically understood as the incoming (primary) wave. G0(r − r0) is the Green´s

function of a free particle, and thus the solution of

(∆ + k2)G0(r − r0) = δ(3)(r − r0), (2.11)

with δ(3) being the tree-dimensional Dirac delta-function. Two forms of the Green´s function are used:

− 1 4π eik|r−r0| |r − r0_| or − i 8π2 Z _d2_k || kz eik(r−r0), (2.12) for dkxdky/kz ≡ d2k||/kz with kz = q k2_{− k}2 x− k2y > 0.

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Therefore we get the solution for the scattered wave according to

E(r) = Ei(r) + Z d3r0G0(r − r0) ˆT (r0)Ei(r0), (2.13) where ˆ T = ˆV + ˆV ˆG0V + ˆˆ V ˆG0V ˆˆG0V + . . .ˆ (2.14)

is the scattering operator (T-operator) that contains the information on the whole illuminated volume. With the explicit expression of the Green function, the scattered wave turns out to be:

Es(r) = − i 8π2Ei Z _d2_k || kz eikr Z d3r0T (rˆ 0)e−i(k−ki)r0 _(2.15)

Which is a superposition of all Fourier transformations of ˆT (r) with all possible wave vectors k having the same length but different directions. This fully describes the scattered electromagnetic field, however, for simulations of scattered intensities, i.e. where this integral is not solvable analytically, approximations providing the possibilities of numerical calculations are necessary.

Fraunhofer approximation

In the Fraunhofer approximation we assume that the region where ˆT (r) differs substantially from zero (i.e. the illuminated sample volume) is much smaller than the first Fresnel zone, i.e. the distance sample/detector is much larger than the sample size. Then we get k|r − r0| ≈ kr − ksr0, where ks = kr_r. Furthermore one can assume that

1

|r−r0_| is a slowly varying function under the integral and can therefore be written as a

prefactor before the integral. With the first form of the Green´s function in Equ.2.12

and these approximations we get:

Es(r, Q) ≈ − 1 4π eikr r Ei Z d3r0T (rˆ 0)e−iQr0 (2.16)

The scattered wave-field depends in this approximation on the scattering vector Q = ks− ki. The direction of ksis fully determined by the detector position r. When we now

map out reciprocal space with some “virtual” point detector with a resolution volume ∆VQ we get an intensity distribution in reciprocal space according to

J (Qd) = Ii 16π2_A Z ∆VQ d3∆Q Z d3r0T (rˆ 0)e−i(Qd+∆Q)r0 2 , (2.17)

as derived in [35]. With Ii being |Ei|2, i.e. the intensity of the incident wave, and A the

irradiated area on the sample’s surface. Apparently, Fraunhofer approximation means that the sample acts as a point scatterer in real space, and the scattered wave has a

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Fig. 2.2: Polarization of the incoming beam with respect to the sample surface

spherical wavefront. The wave vector of the scattered wave-field is then fully determined by the relative positions of the sample and the “virtual” point detector. Qd denotes

Q-discrete, since what is done here is basically convolution the Fourier transformation of ˆT (r) with a certain detector resolution function at predefined positions in reciprocal space, i.e. the positions of the “virtual” point detector. However, if one wants the electromagnetic field (and by J (Q) = |E(Q)|2 _{also the intensity again) in reciprocal}

space along the surface of a 2D detector, we define r = rc+ rd+ ∆r. The vector rc is

then for instance the center of the detector, rd the positions of the detector’s pixels and

∆r the points inside one pixel. Let’s assume that the detector plane is perpendicular to the vector rc, i.e. rc ⊥ (rd+ ∆r). To keep the equations simple, the detector is

assumed to be really far away from the sample, therefore the whole detector opening is small compared to this distance. With this two assumptions one can estimate r ≈ rc.

By defining k0s = kr_rc_c, Q0 = ki− k0s, Qd = kr_rd_c, and ∆Q = k∆r_r_c we then calculate the

electromagnetic field on the detector surface at the positions of every pixel as:

Es(Q0+ Qd) = − 1 4π eikrc rc Ei Z Apixelq d2∆Q Z d3r0T (rˆ 0)e−i(Q0+Qd+∆Q)r0 _(2.18) with Apixel

q being the area of each pixel of the 2D detector in reciprocal space. However,

solving all this problems would be still intractable since the T operator still looks as given in Equ.2.14. This brings us directly to the next section...

2.1.3 Kinematical approximation and Diffraction on perfect

crystals

Here, multiple scattering effects are neglected, which is equal to the first Born approximation in quantum mechanics. Therefore we take only the zeroth _{order of}

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different experimental condition and sample. The dielectric susceptibility χ(r, λ) as seen in Equ.2.7, is related to the electron density by

χ(r, λ) = −λ 2 π relCρ(r), (2.19) where rel = e 2 4π0melc2 ≈ 2.82 × 10

−15 _{is the classical electron radius, and C the linear}

polarization factor. In a perfect crystal the polarizability is a periodic function of the

C S-polarisation 1 P-polarisation cos 2θ

position and can be written as a Fourier series over the reciprocal lattice with reciprocal lattice vectors g. But let us start at the beginning, a perfect crystal means that there exists a certain periodicity in every spacial direction of the arrangement of its building blocks, i.e. the atoms. Crystal lattice vectors a1, a2 and a3 represent the minimum

periodicity in each main crystallographic direction of our perfect crystal. These vectors define the unit cell of the crystal, which contains the minimal group of atoms representing this periodicity. So by starting from the origin, every unit cell in the crystal can be reached by the translation

T = n1a1+ n2a2+ n3a3, (2.20)

with n1, n1, and n1 being integers. The reciprocal lattice is then spanned by the vectors

b1, b2 and b3 given as

bi = 2π

aj × ak

ai(aj × ak)

, (2.21)

and the reciprocal lattice is then:

G = hb1+ kb2+ lb3, (2.22)

and h, k, and l being again integers. With the reciprocal lattice vectors g ∈ G, we can now define the polarizability of the whole crystal as

χcrystal(r) = Ω(r)

X

g

χg(λ, C)eigr, (2.23)

with the Fourier polarizability given by

χq(λ, C) =

λ2C Vcell

Scell(q), for Scell(q) =

Z

Vcell

d3rχ(r)e−iqr, (2.24)

being the structure factor and Vcell the unit cell volume. The shape function Ω(r)

is unity inside the illuminated part of the sample and zero outside. Lets assume the elementary cell contains s Atoms with atomic form factors fj(q), 1 ≤ j ≤ s and

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24

Fig. 2.3: Illustration of Equ.2.27along one dimension of reciprocal space.

positions rj, 1 ≤ j ≤ s. In most cases the electrons within one unit cell are assumed

to belong to the atoms in the same cell, and the influence of electrons from other unit cells is neglected. Therefore, the structure factor of one unit cell is proportional to the sum of the atomic form factors of all the atoms in it.

Scell(q) = − rel π X j Z d3r0ρa(r0)eiqr 0 | {z } fi(q) e−iqrj _(2.25)

As already stressed, only elastic scattering scattering is assumed here, i.e. the energy of the radiation is far from absorption energies or resonances. However, the atomic form factor, defined as the amplitude of the wave scattered by the electronic could of each atom, actually looks like this:

fi(g, λ) = fi,o(g) + fi0(g, λ) + if 00

i (g, λ) (2.26)

By inserting Equ.2.23 in Equ.2.16 (in kinetic approximation) we get: E(r, Q) = πC Vcell eikr r Ei X g Scell(g) Z d3r0 Ω(r0) e−i(Q−g)r0 (2.27)

The momentum transfer Q is determined by the angular positions of the sample and the detector, and obviously the maximum amplitude occurs when the Laue condition

Q = g (2.28)

is fulfilled. When we take the absolute values of both sides of the Laue condition we get

4π λ sin(θ) = 2π d √ h2_{+ k}2_{+ l}2_, _(2.29)

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Fig. 2.4: Sketch of two commonly used coplanar diffraction geometries.

which is eventually Bragg’s law:

2dsin(θ) =√h2_{+ k}2_{+ l}2_λ. _(2.30)

2θ and d are illustrated in Fig.2.4.

Two Structure Factor examples

The structure factor is one of the most important tools for crystallographers: By recording the position as well as the (integrated) intensity of optimally a large number of Bragg reflection of some (more or less) perfectly crystalline material, and then fitting (with of course some a priori information on the crystal structure) the theoretically described intensities to the measurement, the crystal structure can be fully determined. The intensity of the Bragg reflections is, as seen for example in Equ.2.27, mainly described by the structure factor. Let us calculate the structure factor of cubic diamond structure, which is found in Si crystals. The basis cell in this case consists of 8 atoms of the same element. Diamond structure can be described as a cubic lattice with a face centered unit cell (fcc) and a two atomic base. This two atomic base forms two fcc unit cells, where the second fcc unit cell is shifted by a(1₄,1₄,1₄) with respect to the first

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one. The structure factor for atoms with form factor fAforming a diamond crystal with

lattice constant a is then:

Scell(g) = −

rel

π fA(e

−ig0

+ e−ia2g(a1+a2)+ e−i a 2g(a2+a3)+ e−i a 2g(a1+a3)+ e−i a 4g(a1+a2+a3)

+ e−ia4g(3a1+a2+3a3)+ e−i a

4g(3a1+3a2+a3)+ e−i a

4g(a1+3a2+3a3))

Scell(h, k, l) = − rel π fA(1 + (−1) h+k + (−1)k+l+ (−1)h+l+ (−i)h+k+l+ (−i)3h+k+3l + (−i)3h+3k+l+ (−i)h+3k+3l) = −rel π fA[1 + (−1) h+k_{+ (−1)}k+l_{+ (−1)}h+l_{][1 + (−i)}h+k+l_]

With g ∈ G, the structure factor is therefore non-zero when:

• h k l are odd, then the structure factor is Scell = r_πelfA4(1 ± i).

• h k l are even, and h+k+l=0(mod4) resulting in Scell = r_πelfA8.

As a second example we will have a look at the resulting structure factor for the zinc-blende structure. This is similar to diamond structure but the base consists of two different elements with form factors fA and fB. The structure factor is calculated as

above, however only the result will be shown now. Therefore for zink-blende structure we get a non-zero structure factor when the following conditions are fulfilled.

• When h, k, and l are even, and h+k+l=0(mod4), which results in

Scell = 4

rel

π (fA+ fB).

• Then h, k, and l are all odd, giving

Scell = 4

rel

π (fA± ifB).

• And the last non-zero case are weak reflections, where h, k, and l are even and h+k+l=2(mod4). Therefore, one gets a structure factor of

Scell = 4

rel

π (fA− fB).

2.2 Simulations

Standard X-ray diffraction experiments were done as discussed in Chap.5. Therefore, it was necessary to perform simulations of the diffracted intensities from an ensemble of NWs on a large, i.e. in the range of mm2_{, illuminated spot on the sample. In this}

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experiments, as shown in Chap.4 and at the end of Chap.5, are well described using this far field approximation. The simulation of ensemble X-ray diffraction measurements takes more considerations, and will be discussed later. Anyways, for all simulations, Finite Element Method (FEM) simulations had to be done first to get a good approximation of the displacement field in the investigated NWs, and then according diffraction maps around the measured Bragg reflections were calculated from the resulting displacement fields.

2.2.1 Finite Element modelling

To write down the mathematical description for the problem of deducing the strain inside for example a heterostructured NW with given shape, dimensions, and chemical composition, a short introduction into linear elasticity is necessary. Notations and derivations follow the book of Hirth and Lothe [36]. Let xj(j = 1, 2, or 3) be coordinates

in an orthogonal Cartesian coordinate system, and σij the force per unit area acting on

an infinitesimal volume element. In mechanical equilibrium each volume element inside a solid object must be at rest. This means that there can be no net torques, so that

σij = σji. (2.31)

Additionally, no net force can act on the element, so that

∂σij

∂xj

+ fi = 0 (2.32)

(using Einstein´s sum convention), where fi is the body force per unit volume acting

on the body. When stress is introduced to a solid object it responds with deformation which can be expressed by a displacement u(r) at point r. Its i-th component is denoted as ui(r), and the strain inside the material is then defined as

ij = 1 2 ∂ui ∂xj +∂uj ∂xi . (2.33)

One can directly conclude that with this definition

ij = ji. (2.34)

For small distortions Hooke’s law is valid, which means a linear dependence of stress on the deformation. This is written as:

σij = cijklkl, (2.35)

and with Equ.2.31 and Equ.2.34 we find

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By substituting Equ.2.33 and Equ.2.36 into Equ.2.35 one gets σij = cijkl

∂uk

∂xl

, (2.37)

moreover by inserting this equation into Equ.2.32the equation can eventually be written as:

cijkl

∂2uk

∂xj∂xl

= −fi (2.38)

With that we finally arrived at the equation we have to solve inside the simulated object with the according material parameters cijkl and correct boundary conditions. How the

Finite Element Method approach of finding an approximation of the solution for this problem looks like will be described subsequently. Note, only the main ideas will be given, and if the dear reader wants a deeper knowledge of FEM and also the numerical implementations and solving methods I want to refer to the book from Zulehner [37]. The discussion will maybe not be fully correct in a strict mathematical way, just bear in mind that it was compiled by an experimental physicist. First lets define an open bounded connected set Ω in R3 _{with piecewise C}1 _{boundary Γ. Then Γ}

1 is a part of Γ

having strictly positive measure and Γ2 = Γ − Γ1. So the object we want to simulate

is Ω, with some boundary conditions on its surface Γ. For instance, one part of the surface (Γ1) could be fixed, i.e. its displacement is zero, and on the remaining surface

some pressure g could be acting. Additionally we allow for a body force f , which is now more something artificial and could for example be some inertial force. For x ∈ Γ2, let

ν(x) be the outward unit vector normal to Γ2. With the given functions

fi : Ω → R3 and gi : Γ2 → R3, (2.39)

our problem defined in Equ.2.38is written, with given boundary conditions, as: Find a function ui fulfilling cijkl ∂2uk ∂xj∂xl = −fi in Ω, (2.40) ui = 0 on Γ1, (2.41) cijkl ∂2_u k ∂xj∂xl νi = gi on Γ2. (2.42)

With that we describe the displacement field with respect to the natural state of an elastic homogeneous anisotropic solid object to a density force f in Ω, and a density force g on Γ2, with zero displacement on Γ1. For the Finite Element Method approach we

now need plenty of properties of the classical spaces L2_{(Ω) and H}1_{(Ω) (Sobolev space}

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elaborated here, and can be found in textbooks. With the assumptions that fi ∈ L2(Ω)

and gi ∈ L2(Γ2), and considering the space of admissible displacements as

V =v ∈ (H1(Ω))3 : v|Γ1 = 0

(2.43)

the weak formulation, i.e. including only first order derivations of u, of our problem is: Find u ∈ V such that

Z Ω cijkl ∂uk ∂xl ij(v)dx = Z Ω fividx + Z Γ2 gividγ, ∀v ∈ V (2.44)

where dγ denotes the one-dimensional measure of Γ. The left hand side is the bilinear form a : V × V → R and the right hand side the linear form l : V → R. The resulting variational continuous problem is written as:

Find u ∈ V such that : a(u, v) = l(v), ∀v ∈ V. (2.45)

The Finite Element Method Ansatz is then dividing the set Ω into elements, creating a discrete mesh inside the simulated object. This mesh elements can be for one dimensional objects intervals, for two dimensional objects triangles or squares, and for three dimensional objects tetrahedrons, prisms, or cuboids for example. Then, let Vh

be the finite-dimensional subspace of V containing all the continuous functions that are linear within each mesh element. The goal is then finding an approximation uh ∈ Vh

such that the equation above is true for all vh ∈ Vh. Note that the functions in Vh are

determined by coefficient vectors with respect to the finite basis. Thus, the equation a(u, v) = f (v) can be transformed into a linear equation

Au = b, (2.46)

where u is the coefficient vector of the approximation uh, and b stems from l, and A is

the stiffness matrix. To solve the resulting equation system, mainly conjugated gradient method with a geometric multigrid pre-conditioner was used during this work. Here, I want to refer again the reader to the literature for further details. So in the end one gets an approximation of u(x) inside the simulated object, and with that we go on to the next chapters to calculate the scattered intensity of such a simulated displacement field.

2.2.2 Simulating the scattered intensity

Simulating ensemble averages

For standard ensemble XRD-measurements with illumination spot sizes of several µm2 to mm2, the Fraunhofer approximation might, depending on the sample and the sample-detector distance, not be valid. The sample-detector distance can actually be very short for measurements with laboratory diffractometers. Then the radius of the first

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Fresnel zone, approximately √λr (r is the sample-detector distance), ranges typically a few microns. Hence, one needs another method to calculate the amplitude of the scattered wavefield. Therefore, we assume that the scattered wave is a superposition of plane components with different wave vectors having the same length, which is called the homogeneous wavefield approximation. And as the name already says microscopic features on the investigated sample have to be homogeneously distributed. First let us define the mutual coherence function (MCF), with the amplitude of the wave field from Equ.2.15.

Λ(r, r0) = hE(r)E∗(r0)i (2.47) The brackets hi denote an averaging over the statistical ensemble of all macroscopically indistinguishable structure states of the scattering sample. The intensity of the scattered wavefield at some point r is obtained from the MCF by

I(r) = Λ(r, r). (2.48)

We use the coordinate system with x and y-axes in the sample surface and the z-axis parallel to the outward surface normal. When using the Fourier transformation of the scattered wave the MCF can be written as:

Λ(r, r0) = 1 k2 Z d2k|| Z d2k0|| 1 kzk0z Λ(k, k0)ei(kr−k0r0), (2.49) with Λ(k, k0) = hE(k)E∗(k0)i , (2.50) being the Fourier transformation of the MCF. Since the wavefield is homogeneous the MCF only depends on r − r0, i. e. the wave field is independent of the position r: I = Λ(r, r) ≡ Λ(0). It follows that

Λ(k, k0) = δ(2)(k||− k0||)J (k), (2.51)

and J (k) denotes the intensity of the plane component of the wave field with the wave vector k, called the reciprocal-space intensity distribution. With an inverse Fourier transformation we get: J (k) = k 2 zk2 4π2 Z d2(r||− r0||)Λ(r − r 0 )e−ik(r−r0) (2.52)

This equation eventually, by inserting a homogeneous solution for the scattered wave from a laterally infinite sample, becomes to:

J (k) = k 2_I i 16π2_A Z d3r0Tˆ0(r0)e−i(k−ki)r0 2 (2.53)

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Where A is the irradiated area on the sample, and k an arbitrary vector which is, in contrast to Fraunhofer case, not determined by the detector position. In this case the sample acts as a point-like scatterer in reciprocal space, i.e. the scattered waves ks from each microscopic object ˆT0(r) distributed over the sample, as for example

nanowires, have the same direction. So the ensemble averaging can be done in reciprocal space by calculating the arithmetic average of all Fourier transformations of statistically varied (their dimensions for example) microscopic objects ˆT0(r). The calculated intensity distributions of each object, i.e. its Fourier transformation, in the ensemble are at the very same position in reciprocal space. However, to be able to compare measurements with this approximation, the directions of the scattered beams ks have to be determined very precisely, i.e. a very small angular resolution of the

detector is crucial. Unfortunately, this is only possible with analyser crystals, making this approximation not very suitable for comparison with intensity distributions recorded with point detectors or position sensitive detectors. So in reality one detector pixel (for 1D or 2D detectors), or one detector, collects scattered waves with a certain distribution of directions, according to the pixel size, or detector opening, and the sample-detector distance. This corresponds to an averaging of the scattered intensity distributions from each object of the ensemble not around the very same position in reciprocal space, but around a distribution of positions according to the angular resolution of the detector opening (pixel size) convoluted with the size of the X-ray beam’s footprint on the sample. Let us recapitulate Fraunhofer approximation, Equ.2.27, where the illuminated sample size has to be smaller than the first Fresnel zone, which depends on the sample-detector distance. As already discussed, this perfectly holds for nano- or micro-focused synchrotron radiation diffraction. However, for the ensemble X-ray diffraction experiments of Chap.5, a more detailed discussion is needed. The microscopic objects in this case were nanowires with diameters in the 100 nm range and lengths of 2-3 µm, distributed over the sample’s surface having several micrometers interspace. The whole illuminated sample volume does therefore not act as a perfect point scatterer. However, this measurements were done on laboratory diffractometer with a long sample-detector distance (≈ 1 m), and also at synchrotron beamlines with even larger distances. When we now assume that the nano-objects are point scatterer being distributed over the illuminated area then, in Fraunhofer approximation, the intensity distributions from each micro-object would be at slightly different position in reciprocal space, since the direction of the scattered wave ks is given by the detector position with respect to

the position of the nano-object on the sample. This is a good approximation of the actual scattering experiment, and the distances between the microscopic features are in the range om micrometers resulting in a very small shift in reciprocal space. The scattered intensity distribution from the micrometer (or nanometer) sized structures itself are very broad in reciprocal space. Eventually, the effect of this shift in reciprocal space can be neglected. Calculating the scattered intensity from different statistically varied nano-objects (in our case nanowires) using always the same directions of the scattered wave ks, and then calculating the average intensity in reciprocal space, is a

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good approximation for the simulations. If the reader is interested in more details to ensemble averaging considerations, I refer to Chapters 5 and 10 in the book from the “Three Wise Men” (Hol´y, Pietsch, and Baumbach) [38].

Randomly deformed samples

Now we consider a deformed crystal, and write down the polarizability as a distorted Fourier series:

χ0(r) = Ω(r)X

g

[χg+ δχg(r)]eig[r−u(r)] (2.54)

If the disturbance of the sample structure is not too strong the equation can be simplified using Takagi approximation. This means that within the unit cell, relative displacements and possible removements of atoms are neglected, which is only valid when the elements of the strain tensor, see Equ.2.33, be very small compared to unity. Setting δχg(r) =

0 implies that the polarizability of the deformed crystal in point r is equal to the polarizability of a not deformed crystal at point r − u(r), i.e. u(r) is the shift of the whole non-deformed elementary cell at position r. Moreover, the deformation of the sample can be random but has to be statistically homogeneous, i.e. the macroscopic quantities causing the deformation such as defect densities, correlation lengths, etc. are constant in the whole crystal volume. Therefore we divide the displacement field into two parts

u(r) = hu(r)i + δu(r), (2.55) and the average strain tensor has to be constant

huj(r)i = hjkixk. (2.56)

Eventually, by taking Fraunhofer approximation along with the kinematical and the Takagi approximations the scattered intensity of a randomly deformed sample illuminated by a large X-ray beam is:

E(r, Q) = πC Vcell eikr r Ei X g Scell(g) Z

d3r0 Ω(r0) e−igδu(r0) e−i(Q−gdef)r0 (2.57)

With C being the linear polarization factor, Vcell the unit cell volume, and Ei is the

electromagnetic field of the incident wave. The position of the Bragg reflections is now given by the averaged crystal lattice:

g_jdef = hj− hjkihk. (2.58)

This means while approaching the borders where Takagi approximation does not hold any more, the position of the Bragg reflections in this formula will become incorrect. The integrand part on the right hand side gives the diffuse scattering due to strain in the sample. The differences between perfect crystals and randomly deformed crystals

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is seen directly by comparing this equation with Equ.2.27. By inserting now the FEM result of a strained (with strain values according with the Takagi approximation) object with dimensions in the nanometer range, as for example the heterostructured nanowire investigated using nano-focused synchrotron radiation in Chap.5, one can calculated the scattered electromagnetic field, and with Equ.2.17 the scattered intensity. For the simulation of ensemble measurements one has to take the considerations from the previous section into account.

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Chapter 3 Focusing and Diffraction

X-ray diffraction can access several physical properties of a crystalline sample very precisely and easily, as for example the crystal structure, lattice constants, chemical compositions, crystal quality, strain, size, and shapes of nano-scale objects on or embedded inside the sample, crystal orientations, interface properties and dimensions of layer structures, and so on. However, in the past it was mainly used to find these properties averaged over a macroscopic volume of the investigated sample, which is actually one of the big advantages of X-ray diffraction. But with the possibilities of 3rd generation synchrotrons the opportunity of making all this spatially resolved, even with a quite good resolution, showed up. Therefore, focusing of synchrotron radiation in the energy range from soft to hard X-rays was necessary, and technical solutions were found to overcome the problems involved with this task. The following discussion will be done according to the book of Stangl et al. [39] and the lecture on “X-ray scattering in material science” held by Tobias Sch¨uli. Due to the highly-collimated and brilliant synchrotron radiation it became possible to insert focusing devices producing focal spots with dimensions in the several 10 nm regime, and still having enough photons to perform diffraction and scattering experiments. Focusing however, is always a balance between the beam’s spot size and photon flux density against its divergence and coherence. Let us introduce the focusing device as a black box with certain properties, and discuss how the focusing quality is judged quantitatively. A sketch of a synchrotron beamline producing a focused beam is sketched in Fig.3.1 The synchrotron radiation source with the size Σh × Σv (horizontal and vertical direction) is projected downstream onto the

acceptance = a × b

Intensity = I0 Spot

(image of the source) size=sh× sv Transmission (T) I₀ Focusing I=T×I0 device Source size=ΣΣΣΣ h× ΣΣΣΣv L₁ L₂ L₂=f (if L₁=∞) 2αααα 2θθθθ

Fig. 3.1: Figure taken from Ref. [39]. Schematics of focusing the X-rays by the optical element (similar to visible optics).

Strain mapping in semiconductor nanowires using X-ray diffraction / eingereicht von: Mario Keplinger

JKU

Strain mapping in semiconductor nanowires

using X-ray diffraction

DISSERTATION

Doktor

Technischen Wissenschaften

Mario Keplinger

Institute of Semiconductor and Solid State Physics

a.Univ.Prof. Dr. Julian Stangl (Betreuung)

a.Univ.Prof. Dr. Alois Lugstein

Contents

Chapter 1

Introduction

1.1

Structure of the thesis

Chapter 2

Theory

2.1

Diffraction for Dummies

2.1.1

X-rays/Synchrotron Radiation

2.1.2

Scattering

2.1.3

Kinematical approximation and Diffraction on perfect

crystals

2.2

Simulations

2.2.1

Finite Element modelling

2.2.2

Simulating the scattered intensity

Chapter 3

Focusing and Diffraction

_JKU