X-ray based investigations of semiconductor multilayer and microbridges / eingereicht von: Tanja Etzelstorfer

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UNIVERSIT¨AT LINZ

JOHANNES KEPLER

_JKU

Technisch-Naturwissenschaftliche Fakult¨at

X-ray based investigations of semiconductor

multilayer and microbridges

DISSERTATION

zur Erlangung des akademischen Grades

Doktorin

im Doktoratsstudium der

Technischen Wissenschaften

Eingereicht von:

Tanja Etzelstorfer

Angefertigt am:

Institut f¨ur Halbleiter- und Festk¨orperphysik

Beurteilung:

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

Prof. RNDr. V´aclav Hol´y, CSc.

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Eidesstattliche Erklärung

Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne frem-de Hilfe verfasst, anfrem-dere 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 übermittelten Textdokument identisch.

Linz, August 2015

Tanja Etzelstorfer

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Kurzfassung

Die vorliegende Arbeit beschäftigt sich mit der Strukturanalyse verschiedenster Halbleiter-nanostrukturen mittels Röntgenstreuverfahren. Die untersuchten strukturellen Merkmale bestimmen maßgeblich elektrische, thermische und optische Eigenschaften der untersuchten Materialien und sind daher zwingend zu bestimmen, um das Wachstum zu optimieren, bzw. gültige Eingabeparameter für Simulationen und Konstruktionen verbesserter Strukturen zu erhalten.

Es handelt sich dabei um Strukturen, die zukünftig im Bereich erneuerbarer Energie ein-gesetzt werden können oder in integrierten optischen Schaltungen Anwendung finden. Einerseits werden konventionelle Laborexperimente mit Röntgenstrahlgrößen im mm Bereich durchgeführt, andererseits aber neuartige Nanofokus-Aufbauten mit Strahlgrößen von einigen hundert nm verwendet, mit denen es möglich ist, lokal aufgelöste Untersuchungen im sub-µm Bereich zerstörungsfrei durchzuführen.

Der erste Teil der Arbeit beschäftigt sich mit der Untersuchung von SiGe Übergittern. Einerseits werden Übergitter bestehend aus SiGe und Ge Schichten mit Dicken im nm Bereich für thermoelektrische Anwendungen untersucht, andererseits Übergitter, bestehend aus ultradünnen reinen Si und Ge Schichten, konzipiert für optische Anwendungen. Mittels Röntgenbeugungsmethoden werden Perioden, Verspannungen sowie die Zusammensetzung und Dicke einzelner Schichten untersucht. Um Einblick in die Grenzflächenmorphologie und etwaige Interdiffusionsmechanismen zu bekommen, werden Röntgenreflektometrie Unter-suchungen durchgeführt. Im Vordergrund steht dabei die Änderung all dieser Parameter in Abhängigkeit der Wachstumsbedingungen.

Der zweite Teil der Arbeit befasst sich mit der Untersuchung von tensil verspannten Ge Brückenstrukturen. Durch den Transfer eines Elektronenstrahl-Schemas wurde vorverspann-tes Ge weiter verspannt, was große Auswirkungen auf die Bandstruktur hat und eine Mög-lichkeit darstellen könnte, zukünftig optische Bauelemente zur drahtlosen Kontaktierung her-zustellen. Einerseits dienen die Brücken als Modellstrukturen für ein neues Rasterverfahren zur Verspannungsmessung, andererseits werden durch die Untersuchung von verschieden aus-gerichteten Brücken Phononen-Deformationspotentiale angepasst und so Raman Verspan-nungsverschiebungen für den tensil verspannten Bereich von Ge kalibriert.

Der dritte Teil der Arbeit befasst sich mit Si Nanodrähten, die erstmals in hexagonaler Gitterstruktur hergestellt werden konnten. Dazu wurde ein hexagonaler GaP Nanodraht als Vorlage genutzt und Si epitaktisch aufgewachsen. Die so entstandene Hülle aus hexagonalem Si wird mittels Röntgenbeugung eindeutig identifiziert und ebnet so den Weg, um vorallem optische, elektrische und mechanische Eigenschaften dieses völlig neuen Materials zu unter-suchen.

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Abstract

In the herewith presented thesis structural properties of various semiconductor nanostruc-tures are investigated by means of X-ray scattering techniques. The studied structural parameters highly influence electrical, thermal and optical characteristics of the materials. Hence, precise knowledge is mandatory to optimize growth, fabrication and design of op-timized structures which are intended for renewable energy applications or for integrated optical circuits, respectively.

Besides conventional laboratory scattering methods, where X-ray beams with dimensions of a few mm are used, nanodiffraction experiments with focused X-ray beams are carried out in order to conduct spatially resolved investigations in the sub-µm beamsize region.

The first part of the work deals with the investigation of SiGe superlattices. One section is devoted to superlattices composed of SiGe and Ge layers with thicknesses in the nm-range intended for thermoelectric applications. A further section is devoted to superlattices composed of ultra-thin pure Si and Ge monolayers intended for optical applications. The periodicity, thicknesses and composition of the layers as well as the strain state is obtained by X-ray diffraction experiments. In order to gain insight on the interface quality and interdiffusion effects also X-ray reflectivity studies are performed. The influence of the growth parameters on all these structural characteristics is studied in great detail.

The second part of the thesis focuses on the characterization of tensile strained Ge mi-crobridges. An electron beam pattern was transferred onto a pre-strained Ge layer which enhances the strain at certain regions. Strain has dramatic influence on the bandstructure and could hence pave the way for future light emitters integrable in Si-technology. The stud-ied bridge-structures serve on the one hand as model structures for a new scanning strain microscopy technique developed at the synchrotron, on the other hand for studies on differ-ently oriented bridges to obtain the directional dependent Raman strain shift coefficient for the tensile strain region by fitting phonon deformation potentials.

The third part of the work is devoted to Si nanowires where the hexagonal crystal phase is confirmed unambiguously for the very first time. Therefore, hexagonal GaP nanowires served as template for the epitaxial and thus also hexagonal overgrowth of Si shells. The so formed hexagonal crystal growth is confirmed with X-ray diffraction and clears the way for further optical, electrical and mechanical studies on this completely new material.

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Abbreviations

AFM atomic force microscopy

BS beam stop

COM centre of mass COR centre of rotation

CVD chemical vapour deposition DFT density functional theory DOS density of states

FEM finite element method FWHM full width at half maximum FZP Fresnel zone plate

Ge germanium

HH heavy hole

LD Lonsdaleite

LDA local density approximation

LEPECVD low-energy plasma-enhanced chemical vapor deposition

LH light hole

NW nanowire

OSA order sorting aperture

PDP phonon deformation potential

PL photoluminescence

PV photovoltaic

QW quantum well

r.m.s. root mean square RS Raman spectroscopy RS reciprocal space Si silicon SL superlattice Sn tin SO split off TE thermoelectric

TEC thermal expansion coefficient TEM transmission electron microscopy

WZ wurtzite

XRD X-ray diffraction XRR X-ray reflectivity

ZB zinc blende

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A.1 Relevant parameters . . . III A.2 Error propagation in lattice constant determination . . . IV A.2.1 Energy uncertainty . . . V A.2.2 Angular uncertainties . . . V A.2.3 Systematic error due to refraction . . . VI A.2.4 Fitting error . . . VII A.3 Error propagation in strain determination . . . VIII A.4 Exemplary uncertainties for different setups . . . IX

A.4.1 Panalytical MRD, Hybrid Monochromator, PIXCel detector . . . IX A.4.2 Beamline ID13 @ ESRF, nanofocus-endstation, Maxipix detector . . . X

Acknowledgements XIII

Curriculum Vitae XVI

List of publications XVII

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Part I

Introduction & Motivation

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Chapter 1 Introduction

Silicon (Si) is the dominating material in semiconductor industry already for decades due to it’s superior physical and technological properties compared with other semiconductor materials: it is one of the most abundant materials on Earth, thus easily available and cheap and it emerges with it’s native oxide, which can be used as passivation material without adding any other material. Together with the mature growth processes an enormous production infrastructure has been built around Si and the amount of knowledge that has been obtained can not be caught up so easily with any other material. However, within the last decade, the end of Si technology was often forecasted due to the disadvantage of Si’s indirect bandgap, which impede the development of optical components. Optical on-chip data distribution and interconnections are the only real alternative to electrical wiring which limits the performance of the newest generation processors to date [1, 2]. Fig. 1.1 shows the bandwidth density, which is an effective criterion for evaluating the ability to transmit data through a unit width, for optical and copper-based electrical interconnects. As can be seen, optical interconnects increase transmission ability dramatically.

Figure 1.1: Comparison/forecast of the bandwidth density of electrical and optical intercon-nects. Figure reprinted with permission from Ref. [2]

Regardless of all the drawbacks caused by the indirect bandgap of Si, almost all components required for optical data transmission have been demonstrated on the Si platform nonethe-less: photodetectors [3, 4], modulators [5] and nonlinear elements [6]. The one still missing element is an efficient light emitter.

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The combination of Si with other highly compatible group IV materials, such as germa-nium (Ge) or tin (Sn), offers a lot of prospects to explore new and superior features. Ge is discussed as the most promising candidate material to realize the still missing light source [7, 8, 9, 10].

However, Si is not only an important material in optoelectronic research and fabrication, it offers also chances in renewable energy applications. It is already the leading material in photovoltaic (PV) industry: Si-wafer based PV technology accounted for about 92 % of the total PV production in 2014 [11]. Though, in the field of energy harvesting by thermoelectric (TE) applications tellurium-based (rare and often toxic) materials are still the dominating ones, especially for operation at room temperature [12].

From Fig. 1.2 it can be seen that the figure of merit ZT (for both n- and p-type materials), which describes the efficiency of a TE material, reported for SiGe at room temperature is below 0.4, whereas for high temperature applications above 900 K SiGe materials have the best ZT and have been successfully incorporated in generators for space missions.

Figure 1.2: Figure of merit for p- and n-type thermoelectric materials as a function of oper-ation temperature. Figure reprinted with permission from Ref. [12]

Even though the efficiency at room temperature for the to date best SiGe materials is substantially lower than of other materials, the integration with Si-based devices together with the sustainability idea are key factors for the research on SiGe. Integrated cooling and energy harvesting modules could be coupled to electronic devices, solar cells and autonomous systems, such as remote sensors.

Novel properties of Si-based materials are highly linked to their structural parameters. By going to the nano-scale, i.e. designing structures with a few 10 - 100 nm in size, enhanced features can be developed since the physical properties change drastically with respect to the material’s bulk properties. Quantum mechanical effects come into play, which lead to enhanced electrical, thermal and optical performances and even change the band structure [13, 14, 15]. Electrical properties are enhanced due to an increase in the density of states and quantum confinement effects. In addition, thermal properties can be tweaked due to tailored scattering processes in the nm range. The main contribution to enhanced optical properties is predicted by a decrease in the bandgap and eventually by transforming the indirect bandgap into a direct one. Alloying, zone folding and strain engineering are the most promising routes to achieve this. Combining different materials into heterostructures or scrutinise even different crystal phases allow further degrees of freedom.

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5

Within this thesis novel properties of Si-based semiconductor nanostructures intended for optoelectronic or thermoelectric applications have been studied.

Thermoelectric materials composed of thin layers stacked on top of each other, so called superlattices (SL) are investigated. The confinement of the electrons in the 2D layers leads to an enhancement of the Seebeck coefficient, which describes the voltage creation as a response to a temperature difference [13]. This allows the development of new TE materials which create electricity by harvesting thermal energy. The efficiency is enhanced by an increase in electrical conductivity and Seebeck coefficient and a decrease in thermal conductivity. In bulk form tailoring of these quantities independently from each other is not possible, but they can be independently optimized to a certain degree within SL structures [16, 17, 18, 19]: The higher density of states in the thin layers increases the electrical conductivity, while the interfaces of the layers act as scattering centres for the acoustic phonons, leading to a decrease in the thermal conductivity. The developed materials could serve as a more sustainable alternative made from the Si-Ge material system and eventually replace tellurium-based materials.

Different designs are currently under discussion, ranging from pure Si/Ge superlattices [20], to more complex Si/SiGe, Ge/SiGe and SiGe/SiGe structures [21, 22], even with grading approaches within single layers [23]. The SiGe/Ge superlattices studied within this thesis led to the highest reported efficiency for a non-tellurium n-type material at room temperature. Another type of SL, composed of ultra-thin layers of pure Si and Ge, has been studied for optoelectronic applications. The zone folding approach, can lead to a change in the bandstructure [15]. Therefore, it has already been predicted 30 years ago, that when a special sequence of pure Si an Ge monolayers is stacked on top of each other, this special SL forms a direct bandgap material [24, 25]. However, the optical transitions back then were too weak for actual applications and the idea was laid aside. Since computational power increased dramatically since then, more complex SL sequences with increased optical transitions could be identified [26] and the realisation of these materials would bring light emitters integrable in Si-technology within reach.

The search for optical components based on Si-technology also drives the effort of altering the physical properties of Ge, which is, such as Si, an indirect bandgap semiconductor and thus per se an inefficient light emitter/detector. However, different approaches to convert Ge into a direct bandgap material are under discussion [15, 8, 27, 28]. One of the most promising attempts is strain engineering, since for uniaxial tensile strain of around 4 %, a transition to direct bandgap is predicted [29]. Different approaches to reach this high strains are reported [30, 31, 32, 33]. One approach strains Ge in the nanowire form, where tensile strain values up to 10 % are claimed [33], another approach fabricates Ge microbridges from tensile pre-strained material, where the special geometry leads to peak strains at the centre of the constriction [31]. The complex designs lead to inhomogeneous strain distribution across the Ge-microbridges which have been studied in great detail within this thesis.

Another way to bring optical components for the Si-platform within reach is to study poly-types of the crystal structure. Hexagonal crystal phase transitions have been observed in nanowires, which are also predicted to tweak the material’s bandstructure [34, 35, 36]. The special 1D geometry of nanowires allows the epitaxial growth of materials which are usually incompatible in bulk form due to large lattice mismatch. The efficient strain relaxation due to the low dimensionality overcomes this drawback and growth mechanism, where naturally lattice mismatched materials can be combined, are powerful tools to create promising

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build-ing blocks for novel semiconductor devices such as solar cells, transistors or light emitters. The band gap can be tuned in a huge range by either customisation of the composition or alternatively by fabrication of novel crystal structures. Within this thesis, the hexagonal crystal structure of Si nanowire shells has been studied in great detail and the Lonsdaleite phase has been unambigiously confirmed for the very first time.

All the mentioned aspects of creating new desired properties in the Si-Ge material system have in common that they depend purely or at least highly on structural parameters. In order to connect the growth and fabrication conditions to the desired properties extensive and systematic investigations are required. Thus, methods to deduce the physical properties are inevitable. X-ray scattering techniques are the perfect methods of choice to determine structural properties. Depending on the actual choice of technique composition, strain dis-tributions, interface morphologies, sizes and correlations of structures can be determined non-destructively.

X-ray diffraction (XRD) is a powerful tool to investigate the average structural properties within structures made from nanoscale building blocks such as the mentioned SLs or ensem-bles of Si nanowires. Quantitative results can be determined, even in conventional laboratory experiments. With the fabrication of more and more complex nano-scaled objects and sample structures, information on the average properties might not be sufficient anymore. With the availability of 3rd generation synchrotron facilities, which provide very intense synchrotron radiation, experiments with focused beams with a spot size of several 100 nm are possible. Thus, XRD methods can be conveyed into microscopy tools, where spatially resolved ex-periments are possible. Locally resolved scattering exex-periments as well as studies on single nano-scale objects are thus possible. These techniques have been used extensively to study the inhomogeneous strain field of tensile strained Ge-microbridges.

Within this thesis, both laboratory and synchrotron based studies on superlattices, micro-bridges and nanowires, as mentioned above are discussed. Novel structural properties are determined and combined with other structural investigation techniques in order to deepen the knowledge of certain properties.

1.1 Organisation of the thesis

The first two parts of the thesis (I-II) should set the theoretical background needed. Chapters on X-rays and X-ray scattering, as well as on general properties of Si and Ge can be found, including the relevant aspects of band structures and elasticity theory. The subsequent three parts (III-V) are devoted to the actual studies on superlattices, microbridges and nanowires, including all experimental results. In the appendix elaborate calculations on measurement uncertainties are given.

Each chapter starts with a brief overview and relevant publications originating from the discussed experiments followed by a self-contained detailed introduction to the particular topic. The X-ray scattering studies are then discussed in great detail together with the impacts on physical properties. Where applicable and possible, comparisons with other structural investigation techniques are performed. At the end of every chapter the main findings are summarized and prospects for future trends are given.

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1.1 Organisation of the thesis 7

All of the following chapters are results of several collaborations, rather than the work of an individual. Therefore, all contributing partners are acknowledged with great gratitude at the beginning of every chapter.

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Part II

Background & Theory

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Chapter 2 X-rays & X-ray scattering

In this chapter the basic aspects of X-rays and X-ray scattering theory relevant within this thesis are summarized. After a general description of X-ray properties the very basics of scattering theory are summed up in order to pave the way for the different scattering theory calculations used in the simulations in subsequent chapters.

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2.1 General properties

X-rays are electromagnetic waves with their wavelength ranging from 1 pm to 1 nm. Atomic distances are typically in the order of 1 Å which makes X-rays a perfectly suitable probe to study crystal structures. Due to the periodic arrangement of atoms within a crystal the phenomenon of Bragg diffraction occurs [37]: If the path difference of two scattered waves is equal to a multiple of the wavelength λ constructive interference occurs. These intensity accumulation can then be measured. The corresponding Bragg equation reads as

2d sin(θ) = nλ (2.1)

where d corresponds to the lattice spacing in question, 2θ to the diffraction angle and λ to the wavelength of the X-rays. By inspection of the diffraction angle 2θ and the incidence angle ω, which can be measured with extremely high accuracy, the lattice spacing d can also be determined with extremely high precision. Thus the determination of lattice spacings is translated into angular measurements.

Fig. 2.1(a) shows the connection between ω, 2θ, d and the momentum transfer k in the symmetric case, i.e., probing lattice planes parallel to the sample surface. This implies equal incidence and exit angles (ω = θ). If not only the lattice planes parallel to the surface are probed (depicted in Fig. 2.1(b)), also information on the atomic distances parallel to the surface can be obtained. Therefore, typically symmetric and asymmetric reflections are combined to study the 3D arrangement of atoms within a crystal.

k

i

k

f

d

(a) Sketch of probing lattice planes parallel to the surface, i.e. symmetric Bragg diffrac-tion.

k

i

k

f

d

(b) Sketch of probing arbitrary parallel lat-tice planes, i.e. asymmetric Bragg diffrac-tion.

Figure 2.1: Sketches of symmetric and asymmetric scattering geometry.

Conventionally coplanar scattering geometry is used, which means incidence and diffracted waves lie in the same plane as the surface normal of the investigated sample. Thus, actually only two dimensions can be probed. By rotation of the sample around it’s surface normal the third dimension can be accessed though.

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2.2 Basic scattering theory 13

In coplanar scattering geometry not all of the reciprocal space (RS) can be accessed since incidence and exit wave can not propagate through the investigated sample. An exemplary map of the accessible region in reciprocal space for CuKα radiation and the nominal Bragg reflections of Si and Ge are shown in Fig. 2.2.

−5 0 5 Q∥ (Å−1) 0 1 2 3 4 5 6 7 8 Q[001] ( Å − 1 ) (-4-40) (-4-44) (-3-31) (-3-33) (-3-35) (-2-20) (-2-24) (-1-11) (-1-13) (-1-15) (000) (004) (111) (113) (115) (220) (224) (331) (333) (335) (440) (444)

Figure 2.2: Sketch of accessible RS in coplanar scattering geometry together with the nominal Bragg peak positions of Si (red) and Ge (gray). The size of the bullets corresponds to the structure factor, i.e. the scattering strength.

Plotting the measured scattering intensity directly in RS is most convenient, since lattice constants can then be directly calculated from peak positions in RS. The connection between angular and RS coordinates, as well as lattice constants and peak positions are given in the following: Qk = 4π λ sin(θ) sin(ω − θ) (2.2) Q⊥= 4π λ sin(θ) cos(ω − θ) (2.3) a⊥= 2π Q⊥ l (2.4) ak = 2π Qk p h2_{+ k}2 _(2.5)

where Q_⊥,k is the position of the Bragg peak in reciprocal space and h, k, and l are the Miller indices for the chosen reflection. Eq. (2.4) and (2.5) only have this simple form for cubic crystal structures with the surface normal parallel to the (001) direction.

2.2 Basic scattering theory

In this section the very basics of scattering theory are summarized in order to give an overview on which theory is used behind the simulations of X-ray data of multilayers throughout the

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chapters 4 and 5. A general introduction on diffraction can be found in textbooks such as Ref. [38]. Full derivations of various scattering theories, applicable for specific sample types, can be found in Ref. [39] and if not stated otherwise, the following derivations follow very closely this reference.

In conventional X-ray diffraction (XRD), i.e. high angle scattering around Bragg peaks, the

kinematical approximation of the scattering theory is applicable, where multiple scattering

of X-rays is neglected. The resulting scattering intensity is proportional to the square of the Fourier transform of the electron density of the material. This approach is well suited for (almost) perfect crystals, or samples with a small scattering volume, like thin layers or nanowires. However, the main features appearing in the scattering curves are predicted well also for relatively thick structures, thus the equations and mechanisms are used to interpret scattering intensities around Bragg peaks from periodic multilayers.

However, in small-angle scattering geometry, like X-ray reflectivity (XRR) measurements, effects of multiple scattering can not be neglected. These effects are mathematically described using dynamical scattering theory. This theory is mathematically much more elaborate and therefore limited to simple structure models. In the following chapters this theory is needed for fitting radial scans around Bragg peaks of superlattices and for calculating specular reflectivity of ideal and rough or interdiffused multilayers.

If the dynamical effects cannot be neglected but the structure model is not simple enough to apply full dynamical scattering theory, a semikinematical scattering theory is often applied. The equations in the semi-kinematic regime are used to calculate the diffusely scattering intensity of superlattices in small angle geometry (XRR).

Throughout this thesis only elastic scattering of X-rays will be considered, i.e. the en-ergy/wavelength of the incident and scattered waves are equal. From the dispersion relation of electromagnetic waves in vacuum follows that the wave vectors of both waves have the same length: |Ki| = |K| = K = ω c = 2π λ (2.6)

Furthermore, a plane wave approach is assumed and the incident wave can be written as

Ei(r, t) = Eie−i(ωt−Kir). (2.7) Strictly speaking this is not true for a focused beam, where the divergence of the beam broadens the peak. However, since we are mostly interested in the pure Bragg peak position we ignore these effects here.

Due to the assumption of elastic scattering all considered waves have the same frequency, the factor exp(−iωt) remains the same and will be omitted in the following.

The elastically scattered wave is given by the coherent superposition of plane waves with equal frequencies but various directions. The integration has to be done over all different possible directions in real space dΩ ≡ dϕdθ sin θ, where the angles θ and ϕ determine the direction of the scattered wave.

E(r) =

Z

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2.2 Basic scattering theory 15

By using a coordinate system with x and y-axis in the sample surface and the z-axis parallel to the outward surface normal dΩ can be rewritten into dK_xdKy/(KKz) = d2Kk/(KKz) with Kz = q K2_{− K}2 x− Ky2 and (2.8) becomes E(r) = Z d2K k KKz E(K)eiKr (2.9) In real materials the atoms are never exactly defined spatially due to defects or other lattice distortions present in the structure. Their position r = (x, y, z) can be described by random functions and the scattered wave is therefore described best by an ensemble average of lots of microscopic configurations which correspond to the same macroscopic quantity. Such an averaging approach is also valid for many other random effects. The morphology of a rough surface or interface is described by a random function z(rk) = z(x, y) of the local sample position (x, y). The properties are described by a few macroscopic quantities such as the root mean square roughness (r.m.s. roughness) or the roughness correlation length (Λ_L) as used below.

In these averaging cases the mutual coherence function (MCF) of the scattered wave is used to study the properties of ensemble-averaged scattered waves. This MCF is then also the measurable intensity in an experiment and is defined as

I(r) = Γ(r, r0) =

E(r)E∗(r0)

. (2.10)

The hi stands for a statistical average over different microscopic configurations (arrange-ments, material variations, orientations, size distributions, . . . ).

Any scattering process considered here can be described by the scalar wave equation

(∆ + K2)E(r) = ˆV (r)E(r) (2.11)

where ˆV (r) is the operator of the scattering potential and is given by

ˆ

V (r) = (grad div −K2χ(r)). (2.12)

Both can be derived from Maxwell’s equations. Eq. (2.11) can be rewritten into the integral form

E(r) = Ei(r) +

Z

d3r0G0(r − r0) ˆV (r0)E(r0) (2.13) where Ei(r) is the solution of the vacuum wave equation

(∆ + K2)Ei(r) = 0 (2.14)

and the second term consists of all contributions of single point scatterer where the Green’s function G₀(r − r0) is the solution of the wave equation of a free particle (each such point scatterer)

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Since the wave field E(r) appears on both sides of Eq. (2.13) it has to be solved iteratively starting with E(r0) = E_i(r0) and ˆV (r0)E(r0) = ˆT (r0)E_i(r0). ˆT is the symbolic expression of

the scattering operator ˆT (r) defined as

ˆ

T = ˆV + ˆG ˆG0V + ˆˆ V ˆG0V ˆˆG0V + . . .ˆ (2.16)

and includes all multiple scattering processes. Eq. (2.13) reads then

E(r) = Ei(r) +

Z

d3r0G0(r − r0) ˆT (r0)Ei(r0). (2.17) In the simplest case only one iterative step is performed and thus multiple scattering processes are neglected. This is the kinematical approximation or Born approximation and will be discussed in the next section for scattering from multilayers.

2.3 Kinematic scattering from multilayers

As mentioned above, kinematic X-ray scattering theory manifests itself in the assumption that an X-ray photon, after being scattered by an electron, is not scattered by another elec-tron again. Therefore, the scattering potential in Eq. (2.17) is used in the first approximation only ( ˆT = ˆV ).

The validity of the single-scattering approach implies that the amplitude of the single scat-tered beam is very small compared to the amplitude of the incident beam because then the doubly- and multiply-scattered amplitudes are negligible. The probability of scattering into a particular direction is highly dependent on the possibility of scattering on an ordered array of atoms. A diffracted beam will be diffracted again if it passes through another crystal region which meets the condition for Bragg diffraction again. This is only fulfilled in a large single crystal where the lattice is not distorted due to defects, grain boundaries and so on, but is not the case for real structures with all kinds of lattice distortions. For perfect crys-talline structures with a strong cryscrys-talline reflection multiple scattering becomes important at X-ray path lengths of the order of 1 micron [38]. Thus multiple scattering can be neglected for objects with crystalline perfection in the sub-micron range.

A multilayer is a special type of a non-perfect, i.e. deformed crystal, if it can be assumed that the single layers grow pseudomorphically on top of each other. Such an epitaxially layered system consisting of N layers with different chemical compositions and thicknesses is depicted in Fig. 2.3(a). Pseudomorphic growth assumes that every layer thickness is below it’s critical thickness and no relaxation and defect formation occurs. Therefore, the lateral lattice constant in the plane ak is the same in all individual layers while the normal lattice constant a⊥depends on the tetragonal distortion of the individual layer and thus the composition.

A superlattice in turn is a special type of multilayer structure, where a fixed sequence of multilayers, i.e. one period, is repeated many times. A sketch of a superlattice composed of bilayers is shown in Fig. 2.3(b).

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2.3 Kinematic scattering from multilayers 17

(a) Sketch of a general multilayer with arbitrary lay-ers.

(b) Sketch of a multilayer with bilayers.

Figure 2.3: Sketches of ideal superlattices.

Such multilayer structures or superlattices are thick samples and actually multiple scattering can not be neglected strictly speaking. However, the main features, as discussed below, are calculated correctly by using the kinematic scattering approach and the origin of the appearance of scattering maxima is seen quite easily from the equations. Therefore, the following kinematic scattering intensity of a superlattice is derived as in Ref. [39]. The starting point is the scattering intensity of a generic multilayer system:

I = Ii 4(πr_elC)2 Vcell X gk 1 Kgz X gz N X n=1 F_cell(n)(g_k, Kgz− Kiz)G(n)layer(Kgz− ˜Kgz) 2 (2.18)

F_cell(n)(Q) is the structure factor of the unit cell in the n-th layer and

G(n)_layer(Q_z) = zn

Z

zn+1

dze−iQz(z+u(z)) _(2.19)

is the geometrical factor of the nth layer positioned between the interfaces n + 1 and n. An important simplification can be used here: The structure factor F_celland the displacement field u(r) depend only on the z-coordinate.

For periodic multilayers, i.e. a superlattice, M identical sequences stacked on top of each other, Eq. (2.18) can be modified in order to see the main features appearing in the diffraction signal. The easiest of such a periodic multilayer structure consists of M identical bilayers as sketched in Fig. 2.3(b). Nevertheless, the equations could also be extended to more than two layers within one period and serve only as example.

Each of the bilayers consists of layer A and B with the thicknesses TA,B = nA,BaA,B and the structure factor F_cell(A,B)(Q). aA,B are the tetragonally distorted out of plane lattice parameters of the layers A and B, respectively. The period of the multilayer is D = n_AaA+

nBaB and the thickness is given by T = M D. In such a case it is convenient to choose the averaged lattice as a reference lattice and measure the real positions of the atoms as

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displacements from that averaged lattice. The inplane lattice constant of this reference lattice is still the same as the substrate’s inplane lattice constant. The average vertical lattice constant is defined as

hai = nAaA+ nBaB

nA+ nB

(2.20) Eq. (2.18) can then be modified to

I = Ii 4(πr_elC)2 V2 cell X gk 1 Kgz X gz Fperiod(gk, Kgz− Kiz)Gmultilayer(Kgz− ˜Kgz) 2 (2.21) where Fperiod(Q) = F_cell(B)(Q) e−iqBTB _{− 1} −iqB

+ F_cell(A)(Q)e−iqATAe

−iqATA _{− 1}

−iqA

(2.22) is the structure factor of the multilayer. qA,B= Qz−gz+QzδA,Bwith δA,B = (aA,B−hai)/hai is the vertical mismatch of the layers with respect to the reference lattice and TA,B= nA,Bhai is the thicknesses of the layers.

The geometrical factor of the periodic multilayer is

Gmultilayer(q) = eiqD

eiqT − 1

eiqD_{− 1} (2.23)

with q = Kgz− ˜Kgz.

As can be seen, several types of intensity maxima exist:

1. Main maximum if Kgz = ˜Kgz where Gmultilayer has it’s maximum value. This means a peak occurs at a reciprocal lattice point corresponding to the reference lattice. This main maximum is called SL0.

2. Second order maxima at points where eiqD = 1 which means Q_z = q_z+ 2πp/D with p an integer. These maxima are the superlattice peaks and their distance is inversely pro-portional to the multilayer period. They are numbered with respect to SL0 depending on the side at which they occur: SL±m with m = 1, .., n (- left of SL0, + right of SL0 at a plot with increasing Q_z values).

3. Third order maxima between the second order peaks. M − 1 intensity minima exist, where eiqT = 1. These oscillations are the thickness fringes and the distance between neighboring minima/maxima is inversely proportional to the thickness of the whole layer stack.

These three main features are usually exploited for a first characterization of superlattices without simulating diffraction curves. It is only the exact intensity offsets of the scattering maxima and the exact position but not the number of third order maxima which are different from the full dynamical theory. Thus, from the position of the SL0 the average composition can be determined, from the distance of the second order maxima the period length can be calculated and from the distance of the third order maxima the total thickness of the superlattice can be derived. However, these third order peaks can usually be resolved only for a rather small total thickness, i.e. only a few periods.

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2.4 Dynamical and semikinematical scattering from multilayers 19

2.4 Dynamical and semikinematical scattering from multilayers

2.4.1 Simulating X-ray diffraction curves from superlattices

In order to calculate diffraction signals of periodic multilayers more accurately, dynamical theory has to be used. Dynamical theory applied to crystals with a strain gradient per-pendicular to the surface, which is the case for superlattices, was developed by Takagi and Taupin in Refs. [40, 41, 42]. A set of differential equations, the Takagi-Taupin equations (see for example Ref. [43]), was developed, which has to be solved. Taupin has combined the equations for the Bragg case and only one resulting differential equation has to be solved in order to derive a recursion formula for the calculation of the diffraction satellites of a superlattice. The given form with its solution can be found in Ref. [44]:

− idX dT = X 2_{− eηX + 1} _(2.24) XT = η + (η2− 1)1/2[(S1+ S2)/(S1− S2)] (2.25) where S1 = [X0− η + (η2− 1)1/2]e−iT (η 2₋₁₎1/2 (2.26) S1 = [X0− η + (η2− 1)1/2]eiT (η 2₋₁₎1/2 (2.27) The recursion formula (2.25) gives a relation between the amplitude ratio X ∼ I_hkl/I0 at the bottom of a layer X0 and at its top XT. T denotes the crystal thickness and η is the deviation parameter, including the angular deviation from the Bragg angle θB.

The given formulas allow the calculation of radial scans of complex layered structures such as pseudomorpically deposited superlattices. The equations stated here are used in the matlab package simx which was developed by A. Pesek during his PhD (see Ref. [45]) in the years 1992 and 1993 at the institute of semiconductor physics at JKU Linz and is available at the X-ray group of our institute. This matlab package is used in later chapters to calculate diffraction signals of various types of superlattices.

Strictly speaking, the given differential equation (2.24) and its solution is only valid in the neighborhood of a Bragg reflection where the deviation parameter η is small ([42, 38]. This effect becomes important when diffraction profiles of superlattices with very small periods are calculated and the angular separation of the diffraction maxima is rather large. Then a combination with other methods, such as simulations of reflectivity curves, with the very same structural input parameters has to be used in order to obtain a complete picture.

2.4.2 Simulating X-ray reflectivity curves of superlattices

In many cases kinematical scattering theory is sufficient to describe the observed scattering signals but especially in surface sensitive experiments, where the interaction with the material is rather strong, multiple scattering has to be taken into account. However, applicability of fully dynamical scattering theory, i.e. taking all multiple scattering events into account, is limited due to the complexity of the calculations [46].

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For specular reflectivity curves fully dynamical calculations are possible, but for the diffusely scattered intensity usually an intermediate theory is used.

This semikinematical theory is called Distorted Wave Born Approximation (DWBA) and the imperfection of the material (i.e. roughness of the layers) is taken into account as disturbance (c.f. Eq. (2.17)). An approach as developed in Ref. [47] is deployed. The diffusely scattered intensity is proportional to the Fourier transform of an interface correlation function and hence typical correlation lengths, both in-plane and in growth direction, can be determined. The existence of resonant diffuse scattering sheets (see Ref. [46] and below) at finite Qk values is connected to the vertical replication of rough interfaces and hence allows to determine the real interface roughness. Thus it is possible to distinguish between roughness and interdiffusion, which is not possible from specular XRR (and also XRD) itself.

The Parratt formalism as can be found in Ref. [48] is used to simulate the specularly scattered signal.

In the following the main formulas needed in order to calculate specular and diffuse XRR curves of superlattices are summarized.

Specular XRR

In case of small incidence angles (<10° ), which is the case for reflectivity studies, in contrast to high angle Bragg diffraction, refraction effects have to be taken into account. For perfectly smooth surfaces and interfaces only specular reflectivity occurs. The reflected intensity of a single surface is calculated with the well known Fresnel formulas, whereas for multilayers the recursive Parrat formalism is used [48].

For X-rays the refractive index of any material is a complex number and smaller than unity

n = 1 − δ + iβ (2.28)

where δ corresponds to the dispersive term and β to the absorptive term.

δ = λ

2

2πreρe β =

λ

4πµx (2.29)

re is the classical electron radius, ρe the electron density and µx the linear absorption coef-ficient. The real part of n (1-δ) is connected to the phase-lag of the propagating wave, the imaginary part (β) corresponds to the damping of the wave amplitude.

The reflection from an ideal surface is described by well known Snell’s law as depicted and stated in Fig. 2.4.2.

n1cos(αi) = n2cos(αt) (2.30)

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2.4 Dynamical and semikinematical scattering from multilayers 21

The transmitted beam is bent towards the surface due to the smaller refractive index in the material compared to vacuum/air. This means the wave vector k is different inside and outside of the material. At a certain critical angle αc total external reflection occurs. This means the X-ray beam is totally reflected and only an evanescent wave, which decays exponentially over tens of Å, is penetrating into the topmost part of the material. Only for

αi> αc the transmitted wave propagates into the bulk.

This circumstance makes small angle scattering a surface sensitive method and the penetra-tion depth into the bulk can be controlled by adjusting the angle of incidence.

Exemplary refractive indices and the critical angles for Si and Ge for Cu_Kα radiation are given in Tab. 2.4.2:

δ β αc

Si 7.58 · 10−6 1.72 · 10−7 0.22° Ge 1.45 · 10−5 4.43 · 10−7 0.31°

Table 2.1: Refractive indices and critical angle of Si and Ge for λ = 1.54 Å taken from [49, 50]. The amplitude of reflection and transmission are defined by the Fresnel coefficients:

r = kzi− kzt

kzi+ kzt

t = 2kzt

kzi+ kzt

(2.31)

Both Snell’s law and the Fresnel coefficients are derived by imposing continuous boundary conditions at the interface.

In the small angle limit reflectance and transmission are given by R = rr∗ and T = tt∗. For

αi well below αc, R gets close to unity. If αi exceeds the critical angle, R drops off rapidly with α−4_i .

The critical angle as well as the shape of the reflectivity curve depend on characteristic material properties: electron density, atomic form factor and absorption coefficient. They are independent of the crystalline structure or the orientation of crystallites on the surface or interfaces.

The specular reflection of a perfect thin film shows oscillations as a function of the incidence angle when αi > αc. These are Fabry Perot interferences due to interference of waves reflected from the top and the bottom (interface between layer and substrate) and are called Kiessig fringes after H. Kiessig, who first determined the thickness of thin films from those oscillations [51]. The oscillations result from the thickness of the layer, whereas the amplitude depends upon the contrast at both interfaces (difference of the dispersive term δ).

For layered structures the analysis of the reflected intensity requires a calculation of this fringe pattern recursively. Therefore, it is assumed that at each interface one part of the incident beam gets reflected, one part transmitted, which then gets again partly reflected and transmitted at the next interface and so on and so forth. In this formalism the Fresnel coefficients are calculated separately for each interface and the intensity of the reflected wave is calculated recursively from bottom to top, using again continuous boundary conditions at the interfaces and zero reflectivity at the semiinfinite substrate. The obtained recursive formula reads as follows [48]:

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Rn−1,n = a4n−1 " Rn,n+1+ rn−1,n Rn,n+1rn−1,n+ 1 # (2.32) Rn,n+1 = a2n ER n En with E R

n and Enthe refracted and transmitted waves, respectively, anis an amplitude factor and the r_n−1,n are the Fresnel coefficients.

For rough layers the formula for the Fresnel coefficients have to be modified due to the damping of the roughness in the following way:

rrough= re−2kzikztσ

2

(2.33)

r is the Fresnel reflectivity as described above and σ the r.m.s. roughness of the surface. This

altered roughness can be inserted into the Parrat formalism as well and is used in Part III to study experimental specular reflectivity curves of superlattices and reveals the thicknesses, roughnesses/interdiffusion and composition of the individual layers.

As stated above, from the specular reflectivity alone, roughness and interdiffusion can not be distinguished. Therefore, also the diffusely scattered intensity has to be investigated.

Diffuse XRR

For partially correlated interface profiles, as is the case for a superlattice with rough inter-faces, the diffusely scattered intensity is not distributed uniformly in reciprocal space, but peaks at the same Q_z-positions as the satellite peaks in the specularly reflected intensity, forming so called resonant diffuse scattering sheets (see Fig. 2.5). If the interfaces are per-fectly flat, the inplane translation symmetry gives rise to a δ-like intensity distribution at

Qk = 0 Å−1 which means only specular intensity is visible with maxima appearing along

Q⊥ as discussed above. Rough interfaces break this translational symmetry and give rise to diffuse scattering intensity. But if the interfaces are still vertically correlated the diffuse intensity shows also peaks in Q⊥ direction, the so called diffuse scattering sheets. Near the Laue zones where either α_i or α_f are small, scattering is enhanced when either α_i or α_f are close to the critical angle αc. This enhanced intensity distributions are called Yoneda wings named after Y. Yoneda who published this first in 1963 in Ref. [52]. As a result, the resonant diffuse scattering sheets appear to be bent near the Laue zones and are often called Holý

bananas after Václav Holý who described this effect in one of his papers on diffuse scattering

on rough multilayers (see Ref. [53]).

An extensive derivation on the calculation of the diffuse scattering intensity of rough mul-tilayers can be found in Ref. [54]. The aspects relevant to extract parameters from the reflectivity maps are summarized in the following:

The main feature, which can be exploited, is that the diffusely scattered intensity is pro-portional to the Fourier transform of an interface correlation function C(x). This C(x) is a phenomenological quantity where one usually only knows the limits, but not the exact form in-between. The particular shape of the diffuse intensity distribution calculated by DWBA depends on the chosen model for C(x).

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2.5 Summary 23

Qx

Qz

(a) Scattering intensity of perfect interfaces. Only specular reflected intensity is produced.

Qx

Qz

(b) Scattering intensity of rough correlated interfaces. Diffuse intensity is produced which peaks around the maxima of the specularly reflected part. Near the Laue zones the scattering sheets appear to be bent (Yoneda wings).

Figure 2.5: Comparison of the scattering intensity of perfect and rough interfaces.

A typical correlation function resulting from the fractal description of random rough surfaces [55] is given by C(x − x0, y − y0) = σ2e−( ρ ΛL) 2H (2.34) where x and y are in-plane coordinates, ρ =p(x − x0)2_{+ (y − y}0₎2_{, σ is the r.m.s. roughness,} ΛL is the lateral correlation length and H the Hurst parameter, which gives information on the jaggedness of an interface: the larger H, the smoother the interfaces. Λ_Lcorresponds to the average distance within an interface, over which surface features become uncorrelated. However, this correlation function has the disadvantage that its Fourier transform can only be calculated analytically for H = 0.5 and H = 1. Nevertheless, usually quite good fits are obtained with either 0.5 or 1 for H.

2.5 Summary

In order to obtain a full picture of the structural parameters, both X-ray diffraction and X-ray reflectivity (specular and diffuse) signals have to be evaluated and simulated using the very same set of input parameters.

Diffraction signals are sensitive to strain, composition, relaxation and layer thicknesses. However, roughness and interdiffusion can not be distinguished. Reflectivity is sensitive to layer thicknesses and concentrations as well and simulating diffuse XRR allows to determine the real interface roughness. As a consequence, this eventually allows to distinguish between roughness and interdiffusion.

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Chapter 3 Properties of Si and Ge

In this chapter general properties of Si and Ge which are relevant within this thesis and common to the experiments described in the following chapters are summarized.

First, the bandstructure of Si and Ge and their differences are described followed by a summary of possibilities to alter the bandstructure. Following that, relevant thermal and elastic properties are summed up. Thereafter, the concepts of stress and strain for small deformations are introduced and the connection of both within Hooke’s law together with the definition of the elasticity tensor is presented.

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3.1 General

Both Si and Ge are elemental group IV semiconductors crystallizing in diamond cubic struc-ture. Si is the dominating material in semiconductor industry, even though the first transistor was fabricated in Ge. This is because Si is one of the most abundant material in the Earth’s crust, thus easily available and cheap. Ge, however, is much less abundant, but is chemically highly compatible with Si due to its similar lattice constant. The lattice constants are given in Tab. 3.1. The lattice mismatch of Si and Ge is 4.2 %.

aSi (Å) aGe (Å)

5.43104 5.65785

Table 3.1: Lattice constants of Si and Ge, taken from Ref. [56].

3.2 Bandstructure

Both Si and Ge are indirect bandgap materials in bulk form with the valence band maximum at the Γ-point (<000>) consisting of the degenerate light-hole (LH) and heavy-hole (HH) band and the split-off (SO) band a little lower in energy. The main difference occurs in the conduction band: Si has its conduction band minimum at ∼0.85 along the ∆-direction (from Γ to the X-point (<100>)) with a bandgap of Eg = 1.12 eV. The non-degenerate Γ2 state lies 4.2 eV above the valence band, which is 0.7 eV above the lowest conduction band state at the Γ-point. The energy difference between the conduction band minimum and this non-degenerate Γ state is > 3 eV, which makes it extremely challenging to convert Si into a true direct bandgap material.

Ge, however, has its conduction band minimum a the L point (<111>) with a bandgap of

Eg = 0.66 eV which is only slightly lower in energy than the direct gap at the Γ point, which is E_Γ1 = 0.8 eV. The bandstructures for Si and Ge with the corresponding energy gaps at room temperature are depicted in Fig. 3.1.

3.2.1 band structure engineering

When designing new structures from the Si/Ge material system, it is important to understand how the valence- and conduction-band change with respect to the bulk form. Energy gaps determine the wavelength of potentially extractable photons, whereas the curvature of the bands determines the effective masses and thus transport properties. The band structure of Si and Ge can be manipulated in various ways which are described e.g. in Refs. [15, 27, 28].

Alloying

The oldest concept of band structure manipulation is that of alloying Si and Ge into Si_1−xGe_x. For unstrained Si1−xGex the bandgap is continuously decreased with increasing Ge concen-tration x and a discontinuity appears at x = 0.85, where the conduction band changes from Si-like (along ∆) to Ge-like (at Γ) [27, 57]. This approach is appropriate to tune the

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3.2 Bandstructure 27

(a) Band structure of bulk Si. (b) Band structure of bulk Ge.

Figure 3.1: Band structure of bulk Si and Ge at room temperature (reprinted with permis-sion from Ref. [56]).

bandgap for fiber optic communications (1200-1700 nm). The bandgap stays indirect for this approach.

Another possibility is to change the bandgap by alloying with a third material, e.g. α-Sb or

C. The most promising one is α-Sb, which is a zero-gap semiconductor with a 14 % larger

lattice constant than Ge [15]. Calculations of the bandgap depending on the α-Sb content in relaxed Ge_1−xSn_x and Si_1−xSn_x alloys can be found in Ref. [58]. There, it is predicted that the bandstructure of Ge_1−xSn_x for x > 0.17 changes from indirect to direct. For Si_1−xSn_x this transition is not so clear and predictions vary between x > 0.25-0.55.

Strain

Straining Si, Ge and Si_1−xGe_x is a powerful tool to manipulate the band structure. Com-pressively strained Si_1−xGe_x grown epitaxially on relaxed Si lowers the bandgap even more than by alloying and leads to a split of LH and HH bands [27]. Contour plots of the bandgap, minimum conduction- and valence-band offsets of all combinations of strained Si_1−xGe_x epi-taxially deposited on a relaxed Si_1−yGe_y substrate can be found e.g. in Refs. [57, 58]. Similar figures for strained Ge on relaxed Ge1−x−ySixSny and strained Ge1−xSnx on relaxed Ge_1−ySn_y can also be found in Ref. [58].

For Ge, epitaxially grown in [001] direction on substrates with larger lattice constants, and therefore exhibiting a biaxial tensile strain, the energy of the direct band gap is lowered below the indirect conduction-band minimum at the L-point [15] and a strain > 1.7-1.8 % is predicted to form a direct bandgap [29, 59]. The corresponding uniaxial strain for the indirect-direct transition is predicted to be > 4.6 % [29].

[111] grown Ge acquires a direct gap at biaxial or uniaxial strain along [100] for strains > 4.5 %, whereas [111] Ge never becomes direct for any type of strain [29].

In Si the situation is much harder: in contrast to Ge, both inplane and out-of-plane strains have to be tensile with about 10-13 %, which is extremely unlikely to stabilize [15]

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Zone folding

In superlattices, i.e. periodic repetitions of layered sequences, the new periodicity along the growth direction (d) results in a smaller Brillouin zone of size ±π/d compared with that of the original lattice ±π/a (with a the original lattice constant). The electronic band structure is then folded back into this new reduced Brillouin zone.

For this simple model, the minimum of the conduction band in bulk Si is folded into the Brillouin zone center for d ∼ 5a/2, which corresponds to 10 monolayers of Si, and a direct gap is formed [28, 15].

This naive picture has to be somewhat altered if strains occur in the layers constituting the superlattice. For single repeat SinGep superlattices a direct bandgap is only expected for tensile strain [15]. Strain together with the band offsets at the heterointerfaces makes calculations of the resulting mini-bands and band offsets more complex.

However, the predictions of the direct bandgap of a special [Si_n₀Ge_p₀/Si_n₁Ge_p₁/. . . /Si_n_NGe_p_N]∞ superlattice with special strain conditions, studied in Chap. 5, are based on the zone folding approach.

3.3 Thermal properties

Despite by epitaxial growth, as discussed above, a tensile strain can also be achieved through a mismatch in thermal expansion coefficients (TEC). If Ge is grown on Si it will initially grow epitaxially and relaxes after exceeding the critical thickness of 4-10 nm by forming dislocations [60]. At high temperature both Si and Ge have a larger lattice constant as at room temperature due to the different TEC: αGe = 5.9 × 10−6K−1, αSi = 2.6 × 10−6K−1 [56]. Thus, Ge expands more than Si during growth and epitaxy and relaxation happens in this regime. When the system is cooled down to room temperature eventually, Ge contracts more than Si, but since the two materials are in rigid contact, the thinner Ge layer has to follow the in-plane contraction of the Si and is thus tensile strained.

At a growth temperature of 500 °C a tensile strain of 0.2 % can be achieved [61]. This approach is used to pre-strain Ge layers for the fabrication of Ge microbridges studied in Part IV.

3.4 Elastic properties

Elastic properties play an important role in strained structures, which are designed to alter the bandstructure, as discussed above. For anisotropic materials, such as Si and Ge, many material parameters depend on the crystal orientation. The stiffness tensor Cij relates an applied stress to the deformation of the crystal and hence the strain. The derivation of this relation and successive implications on crystal orientations are discussed in this section and follow the books Mathematical Theory of Elasticity by I. S. Sokolnikoff [62] and Theory of

dislocations by J. P. Hirth and J. Lothe [63]. If applicable Einsteins sum convention is used,

unless stated otherwise.

The stress on an infinitesimal volume element is the force per unit area on its facets. Within an orthogonal Cartesian coordinate system x_i (i = 1, 2, 3) it is defined by σ_ij on a plane

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3.4 Elastic properties 29

with the surface normal x_j. In mechanical equilibrium no net torques and forces act on the elements, causing σij = σji (3.1) and ∂σij ∂xj + f_i = 0, (3.2)

also called equilibrium equations of classical elasticity.

A body deforms when stress is applied and the displacement components of this deformation are ui. The strain is then defined as

εij = 1 2 ∂ui ∂xj +∂uj ∂xi ! . (3.3)

Eq. (3.3) contains only first order derivatives of u and hence is only valid for small displace-ments. In this regime Hooke’s law is valid, describing the stress dependence on strain

σij = Cijklεkl (3.4)

with the elastic or stiffness tensor C_ijkl. The stiffness tensor has 34= 81 components, but due to the symmetry of σ_ij and hence ε_kl (Eq. (3.1)) the independent components are reduced to 36:

Cijkl= Cjikl= Cijlk= Cjilk (3.5) and can be written as a 6 × 6 matrix Cij. The corresponding stress and strain matrices then have also only 6 independent components and hence can be written as vectors. This notation is named after W. Voigt (1850-1919) and the conversion rules are given in Tab. 3.2 where the m and n correspond to a pair of indices ij or kl.

ij/kl 11 22 33 23 13 12

m/n 1 2 3 4 5 6

Table 3.2: Conversion to the one–suffix Voigt notation for the elasticity matrix and stress and strain vector.

Eq. (3.4) then reads          σ11 σ22 σ33 σ23 σ13 σ12          =          C11 C12 C13 C14 C15 C16 C12 C22 C23 C24 C25 C26 C13 C22 C33 C34 C35 C36 C14 C24 C34 C44 C45 C46 C15 C25 C35 C45 C55 C56 C16 C26 C36 C46 C56 C66                   ε11 ε22 ε33 γ23 γ13 γ12          (3.6)

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with

γij = 2εij (3.7)

for i 6= j.

For most crystals the number of independent elastic constants can be reduced further due to crystal symmetry. For cubic crystals, including the diamond configuration, which is the case for Si and Ge, only three parameters are independent. In Cartesian coordinates the stiffness tensor Cmn is then given by

(Cmn) =          C11 C12 C12 0 0 0 C12 C11 C12 0 0 0 C12 C12 C11 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44          . (3.8)

The numerical values of the elastic constants of Si, Ge and SiGe alloys, were usually a linear interpolation between Si and Ge values is used, are given in Table 3.3.

Si Ge SiGe

C11 (GPa) 165.8 128.5 165.8-37.3x

C12 (GPa) 63.9 48.3 63.9-15.6x

C44 (GPa) 79.6 66.8 79.6-12.8x

Table 3.3: Elastic constants of Si, Ge and SiGe, taken from Ref. [64].

Together with Eq. (3.4) the following independent σ_mn follow:

σ1 = C11ε1+ C12ε2+ C12ε3 (3.9) σ2 = C12ε1+ C11ε2+ C12ε3 (3.10) σ3 = C12ε1+ C12ε2+ C11ε3 (3.11) σ4 = 2C44ε4 (3.12) σ5 = 2C44ε5 (3.13) σ6 = 2C44ε6 (3.14)

In case of pseudomorphic growth only the in-plane stresses have (equal) finite values (σ₁ =

σ2) and the the out-of-plane (σ3) and shear stresses (σ4 = σ5 = σ6) are zero. If this is the case, then the useful relationship

ε3 = −

2 ∗ C12

C11

ε1,2 (3.15)

follows from Eq. 3.11.

The representation of elastic constants and strain in the given form is only useful if the forces act along the principal axes ([100], [010] or [001]) and are also measured along these

X-ray based investigations of semiconductor multilayer and microbridges / eingereicht von: Tanja Etzelstorfer

JKU

X-ray based investigations of semiconductor

multilayer and microbridges

DISSERTATION

Doktorin

Technischen Wissenschaften

Tanja Etzelstorfer

Institut f¨ur Halbleiter- und Festk¨orperphysik

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

Prof. RNDr. V´aclav Hol´y, CSc.

Eidesstattliche Erklärung

Kurzfassung

Abstract

Abbreviations

Contents

Part I

Introduction & Motivation

Chapter 1

Introduction

1.1 Organisation of the thesis

Part II

Background & Theory

Chapter 2

X-rays & X-ray scattering

2.1 General properties

k

k

k

k

2.2 Basic scattering theory

2.3 Kinematic scattering from multilayers

2.4 Dynamical and semikinematical scattering from multilayers

2.5 Summary

Chapter 3

Properties of Si and Ge

3.1 General

3.2 Bandstructure

3.3 Thermal properties

3.4 Elastic properties

_JKU