Effect of Ar implantation energy and angle on the reflection pattern of nitrides

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Effect of Ar

implantation energy

and angle on the

reflection pattern of

nitrides

João Salgado Cabaço

Masters degree in Physics

Departamento de Física e Astronomia

2019

Orientador

Professor Doutor João Pedro Esteves de Araújo, Faculdade de Ciências, Universidade do Porto

Coorientador

Doutor Sérgio Nuno Canteiro de Magalhães, Campus Tecnológico e Nuclear, Instituto Superior Técnico

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O Presidente do Júri,

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M

ASTER

’

S

T

HESIS

Effect of Ar implantation energy and angle

on the reflection pattern of nitrides

Author:

Jo˜ao SALGADO CABAC¸O

Supervisor:

Professor Doutor Jo˜ao Pedro ESTEVES DE ARAUJO

Co-supervisor:

Doutor S´ergio Nuno CANTEIRO

DEMAGALHAES˜

A thesis submitted in fulfilment of the requirements for the degree of Mestrado em F´ısica

at the

Faculdade de Ciˆencias

Departamento de F´ısica e Astronomia

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First and foremost, I have to thank Sérgio Magalhães for guiding me through this process. Without his endless patience and valuable insights the work presented here wouldn’t have been possible. I also have to thank João Pedro Ara újo for providing me the opportu-nity to do my master’s thesis at CTN, in Lisbon, and allowing IFIMUP to fund the travel expenses. I would also like to thank Katharina Lorenz for helping with the XRD, Jorge Rocha for the ion implantation, and many others who made my stay at CTN so pleasant.

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Acknowledgements iii

Contents v

List of Figures vii

List of Tables xi

Abbreviations xiii

1 Introduction 1

1.1 State of the Art. . . 1

1.2 Objectives and outline . . . 3

2 Overview of the thesis 5 2.1 Properties of Group-III nitrides . . . 5

2.1.1 Crystal structure . . . 5

2.1.2 Samples and growth process . . . 7

2.2 Ion implantation. . . 8

2.2.1 Theoretical description . . . 8

2.2.2 Experimental setup . . . 8

2.2.3 Parameter choice and SRIM simulations . . . 9

2.3 RBS/Channeling . . . 12

2.3.1 Experimental setup and data acquisition . . . 12

2.3.2 Channeling . . . 13

2.3.3 Spectra analysis . . . 14

2.3.4 Data for AlGaN samples . . . 14

3 Results for as-grown samples (virgin AlGaN) 17 3.1 X-ray diffraction . . . 17

3.2 Different types of scans . . . 18

3.3 Experimental setup . . . 19

3.4 Symmetric and asymmetric reflections . . . 21

3.5 Bond’s method . . . 22

3.6 Vegard’s law and determination of composition . . . 24

3.7 Reciprocal space maps . . . 27

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3.8 Mosaicity . . . 29

3.8.1 Williamson-Hall approach. . . 30

3.8.2 Graphical method . . . 31

3.8.3 Analytical method . . . 33

3.9 Converging two models for calculating defects/mosaicity . . . 37

3.9.1 Assumptions and approach . . . 37

3.9.2 Genetic algorithm. . . 37

3.9.3 Results obtained . . . 39

4 Dynamical theory of X-Ray diffraction 43 4.1 Structure factor . . . 43

4.1.1 Example: Calculating full structure factor . . . 44

4.2 Dynamical theory of X-ray diffraction . . . 48

5 Results for implanted samples 51 5.1 Lattice strain and crystal degradation . . . 51

5.1.1 Implantation with increasing energy . . . 51

5.1.2 Same penetration depth with different implantation conditions . . . 55

5.2 Simulating a 2θ−ωprofile . . . 57

5.2.1 Starting from scratch . . . 57

5.2.2 Depth resolution . . . 61

5.2.3 Comparing simulation parameters for all the samples . . . 64

5.3 Reciprocal space maps (RSM) . . . 69

6 Conclusions 71

A Results obtained for mosaicity 75

B Rocking curve simulations 77

C Simulations of2θ−ω 81

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1.1 2θ−ωprofiles for GaN (a) and Al0.53Ga0.47N [23]. . . 2

2.1 Crystalline structure of AlGaN . . . 6

2.2 Illustration of the several layers of the samples studied. AlGaN grown on

top of a sapphire substrate (Al2O3) with a sacrificial (buffer) layer between

them. . . 7

2.3 Danfysik 1090, high current implanter, equiped with a CHORDIS 920 ion

source, schematics adapted from [54], courtesy of Jorge Rocha. . . 9

2.4 Simulation obtained with SRIM of the distribution of ions with depth at

dif-ferent implantation energies/angles. The arrows on top indicate the region with the highest concentration of ions and the dashed lines at the bottom

the maximum penetration depth. . . 10

2.5 Simulation obtained from SRIM, energy of implantation 200 keV for a

sam-ple of Al0.17Ga0.83N. The white lines correspond to the trajectories of the

Ar+_{ions in depth (a) and in tranverse view (b).} _{. . . 11}

2.6 Vacancy profile obtained with SRIM. . . 11

2.7 Schematics of the RBS at CTN. (1) Van de Graaff accelerator; (2) Deflecting

magnet; (3) Chamber; adapted from [24]. TM- Turbomolecular vacuum

pump; VM - Vacuummeter; V - vacuum valves; T - beam-stopper Tantalum;

S1,2- stabilizing slits; C1,2- Beam collimators. . . 12

2.8 Illustration of how the ion beam propagates along a major crystallographic

axis of the crystal, phenomenon of channeling. Image adapted from [48]. . . 13

2.9 RBS spectra for an AlGaN sample (black points) and NDF simulation (red

line). The vertical arrows indicate the position of the elements, calculated using eq. 2.3, and the horizontal line the width of the AlGaN layer obtained

by simulation. The aligned spectra < 0001 > can also be seen (purple

triangles).. . . 15

3.1 Geometrical interpretation of Bragg’s law, d is the interplanar distance [61]. 18

3.2 The different movements the goniometer can execute when doing XRD

measurements, adapted from [41]. . . 19

3.3 Schematics of the XRD used in this thesis. . . 20

3.4 Photo indicating the several components of the diffractometer used. . . 20

3.5 Symmetric reflection (left), the measured planes are parallel to the sample

surface. Asymmetric reflection (right), the measured planes are at an angle

ψ6=90◦ with respect to the normal. Figure adapted from [48]. . . 21

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3.6 Rocking curves measured for Bond’s method, corresponding to sample 1. The numbers in each figure correspond to the fitting parameters of the

Pseudo-Voigt function. a1 is the incident angle, which, depending on the

reflection, can be ω+_i or ω_i−. . . 24

3.7 Reciprocal space maps for two virgin samples in symmetric (0004) and asymmetric (10¯15) reflections. The coordinates in the lower left region of each graph are for the center (indicated by the cross lines), using this set of coordinates the lattice parameters are calculated are represented in table 3.4. 28 3.8 Different types of defects encountered in Group-III nitrides. . . 30

3.9 Symmetrical reflections (0002) and (0004) for sample 1. . . 32

3.9 Reflection (0006) and Williamson-Hall plot for sample 1. . . 32

3.10 Results obtained using the analytical method [12], for the coherence lengths, tilt and strain of the six AlGaN samples studied. The vertical error bars are represented, since the results obtained for each reflection are similar they are enclosed by a box. . . 36

3.11 A simplified flowchart for the steps necessary when performing the analy-sis with the genetic algorithm software. . . 38

3.12 Example of bad fits obtained for (0002) with the software. . . 39

3.13 Simultaneous fit for the three reflections of sample 1 using the genetic al-gorithm (a-c), the last graph shows the Williamson-Hall plot. . . 40

3.14 Representation of the values obtained for the tilt (red circles and righ y-axis) and lateral coherence lengths (black squares and left y-y-axis). . . 40

4.1 Representation of f’ and f” as a function of energy for each of the elements of the AlGaN ternary, values obtained from [70]. . . 46

5.1 The diffractograms correspond to 2θ−ωscans for the Ar+implanted Al-GaN samples in three different symmetric reflections. For the secondary peaks the perpendicular strain can be seen in the upper horizontal axis. . . 53

5.2 2θ−ω diffractogram that shows a virgin and an implanted profile for the same sample. The area that is represented by the dashed lines can be seen in tab. 5.1 for all the samples. . . 54

5.3 Strain as a function of energy for the three reflections in fig. 5.1, using the data from table 5.1. . . 55

5.4 Diffractograms for 2θ−ω scans of two samples, the blue line represents Ar+implantation at 200 keV. The pink line represents Ar+implantation at 250 keV and 38 deg. tilt with respect to the normal. According to the SRIM simulations [29], the region with the highest concentration of ions should be the same for both samples. . . 56

5.5 MROX simulations, steps (a) and (b) in this section. . . 58

5.5 MROX simulations, steps (c) and (d) in this section. . . 59

5.5 MROX simulations, steps (e) and ()f) of this section. . . 61

5.6 Interchanged layers. . . 63

5.7 Parameters used in the 2θ−ωsimulations for (0002) obtained with MROX. In (a) the strain as a function of depth and in (b) the Debye-Waller factor as a function of depth. . . 65

5.8 Parameters used in the 2θ−ωsimulations for (0004) obtained with MROX. In (a) the strain as a function of depth and in (b) the Debye-Waller factor as a function of depth. . . 66

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5.9 Parameters used in the 2θ−ωsimulations for (0006) obtained with MROX.

In (a) the strain as a function of depth and in (b) the Debye-Waller factor as

a function of depth. . . 67

5.10 Areas of the strain profile integrated at 50 nm intervals in the whole im-planted depth for (0002). . . 68

5.11 Reciprocal space maps of sample 1 for(0004), as-grown (left) and implanted at 25 keV (right). The vertical line indicates the center, there is a slight ex-pansion to lower values of Qz after implantation (right). . . 69

5.12 Reciprocal space maps for sample 5 and (0004), as-grown (left) and im-planted at 250 keV (right). The vertical line indicates the center, the expan-sion to lower values of Qz after implantation (right), is even more notice-able than for 5.11. . . 69

5.13 Reciprocal space maps of sample 5 for 10¯15), as-grown (left) and implanted at 250 keV (right). . . 70

B.1 Simultaneous fit for the three reflections of sample 2 using the genetic al-gorithm, the last graph shows the Williamson-Hall plot.. . . 77

C.1 Simulations using MROX. . . 82

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2.1 Atom positions in the unit cell of AlGaN. . . 6

3.1 Lattice parameters obtained from Bond’s method. . . 23

3.2 Reference values for binaries [34,71–74].. . . 25

3.3 Values obtained for the composition of the AlGaN samples. . . 26

3.4 Lattice parameters obtained from RSM. . . 28

3.5 Values obtained for the graphical Williamson-Hall method. . . 33

3.6 Values obtained using the analytical method for sample 1 . . . 34

3.7 Values obtained for coherence length (parallel to the sample surface) and crystalline tilt using the genetic algorithm . . . 41

4.1 Cromer-Mann coefficients used in eq. 4.6, Al (upper left), Ga (upper right) and N (lower left), values from [70]. . . 45

5.1 Strain calculated graphically for implanted samples (2nd, 3rdand 4thcolumns). Comparison of the areas from which the implanted sample deviates from the virgin sample in the 2θ−ω peaks, corresponding to the ”area of im-plantation” (column 5). . . 54

A.1 Values obtained using the analytical method for sample 1. . . 75

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HEMT High Electron Mobility Transistor LED Light Emitting Diode

RC Rocking Curve XRD X-Ray Diffraction

RBS/C Rutherford Backscattering / Channeling RSM Reciprocal Space Map

D.W. Debye-Waller P.V. Pseudo-Voigt

MROX Multiple Reflection Optimization Package for X-ray diffraction FWHM Full Width at Half Maximum

MOCVD Metal-Organic Chemical Vapour Deposition NDF Nuno’s Data Furnace

SRIM Stopping and Range of Ions in Matter

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Introduction

1.1 State of the Art

The development of the first commercially available long-lived blue LED and blue lasers, in the early 1990s, launched the interest in III-nitrides. A group that includes semicon-ductors such as AlN, GaN and InN, which have band gaps spanning the entire UV and visible ranges [1–6]. Thin films of III-nitrides can be used to make UV, violet, blue and green LEDs, as well as solar cells, high-electron mobility transistors (HEMT), among other devices [7, 8]. Since the millennium, the optical and electrical properties of III-nitride ternaries have been explored extensively in the fields of opto- and microelectronics [8,9]. III-nitrides crystallize in the hexagonal wurtzite (HZ) or in the cubic zinc blend struc-ture, and have a wide range of bandgap energies, depending on the composition [10]. Despite the interest and commercial applicability, the information available in the liter-ature regarding many of its physical properties are still in the process of evolution and subject of some controversy [11]. In epitaxial layers of group-III nitrides the film growth process gives rise to unusually high strain and high defect densities, affecting the device performance [12,13]. The measurement of the full width at half maximum (FWHM) of the rocking curve with a double-crystal diffractometer is a well established method of proving the epitaxial quality of GaN films [14]. The heterogeneous strain, the correlation lengths normal and parallel to the substrate surface and the degree of mosaicity expressed by the tilt and twist angles are key issues in characterizing the quality of epitaxial films with a large lattice mismatch to the surface [12]. X-ray diffraction provides information on crystalline lattice parameters, from which strain and composition can be determined, and misorientation, from which defect types and densities can be deduced [15]. In order

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to determine with precision the lattice parameters, in a relatively fast way, Bond’s method can be used [16].

Ion implantation is an important technique in the silicon industry and has had applica-tions for selective area doping and implant isolation in AlxGa1−xN HEMT. Its main

ad-vantages are the possibility to control dopant concentrations and depth profile, and the high purity through mass selection [17–19]. The nature of the process, in which energetic ions collide and penetrate into the semiconductor, results in lattice damage [20–22]. While implantation damage formation in the binaries GaN and, to a lesser extent, AlN, has been studied by several groups, fewer studies exist for ternary compounds [23].

In a study by Faye et al. [23], AlxGa1−xN alloys in the (0 6 x 6 1) region were

implanted with 200 keV argon at increasing fluences. In the 2θ−ω scans in fig. 1.1,

obtained by XRD, the 2-peak profile is evident, resulting from the c-lattice expansion in the damaged region (secondary peak). As fluence increases the peaks tend to approach ever lower angles, increasing the c-lattice parameter.

FIGURE1.1: 2θ−ωprofiles for GaN (a) and Al0.53Ga0.47N [23].

XRD also shows that implantation leads to the incorporation of large lattice strain in the implanted layer, which increases with increasing fluence. By examining the measure-ments obtained by Faye et al. in fig.1.1(a) and (b), for different fluences and composition, a fluence that yields a peak far enough from the main diffraction peak, but not so far as to be indistinguishable from the shape of the curve can be chosen. A fluence around 1014 Ar+_/cm2 _{corresponds to a secondary peak that is far enough from the main peak. But}

not so far as to be very broad (> 1015 Ar+/cm2). Several studies of defect accumulation in III-nitrides exist, but are usually based on increasing fluence or varying geometry of

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implantation, [17, 21, 23]. Other works were also found, where the techniques of XRD and RBS/C are used to study ion implantation on nitrides, [1,24–28].

1.2 Objectives and outline

Building upon the methods and results outlined in the previous section, the aim of this work is to study and characterize the damage profiles of AlGaN samples implanted at increasing energies and angles. Starting with a study of the virgin samples (before im-plantation), the growth quality is characterized with XRD (X-ray diffraction) and RB-S/C (Rutherford Backscattering / Channeling). After implantation a simulation software, MROX (Multiple Reflection Optimization Package for X-ray diffraction) [9] will be used to simulate the deformed profiles and study strain propagation in depth. The structure of the thesis has 6 chapters, the following provides an outline of what is discussed in each of them:

• Chapter 2: the ion implantation technique is presented, as well as the software that allows the simulation of depth profiles, SRIM [29]. Simulations are made using SRIM, in order to choose the implantation parameters. A brief description of the RBS/C technique is made, focusing on how it can be used to determine the thickness of a thin film (NDF simulation [30] - Nuno’s Data Furnace), and to evaluate the crystal quality.

• Chapter 3: the virgin samples of AlGaN are studied, starting with the calculation of lattice parameters from Bond’s method [16], then the composition is determined using Vegard [31,32] and Poisson’s law [33]. Reciprocal space maps of two reflec-tions are also used to confirm the previously determined lattice parameters. The chapter comes to an end with the study of mosaicity for the virgin samples, based on the treatment made by Metzger et al. [12]. A new approach, based on a genetic algorithm, is developed to make the results of each method converge.

• Chapter 4: a brief description of the structure factor, including the anomalous con-tributions. The full structure factor is calculated for the (0002), (0004) and (0006) reflections. In this chapter there is also a description of the dynamical theory of X-ray diffraction, used in the software MROX [9], to simulate the 2θ−ω scans of

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• Chapter 5: the main results obtained for the damage profiles of ion implanted Al-GaN, as seen in 2θ−ω scans and RSM. A discussion of the results obtained for

deformation and crystal degradation as a function of depth using the simulation software MROX [9].

• Chapter 6: an outline of the main results obtained, and the conclusions derived from them, as well as some ideas for future work.

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Overview of the thesis

A description of the properties of III-nitrides and in particular of the AlGaN samples stud-ied in this thesis. The growth process and crystal structure are specifstud-ied. Followed by an explanation on how the SRIM software [29] is used to simulate the penetration depth and the vacancy profile of Ar+implantation on virgin AlGaN. The technique of RBS/Chan-neling is also used to acquire more information about the thickness and composition of the thin film.

2.1 Properties of Group-III nitrides

Group-III nitride semiconductors have been the subject of intense research in the last cou-ple of decades [15]. Their unique physical properties and high potential for numerous electronic and optoelectronic applications make them attractive for new technologies. These compounds are formed due to bonding of one of the group-III elements, such as aluminum, boron, gallium or indium, with nitrogen, from group V [15,34,35]. The III-nitrides and their alloys are direct band gap semiconductors with a band gap that varies from 0.7 eV, for InN [36–38], to 3.5 eV for GaN [39] and 6.1 eV for AlN [39]. Properties that justify the interest of these compounds in optoelectronic devices, for example, in building LED’s, detectors, HEMT [34] and even medical devices [40].

2.1.1 Crystal structure

Nitrides from group-III share three crystal structures, wurtzite, zincblende and rocksalt, the thermodynamical stable phase is wurtzite for bulk AlN, GaN, and InN [11,41]. The AlGaN films studied in this thesis have the hexagonal wurtzite structure (HZ), which can

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be seen in fig. 2.1(A) and (B). The wurtzite is a non-centrosymmetric crystal structure, and consists of interpenetrating hexagonal close packed (HCP) sublattices, each with one type of atom [42]. This structure belongs to the space group P63mc in the Hermann-Mauguin

notation and C_6v4 in the Schoenflies notation. The position of the atoms in the unit cell can be seen in table2.1. For the ternary Al1−xGaxN that is studied the composition should be

around 17% of AlN and 83% of GaN, a fact later confirmed by XRD in (Ch.3) and by RBS in the final section of this chapter. The following subsection will provide some information about the samples studied.

Element x y z Al 0 0 0 Al 0.33 0.666667 0.5 Ga 0 0 0 Ga 0.33 0.666667 0.5 N 0 0 0 N 0.33 0.66667 0.1250 TABLE2.1: Atom positions in the unit cell of AlGaN.

/

3+ 3+ 3−

(A) Wurtzite structure of AlGaN, in grey the N atoms and in yellow the Ga/Al.

(B) Unit cell of AlGaN, in yellow the Al/Ga atoms

and in blue the N atoms. FIGURE2.1: Crystalline structure of AlGaN

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2.1.2 Samples and growth process

The samples used were produced by NOVAGAN [43], they are composed of an AlGaN layer grown on top of a sapphire substrate (Al2O3), with a thin sacrificial layer of AlN

between them, fig. 2.2. The sacrificial (also called buffer) layer is very thin and inserted to reduce lattice mismatch between the film and the substrate [44]. The AlGaN thin films are c - grown, and the thickness of the layer is expected to be around 650 nm, a fact later confirmed by RBS (Rutherford Backscattering) measurements in the end of the chapter.

0.17 0.83

−

2 3 Sacriﬁcial layer Substrate 650 nm < 10 nm

FIGURE2.2: Illustration of the several layers of the samples studied. AlGaN grown on top of a sapphire substrate (Al2O3) with a sacrificial (buffer) layer between them.

The process of growing high quality group-III nitride epitaxial layers faces several challenges, [12, 41]. The main problem arises from the lattice mismatch and the coeffi-cients of thermal expansion that differ between III-nitride layers and the commonly used substrates. The most often used substrate to grow III-nitride films is sapphire, due to its availability and lower cost, chemical and thermal stability at high temperatures [45]. The growth method used for the NOVAGAN samples was MOCVD (Metal-Organic Chemical Vapour Deposition), a method that allows the growth of good quality epitaxial layers on c-sapphire substrates. The first step consists in growing a thin layer of AlN at low tem-perature. This is followed by the growth of a main layer at a higher temperature, a more thorough discussion of the process can be found in [41,46,47]. The defects that appear after growth will be studied in Ch.3 using XRD.

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2.2 Ion implantation

2.2.1 Theoretical description

Ion implantation is a process by which ions of one element are accelerated into a solid target, thereby changing the physical, chemical, mechanical or electrical properties of the target [48–51]. With the ability to introduce impurity ions in the lattice, this technique finds numerous applications in the semiconductor industry to dope materials [20]. The nature of the interaction between incident ions and the atoms of the target material depends on the energy, charge and mass of the ion, as well as the chemical composition, structure and temperature of the target atoms. The damage that the incident ion beam inflicts on the target material is an important topic of study.

As the ion beam goes through a solid it loses energy due to the energy transfer be-tween the ion and the electrons of the target, leading to ionization or excitation. This energy transfer reduces the incident ion energy, however, it’s trajectory is mainly unal-tered. The incident ions will eventually suffer high angle collisions with target nucleus, a phenomenon that leads to a cascade variation of the target atoms. The incident ions will come to a stop after a penetration of a few hundred nanometers inside the target [41]. In order to simulate the interaction of ions with matter, the Monte Carlo based software SRIM [29] can be used. Information such as penetration depth, spread of ions, concentra-tion of vacancies, ionizaconcentra-tion and phonon producconcentra-tion in the target material can obtained [29,52]. The following subsections provide information about the experimental setup for the implantation and how SRIM simulations are used to choose the parameters.

2.2.2 Experimental setup

In fig. 2.3the schematics of the ion implanter at CTN/IST1_{, used in this thesis, is shown.}

A sample holder allows the rotation around the axis perpendicular and parallel to its surface. The following provides a brief description of the steps needed to achieve ion implantation, with the main components in bold:

1. A positively charged ion beam can be produced from elemental or compound gases (hot filament ion source) or from sputtering. Both gases, vapours and solids may be used as charge materials [53]. Beam extraction is performed with an accelerated

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voltage in the 15-50 keV range. The ion source can produce ions from all stable elements of the periodic table and is powered by the power generators.

2. An analyzing magnet will perform the mass selection with a mass resolution in the 150-250 _∆mm . The beam passes through an accelerating tube with maximum voltage of 160 keV (using the acceleration supply), the beam will reach three focusing mag-nets. The quadrupole magnet controls the beam shape. Before entering the target chamber the sweeping magnets are responsible for the scanning process through-out the whole area of implantation [54].

3. The target chamber has an entrance with 40×40 cm2, the scanning will depend on the ion beam used. The sample substrate can be heated up to 600 ◦C, in order to perform high temperature implantation.

Target Chamber SweepingMagnets

Focusing Magnets Acceleration Supply Power Generators Ion Souce High Voltage Terminal Analyzing Magnet 0 1 2 m

FIGURE2.3: Danfysik 1090, high current implanter, equiped with a CHORDIS 920 ion source, schematics adapted from [54], courtesy of Jorge Rocha.

2.2.3 Parameter choice and SRIM simulations

The main purpose of the thesis is to study the damage induced by ion implantation, in the current subsection the parameters used for the process are presented. Following the study made by Faye et al [23], and considering a composition Al0.17Ga0.83N, an implantation

fluence of 1×1014 at/cm−2 is chosen. Argon is used as the implantation ion, this is due to its chemical properties and the fact that it is an inert element. In order to simulate the

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effects of implantation, the three layers of the material are inserted in SRIM [29], as well as the composition. For the study in this thesis it is interesting to analyze the damage at different energies of implantation. After trying several parameters, the ones shown in fig.

2.4 were chosen. The samples 1 - 5 were implanted at increasing energies (25 - 250 keV) and zero deg. to the sample normal. For sample 6 the parameters are 250 keV and 38 deg. to the normal, a combination that corresponds to the same maximum ion concentration depth as implantation at 200 keV and 0 deg. to the normal (sample 4). The ion ranges are also shown, in fig. 2.4, the region with the highest concentration of ions can be seen by the arrows on top of each curve and the maximum penetration depth(x) by the dashed arrows on the bottom. The statistical process of the interactions is typically described by a Gaussian function, where its central value is the projected mean range Rpwith a standard

deviation of∆Rp. The depth distribution of the implanted ions can be expressed as [55]:

n(x) = √ Φ 2π∆Rp exp[−(x−Rp) 2 2∆R2 p ] (2.1)

where Rpand∆Rpare the parameters of the distribution, projected range and range

strag-gling, respectively. The ion fluence in(ion/cm2)is given byΦ.

0 nm _{100 nm} 200 nm 300 nm

Target depth (nm)

(A

toms/

𝑐𝑚

3

)/

(Atoms

/𝑐

𝑚

2

)

17.6 nm 60.7 nm 90.5 nm 120 nm 152.7 nm 4 × 104 6 × 104 8 × 104 10 × 104 12 × 104 14 × 104

Sample 1, E=25 keV Sample 2, E=100 keV Sample 3, E=150 keV Sample 4, E=200 keV Sample 6, E=250 keV 38 deg. to the normal Sample 5, E=250 keV

50 nm 175 nm 211 nm 280 nm

290 nm

320 nm

FIGURE 2.4: Simulation obtained with SRIM of the distribution of ions with depth at different implantation energies/angles. The arrows on top indicate the region with the highest concentration of ions and the dashed lines at the bottom the maximum

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In fig.2.5the collision cascades are shown, it is unlikely that the ions keep their initial trajectory after collision. During the collision cascade the lattice atoms suffer displace-ments and depending on factors such as the ion mass, energy, implanted fluence and binding energy of lattice atoms, can create several types of defects beside vacancies and interstitials, resulting of atom displacements [41]. In fig. 2.6the vacancy profile is repre-sented for the samples studied.

(A) Depth vs Z-Axis (B) Transverse View

FIGURE2.5: Simulation obtained from SRIM, energy of implantation 200 keV for a sam-ple of Al0.17Ga0.83N. The white lines correspond to the trajectories of the Ar+ ions in

depth (a) and in tranverse view (b).

0 nm _{100 nm} 200 nm 300 nm

Target depth (nm)

(N

um

b

er

)/

(𝑛𝑚

−

𝐼𝑜𝑛

)

Sample 1, E=25 keV

Sample 2, E=100 keV Sample 3, E=150 keV Sample 4, E=200 keV

Sample 6, E=250 keV 38 deg. to the normal Sample 5, E=250 keV

vvvv

Vacancies produced

0.1 0.08 0.06 0.04 0.02

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2.3 RBS/Channeling

The technique of Rutherford backscattering spectrometry / channeling (RBS/C) allows the determination of composition, depth profile and crystal quality of a material. It consists on the bombardment of a sample with a high energy positively charged ion beam (1 -3 MeV) and collecting the number and energy distribution of the backscattered ions after the collision with the target atoms [56]. A detailed analysis of the physics of this process can be seen in [20, 56–59]. In this section a brief description of the experimental setup used and underlying theory of the technique is made. The last subsection focuses on the results obtained for the AlGaN samples studied in this thesis.

2.3.1 Experimental setup and data acquisition

In fig. 2.7 the schematics of the RBS/C setup is presented. A beam of 2 MeV4He+ is produced in (1), a Van de Graaff accelerator, and directed to the respective line Pixe/NRA, Microbeam (are not used in this thesis) or RBS, this is done using an analyzing magnet (2), which selects the energy and mass of the ion beam [41]. The beam is stabilized through the current measurements on slits S1 and S2. Before entering the chamber, the beam is

collimated using C1and C2, the final cross section is 1 mm2. A very low pressure, as low

as 2×10−6mbar is kept in the beam line during the measurements, in order to minimize the beam spread due to collision with air molecules.

FIGURE2.7: Schematics of the RBS at CTN. (1) Van de Graaff accelerator; (2) Deflecting magnet; (3) Chamber; adapted from [24]. TM Turbomolecular vacuum pump; VM -Vacuummeter; V - vacuum valves; T - beam-stopper Tantalum; S1,2- stabilizing slits; C1,2

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In (3) the beam reaches the small chamber where samples are placed, three photodi-ode detectors are placed at+/−165 deg. and 140 deg., with respect to the incident beam. The samples are introduced in a small chamber, a motorized goniometer allows rotation in two axis, the vertical plane φ and the horizontal plane, θ. A tilt of 5 deg. with respect to the incident ion beam is made to minimize the channeling phenomenon. A rotation in

φduring the measurements is made to avoid planar channeling [41]. When the

backscat-tered particles hit the detector an electric pulse with an amplitude proportional to the particle energy is generated. In the following subsection the channeling phenomenon is briefly described.

2.3.2 Channeling

In a monocrystal channeling occurs when a beam of charged particles propagates along a line/plane of atoms. Feeling the effect of the potential of the crystalline structure, these lines/planes constitute channels that facilitate the propagation of the beam deep into the sample [60]. In a channeling direction the backscattering yield significantly decreases and is very small compared to the random direction [57]. The trajectory of a channeled ion corresponds to a series of successive collisions with very small angles ψ, represented in fig. 2.8. As long as the beam doesn’t get too close to the extremes,Ψ_critical or above, the particles will remain in the channeling condition. If the angle is higher thanΨcritical the

particles will be backscattered, reaching the detector and increasing the yield.

Channeled beam Random beam −θ +θ ψ ψcritical Z2 Z1

FIGURE2.8: Illustration of how the ion beam propagates along a major crystallographic axis of the crystal, phenomenon of channeling. Image adapted from [48].

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2.3.3 Spectra analysis

An RBS spectra is a plot of yield vs channel, in order to convert between channel and energy a calibration equation, in this case eq. 2.2is used.

E(keV) =2.18 keV Channel ×Channel+84(keV) (2.2) The RBS interaction, where the 4He+ particles hit the sample and some particles are backscattered can be described as a classical elastic two-body collision. An assumption that is valid since the energy of the4He+_{beam is much higher than the binding energy of}

the target atoms, but not so high as for a nuclear reaction to occur [41,56]. The interaction can be described by the kinematic factor in eq. 2.3. E0and E1represent the energy of the

incident and backscattered particles, respectively. If the backscattering angle θ and M1

the mass of the incident beam, M2the mass of the backscattering particles are know, eq.

2.3can be used to determine the energy of each element barrier in an RBS spectra.

K = E1 E0 =   cosθ+ q (M2 M1) 2₋_sinθ 1+ M1 M2   2 (2.3)

In a channeling direction the backscattering yield is minimized, the ratio of the area between the channeling yield (minimum yield) and the random yield is typically around 2−3% for excellent quality III-nitride, 3−5% for very good quality,<10% for good and <20% for reasonable quality. This quantity provides the fraction of displaced atoms from their regular sites, therefore, it is a good way to probe the crystal quality.

2.3.4 Data for AlGaN samples

In fig. 2.9 a RBS spectra is shown for one of the AlGaN samples. Using eq. 2.3 the position of the barriers corresponding to the elements Ga, Al, N and O are calculated and represented by arrows in the graph. Gallium is the heaviest of the elements present and is at the surface, the barrier is clearly seen. For Al, present in the three layers(AlGaN/AlN /Al2O3), from E=1106 keV down to the lowest values of energy in the scale it is always

seen. The great decrease in yield between energy values of 1100 keV down to 1000 keV, and the fact that the yield isn’t zero at lower values is an indication of the presence of Al. In the AlGaN layer the concentration of AlN is around 17%, however, it is much higher at the substrate, a fact justified by the increase in yield from E = 800 keV down to 500 keV.

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1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

0 3 0 0 0

6 0 0 0

9 0 0 0

1 2 0 0 0

1 5 0 0 0

_{R a n d o m}

_{θ = 5}

_{d e g .}

S i m u l a t i o n

A l i g n e d < 0 0 0 1 >

Y

ie

ld

C h a n n e l

5 0 0

7 0 0

9 0 0

1 1 0 0

1 3 0 0

1 5 0 0

1 7 0 0

N

6 5 0 n m

G a

A l

A l , s u b s t r a t e

E n e r g y ( k e V )

O , s u b s t r a t e

FIGURE2.9: RBS spectra for an AlGaN sample (black points) and NDF simulation (red line). The vertical arrows indicate the position of the elements, calculated using eq. 2.3, and the horizontal line the width of the AlGaN layer obtained by simulation. The aligned

spectra<0001>can also be seen (purple triangles).

In fig. 2.9 the red line is obtained using the simulation software NDF [30]. The values obtained by the simulation are the thickness of the AlGaN film, 650 nm, and composition, 17% of AlN and 83% of GaN. The notorious slope between channels (700 - 400 ) might be due to variations of gallium concentration in depth, and deviates slightly from the simu-lation. In fig.2.9an aligned spectra (purple triangles) is also represented. The region with the highest yield in the aligned spectra, seen around 1100 keV, corresponds to the transi-tion between the layers of AlGaN and AlN. It is interesting to note that between the bar-riers of Al and Ga the spectra yield increases with decreasing energy (increasing depth), this is due to dechanneling effects, but also showing the density of defects is higher at the AlGaN/AlN interface, rather than at the surface.

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χmin =

Yaligned

Yrandom

(2.4) By looking for the minimum in the data (aligned spectra), near the surface of gallium (box in blue), and integrating a small window in the random (Y_random) and in the aligned spectra (Yaligned), using eq. 2.4 a minimum yield of 3% is obtained. Values of χmin ≤ 3%

indicate excellent crystal quality, characteristic of III-nitrides [41,48].

The thickness of the AlGaN layer was obtained, as well as the concentration of Al and Ga. Using the aligned spectra it was possible to show that the samples have good crystal quality and are state of the art. In the next chapter further analysis using XRD will confirm the concentration and crystal quality of the samples.

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Results for as-grown samples (virgin

AlGaN)

In order to characterize the six samples of virgin AlGaN, grown on top of a sapphire substrate, XRD is used. XRD is an established tool for the structural characterization of crystalline substrates and thin films. The lattice parameters are determined using Bond’s method, measuring only four rocking curves, and also by RSM. The molar fraction of the components that make up the ternary alloy are obtained, using Vegard and Poisson’s law. Two models are used to study the mosaicity of the virgin samples, Williamson-Hall and the analytical method. An attempt to make the two methods converge is made, using a software developed for this purpose.

3.1 X-ray diffraction

Bragg’s diffraction condition occurs when radiation with a wavelength compared to atomic spacings is scattered in a specular way by atoms of a crystalline system and undergoes constructive interference. In a crystalline solid, the waves are scattered from lattice planes separated by the distance d, the interplanar distance. When the scattered waves inter-fere constructively, they remain in phase since the difinter-ference between the path lengths of the two waves is equal to an integer multiple of the wavelength. The path difference be-tween two waves undergoing interference is given by 2dsinθ, θ is the scattering angle in fig.3.1. This leads to Bragg’s law, which describes the condition on θ for the constructive interference to be at its strongest.

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FIGURE3.1: Geometrical interpretation of Bragg’s law, d is the interplanar distance [61].

2dsinθ =nλ (3.1)

n is a positive integer and λ is the wavelength of the incident wave. The material studied in this thesis has an hexagonal crystal structure, the interplanar distance d, can related with a and c, the lattice parameters for the crystal as:

1 d2 = 4 3( h2₊_hk₊_k2 a2 ) + l2 c2 (3.2)

h, k and l represent the Miller indices of the Bragg plane [62–65].

3.2 Different types of scans

Several types of scans can be made using XRD, the following is an outline of the most common ones, fig. 3.2illustrates the different axis mentioned.

Rocking curve (RC): Only the ω axis of the goniometer moves, all others remain sta-tionary. This scan measures film quality, detecting broadening by dislocations and wafer curvature [15].

Detector scan (2θ2θ2θ): Only the 2θ axis moves, either in the positive or negative direction (according to the convention). The sample and the source remain stationary, only the de-tector moves. Dede-tector scan provides information of the lattice parameter, therefore the fingerprint of the studied material. Morever, detector scans constitute valuable probes for system calibration.

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2θ−ω

2θ2θ−−ωω scans: The sample is rotated by ω and the detector is rotated by 2θ with an

an-gular ratio of 2 : 1. It provides information about crystal thickness, strain and mosaicity parameters, which depend on the 2θ such as the vertical coherence length.

Reciprocal space maps (RSM): Performs a mapping over a 2D region of reciprocal space. Carried out using a software that allows the combination of 2θ−ω scans over all

pos-sibilities of ω and 2θ in a pre-defined range. By definition, in a RSM all measurements with the exception of χ and φ can be obtained. Rocking curves trace an arc along the Qx

direction as the major axis, and 2θ−ωare cut crossing the origin of the reciprocal lattice

point and the measured centroid.

φ

φφ scan: Rotation of the sample about the φ axis (usually in the plane of the sample). On φ

scan all crystallographic effects that can perturb the a-lattice parameter can be obtained. Edge threading dislocations are an example of such defects, although these defects are not in the scope of this work [15].

Goniometer

X-rays

FIGURE3.2: The different movements the goniometer can execute when doing XRD mea-surements, adapted from [41].

3.3 Experimental setup

The diffractometer used is a Bruker D8 Discover, avaliable at LATR, in CTN. As an X-ray source it has a Cu anode, a double crystal monochromator, Ge(220), a scintillation detector and a seven axis goniometer, fig.3.3and3.4[41,48,66,67].

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Sample (4) Slits 0.2 mm (1) Detector (2) Slits 0.1 mm (7) X-ray source (6) Parabolic Göbel mirror (5) Monochromator

FIGURE3.3: Schematics of the XRD used in this thesis.

The device has seven axis, the ω, φ, χ and 2θ for the scanning process, and the x, y and z for the alignment of the sample with the beam.

FIGURE3.4: Photo indicating the several components of the diffractometer used.

The numbers in the figure above represent the following components: 1. Scintillation detector.

2. Place to insert the slit. 3. 7-axis goniometer.

4. Two-crystal Germanium (220) monochromator to control the vertical divergence of the beam and select only the CuKα1emission line.

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5. 0.2 1mm width slit, or other, depending on the sample studied.

6. G ¨obel mirror, for the focusing process of the beam and to make it parallel. 7. X-ray Cu source with a wavelength of 0.15406 nm.

3.4 Symmetric and asymmetric reflections

The planes that contribute to Bragg’s condition are parallel with each other. When they are parallel with the surface of the crystal, the reflections are referred to as symmetric reflections. If the planes are at an angle with the normal to the surface that is different from 90 deg., then the reflections are asymmetric [16, 48]. An analysis of fig. 3.5 shows that the incidence angle is ωi = 2θ₂ +(ψ− π₂), in the symmetric case the second term goes

to zero, since ψ is 90o. In fig 3.5~ki and~kf represent the incident and diffracted vector,

and they are at an angle ω_i+and ω+

e with the sample surface. ψ is the angle between the

measured plane and the normal to the surface [16].

+ + 2 + + 2

Symmetric

reﬂection

Asymmetric

reﬂection

Atomic Planes ⃗ ⃗ = − ⃗ ⃗ ⃗ = − ⃗ ⃗ ⃗ ⃗ ⃗ [0001]

FIGURE3.5: Symmetric reflection (left), the measured planes are parallel to the sample surface. Asymmetric reflection (right), the measured planes are at an angle ψ6=90◦with

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In symmetric reflections it is possible to access planes with h, k = 0, from eq. 3.5, comes a relation between the c-lattice parameter and Qz of the reciprocal space, d000l =

c

l ←→ c = 2πlQz. Whereas, for asymmetric reflections the information is related with the

three components of the unit cell a, b and c.

3.5 Bond’s method

Now that some of the underlying principles regarding the XRD theory and experiment have been presented, it is time to proceed to actually analyzing the virgin samples of AlGaN. In the current section the lattice parameters are obtained, using Bond’s method. A method that relies on the measurement of rocking curves, by measuring a symmetrical and an asymmetrical reflection on the grazing incidence and reflection geometry, four RC in total. This method has an accuracy on the order of a few ppm [16]. According to [12] the function that best describes a RC is a convolution of a Gaussian and a Lorentzian function, usually referred to as Pseudo-Voigt (P.V.), eq. 3.3. a0is the maximum intensity,

a1the center of the P.V., a2half the width of the Gaussian, which is equal to the width of

the Lorentzian. And a3 is the fraction of Lorentzian component present in the P.V., if it

tends to zero then the P.V. converges to a Gaussian, on the other hand, if it tends to one it converges to a Lorentzian. I(ω) = (1−a3)G(ω) +a3L(ω) =a0[(1−a3)exp(−log(2)( ω−a1 a2 )) + a3 1+ (ω−a1 a2 ) 2] (3.3)

The reflections(10¯14)+/−and(0004)+/−are used in Bond’s method. Using eq. 3.3to fit the experimental data, the center values of the RC are obtained, the parameters used for the fit can be seen in fig.3.6. The geometric relations in3.4[16,48] can be used to express

θB as a function of the incidence angles in the measured R.C..

ω−_i = π−2θB+ω_i+ 2θB = ω+_i +π−ω_i− θB = ω_i++π−ω−_i 2 (3.4)

As an example, since the samples studied shouldn’t have a lattice parameter that deviates a lot from GaN (a = 3.160 ˚A and c = 5.125 ˚A [68]). Having said values, it is easier to look for the corresponding peaks in AlGaN. For (0004), using Bragg’s law, 2θB = 73.92

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deg., and so ω−_i = (180−73.92+36.96)deg. = 143.04 deg. Using θB in Bragg’s law and

3.4, the interplanar distance, d, turns into3.5, and the c-lattice is3.6[81]. In3.4 it is also possible to see a geometric representation of the incident and diffracted X-rays. Finding the a-lattice spacing is simply a matter of replacing the already known values in3.7. For the case of a symmetric reflection,(000l), ω_i+ =ω_000l+ and ω−_i = ω_000l− , in the asymmetric

case (h0¯hl), ω_i+=ω+_h0¯hland ω_i−=ω−_h0¯hl. The values obtained for the lattice parameters of

the six samples are represented in table3.1.

d= λ 2 sin(π−|ω + 104−ω−104| 2 ) (3.5) c= λl 2sin(π−|ω + 004−ω − 004| 2 ) (3.6) a= v u u t 4(h2₊_hk₊_k2₎ 3 1 d2 − l 2 c2 (3.7) Sample a ( ˚A) c ( ˚A) 1 3.167 5.156 2 3.167 5.160 3 3.166 5.156 4 3.168 5.156 5 3.167 5.157 6 3.167 5.156

TABLE3.1: Lattice parameters obtained from Bond’s method.

The uncertainties can be calculated using 3.8 and 3.9, where ∆λ = 3.8×10−6 A˚ represents the absolute error in the wavelength.∆ω+₀₀₀₄/−and∆ω+_10¯14/−represent the uncer-tainties in the determination of the center for the rocking curves.

∆c= s (∂c ∂λ∆λ )2_{+ (} ∂c ∂ω₀₀₀₄+ ∆ω + 0004)2+ ( ∂c ∂ω−₀₀₀₄∆ω − 0004)2 (3.8) ∆a= s (∂a ∂λ∆λ) 2_{+ (} ∂a ∂ω_10¯14+ ∆ω + 10¯14)2+ ( ∂a ∂ω−_10¯14∆ω + 10¯14)2+ ( ∂a ∂c∆c) 2 _(3.9)

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+ 2 = 3023 0 = 16.3491 ± 0.0003 . 1 = 0.0482 ± 0.0484 . 2 = 0.7043 ± 0.7039 3 = 0.0961 . (A)(10¯14+) − 2 = 737.94 0 = 113.7051 ± 0.0002 . 1 = 0.0624 ± 0.0003 . 2 = 0.7072 ± 0.0146 3 = 0.1264 . (B)(10¯14−) + 2 = 8253.36 0 = 36.8861 ± 0.0001 . 1 = 0.0382 ± 0.0002 . 2 = 0.8941 ± 0.0151 3 = 0.0772 . (C)(0004+) − 2 1= 143.4871 ± 0.0001 . = 8910.76 0 = 0.0343 ± 0.0001 . 2 = 0.9582 ± 0.0138 3 = 0.0693 . (D)(0004−)

FIGURE3.6: Rocking curves measured for Bond’s method, corresponding to sample 1. The numbers in each figure correspond to the fitting parameters of the Pseudo-Voigt

function. a1is the incident angle, which, depending on the reflection, can be ω+i or ω − i .

For the lattice parameter c,∆c' 10−5_A_˚ _{for all the samples and likewise for a,}_∆a_'

10−5 A. The values obtained for the lattice parameters a and c are satisfactory and in˚ good agreement with similar studies made in [41,76].

3.6 Vegard’s law and determination of composition

In order to determine the composition of the virgin samples Vegard [31,32] and Poisson’s [33] equations can be used. Considering Vegard’s equation valid, the molar fraction of GaN in the Al1−xGaxN ternaries can be determined according to3.10and3.11.

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cAl1−xGaxN =cGaNx+cAlN(1−x) (3.11)

The previous expressions show how the lattice parameters for the ternary compounds can be obtained by linear interpolation of the lattice parameters of the binaries they are made out of. The expressions3.10and3.11assume the thin film of Al1−xGaxN is relaxed,

which is not usually the case. AlGaN thin films grown on top of layers are usually subject to strain [15], the components of parallel and perpendicular deformation, exxand ezzare

given by3.12and3.13, where a0and c0are the relaxed lattice parameters. The values used

for the lattice parameters and stiffness coefficients of binaries AlN and GaN are given in table3.2.

Binary a ( ˚A) c ( ˚A) C13 C33

AlN 3.110 4.9800 99 389 GaN 3.1896 5.1855 103 405 TABLE3.2: Reference values for binaries [34,71–74].

exx= aAl1−xGaxN−a0(x) a0(x)) (3.12) ezz = cAl1−xGaxN−c0(x) c0(x)) (3.13) D= −ezz exx = − 2ν 1−ν = −2 C13 C33 (3.14) According to the biaxial strain model, the components of parallel and perpendicular deformation are connected through the distortion factor3.14, ν is the Poisson coefficient, C13 and C33 are the tensor of strains. The presence of biaxial tension leads to tetragonal

distortion of the unit cell. The composition and strain induce variations in the spacial distribution of the atomic planes, the effect of the lattice parameters can be separated. Using3.12and3.13the following is obtained:

c_Al1₋_{x Gax N}−c0(x) c0 a_Al1₋_{x Gax N}−a0(x) a0 = −C13 C33 (3.15) By substituting a0and c0, the relaxed lattice parameters, given by Vegard’s, for the case of

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cAl_Al1₋_{x Gax N}− [cGaNx+ (1−x)cAlN+

2cGaNx+cAlN(1−x) aGaNx+aAlN(1−x)

C13

GaNx+C13AlN(1−x)

C33_GaNx+C33_AlN(1−x)(−aAl1−xGaxN+aGaNx+ (1−x)aAlN)] =0

(3.16) aAl1−xGaxN and cAl1−xGaxN are the measured lattice parameters in the thin film, C

jN

i3 are the

stiffness coefficients, (i=1,3) of the AlN and GaN binaries, respectively. Equation3.16must be solved numerically, to determine the x it must be guaranteed that only one solution exists between 0 < xGaN < 1. Numeric methods were used to solve the cubic equation

3.16, the solution of interested is summarized in table 3.3 for the six samples of virgin AlGaN. Sample x 1 0.823 2 0.828 3 0.821 4 0.822 5 0.833 6 0.819

TABLE3.3: Values obtained for the composition of the AlGaN samples.

The values in table 3.3 have an average of x = 0.824 and a standard deviation of

σ =0.005. Even though all the samples were cut from the same slice, during the growth

process the compositions might vary depending on the position of the different regions of the sample. The small variations obtained in composition might be attributed to this phenomenon. The uncertainty in the determination of composition could be determined using the maximum and minimum values of the lattice parameters a+∆a and c+∆c, but considering how small the deltas are this was not done.

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3.7 Reciprocal space maps

An alternative way to determine lattice parameters is through RSM, however, this tech-nique is not as accurate as Bond’s method and the results obtained in this section will serve only as a comparison. Bragg’s law in reciprocal space can be written as3.17.

|| ~Q|| = 2π d (3.17) || ~Qx|| = 2π a r 4(a2₊_hk₊_k2₎ 3 (3.18) || ~Qz|| = 2π c l (3.19)

The pair of coordinates (Qx, Qz) that maximize the intensity are related to the lattice

pa-rameters of a unit hexagonal cell by the expressions3.18and3.19. Equations that can be obtained using3.17and3.5, giving two orthogonal components, one that depends on a and another that depends on c. For symmetric reflections like (0002), (0004) and (0006), a loss of intensity can be seen as l increases. Since symmetric reflections depend only on l, this means they are sensitive to the c-lattice parameter, assuming the sample is grown in the [0001] direction, in this section the RSM of (0004) is measured. For asymmetric reflec-tions, sensitive to a and c, the (10¯15) reflection is measured. The 2θ−ωscans that make

the reciprocal space maps can be obtained by grazing incidence,(10¯15+)/(0004+) or by grazing exit (10¯15−)/(0004−), however, all the RMS obtained in this thesis are for the for-mer, so the+index will be omitted. Figure 3.7represents the RSM for the two samples measured. After acquiring the(Qx, Qz)coordinates in the center and using3.18and3.19,

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-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Q

_x

(1/nm)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Q

_x

(1/nm)

47.7

47.9

48.1

48.3

48.5

48.7

48.9

49.1

49.3 Q

z

(1/nm)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 (Q_x, Q_z)=( 0, 48.74) nm-1 _(Q x, Qz)=(0, 48.76) nm-1

(0004)

22.2 22.4 22.6 22.8 23 23.2 23.4

Q

_x

(1/nm)

60.4 60.6 60.8 61.0 61.2 61.4 61.6 61.8

Q

z

(1/nm)

22.3 22.5 22.7 22.9 23.1 23.3 Q_x (1/nm) (Q_x, Q_z)=( 22.89, 60.92) nm-1

Sample 5

(1015)

_

(1015)

_

Sample 5

Sample 1

(Qx, Qz)=( 22.90, 60.83) nm-1

FIGURE 3.7: Reciprocal space maps for two virgin samples in symmetric (0004) and

asymmetric (10¯15) reflections. The coordinates in the lower left region of each graph are for the center (indicated by the cross lines), using this set of coordinates the lattice

parameters are calculated are represented in table3.4. Sample a ( ˚A) c ( ˚A)

1 3.164 5.156 5 3.165 5.154

TABLE3.4: Lattice parameters obtained from RSM.

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vertical width or coherently diffracting domains, vertical strain or compositional fluctu-ations [15]. Horizontal broadening might be connected with small lateral width of co-herently diffracting domains (high dislocation density), or fluctuations in lateral strain or composition [15, 77]. The substrate is not present in the RSM measured because the lattice parameters of sapphire are very different from AlGaN. For (0004) the coordinates in reciprocal space would be(Qx, Qz) = (0, 1.935)nm−1and for(10¯15)they would be

(Qx, Qz) = (1.516, 2.418)nm−1. Since the experimental process to measure RSM is so

time consuming, it would be impossible to have a range so wide as to see both the AlGaN and the substrate. The lattice parameters calculated with the RSM are very similar to the ones obtained using Bond’s method in section3.5. The absolute error for the calculation of c is just∆c=0.003 ˚A and∆c=0.002 ˚A, in sample 1 and 5, respectively. For the lattice parameter a,∆a= 0 ˚A and∆a= 0.003 ˚A. With errors in the order of 10−3 A˚ or 10−13 m, the results obtained in this section are in good agreement with Bond’s method.

3.8 Mosaicity

State of the art epitaxial layers of Group-III nitrides intended for use in electronic and optoelectronic devices exhibit a high density of structural defects [12]. The presence of high strain and high defect densities affect the device performance [12,15], hence the im-portance of characterizing the defects. High dislocation densities, mosaicity, tilted and twisted small crystallites are well known characteristics of the heteroepitaxial layers [78]. The most common type of defect is a dislocation, characterized by a Burgers vector [15]. There are three types of defects: edge, screw and mixed. For an edge dislocation, the Burgers vector is perpendicular to the direction of dislocation. In the screw case the Burgers vector is parallel to the direction of the dislocation. Depending on the type of dislocation, the Burgers vector will take different vector coordinates, for the screw dislo-cation~b =< 0001 >, for the edge~b = 1₃ < 11¯20 >, and lastly, in a mixed dislocation, ~_b₌ 1

3 <11¯23>. Coordinates that are valid for wurtzite structure materials grown along

the c-plane. Each type of dislocation has an associated local lattice distortion, for the screw dislocation the tilt of the crystallites, for the edge dislocation a rotation, while in the mixed case both of them are present. In fig. 3.8the Burgers vectors associated with each distortion can be seen.

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c axis Inclination and rotation αϕ αθ Rotation Inclination

l⃗

Screw dislocation = < 11 3 > b⃗ 1 3 2¯ = < 11 0 > b⃗ 1 3 2¯ =< 0001 > b⃗ Mixed dislocation Edge dislocations

l⃗

FIGURE3.8: Different types of defects encountered in Group-III nitrides.

The fundamental parameters that characterize the crystalline mosaicity of the thin films are the heterogeneous strain (e⊥), the lateral and perpendicular coherence lengths

(L||, L⊥), the inclination of the crystallites (αθ) and the angle of rotation between

crys-tallites (αφ). Several methods are found in the literature to obtain the aforementioned

quantities, in the current section the Williamson-Hall approach will be described [79]. In the following subsections a modified W.H. method [12] is used to analyze the samples, as well as the analytical method [12].

3.8.1 Williamson-Hall approach

The Williamson-hall approach [79] relies on the principle that the approximate formulae for size broadening βL, and strain broadening, βe, vary differently with respect to Bragg

angle, θ.

βL=

Kλ

Lcosθ (3.20)

βe=Cetanθ (3.21)

K depends on the assumptions made by the theory, λ is the wavelength of the incident radiation and L the size. For eq. 3.21 C is a constant that depends on the assumptions concerning the nature of the inhomogeneous strain e. If both contributions in eq.3.20and

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simplification of the W.H. approach is to assume the convolution is either a simple sum or sum of squares. Using the former:

βtotal = βe+βL= Cetanθ+ Kλ

Lcosθ (3.22)

and multiplying the equation above by cosθ we get:

βtotalcosθ=Cesinθ+

Kλ

L (3.23)

By plotting βtotalcosθ versus sinθ it is possible to obtain the strain component from the

slope Ce and the size component from the intercept(Kλ/L).

3.8.2 Graphical method

For the rocking curve measurements, as the broadening in reciprocal space due to tilted crystallites is proportional to the scattering order, and the broadening in reciprocal space due to a small correlation length is independent of the scattering vector [12]. It is pos-sible to do a graphical separation of the two effects, by recording higher order reflec-tions [80]. Assuming a linear supposition of both effects, a separation analogous to the Williamson-Hall plot can be performed, plotting FW HMsinθ_λ against sinθ_λ for each reflec-tion [12]. FWHM represents the full width at half maximum of the measured RC. The function that best describes a RC is the Pseudo-Voigt [12], discussed earlier in section3.5. Doing a linear fit for the W.H. method, the slope gives a direct measure of tilt, the incli-nation of the crystallites (αθ), and the intercept with the y0 axis a measure of the parallel

coherence length, L|| = _2y0.9₀ [12,48]. There is an alternative approach, instead of using the

FWHM, using β_Ω, the integral width of the rocking curves, eq.3.24[81]. In fig.3.9 (A)-(C), it is possible to see the RC for three reflections fitted with a P.V. function. The parameters used in the fit can be seen in each figure. In fig 3.9(D) the W.H. plot using the FWHM and in 3.9(E) using β_Ω, from eq.3.24. For the latter the linear fit is better (higher r2), so β_Ωwill be used for the rest of the current chapter.

β_Ω = [a3π+ (1−a3)

√

πln 2]FW HM

2 (3.24)

In the radial-scan direction (2θ−ωscan), of the reflections (0002), (0004) and (0006), a

small correlation length normal to the substrate surface and a heterogeneous strain along the c-axis leads to a broadening of the Bragg reflections [12]. Separating the two effects,

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this time plottingβΩcos θ_λ as a function ofsinθ_λ , and doing a linear fit, L⊥=0.9/(2y0)and the

slope is 4e⊥. This procedure is usually termed the graphical or Williamson-Hall method,

and the values obtained for the six samples are summarized in table3.5. The errors were calculated with the standard propagation of uncertainties formulas. The values obtained for L||(parallel coherence length) are negative and that doesn’t seem to have any physical

meaning [48]. However, for αθ, represented in the same table, the values obtained are

between (0.149 - 0.229) deg., close to the ones mentioned in [40] for AlGaN with a similar composition. For vertical strain the values obtained are between(3−4×10−4), also re-markably close to [40]. For L⊥the values (107 - 236) nm are around 4 - 5 times small than

the ones found in the literature [40,82].

1 7 . 0 1 7 . 5 1 8 . 0 0 2 0 0 0 0 4 0 0 0 0 6 0 0 0 0 Experimental data Pseudo-Voigt fit C P S ω (deg.) a0=77469.6 CPS a₁=17.5743 ± 0.0001 deg. a₂=0.0361 ± 0.0001 deg. a3=1.000 ± 0.012 FWHM=0.0721 deg. (0002) (A) 0002 3 6 . 5 3 7 . 0 3 7 . 5 0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 (0004) C P S ω (deg.) a₀= 8253.36 CPS a₁= 36.8863 ± 0.0001 deg. a₂= 0.0382 ± 0.0002 deg. a₃= 0.894 ± 0.015 FWHM= 0.0771 deg. (B) 0004 FIGURE3.9: Symmetrical reflections (0002) and (0004) for sample 1.

6 3 . 0 6 3 . 5 6 4 . 0 6 4 . 5 0 7 0 0 1 4 0 0 2 1 0 0 2 8 0 0 (0006) C P S ω (deg.) a₀= 3001 CPS a1= 63.8871 ± 0.0004 deg. a2= 0.6676 ± 0.0005 deg. a3= 0.704 ± 0.031 FWHM= 0.1341 deg. (C) (0006) 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5 0 . 4 0 0 . 4 5 0 . 5 0 0 . 5 5 0 . 6 0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 ( 0 0 0 6 ) ( 0 0 0 4 ) ( 0 0 0 2 ) E x p e r i m e n t a l d a t a L i n e a r f i t F W H M s in θ/ λ s i n θ/λ r2 = 0 . 8 4 3 (D) W.H. plot FIGURE3.9: Reflection (0006) and Williamson-Hall plot for sample 1.

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0 . 1 5 0 0 . 2 2 5 0 . 3 0 0 0 . 3 7 5 0 . 4 5 0 0 . 5 2 5 0 . 6 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 ( 0 0 0 6 ) ( 0 0 0 4 ) r2 = 0 . 8 7 7 E x p e r i m e n t a l d a t a L i n e a r f i t β s in θ/ λ s i n θ/λ ( 0 0 0 2 )

(E) Williamson-Hall plot for sample 1, using β_Ω instead of FWHM, the

linear fit is better than in fig 3.9(D).

Sample L||(nm) Angle (deg.) L⊥(nm) e⊥(×10−4)

1 -104.651±0.385 0.206±0.048 160.714±0.431 3.1±0.1 2 -187.521±0.201 0.149±0.035 173.077±0.241 3.0±0.1 3 -300.312±0.652 0.218±0.025 125.321±0.641 3.2±0.1 4 -109.756±0.221 0.229±0.011 236.333±0.131 4.1±0.1 5 -93.725±0.512 0.206±0.034 107.142±0.241 3.1±0.1 6 -97.826±0.391 0.212±0.022 236.842±0.431 4.0±0.1

TABLE3.5: Values obtained for the graphical Williamson-Hall method.

Considering that the lateral coherence length obtained here gives non-physical results, a new approach will be used in the last section of this chapter. In the following subsection a different method will be used to calculate the same quantities.

3.8.3 Analytical method

The second method that was used to study mosaicity is the analytical, it is based on the assumption that a Gaussian profile in the experimental rocking curve is related with a high inclination of the cristallytes. On the other hand, a Lorentzian profile is related with a low coherence length [12, 48]. By adjusting the experimental data with Pseudo-Voigt curves3.3, the weight of both components is reflected on the variable a3. 3.25gives the

lateral coherence length in this method, θBis the Bragg angle of the symmetrical reflection

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L|| =

0.9λ

β_Ω[0.017475+1.500484a3−0.534156a32]sin(θB(000l))

(3.25)

3.26gives the inclination of the crystallites.

αω =βω[0.184446+0.812692(1−0.998497a3)1/2−0.659603a3+0.445542a23] (3.26)

The same follows for L⊥(3.27) and e⊥(3.28), in this case the measured curves correspond

to the 2θ−ωscans. The results obtained are summarized in table3.6for sample 1 and in

appendixA.1toA.6for all the samples.

L⊥= 0.9λ

β_2θ−θ(0.017475+1.500484a3−0.534156a2₃)cosθ

(3.27)

e⊥=

β2θ−θ[0.184446+0.812692

√

1−0.998497a3−0.659603a3+0.445542a23]

4tanθ (3.28)

For screw dislocation the density N(0001) can be obtained from3.29, where bc is the

Burgers vector of the dislocation,|bc| =0.515 nm. The value used here for the tilt αΩ is

the one obtained using the graphic method of the previous section. As a comparison, the value mentioned in [40] is 5.5×108cm−2for AlGaN with a similar composition.

Nscrew =

α2_Ω

4.35b2_c =9.037×10

8 _cm−2 _(3.29)

The value obtained for Nscrew is of the same order of magnitude of the values found in

[35,40,82].

Reflection L||(nm) α(deg.) L⊥(nm) e⊥(×10−4)

(0002) 236.637±1.438 0.00021±0.00003 430.131±0.149 5.1±0.1 (0004) 125.689±1.161 0.02471±0.00054 193.494±0.859 3.0±0.1 (0006) 61.803±1.769 0.06812±0.00265 136.844±1.059 2.2±0.1

TABLE3.6: Values obtained using the analytical method for sample 1

The values obtained for all the quantities mentioned in this section can be seen repre-sented in figs. 3.10(A)-(D). The errors were calculated with the standard error propagation formulas, and are also represented in the graphs.

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1 0 2 0 3 0 4 0 5 0 6 0 7 0 6 0 9 0 1 2 0 1 5 0 1 8 0 2 1 0 2 4 0 2 7 0 3 0 0 ( 0 0 0 6 ) ( 0 0 0 4 ) S a m p l e 1 S a m p l e 2 S a m p l e 3 S a m p l e 4 S a m p l e 5 S a m p l e 6 P a ra lle l c o h e re n c e l e n g th L || (n m ) A n g l e ( d e g . ) ( 0 0 0 2 )

(A) Coherence length parallel to the sample surface L||, represented for the six samples analyzed.

1 0 2 0 3 0 4 0 5 0 6 0 7 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 ( 0 0 0 6 ) ( 0 0 0 4 ) T ilt α ( d e g .) A n g l e ( d e g . ) ( 0 0 0 2 )