Theoretical study of the stability and the mobility of noble gas atoms in silicon and silicon carbide

TH `ESE

Pour l’obtention du grade de

DOCTEUR DE L’UNIVERSIT ´E DE POITIERS

(Facult´e des Sciences Fondamentales et Appliqu´ees) (Diplˆome National - Arrˆet´e du 7 aoˆut 2006)

´Ecole Doctorale : Sciences et Ing´enierie en Mat´eriaux, M´ecanique, Energ´etique et A´eronautique (SI-MMEA)

Secteur de Recherche : Milieux denses, mat´eriaux et composants Pr´esent´ee par

Azzam CHARAF EDDIN

Theoretical study of the stability and the mobility of noble

gas atoms in silicon and silicon carbide

Directeur de th`ese : Laurent PIZZAGALLI Soutenue le 16 novembre 2011

JURY

C. BECQUART Professeur, UMET, Lille Rapporteur

M-F. BARTHE Directrice de recherche CNRS, CEMHTI, Orl´eans Rapporteur

M. BERTOLUS Ing´enieur chercheur CEA, Cadarache Examinatrice

E. OLIVIERO Ing´enieur chercheur CNRS, CSNSM, Orsay Examinateur

M-F. BEAUFORT Directrice de recherche CNRS, Institut P’, Poitiers Examinatrice

List of Tables v

List of Figures vii

Introduction 1

1 State of the art 5

1.1 Overview . . . 5

1.2 Experimental scenario . . . 7

1.2.1 The implanted state . . . 7

1.2.2 Post-annealing state . . . 8 1.2.2.1 Low fluence . . . 9 1.2.2.2 High fluence . . . 9 1.2.2.3 Medium fluence . . . 10 1.2.3 Other systems . . . 10 1.2.3.1 He:SiC . . . 10 1.2.3.2 Other gas . . . 11

1.2.4 Model for defects formation and evolution . . . 11

1.3 Atomic and nano-scale quantities . . . 12

1.3.1 Intrinsic silicon defects . . . 12

1.3.1.1 Vacancies . . . 13

1.3.1.2 Interstitials . . . 14 i

ii CONTENTS

1.3.2 Intrinsic SiC defects . . . 14

1.3.3 Small NG-related defects . . . 15

1.3.3.1 Interstitials . . . 15

1.3.3.2 Small NG-V complexes . . . 18

1.3.3.3 Large NG-V complexes . . . 19

1.3.4 NG bubbles . . . 20

1.4 Objectives of the PhD . . . 21

2 Methods of calculations and system modeling 23 2.1 Calculations Methods . . . 23

2.1.1 Density Functional Theory . . . 24

2.2 Determination of calculation parameters . . . 27

2.2.1 Plane-wave cutoff energy . . . 27

2.2.2 k points . . . 27

2.3 Search for the atomic configuration . . . 28

2.4 Defects modeling . . . 30

2.4.1 Size of simulation cell . . . 30

2.4.2 Silicon self defects . . . 31

2.4.3 3C-SiC self defects . . . 33

2.4.4 Surface modelling . . . 34

2.5 Noble gas Si and SiC interaction . . . 35

3 Stability of noble gas atoms in Silicon 39 3.1 Noble gas in perfect Si bulk . . . 39

3.1.1 Stability . . . 40

3.1.2 Formation energies . . . 42

3.1.3 Quantification of the lattice deformation . . . 44

3.2 Noble gas monovacancy complex . . . 46

3.3 Noble gas divacancy complex . . . 51

3.4 Discussion . . . 53

4 Mobility of isolated noble gas atoms in Si crystal 57 4.1 Noble gas migration in perfect Si bulk . . . 58

4.2 Migration of a noble gas monovacancy complex . . . 61

4.4 Discussion . . . 65

5 NG desorption from bubbles in silicon 71 5.1 Modeling . . . 72

5.1.1 Silicon surface . . . 73

5.1.1.1 State of the art . . . 73

5.1.1.2 Modeling . . . 74

5.1.2 Noble gas state in bubbles . . . 75

5.1.2.1 Experimental facts . . . 76

5.1.2.2 Theoretical description . . . 77

5.2 Results . . . 78

5.2.1 Exploring the system . . . 78

5.2.2 Getting out of (or in) a bubble . . . 82

5.2.3 Considering temperature and pressure . . . 88

5.3 Discussion . . . 90

6 Stability of noble gas atoms in 3C-SiC 95 6.1 Noble gas in perfect 3C-SiC bulk . . . 95

6.1.1 Stability . . . 96

6.1.2 Formation energies . . . 97

6.1.3 Quantification of the lattice deformation . . . 100

6.2 Noble gas monovacancy complex . . . 101

6.3 Noble gas divacancy complex . . . 105

6.4 Discussion . . . 106

Summary and Outlook 109 A Materials 115 A.1 Silicon . . . 115

A.1.1 Properties . . . 115

A.1.2 Structure . . . 115

A.2 Silicon Carbide-3C . . . 117

A.2.1 Properties . . . 117

A.2.2 Structure . . . 117

iv CONTENTS

C The Nudged Elastic Band method (NEB) 123

D Noble Gas in fluid state 127

D.1 Perfect gas case . . . 127

D.2 High-pressure model . . . 128

D.2.1 Interaction energy . . . 129

D.2.2 Covolume . . . 129

D.2.3 Equation of state and chemical potential . . . 130

Bibliography 131

2.1 Formation energies of different silicon self defects . . . 32

2.2 Formation energies of different silicon self defects in larger cells . . . 32

2.3 Formation energies of different 3C-SiC self defects . . . 34

2.4 Formation energies of different 3C-SiC self defects in larger cells . . . 35

2.5 Equilibrium bond lengths of NG dimers . . . 37

2.6 Binding energies of NG dimers . . . 37

2.7 Equilibrium bond lengths and binding energies of NG-Si dimers . . . 38

3.1 Formation energies of various NG interstitials . . . 42

3.2 Formation energies of different NG monovacancy complexes . . . 46

3.3 Formation energies of different NG divacancy complexes . . . 52

4.1 Migration energies for THT mechanism . . . 60

4.2 Migration energies for exchange mechanism . . . 62

4.3 Migration energies for di-exchange mechanism . . . 64

5.1 Initial energy barriers to get out of (or in) a bubble . . . 83

6.1 Formation energies of various NG interstitials in 3C-SiC bulk . . . 99

6.2 Formation energies of different NG monovacancy complexes in SiC . . . . 102

6.3 Characteristics of different NG divacancy complexes . . . 106

A.1 Nearest neighbors numbers and distances for Si . . . 116 v

vi LIST OF TABLES

1.1 Micrographs (XTEM) of helium implanted silicon . . . 5

1.2 A view of the damage zone and He distribution . . . 8

1.3 Cross-section TEM images of Si after helium implantation . . . 9

1.4 The most common NG interstitials in silicon . . . 15

1.5 The most common NG-V complexes . . . 18

2.1 Total energy for bulk Si as a function of Ecut . . . 27

2.2 Total energy for bulk SiC as a function of Ecut . . . 28

2.3 Total energy for bulk Si as a function of k points . . . 29

2.4 Effect of periodic boundary conditions on defect modelling . . . 30

2.5 The most common silicon self defects . . . 31

2.6 The most common 3C-SiC self defects . . . 33

2.7 Surface model supercell . . . 36

3.1 High symmetry NG configurations in Si . . . 40

3.2 Geometry of the T interstitial after relaxation . . . 41

3.3 Formation energies of various NG interstitials . . . 43

3.4 Variations in distance and angle for all NG atoms in T . . . 44

3.5 Local volume variation as a function of NG formation energies in T . . . . 45

3.6 Formation energies of a He atom in different relaxed configurations . . . . 47

3.7 Formation energies of NG-monovacancy complexes . . . 48

3.8 Available volume variation as a function of formation energies . . . 49 vii

viii LIST OF FIGURES

3.9 A NG atom in V_C2 configuration . . . 51

3.10 Formation energies of NG defects . . . 54

3.11 Formation energies difference of NG defects . . . 55

4.1 NG interstitial migration path . . . 58

4.2 THT paths after structural relaxation . . . 59

4.3 Migration energy as a function of NG core radius . . . 61

4.4 NG monovacancy complex migration path . . . 62

4.5 Illustration of the exchange mechanism . . . 63

4.6 Exchange mechanism migration energy after structural relaxation . . . 64

4.7 NG divacancy complex migration path . . . 65

4.8 Illustration of the di-exchange mechanism . . . 66

4.9 Di-exchange mechanism migration energy after structural relaxation . . . . 67

4.10 A possible NG divacancy complex migration path . . . 68

5.1 Scale separation used for modeling NG desorption from bubbles . . . 72

5.2 Top view of the unreconstructed (ideal) Si(001) surface and four recon-structions . . . 73

5.3 Calculation supercell after structural relaxation of Si(001)-(2 × 1) . . . 75

5.4 He atoms in various T locations in the surface . . . 78

5.5 He and Ne close to the surface . . . 79

5.6 Ne atoms in various T locations . . . 80

5.7 He and Ne formation energies in T configurations . . . 81

5.8 Different possible NEB calculated barriers for He and Ne . . . 82

5.9 Initial saddle configuration for He for B and C paths . . . 83

5.10 Initial saddle configuration for Ne for A path . . . 84

5.11 Initial saddle configuration for Ne for B and C paths . . . 85

5.12 He and Ne migration energies using THT mechanism in the direction ”out” 86 5.13 He and Ne migration energies using THT mechanism in the direction ”in” . 87 5.14 Chemical potential and pressure as a function of He concentration . . . 88

5.15 Energy variation as a function of He position for the C path . . . 91

6.1 High symmetry NG configurations in 3C-SiC . . . 96

6.2 Geometry of the TSi_{interstitial configuration after relaxation . . . 97}

6.4 Variations in distance and angle for all NG atoms in T site . . . 100

6.5 Local volume variation as a function of NG formation energies in TSi . . . 101

6.6 Formation energies of NG monovacancy complexes in SiC . . . 103

6.7 Local volume variation as a function of formation energies in VSi S . . . 104

6.8 A NG atom in V2 Cconfiguration . . . 105

6.9 Formation energies of different NG complexes in silicon carbide . . . 107

A.1 Unit cell of a cubic diamond lattice . . . 116

A.2 Unit cell of a zinc-blende lattice . . . 117

A.3 Crystalline structure of SiC polytypes . . . 118

INTRODUCTION

Noble gas (NG), also sometimes called rare gas or inert gas, are chemical elements (He, Ne, Ar, Kr, Xe, and the radioactive Rn) characterized by several common properties. Maybe the most spectacular one is their very low chemical reactivity. In fact while several com-pounds have been found, in particular for the heaviest species, the completeness of their valence electron shells makes the formation of chemical bonds unlikely. This specific prop-erty explains why they can provide a quasi inert atmosphere, which is used for instance for the growth process of semiconductors in electronics. However, despite this extremely low reactivity, they have been shown to be able to induce significant structural modifications when they are introduced into many solid materials. The results appear to be similar in most of the cases. Depending on the conditions and the amount of introduced NG atoms, these tend to agglomerate, ultimately leading to the formation of extended defects such as platelets (disc shape) or bubbles (spherical shapes). The generally accepted driving force for extended defects formation is the fact that NG atoms are insoluble in most materials, their heat of solution being positive. As a result, the formation of extended defects such as platelets and bubbles, segregating the NG atoms, is favored. Those can then be sources of several mechanisms such as swelling, surface blistering, plane cleavage, which will irre-versibly change the materials properties.

There has been a large amount of studies devoted to the whole process, the earliest ones being more specifically focussed on helium in metals [1,2]. This interest was largely driven by the nuclear materials research. In fact, helium production by neutron capture reactions is an important source of helium in the metal containers or reactor fuels in fission-type

actors. Helium generation in plasma fusion devices was another reason for this research. In all cases, the accumulation of helium in structural materials like steel or bcc metals leads to the formation of He-filled cavities. Other NG species in metals have been investigated more recently [3, 4], resulting in the similar formation of small NG-filled cavities. Next, several scenarios are possible, including the expansion of cavities into bubbles, the desorption of He leaving empty cavities, the coalescence of bubbles or platelets leading to the initia-tion of cracks,...These transformainitia-tions may have a dramatic influence on the mechanical properties of the materials. The whole process itself can be clearly labelled ”multi-scale”, ranging from the atomic scale mechanisms to the possible exfoliation of the materials at the macroscale.

Besides, the metallic character of the matrix materials is not an essential feature, since it has been shown that a similar process could be observed in covalent systems like silicon. Again, most of the studies have been focussed on He [5, 6], although works on Ne [7], Kr [8] and Xe [9] have been performed too. A light NG element like He has been demon-strated to be potentially useful for the gettering process in electronics industry [10]. NG atoms are initially implanted in the material, far from the active layers of the future device. Subsequent treatments lead to the formation of cavities [11], which are then used to trap undesirable impurities, mostly metallic, migrating from active layers. Other materials have been investigated too, such as silicon carbide [12–17], diamond [18], or oxides [19]. The interest for silicon carbide is especially motivated by potential nuclear applications, as a confining material. Also, NG atoms embedded into a silicon carbide matrix are also found in interstellar dusts [20]. The characterization of such systems is important for models at-tempting to describe stars evolution.

As a side effect, the inertness of NG atoms makes their experimental investigation ex-tremely challenging after insertion into a material. Therefore most of the available results concern the modification of the materials properties following the NG insertion. Electronic microscopy is typically used for characterizing already formed extended defects such as platelets and cavities. Additional information can be obtained using techniques such as photo-luminescence, positron beam analysis, X-ray diffraction, and deep level transient spectroscopy. Nevertheless, other techniques such as Rutherford backscattering, nuclear reaction analysis, neutron depth profiling, elastic recoil detection, thermal desorption spec-trometry allow for the determination of several averaged properties such as the amount or

INTRODUCTION 3

location of NG. What is clearly missing is information at the atomic scale, i.e. the in-teraction of few NG atoms with the host lattice, and structural defects such as vacancies and interstitials. Atomistic calculations appear as an interesting alternative approach for tackling this issue, but very few studies have been done. Another unknown concerns the formation and evolution of the extended defects. Several scenarios have been proposed, but their validation also requires inputs at the atomistic level such as stability and mobility of the smallest vacancy-NG complexes.

The objective of this PhD is the acquisition from first-principles calculations of data relative to the stability and mobility of NG atoms in semiconductors like silicon and silicon carbide, and their interactions with vacancies, in order to improve our knowledge on the formation and evolution of extended defects such as platelets and bubbles. The manuscript is organized as follows. The first chapter is a review of the available information regarding NG in silicon, covering extended defects at the mesocale as well as available information at the atomistic scale. In the second chapter, the method and calculation techniques are described. The third and fourth chapters report on the stability and mobility of NG atoms in silicon respectively, with an emphasis on the interaction with vacancies. In the fifth chapter, the mechanism of He escape from an bubble is investigated. A sixth chapter reports on the stability of NG atoms in 3C silicon carbide. Then, our results are discussed in relation with available information in the conclusion. Additional information about technical details and useful data are reported in appendixes.

CHAPTER 1

STATE OF THE ART

1.1 Overview

Figure 1.1 – Micrographs (XTEM) of helium implanted silicon (1×1017 _cm−2_{; 1.55 MeV) after}

annealing at 1073 K for 1000 (a) and 6.1×104_{s (b), respectively (Figure from [5]).}

Due to their very low solubility, NG atoms are typically introduced into materials through highly energetic processes such as ion implantation. With this technique, NG particles with high kinetic energy impinge on a target material. These particles are first slowed down by inelastic collisions with electrons, the so-called electronic stopping. This

first phase is followed by elastic collisions of the energetic particles with the atoms of the material, the nuclear stopping, until the particles come at rest. During this phase, a certain amount of damage is created in the host material since atoms can be displaced from their original lattice sites in cascades. The process of ion implantation then results in a distribu-tion of impinging particles and a damaged zone in the target materials. Their characteristics (extent, distance from the surface, nature of the damage) depend on many parameters such as the energy, the flux and the nature of the particles, as well as the implanted dose, the temperature, and the nature of the implanted material.

The implantation of NG atoms in many materials leads to the formation of extended defects such as bubbles, and in some cases platelets. Those are obtained during the implan-tation phase, or during post-annealing treatments. This is the case for covalent materials like silicon. For instance, the Fig. 1.1 shows a distribution of such defects, formed after He implantation in silicon. These defects then evolve in time and depending on the tem-perature, according to several possible scenarios. Both platelets and bubbles tend to grow, and the former usually transforms into the latter. Eventually, at high temperatures, the NG atoms can leave the defects, thus leaving empty cavities. Finally, due to the presence of these extended defects in the vicinity of the material surface, several mechanisms such as blistering, swelling, or even exfoliation may be observed.

In order to go beyond the phenomenological description, it is necessary to understand each step of the whole process from individual impinging particles interacting with the damage due to implantation, to their aggregation, formation and evolution of large defects. However, this goal is difficult to reach for two reasons. The first one, the most difficult to circumvent, is associated with the possible dynamic aspect of the process. Depending on the temperature, the fluence, and the energy of the impinging particles, some of the steps previously described can readily occur during implantation. For instance, implanted particles can aggregate or even bubbles may be formed without the help of post-annealing treatments. Therefore, it may be not satisfactory to examine each step independently from each other. Such an issue brings up the need for in situ experiments, which remain difficult to perform [21]. Usually, one tries to work within conditions where formation and evolution of the defects can be decorrelated of the implantation step. The second reason is the intrin-sic multiscale aspect of the process. In fact, it starts at the atomic scale with the interaction of NG atoms with the host lattice and defects like vacancies, and ultimately reaches the

1.2. EXPERIMENTAL SCENARIO 7

macroscale, when exfoliation or blistering occurs. Whenever possible, the usual approach consists in decoupling the different scales. This allows for using experimental techniques and theoretical models well suited for each scales.

In the following, we describe the different steps of the formation of the extended de-fects, starting from the initial implanted state to the last stage of the process. We essentially consider silicon in this description, although many of the reported features could be gener-alized to other materials. Finally, the last section is especially dedicated to the state of the art at the atomic scale, the main objective of this PhD work.

1.2 Experimental scenario

Now we describe with more details the process of formation of extended defects such as bubbles or platelets following implantation of NG atoms. Since most of the available data concern the He:Si system, we will use it as a model for the general case, and describe what is known for other gas and SiC in a separate section.

1.2.1 The implanted state

Although ion implantation is a complex, dynamic, out-of equilibrium process, it is possible to measure the depth profiles of the implanted species using techniques such as Electron Recoil Detection (ERD), Secondary Ion Mass Spectroscopy (SIMS), or Neutron Depth Pro-filing (NDP). The displacement field in the sample can also be determined using Rutherford Backscattering Spectroscopy (RBS) and X-Ray Diffraction (XRD). Other techniques such as Positron Annihilation Spectroscopy (PAS) probe vacancy-like defects left by the implan-tation process. Besides, dedicated computer codes such as SRIM [22], which are based on a Monte-Carlo approach and the binary collision approximation, allows for the calculation of approximate depth profiles of both the implanted species and the created damages. As an example, the Fig. 1.2 shows the distribution of He in Si.

In the case of the He:Si system, it has been shown that the He depth profiles computed by SRIM are not in good agreement with ERD measurements, a possible explanation being the condensation of He atoms in the implanted state [23]. The displacement field measured by XRD and RBS is usually centered on the depth of maximum He concentration. The

Figure 1.2 –A view of the damage zone and He distribution from the surface up to 4 μm below the surface.

amount and depth of the vacancy-type defects distribution depend on the fluence and the implantation energy, but the depth is typically closer to the surface than the maximum He concentration.

1.2.2 Post-annealing state

Common experimental techniques used for examining the evolution of the implanted state include Transmission Electron Microscopy (TEM) and Thermal Desorption Spectroscopy (TDS). It has been reported that the system remains in the as-implanted state without no-table modifications as long as the annealing temperature remains below 200◦C [23]. How-ever, when the temperature is greater than about 250◦C, the He distribution drifts towards the vacancy defects maximum [24]. Thereafter, the following steps depend on the fluence used during implantation.

1.2. EXPERIMENTAL SCENARIO 9

1.2.2.1 Low fluence

For fluences typically below 1× 1015 _cm−2_{, annealing first leads to the disapperance of}

signals associated with vacancy-type defects in PAS experiments, indicating a strong in-terplay between He and vacancies. However, effusion of He continues and increasing the temperature allows for a progressive recovering of the interaction with positrons. For the highest temperatures, above 500◦C, the vacancy-type defects are annealed until full disap-pearance at 900◦C [23]. The temperatures required for a complete annealing depends on the initial concentration of defects (and therefore on the fluence).

At very low fluence (8× 1011 _cm−2_{), the vacancy-type defects have been identified to}

be essentially divacancies or complexes including a monovacancy and one impurity [25]. In all cases, there is no formation of extended defects such as bubbles, due to the limited supply of He into the materials.

1.2.2.2 High fluence

Figure 1.3 – Cross-section TEM images of Si after helium implantation with 1×1017 _cm−2 _at

1.6 MeV and heat treatment for 1000 s at 800◦C. (a) Damage visualization: F (Frank loop), I ({113} defects), B (small-bubbles). (b) Observation of the faulted loops by tilting the sample towards the(312) plane. As can be seen, all the Frank loops are associated with bubbles, even in front of the bubble band (Figure from [26]).

evolu-tion. For temperatures in the range 400◦C - 500◦C, post-mortem microscopy experiments clearly show the presence of bubbles [23, 25]. These bubbles are contained in a strip typ-ically centered on the vacancy-type defects distribution maximum. The shape, size, and density of the bubbles strongly depend on the experimental conditions. In particular, the sizes of the bubbles tend to increase with the temperature. The internal surfaces of large bubbles can be well defined, leading to facetted shapes. Finally, at very high temperature, He effusion is complete and empty cavities are left in the system.

As shown in Fig. 1.3, when implantations are done at high energy, in the MeV range, it is found that He effusion is more difficult to perform, since the formed bubbles are located deeper below the surface. Also, extended defects such as {113} interstitials clusters and Frank loop dislocations have been observed in the vicinity of the bubbles [26].

1.2.2.3 Medium fluence

More complicated structures have been discovered for fluences ranging from about 1× 1015 _cm−2 _{to 5× 10}16 _cm−2 _{[21, 25, 27, 28]. The limited amount of available He leads}

to the formation of a set of diluted defects. For low temperature annealing, 300◦C, there is formation of both isolated bubbles and platelets (aggregates of He in {100} or {110} planes). When the temperature reaches 400 ◦C, large platelets with a lenticular shape are observed. Finally, at 600 ◦C, a complex ”planetary-like” structure is obtained, including large cavities surrounded by smaller spherical bubbles and interstitial dislocation loops.

1.2.3 Other systems

1.2.3.1 He:SiC

In SiC, as for silicon, the effect of He was much more studied than other NG species in implantation studies. This can be explained by the large amount of data available for He in other materials, as well as by the potential use of silicon carbide for nuclear applications. What is known for the He:SiC system is not so different from He:Si. In fact, experimental works showed that depending on the dose or the annealing treatment, several kind of defects can be observed: platelets, small bubbles, disk of bubbles, dislocation loops [12,14,29,30]. The interaction of the as-created defects with the microstructure of the material has also been investigated [16, 30]. One of main difference with silicon is the highest amount of

1.2. EXPERIMENTAL SCENARIO 11

damage created in SiC, which has triggered previous studies of the amorphization and recrystallization of the material [25].

1.2.3.2 Other gas

There have been much less investigations of the behavior of other NG species than He in silicon. However, it has been reported that implantation of Ne, Ar, Kr, and Xe leads to the formation of precipitates in the silicon matrix [7–9, 31, 32]. One of the main differences with He is that higher temperatures for implantation or post-annealing are required in order to prevent the amorphization of the sample. When the amorphization can not be avoided, the matrix can be recristallized during the annealing stage. For the largest NG elements like Ar, Kr and Xe, it is found that the formed precipitates are typically much larger than in the He case. It allowed for determining the state, liquid or solid, of the NG bulk into the cavities, depending on the conditions [8, 9].

Very few works are available regarding the formation of extended defects after the implantation in silicon carbide of NG atoms other than He. Nevertheless, it is known that post-annealing treatements of Ne- and Xe-implanted SiC lead to a large concentration of small bubbles in the highly damaged region [15].

1.2.4 Model for defects formation and evolution

It has already been mentioned that the whole process described above is inherently mul-tiscale. In fact, in the as-implanted state, the smallest active component is the NG atom, which interacts with the host lattice, with intrinsic atomic-scale defects such as vacancies or interstitials, with clusters of intrinsic defects, or even with other NG atoms. These inter-actions have been shown to be a prerequisite for the formation of bubbles in silicon [33]. A larger scale is used for describing the formation and evolution of extended defects such as bubbles and platelets, typically in the sub-micron range. Finally, these defects can further grow, accompanied by many possible effects such as swelling or blistering at the surface, occuring at an even larger scale. A full description of the process then requires a model encompassing many scales.

The complexity of the phenomenon lies in the competition of many mechanisms, which can be occuring simultaneously. For instance, in the damaged zone, the evolution of

bub-bles depends not only on the flux of NG atoms coming from the implanted region, but also on the flux of NG atoms leaving the zone to reach the surface, or on the interactions between bubbles, with mechanisms such as Ostwald-Ripening or migration-coalescence [34]. Other mechanisms such as spontaneous emission of interstitials from the bubbles by the so-called dislocation loop punching are also possible [1]. The difficulty of separating the different possible mechanisms explains why the understanding of the process is still fragmented.

In order to build a theoretical description of the process, the model has to take into account several orders of magnitude in the length scale, and a huge gap in timescale, from atomic scale events (occuring roughly at the Debye frequency 10−12− 10−14 _{s) to usual}

annealing times in thousands of seconds. These requirements exclude atom-based descrip-tions such as molecular dynamics. The more suited theoretical framework is probably kinetic theory, where all competing mechanisms are described by a set of differential equa-tions [23]. Although such a description remains rather simplistic in regards of the complex-ity of the involved mechanisms, it allows one to provide a full description of the process.

Nevertheless, it requires that all occuring mechanisms have been identified, and that the associated parameters such as the activation energies are known. This is a crucial issue since the validity of the final results can be critically dependent on these parameters. In the following sections, we summarize the available knowledge on atomic- and nano-scale mechanisms.

1.3 Atomic and nano-scale quantities

1.3.1 Intrinsic silicon defects

There have been a large number of works dedicated to the determination of the properties of intrinsic defects like the vacancy and the interstitial in silicon. This is essentially due to the role of this material in electronics industry. Another reason is the use of the single vacancy as a model case for evaluating new computational methods. An exhaustive bibliography of the topic is out of the scope of the present PhD work, and only recent works and well established data will be reported.

1.3. ATOMIC AND NANO-SCALE QUANTITIES 13

1.3.1.1 Vacancies

The single vacancy, or monovacancy, can have different charge states. Although this aspect is important for doped silicon, we only considered here the most studied neutral state. In that case, it is known that the symmetry of the relaxed vacancy is D2d, due to a Jahn-Teller

distortion. The formation enthalpy has been measured via positron lifetime measurements to be 3.6 ± 0.2 eV [35]. From the theory side, many values have been proposed in the past [36–41]. The general consensus is that the formation energy is ranging between 3.3 eV and 3.6 eV [42]. The migration energy of the neutral monovacancy has been determined to be 0.40 eV, in very good agreement with experiments [23]. The migration mechanism is simple, and involved a neighbor Si atom, which moves to the vacancy, leaving a new vacancy behind [43]. As a consequence of the small barrier, the monovacancy is a fast diffuser even at low temperatures.

It has been known for a long time that a substantial energy gain is obtained by aggre-gating vacancies in silicon. For instance, the hexavacancy V6has been early identified as a very stable complex [37]. This result has opened the way to numerous investigations trying to determine the most stable configurations for a complex Vnof n vacancies [41, 44]. Ex-perimentally, it can be difficult to detect such large vacancies aggregates since they appear to be electrically and optically inactive. It is therefore not clear whether they are present in substantial amount in the damaged zone after irradiation.

The situation is different for the smallest aggregate, the divacancy V2. Its calculated formation energy is 5.9 eV [45], i.e. there is an energy gain of about 0.7-1.3 eV compared to two isolated vacancies. Divacancy centers can be detected and they have been shown to be present in irradiated sample [46]. Theoretical and experimental investigations of migration are in agreement, with a migration energy of 1.2-1.3 eV [46–48]. Divacancies are expected to play a role in the formation of extended defects. Nevertheless, even if it is energetically favorable to form divacancies from monovacancies, the relative amounts of both species have been shown to be dependent on temperature and the initial vacancy concentration [48].

1.3.1.2 Interstitials

All investigations revealed that in its most stable configuration, a single interstitial forms a dumbbell along the 110 with a lattice atom, a structure commonly named 110-split interstitial [36, 49]. As in the case of the monovacancy, there is a large dispersion in the calculations of formation energies [36, 49]. The most recent investigations are expected to be the most accurate, yielding a value of 3.4 eV [40]. Interstitials are highly mobile, with a migration energy of 0.45 eV [50].

Like vacancies, interstitials have been shown to be prone to aggregation, with a sub-stantial energy gain for di-interstitials and larger clusters [51–53]. Ultimately, interstitials can arrange themselves to form ordered structures such as the rodlike{113} defects [52]. Those defects can be observed in irradiated samples depending on the conditions. How-ever, their possible role in the formation of NG-related extended defects such as bubbles or platelets remains unclear.

1.3.2 Intrinsic SiC defects

There have been many investigations focussed on intrinsic defects in silicon carbide. Com-pared to silicon, the situation is somewhat more complicated. In fact, in addition to charge effects on the properties of defects, there are more potential defects due to the binary nature of SiC. For instance, there are two kinds of vacancies, and a rich variety of possible inter-stitials. Moreover, the stability and mobility of defects can depend on the stoichiometry of the material. In the following, we just recall the main well-established results. Note that we discuss solely defects in the cubic polytype of SiC, but that there is a correspondence with defects in the hexagonal ones [54].

The silicon vacancy is a high energy defect, which can be expected to appear only dur-ing irradiation. Several charge states are possible. The formation energy for a neutral defect in a stoichiometric material has been computed to range from 6.5 to 8.5 eV [55–58]. It has been shown that due to this high formation energy, the silicon vacancy was a metastable defect, which could evolve to a more stable carbon vacancy - antisite complex [59]. Note, however that the associated activation energy is not negligible, being larger than 2 eV [60]. Regarding the carbon vacancy, the formation energy is lower, with calculated values rang-ing from 3.4 to 5.9 eV [55–58].

1.3. ATOMIC AND NANO-SCALE QUANTITIES 15

The situation is more complicated for interstitials, since several possible configurations have been identified. It is now well established that the more stable geometries for a silicon interstitial are the tetrahedrally carbon-coordinated configuration and the 110-oriented split interstitial [56, 58, 61]. A large range of formation energies has been published for these defects [55–57, 62], because this quantity appears to be very dependent on the size of the computational cell, as well as the convergence of the electronic structure [63]. In the current state of knowledge, it seems that both configurations can coexist. For the car-bon interstitial, the most stable configurations are obtained for split-interstitial geometries. However, there are several possible types, with competing energies [56, 58, 62]. Again it seems that different defect structures could be formed in irradiated SiC.

1.3.3 Small NG-related defects

1.3.3.1 Interstitials

Figure 1.4 –The most common NG interstitials in silicon shown along two different directions. Si atoms are shown as yellow balls, where other colored balls represent interstitial NG atoms.

Although it is possible to measure the profiles of implanted NG, using techniques such as ERD for instance, it is more difficult to determine experimentally the exact location of

one individual NG atom relatively to the host lattice. The main reason is the very low re-activity of NG species, which make them non sensitive to many probing techniques. While information can be gained with RBS, it is not always possible to determine the NG position with accuracy [64].

Possible sites for a NG interstitial in a cubic diamond lattice are shown in the Fig. 1.4. When a NG atom is located in the tetrahedral T site, it has 4 Si neighbors, at a distance of √

3a0/4, i.e. the Si–Si first-neighbors distance. Instead, the hexagonal H site is

character-ized by 6 Si neighbors, at a distance of√11a0/8. Another high-symmetry position is the

Bond-Center (BC) site, where the NG atom is located between two first-neighbors Si atoms. This configuration is the most stable for atomic hydrogen in silicon for instance [65]. It is also possible to imagine a low symmetry configuration with the impurity forming a dimer with a lattice atom. The zinc-blende lattice of 3C-SiC is very close to the cubic diamond lattice, and similar locations for interstitials are expected. The only differences are that two possible sites are possible for the tetrahedral T site, depending whether the NG interstitial is surrounded by 4 Si or C atoms.

Nevertheless, it has been early postulated that since NG atoms are characterized by fully occupied electronic shells, the most favorable locations in the host lattice should be empty spaces where the electronic density is as low as possible. This would exclude the BC and the dimer configurations, thus leaving the T and H sites as candidates. These con-figurations are calculated by numerical simulations using desity functional theory almost twenty years ago by Alatalo et al for the He:Si system [66]. They found that the T site was the most stable location, with a formation energy of 0.77 eV (in that case the formation en-ergy is a solubility enen-ergy). The H site was found to be stable too, with a larger formation energy of 1.59 eV. The authors proposed that the He migration would follow a T-H-T path, with a barrier approximately given by the formation energy differences between T and H, i.e. 0.82 eV. The issue has been revisited few years later by Estreicher and co-workers with Hartree-Fock calculations. They confirmed that a single NG atom, He or the others, would be located at T site, whereas the H site exhibits larger formation energies. However, they computed larger formation energies overall, with a steep increase from He (1.3 eV) to Xe (25 eV). Finally, a recent study also indicated T as the most favorable site for He in Si, with a formation energy of 1 eV [67].

1.3. ATOMIC AND NANO-SCALE QUANTITIES 17

In the case of silicon carbide, the few available works all focussed on the He:SiC sys-tem. Experimentally, it has not been possible to determine whether He was located in a vacancy or in an interstitial site [64]. The relative stability of C and Si-tetrahedral intersti-tial configurations has been computed with first principles calculations [68, 69]. It is found that the tetrahedrally Si-coordinated configuration has the lowest formation energies, al-though there is a disagreement regarding the magnitude of these energies. Other possible geometries, such as the H and BC sites, have not been tested yet.

Experimental measurements of the diffusivity have been made, but only for He as far as we know. In silicon, A. Van Wieringen and N. Warmoltz proposed a migration energy of 1.35 eV [70], whereas more recents TDS data yield a value of 0.8 eV instead [71]. Theoret-ically, the few available works used the assumption from Alatalo et al that the He migration energy is simply the difference between T and H formation energies (as already mentioned above, they found 0.82 eV). Estreicher et al applied this idea to obtain migration energies for He (2.1 eV), Ne (3.0 eV), Ar (6.3 eV), and Kr (8.3 eV) [72]. Combining all available data for He, the migration energy ranges from 0.8 to 2.1 eV, giving a rather large uncer-tainty. Nevertheless, one can safely considers that He diffusion in silicon is associated with a much larger barrier than in typical bcc metals like Fe [73, 74] and W [75], but is easier than in materials like diamond [18], and uranium dioxide [76]. The diffusivity of He in hexagonal SiC has been investigated by measuring desorption, yielding contrasting results. On the one hand, Jung determined an activation energy for He diffusion of 1.14 eV [77]. On the other hand, Oliviero et al measured an activation energy of 1.5 eV which they associate with single or clustered interstitial He, and a second data of 3.2 eV which could be related to detrapping of He from He-vacancy clusters, i.e. bubbles precursors [13]. Theoretically, the only work leads to an activation energy for diffusion of 2.5 eV [68].

Regarding the formation of bubbles, it has been early questioned whether vacancies were an absolute necessity. Hence, Wilson an co-workers suggested thirty years ago from empirical potential calculations that He clustering in metals would be a driving force strong enough for forming bubbles without initial vacancies [78]. This has triggered several inves-tigations of the He clustering effect. In particular, it was found that there is a large energy gain by aggregating He in bcc metals [73, 75]. However, it seems that in cubic silicon car-bide there is no significant bonding between He interstitials in neighboring locations [69]. For silicon, the only data available comes from the calculations of Alatalo et al who

pro-posed a binding energy of 0.08 eV between two He atoms [66]. To our knowledge, no information exists for other NG species.

1.3.3.2 Small NG-V complexes

Figure 1.5 –The most common NG-V complexes in cubic diamond lattice. Si atoms are shown as yellow balls, V as black balls, whereas other colored balls present NG atoms.

The presence of NG atoms in materials essentially results from implantation / irradia-tion processes, which are usually accompanied with the creairradia-tion of damage in the lattice. Obviously, vacancy-type defects are expected to play an important role in the formation and evolution of defects like bubbles. As a first step, one can consider the interaction of NG atoms with mono (V) and divacancies (V2) in silicon. The most simple and trivial configuration for a NG-V complex is obtained when the NG atom is located in the center of the monovacancy, i.e. in substitution. This configuration is called VSin the following.

Other configurations which have been studied include a NG atom in interstitial T site in the vicinity of the monovacancy. VT1 corresponds to the NG atom initially located in the T

site first-neighbor of the vacancy, VT2to the NG atom in the T site second-neighbor of the

vacancy, etc... In the case of the divacancy, important configurations are V_C2 where the NG atom is located in the center of the divacancy, and V2_Swhere the NG atom occupies one of the two monovacancy centers. All these configurations are shown in the Fig. 1.5.

1.3. ATOMIC AND NANO-SCALE QUANTITIES 19

Existing information on NG-V complexes in silicon is rather scarce, and is essentially concerning He. Earlier investigations by Alatalo et al suggested that the VS position was

unstable, the He atom relaxing to a VT1configuration [66]. This new structure was

associ-ated with a formation energy of 1.03 eV, i.e. 0.15 eV higher than an interstitial configura-tion, thus suggesting that a monovacancy did not act as a trapping center for He atoms. The same authors also reported the formation energy when a second He atom was inserted into the monovacancy. Estreicher and co-workers also found that the VS position was unstable

for He, the latter relaxing toward the VT1site. However, they obtained a very different

for-mation energy of about 3.5 eV [72]. More recently, Zavodinsky et al [67] reported instead a formation energy of 1.4 eV for He in VS, thus inferring that this configuration is stable.

Regarding He in a silicon divacancy, only one investigation has been performed, which indicates that the formation energy would be about 1.4 eV [72]. However, the authors did not describe the exact location of He in the divacancy.

Therefore, from the available data, there is a large discrepancy on the formation ener-gies. Nevertheless, it seems that a He atom can bind to a divacancy, but not to a mono-vacancy. This is in agreement with experimental findings [79]. This situation is clearly contrasting with other helium-materials systems. In fact, He has been shown to be more stable in a monovacancy than in interstitial position for many materials, diamond [18], Fe [73], and W [80]. Also, in silicon carbide, first-principles calculations suggest that He could bind to monovacancies [68]. Nevertheless, it is not clear whether the NG impurity would be located in the vacancy, or in close vicinity.

For other NG atoms in silicon, only the study by Estreicher et al provides some in-sights [72]. They found that in silicon Ne, Ar and Kr in the vicinity of a monovacancy relaxed to a VT1position like He, with increasing and large formation energies. Besides,

according to their results, it is easier to insert NG atoms into a divacancy, with a formation energy of about 3 eV for Kr for instance.

1.3.3.3 Large NG-V complexes

When the numbers of both NG atoms and vacancies increase, it becomes impossible to determine the most stable configuration for each case, because the amount of possibilities increases very quickly. Besides, such a determination is not necessarily relevant since the

NG atoms could have a high mobility in vacancy clusters at finite temperature. Therefore, previous investigations focussed mostly on trends, and tried to understand the behavior of the large NG-vacancy complexes as a function of the ratio between the number of NG atoms and the number of vacancies.

The available data essentially concerned He. In the case of the He:Fe system, it has been shown that inserting He atoms into a vacancy cluster would reduce the emission of vacancies, thus stabilizing the cluster [81]. For a high Fe:V ratio, there is a competition between the emission of He from the clusters, and the emission of interstitials (hence in-creasing the number of vacancies and enlarging the cluster) [82]. An optimal ratio of 1.3 was determined for the same system [73], although it has been later shown that this ratio would depend on the size of the considered cluster [83]. Seletskaia and co-workers pro-posed that a ratio of 6 is the upper threshold above which self-interstitials are expelled from the NG-vacancy clusters [84]. Similar trends have been obtained for He in Ga-Pu alloys, with an optimal ratio of 2 He for one vacancy in nanometer-sized clusters [85]. In silicon carbide, semi-empirical free energy calculations showed that 3-4 He atoms could be con-tained in a single vacancy [17]. For larger clusters, the same work shows that the He:V ratio tends to a value slightly above 1. However, the emission of self-interstitials does not occur during simulations. Unfortunately, to our knowledge, there have been no investigations of large NG-vacancy complexes in silicon.

1.3.4 NG bubbles

After their formation, the NG-V complexes will evolve depending on conditions, eventu-ally leading to large extended defects such as platelets or bubbles as described in a previ-ous section. These defects can be quite large, preventing atomic-scale calculations such as molecular dynamics. It is therefore more appropriate to use models based on a continuum description of matter. The foundations for such a description have been summarized by Trinkaus [86]. The system energy is divided into three parts: a NG free energy inside the bubble, a bubble-matrix free energy interface, and an elastic energy in the matrix. There have been several extensions or modifications of this model, either to take into account the specific shape of the defects [87], or growth mechanisms. Several possible mechanisms for growth have been identified, such as Ostwald-ripening, trap mutation, dislocation loop punching, migration coalescence [1, 88].

1.4. OBJECTIVES OF THE PHD 21

Nevertheless, although many scenarios are possible, a key quantity in all models is the pressure into the defect, which needs to be determined. There have been many attempts to measure it, using several experimental techniques [8, 9, 31, 89–93], or atomistic calcu-lations [94]. Depending on the local pressure and the temperature, the bubble interior can be solid, liquid, or in a gas state. The solid state is more easily obtained for the heaviest NG species, or in small bubbles where the pressure can be quite large. For light NG atoms, medium-sized bubbles, or high temperatures, it is likely to observe a confined fluid. In that case, appropriate equation of states, or models [95], can be used to determine the free energy as a function of temperature and pressure.

In this PhD work, we mainly focus on elementary mechanisms and their associated activation energies. In the case of large defects, an important elementary mechanism to describe the evolution of bubbles is the effusion process, when a NG atom escapes from the defect and leaves the system by migrating towards the surface. The associated activation energy depends on the free energy of NG atoms in bubbles, and therefore it obviously depends on the pressure and the temperature. TDS measurements indicated that the He effusion energy was about 1.8 eV in Si [5, 6] and 3.2 eV in SiC [13]. An approximate value for this energy is usually obtained as the sum of formation and migration energies of the NG atom in the host lattice. However, the escape mechanism as well as the associated activation energies have not been studied at the atomistic level.

1.4 Objectives of the PhD

The analysis of the available information concerning the behavior of NG atoms in silicon and SiC at the atomic scale revealed an important lack of data. The objective of this PhD work is to fill the gap as much as possible. In particular, the following issues can be highlighted:

• What are the formation energies for a NG interstitial in silicon and SiC? For He, the range of proposed values is much too wide. For the other species, few data are available.

• Is the T-H-T migration path the correct one for a NG interstitial? This has never been checked to our knowledge.

• What are the stable configurations for a NG atom in interaction with vacancies? Again, the available information is scarce.

• How do these possible complexes migrate in silicon and SiC (if they can)? Nothing is known on this topic.

• Is the aggregation of interstitial NG atoms an important mechanism?

• What are the mechanism and the associated activation energy for detrapping a NG atom from a formed bubble? In particular, is it correct to consider that this energy is simply the sum of formation and migration energies for an interstitial.

• Is it possible to determine the pressure from atomic-scale calculations?

In the following chapters, we bring answers to some of these questions, with a particular focus on helium.

CHAPTER 2

METHODS OF CALCULATIONS AND

SYSTEM MODELING

We will show how answers to the various questions described in the previous chapter can be obtained by atomistic simulations. We have performed ab initio calculations in the framework of the density functional theory. All calculation parameters have been optimized to obtain a good accuracy. Next, we explain how we have modeled our system using periodic boundary conditions. Finally, our choice of method and associated parameters are validated by comparing our calculations for Si and 3C-SiC self defects with existing literature data.

2.1 Calculations Methods

Many types of atomistic modelling methods are used in physics and materials sciences. These methods are characterized by the way they describe atomic and electronic interac-tions, the number of atoms which can be treated, expected accuracy, computational effort, etc... The choice of the calculation method depends on its ability to give an accurate de-scription of the problem investigated.

Three of the most famous methods in the field of atomistic simulation of matter are: empirical potential approach, Tight-binding (TB) approximation, and ab initio methods.

In this study, as it is explained in the previous chapter, we are interested by the NG im-purities in Si and 3C-SiC crystals. For the first approach, empirical potentials describing the interactions between NG and Si (C) atoms are needed. Unfortunately such potentials are not available. Other choices were then electronic structure calculations, either TB or

ab initio. The TB method works by replacing the many-body Hamilton operator with a

parameterized Hamiltonian matrix. However, such a parameterization is not available for NG-Si(C) system to our knowledge, therefore we decided to use ab initio methods.

2.1.1 Density Functional Theory

Ab initio methods allow one, in theory, to determine the exact electronic structure of sys-tems as the solution of the Schr¨odinger equation. The term ab initio indicates that the calcu-lation is based on first principles and that no empirical data are used. Since the Schr¨odinger equation is solvable only for simple atoms, like hydrogen, approximations are needed for more complex systems. One of the most successful approximate approaches is the Density Functional Theory (DFT), which results from the works of Hohenberg, Kohn ( Nobel Prize in Chemistry 1998 for his development of the DFT) and Sham [96, 97]. Most electronic structure calculations for solids are based on DFT, and it has become the standard technique for the calculation of matter at the atomistic level with a first-principles accuracy. In DFT the total energy of a system is given as a functional E[ρ] of the electronic density ρ, and is minimized to find the system ground state [98].

According to the equations of Kohn and Sham [97], the many body complexity of the electronic structure is hidden into one term of the energy functional E[ρ], which is not known and called exchange-correlation energy EXC[ρ]. It is defined to include all the

quantum mechanical effects such as exchange and correlation that can not be easily com-puted. Many approximations of this term are available, the most used being Local-Density Approximation (LDA) [97] or the different formulations based on Generalized Gradient Approximation (GGA) [99].

LDA uses only the local density to define the exchange-correlation functional. It con-siders that the exchange-correlation energyεXC(ρ) of an elementary volume at space point

r, at given density ρ(r), is equal to the exchange and correlation energy of an interacting but homogeneous electron gas of constant densityρ0= ρ(r).

2.1. CALCULATIONS METHODS 25

The other commonly used approximation is the GGA, where the local density as well as its gradient are used in order to describe the variation of the electronic density (the electronic density in most systems being not homogeneous), i.e. εXC(ρ) is replaced by

εXC(ρ,∇ρ). Various GGA functionals are possible, depending on the choice of εXC(ρ,∇ρ)

function. Two of the most widely used functionals in calculations involving solids are the Perdew-Wang functional (PW91) [100] and the Perdew-Burke-Ernzerhof functional (PBE) [99].

In this work, we have used PBE GGA. There are several reasons for which this ap-proximation was chosen. First, it is widely used in DFT calculations, and is considered a standard functional. Today, more accurate forms have been developed, like hybrid func-tionals, but they are difficult to use for the systems considered here as their computational cost is very large. The PBE functional is also easy to use, since it is available in many computational codes. Finally, PBE GGA has been successfully applied to NG dimers, as it will be shown in the next section.

The numerical calculations of electronic structure require the description of the wave functions, which are usually represented as a combination of simple functions. Several kind of basis functions are available, and more or less suited depending on the materi-als investigated. Plane waves enjoy great popularity in solid state physics for which they are particularly adapted, because they implicitly involve the concept of periodic boundary conditions. They are not centered at the nuclei but extend throughout the complete space. Every wave function is written as an infinite sum of plane waves. To make the calculations feasible, it is necessary to limit the sum to a finite number of plane waves by including only those corresponding to kinetic energies lower than some value Ecut, usually called the

cutoff energy. As Ecut gets larger, the number of plane waves used increases and the

cal-culation becomes more accurate, but takes more time. This is why a compromise must be found between the size of the basis set and the aimed accuracy of the results [101].

From a physical point of view, core electrons, for many elements, are not especially im-portant in defining chemical bonding and other physical characteristics of materials. From the earliest developments of plane wave methods, it was clear that there could have great advantages in calculations that approximate the properties of core electrons in a way that

could reduce the number of plane waves necessary in a calculation. Pseudopotentials (PP) are used to exclude the inactive atomic core electrons from an explicit treatment in quan-tum chemical calculations [102]. Historically, many pseudopotential approachs have been proposed, like BHS (Bachelet, Hamann, Schl¨uter, 1979-1982), KB (Kleinman, Bylander, 1982), MT (Martin, Troullier, 1990), ultrasoft (Vanderbilt, 1991), or the PAW Method (Bl¨ochl, 1994).

A pseudopotential must be soft enough to be usable with a limited number of plane waves, and transferable, i.e. able to describe an atom whatever its chemical environ-ment. In this study we used ultrasoft pseudopotentials as proposed by Vanderbilt and co-workers [103–105]. Unlike usual pseudopotential approachs, ultrasoft pseudopotentials do not require the charge conservation condition, resulting in a more complicated formulation but allowing accurate calculation with fewer plane waves.

The plane wave code PWSCF in the Quantum Espresso package [106] was employed to carry out our DFT calculations. First of all, it is an efficient and well tested open source (thus freely available) code. Secondly, this code permits one to determine the transition states between two stable positions and the minimum energy path using the Nudged Elas-tic Band technique (NEB) [107], explained in detail in appendix C. Quantum Espresso is an integrated suite of computer codes for electronic structure calculations and materials modeling at the nanoscale. It is based on plane waves, and includes several kind of pseu-dopotentials, among which the ultrasoft ones.

Finally, it is of interest to note that the downside of DFT calculations is their com-putational cost. They often take enormous amounts of computer time, memory, and disk space, since a DFT calculation scales nominally as N3_{(N being the number of atoms), i.e.}

a calculation twice as big takes 8 times as long to complete. With common computational tools, only systems of few hundreds of atoms can be modeled and calculated. Neverthe-less, sometimes it is better and adequate for some properties to have accurate results from a small system, than inaccurate from a larger one [108].

2.2. DETERMINATION OF CALCULATION PARAMETERS 27

2.2 Determination of calculation parameters

Before doing calculation on the system of interest, it is mandatory to determine two key parameters in the DFT calculations, the plane wave cutoff and the k-point sampling. Ap-propriate values for both parameters are obtained by investigating model simple systems such as bulk Si and SiC.

2.2.1 Plane-wave cutoff energy

Figure 2.1 –Total energy (per atom) for bulk Si calculated as a function of cutoff energy.

Plane-wave cutoff energies Ecut for Si and SiC were determined using a system of 64

atoms and several Ecutvalues. The system total energy variation has been plotted in Fig. 2.1

and Fig. 2.2. We found that the system total energy becomes constant when Ecut≥ 22 Ry

for Si and Ecut ≥ 30 Ry for SiC, indicating that the electronic structure was converged.

We decided to use the same value Ecut = 30 Ry in Si and SiC calculations for consistency

reasons. Thus comparisons between NG configurations in Si and SiC crystals will be easier.

2.2.2 k points

To have an accurate description of the electronic structure, it is necessary to sample the Brillouin zone with a large enough number of k points. The Monkhorst-Pack (MP) method

Figure 2.2 –Total energy (per SiC pair atoms) for bulk SiC as a function of cutoff energy.

[109] is used to generate sets of k points in the Brillouin zone. Note that as the supercell gets larger, less k points are needed in the reciprocal space.

To determine how many k points we need, we have performed several calculations us-ing different numbers of k points with a supercell of 64 Si atoms. Lookus-ing at the energies plotted as a function of the number M, giving the size of the MP grid in Fig. 2.3, it is clear that when M ≥ 3 the total energy becomes independent of the number of k points. Thus we can estimate that our calculations will be well converged with a 3× 3 × 3 grid of

k points for supercells of 64 atoms. We found that the single k pointΓ (M = 0) is sufficient for a 216 atoms supercell. The same parameters have been used for 3C-SiC.

Using these parameters, we have determined the lattice parameter of silicon a0 to be

equal to 5.468 ˚A, close to the experimental value of 5.43 ˚A. For a 3C-SiC crystal, we have found that the lattice parameter a0 is equal to 4.376 ˚A, in good agreement with the

experimental value of 4.359 ˚A.

2.3 Search for the atomic configuration

Now that we have determined all necessary parameters for performing accurate electronic structure simulations, we can focus on the next issue, i.e. the calculations of atomic

con-2.3. SEARCH FOR THE ATOMIC CONFIGURATION 29

Figure 2.3 –Total energy (per atom) for bulk Si calculated using a M× M × M grid of k points.

figurations. In most of the cases, we are interested in the determination of stable configura-tions. An atomic configuration will be considered stable when all forces Fiacting on each

atom i are<10−3_{(eV/ ˚A). This corresponds to the system equilibrium, i.e. the system total}

energy E is minimum compared to the atomic positions. The force applied to an atom is given by:

− →_F

i = −−→∇iE

Thus to reach a stable state, atoms will be allowed to move in a way to cancel all forces, according to the conjugate gradient method [110]. This method is based on the fact that the total energy is approximately quadratic near the minimum.

Another aspect of this study was the determination of transition mechanisms and the associated energies. These mechanisms are useful to study diffusion properties of NG atoms in Si and 3C-SiC. To do this, it is possible to use molecular dynamics method but it has a high computational cost. Here, we used the Nudged Elastic Band (NEB) method described in detail in appendix C. This method allows one to determine the minimum energy path and find the saddle point (transition state) between two known stable positions. It works by optimizing a number of intermediate images along an initial migration path. Each image reaches the lowest possible energy while maintaining equal spacing between neighboring images. After all optimization steps, the migration energy Emig will thus be

2.4 Defects modeling

2.4.1 Size of simulation cell

Figure 2.4 –A defect modeled in the center of a simulation supercell, and its replicated images due to the periodic boundary conditions.

Ideally, we would like to model an isolated defect in a crystal, thus a very large cell should be used. But this is difficult, as the number of atoms which can be treated by todays computer and first-principles calculations remains limited to a few hundreds. A possible solution is the use of periodic boundary conditions (PBC). By using PBC our crystal will be replicated in the three dimensions to form a large supercell, as in Fig. 2.4.

The size of the simulation supercell must be large enough to prevent unphysical defect-defect interactions occurring from the created periodic images. On the other hand, DFT-GGA calculations have high computational costs, which severely limit the number of su-percell atoms.

As a consequence, we have to find the best compromise. We chose to use a supercell of 64 atoms, thus with a side equal to two times the lattice constant (2a0× 2a0× 2a0). We

2.4. DEFECTS MODELING 31

interstitial NG atom and its replicated images. Additionally in most of the cases we have perfomed some calculations with larger 3a0× 3a0× 3a0supercell, i.e. with 216 atoms.

In order to validate our calculation method and parameters, we have studied intrinsic defects in Si and 3C-SiC. These defects were already the subject of several works available in literature, with which our results are compared and discussed.

2.4.2 Silicon self defects

Figure 2.5 –The most common silicon self defects shown in the plane of the [110] zigzag chains of the diamond cubic structure. Si atoms are shown as full circles, and Si vacancies as empty dashed line circles.

We have modelled different silicon self defects, using a supercell of 64 and 216 Si atoms. Many self defects are possible, but here we have only focussed on the most sim-ple silicon ones, shown in Fig. 2.5: vacancy (V), divacancy (V2), tetrahedral interstitial (T), split interstitial SiSi_110 where two Si atoms share one lattice site, and finally the hexagonal interstitial (H).

Defect This work [40] [111] [112] [113] V 3.65 3.53 4.1 V2 _5.50 T 3.90 4.35 SiSi_110 3.62 3.40 3.3 2.16 4.19 H 3.67 3.45 2.45 4.23

Table 2.1 – Formation energies (in eV) of different silicon self defects calculated with a 64 Si atoms supercell, GGA-PBE, and 3× 3 × 3 k points. Our results are compared to other 64 atoms studies which used LDA.

Defect This work [39] [49] [114]

216 at. 216 at. 128 at. 128 at.

V 3.75 3.17 (3.56)

V2 5.15

T 3.61 4.01 (3.43) 4.07 (3.43)

SiSi_110 3.25 3.31 (2.88) 3.84 (3.31) 3.84 (3.31) H 3.27 3.31 (2.87) 3.80 (3.31) 3.80 (3.31)

Table 2.2 – Formation energies (in eV) of different silicon self defects calculated with a 216 Si atoms supercell, and a single k points and PBE-GGA. Our results are compared to other studies which used PW91-GGA or LDA (in parentheses).

Our calculations made in a 64 atoms supercell indicate that all tested defect configura-tions are stable. The computed formation energies are reported in the Table 2.1. Available published results for calculations made with similar cell sizes show a significant dispersion of the energies. Our results are in the range of these data. In agreement with the other studies, we found that SiSi_110is the most stable silicon interstitial.

We also checked the influence of the system size, by performing calculations in a 216 atoms cell. The results are reported in the Table 2.2, together with data from previous studies. It appears that all our computed formation energies are reduced by approximately 0.4 eV in the larger cell, probably because of a better relaxation. Nevertheless, the energy ordering of the different configurations is not modified. Besides, our results are again in the range of previously published data.

2.4. DEFECTS MODELING 33

2.4.3 3C-SiC self defects

Figure 2.6 – The most common 3C-SiC self defects shown in the plane of the [110] zigzag chains. Si atoms are shown as black circles, and grey circles for C atoms. Si and C vacancies are presented by empty dashed line circles.

In SiC, there are more simple self defects since one has to distinguish between Si and C point defects. Different types of interstitials (tetrahedral, split [100], split [110], hexagonal) and monovacancies have been first modelled with a supercell of 64 atoms and 3× 3 × 3 k points, all defect configurations being shown in Fig. 2.6.

We found all configurations to be stable after relaxation unlike other works [62, 115], where some tetrahedral interstitials convert to a split configuration. Generally, our cal-culated formation energies with a 64 atoms supercell, shown in Table 2.3, are close to available data [62, 115, 116]. The most stable C interstitial configuration was the CC_100 with a formation energy of 6.86 eV. For Si, the most stable configuration is SiSi_110.