Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...
Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...
Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...
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<strong>Analysis</strong> <strong>of</strong> <strong>the</strong> Extended Defects<br />
<strong>in</strong> <strong>3C</strong>-<strong>SiC</strong><br />
by<br />
Ezra Jacobus Olivier<br />
Submitted <strong>in</strong> partial fulfilment <strong>of</strong> <strong>the</strong><br />
requirements for <strong>the</strong> degree<br />
Magister Scientiae<br />
<strong>in</strong> <strong>the</strong> Faculty <strong>of</strong> Science at <strong>the</strong><br />
<strong>Nelson</strong> <strong>Mandela</strong> Metropolitan University<br />
Supervisor: Pr<strong>of</strong> J.H. Neethl<strong>in</strong>g February 2008<br />
PDF processed with CutePDF evaluation edition www.CutePDF.com
Dedicated to my wife and parents
My s<strong>in</strong>cere thanks to <strong>the</strong> follow<strong>in</strong>g people:<br />
ACKNOWLEDGEMENTS<br />
- My supervisor, Pr<strong>of</strong>essor J.H. Neethl<strong>in</strong>g for his guidance and encouragement<br />
throughout this project.<br />
- Any <strong>of</strong> <strong>the</strong> staff or students from <strong>the</strong> NMMU Physics Department whom<br />
played a contribut<strong>in</strong>g role <strong>in</strong> any way dur<strong>in</strong>g my research<br />
- SANHARP for <strong>the</strong>ir f<strong>in</strong>ancial assistance<br />
- My parents, bro<strong>the</strong>r and sister for encouragement and support throughout<br />
- My wife for her understand<strong>in</strong>g, support and love<br />
The assistance <strong>of</strong> <strong>the</strong> follow<strong>in</strong>g people is also gratefully acknowledged:<br />
- Dr Anna Carlsson at FEI Company <strong>in</strong> E<strong>in</strong>dhoven for <strong>the</strong> HRTEM results<br />
- Dr Chris Franklyn at NECSA for do<strong>in</strong>g <strong>the</strong> proton implantation<br />
- Nova<strong>SiC</strong> for supply<strong>in</strong>g <strong>the</strong> <strong>SiC</strong> used <strong>in</strong> <strong>the</strong> study<br />
- Johan Westraadt for <strong>the</strong> anneal<strong>in</strong>g <strong>of</strong> <strong>the</strong> samples at <strong>the</strong> University <strong>of</strong> <strong>the</strong><br />
Witwatersrand
SUMMARY<br />
The dissertation focuses on <strong>the</strong> analysis <strong>of</strong> <strong>the</strong> <strong>extended</strong> <strong>defects</strong> present <strong>in</strong> as-grown<br />
and proton bombarded β-<strong>SiC</strong> (annealed and unannealed) grown by chemical vapour<br />
deposition (CVD) on (001) Si. The proton irradiation was done to a dose <strong>of</strong> 2.8 × 10 16<br />
protons/cm 2 and <strong>the</strong> anneal<strong>in</strong>g took place at 1300°C and 1600°C for 1hr. The ma<strong>in</strong><br />
techniques used for <strong>the</strong> analysis were transmission electron microscopy (TEM) and<br />
high resolution TEM (HRTEM).<br />
From <strong>the</strong> diffraction study <strong>of</strong> <strong>the</strong> material <strong>the</strong> phase <strong>of</strong> <strong>the</strong> <strong>SiC</strong> was confirmed to be<br />
<strong>the</strong> cubic beta phase with <strong>the</strong> z<strong>in</strong>c-blende structure. The ma<strong>in</strong> <strong>defects</strong> found <strong>in</strong> <strong>the</strong> β-<br />
<strong>SiC</strong> were stack<strong>in</strong>g faults (SFs) with <strong>the</strong>ir associated partial dislocations and<br />
microtw<strong>in</strong>s. The SFs were uniformly distributed throughout <strong>the</strong> foil. The SFs were<br />
identified as hav<strong>in</strong>g a fault vector <strong>of</strong> <strong>the</strong> type 1/3 with bond<strong>in</strong>g partial<br />
dislocations <strong>of</strong> <strong>the</strong> type 1/6 by us<strong>in</strong>g image simulation. The SFs were also<br />
found to be predom<strong>in</strong>antly extr<strong>in</strong>sic <strong>in</strong> nature by us<strong>in</strong>g HRTEM analysis <strong>of</strong> SFs<br />
viewed edge-on. Also both bright and dar-field images <strong>of</strong> SFs on <strong>in</strong>cl<strong>in</strong>ed planes<br />
exhibited symmetrical and complementary fr<strong>in</strong>ge contrast images. This is a result <strong>of</strong><br />
<strong>the</strong> anomalous absorption ratio <strong>of</strong> <strong>SiC</strong> ly<strong>in</strong>g between that <strong>of</strong> Si and diamond.<br />
The analysis <strong>of</strong> <strong>the</strong> annealed and unannealed irradiated β-<strong>SiC</strong> yielded no evidence <strong>of</strong><br />
radiation damage or change <strong>in</strong> <strong>the</strong> crystal structure <strong>of</strong> <strong>the</strong> β-<strong>SiC</strong>. This confirmed that<br />
β-<strong>SiC</strong> is a radiation resistant material. The critical proton dose for <strong>the</strong> creation <strong>of</strong><br />
small dislocation loops seems to be higher than for o<strong>the</strong>r compound semiconductors<br />
with <strong>the</strong> z<strong>in</strong>c-blende structure.
OPSOMMING<br />
Hierdie dissertasie handel oor die uitgebreide defekte <strong>in</strong> β-<strong>SiC</strong> gegroei deur middel<br />
van chemiese damp neerslag op (001) silikon substrate. Transmissie<br />
elektronmikroskopie is ook gebruik om die defekte <strong>in</strong> proton bestraalde beta<br />
silikonkarbied (nie uitgegloei en uitgegloei) te ondersoek. Die beta silikonkarbied is<br />
bestraal met 400 keV protone tot ’n dosis van 2.8 × 10 16 protone/cm 2 .<br />
Die uitgloei<strong>in</strong>g van die materiaal het plaasgev<strong>in</strong>d by 1300°C en 1600°C vir een<br />
uur. Hoë resolusie transmissie elektronmikroskopie is ook aangewend om die<br />
stapelfoute <strong>in</strong> <strong>SiC</strong> te analiseer.<br />
Met behulp van elektron diffraksie en hoë resolusie transmissie<br />
elektronmikroskopie is die fase van die silikonkarbied bevestig as die kubiese<br />
beta fase. Die ho<strong>of</strong> defekte wat gev<strong>in</strong>d is <strong>in</strong> die <strong>SiC</strong> was stapelfoute met hul<br />
geassosieerde parsiële ontwrigt<strong>in</strong>gs en mikro-tweel<strong>in</strong>ge. Die stapelfoute was uniform<br />
gespasieer deur die material. Dit is bevestig deur die gebruik van beeldsimulasie dat<br />
die stapelfoute ‘n fout vektor van die tipe 1/3 en<br />
b<strong>in</strong>d<strong>in</strong>gs parsiële ontwrigt<strong>in</strong>gs van die tipe 1/6 het. Verder is ook vasgestel, met<br />
behulp van hoë resolusie transmissie elektronmikroskopie dat die stapelfoute<br />
ho<strong>of</strong>saaklik ekstr<strong>in</strong>siek is. Beeld simulasie van stapelfoute het aangetoon dat die<br />
ligveld en donkerveld beelde van die foute semetries en komplimentêre is. Dit is as<br />
gevolg van die waarde van die anomale absorpsie koefficient van die silikonkarbied<br />
wat tussen die van silikon en diamand lê.<br />
Die analise van die uitgegloeide en nie-uitgegloeide bestraalde beta silikonkarbied het<br />
geen tekens van stral<strong>in</strong>gskade <strong>of</strong> fase verander<strong>in</strong>g getoon nie. Dit het net weereens<br />
bewys dat beta silikonkarbied ‘n bestral<strong>in</strong>gs weerstandige materiaal is en dat die dosis<br />
van protone wat nodig is vir die vorm<strong>in</strong>g van kle<strong>in</strong> ontwrigt<strong>in</strong>gs lussies heelwat hoër<br />
is vir die materiaal as vir ander saamgestelde halfgeleiers met die s<strong>in</strong>k-blende<br />
struktuur.
CONTENTS<br />
Chapter 1: INTRODUCTION 1<br />
Chapter 2: OVERVIEW OF CRYSTAL DEFECTS IN β-SIC 3<br />
Page<br />
2.1. Introduction 3<br />
2.2. Cubic and Z<strong>in</strong>c-Blende Crystal Structures 3<br />
2.2.1. The Face-Centred Cubic Structure 3<br />
2.2.2. The Z<strong>in</strong>c-Blende Structure 5<br />
2.3. Po<strong>in</strong>t Defects, Defect Migration and Anneal<strong>in</strong>g Behaviour 7<br />
2.3.1. Introduction 7<br />
2.3.2. Po<strong>in</strong>t Defects 7<br />
2.3.3. Defect Migration and Anneal<strong>in</strong>g Behaviour 8<br />
2.4. Dislocations 10<br />
2.4.1. Introduction 10<br />
2.4.2. The Dislocation L<strong>in</strong>e and Burgers Vector 11<br />
2.4.3. Motion <strong>of</strong> Dislocations 13<br />
2.5. Partial Dislocations and Slip 15<br />
2.6. Dislocations <strong>in</strong> <strong>the</strong> Z<strong>in</strong>c-Blende Structure 16<br />
2.7. Stack<strong>in</strong>g Faults 18<br />
2.8. Tw<strong>in</strong>n<strong>in</strong>g 21<br />
2.9. Vacancy and Interstitial Loops 22<br />
Chapter 3: ION IMPLANTATION AND RADIATION DAMAGE 24<br />
3.1. Introduction 24<br />
3.2. Ion Ranges 24<br />
3.2.1. Introduction 24<br />
3.2.2. Range Calculations 25<br />
3.2.3. The L<strong>in</strong>hard, Scharff and Schiott (LSS) Theory 26<br />
3.3. Defects Created by Ion Implantation 30<br />
3.3.1. Introduction 30<br />
3.3.2. Theory <strong>of</strong> Displacement Damage 30<br />
3.3.3. Defects 34<br />
3.3.3.1. Defects Created Dur<strong>in</strong>g Irradiation 34<br />
3.3.3.2. Aggregation <strong>of</strong> Po<strong>in</strong>t Defects and Growth <strong>of</strong> Defect 34<br />
Clusters dur<strong>in</strong>g Heat Treatment<br />
Chapter 4: DYNAMICAL THEORY OF ELECTRON DIFFRACTION 36<br />
AND IMAGE SIMULATION<br />
4.1. Introduction 36<br />
4.2. Theory <strong>of</strong> Electron Diffraction 36<br />
4.2.1. The Direct Space Bragg Equation 36<br />
4.2.2. The Bragg Equation <strong>in</strong> Reciprocal Space 38<br />
4.2.3. Darw<strong>in</strong>-Howie-Whelan Equations 40<br />
4.2.4. DHW Equations: Two-Beam Case 42
4.2.5. Two-Beam Scatter<strong>in</strong>g Matrix 44<br />
4.2.6. Crystal Defects and Displacement Fields 45<br />
4.3. Image Contrast for Selected Defects 47<br />
4.3.1. L<strong>in</strong>e Defects 47<br />
4.3.2. Stack<strong>in</strong>g Faults 47<br />
4.3.3. Fr<strong>in</strong>ge Contrast Observed from Tw<strong>in</strong>n<strong>in</strong>g 51<br />
Chapter 5: LITERATURE REVIEW OF <strong>SiC</strong> 53<br />
5.1. Introduction 53<br />
5.2. CVD Growth <strong>of</strong> β-<strong>SiC</strong> on Si 53<br />
5.3. Crystal Defects <strong>in</strong> β-<strong>SiC</strong> 56<br />
5.3.1. Introduction 56<br />
5.3.2. Accommodation <strong>of</strong> Misfit and Interfacial Tw<strong>in</strong>n<strong>in</strong>g 57<br />
5.3.3. Mechanisms for <strong>the</strong> Formation <strong>of</strong> Stack<strong>in</strong>g Faults 58<br />
5.3.4. Stack<strong>in</strong>g Fault Energy <strong>in</strong> β-<strong>SiC</strong> 61<br />
5.4. Ion Implantation and Radiation Damage <strong>in</strong> β-<strong>SiC</strong> 61<br />
Chapter 6: EXPERIMENTAL DETAILS 64<br />
6.1. Introduction 64<br />
6.2. CVD Growth <strong>of</strong> β-<strong>SiC</strong> 64<br />
6.3. Proton Implantation <strong>in</strong> β-<strong>SiC</strong> 64<br />
6.3.1. Ion Ranges and Defect Production 65<br />
6.4. Anneal<strong>in</strong>g Procedure 67<br />
6.5. Sample Preparation 68<br />
Chapter 7: RESULTS AND DISCUSSION 69<br />
7.1. Introduction 69<br />
7.2. Extended Defects <strong>in</strong> β-<strong>SiC</strong> Grown by CVD on (001) Si 69<br />
7.2.1. Crystal Structure and <strong>the</strong> <strong>SiC</strong>/Si Interface 70<br />
7.2.2. Stack<strong>in</strong>g Faults and Partial Dislocations 74<br />
7.2.3. Tw<strong>in</strong>n<strong>in</strong>g 82<br />
7.3. Hydrogen Implanted β-<strong>SiC</strong> 83<br />
7.3.1. Unannealed 83<br />
7.3.2. Annealed at 1300°C 84<br />
7.3.3. Annealed at 1600°C 85<br />
7.4. Reassessment <strong>of</strong> Unimplanted Sample 94<br />
7.4.1. Overview <strong>of</strong> Investigation 94<br />
7.4.2. Results 95<br />
Chapter 8: CONCLUSION 96<br />
8.1. Introduction 96<br />
8.2. TEM <strong>Analysis</strong> <strong>of</strong> β-<strong>SiC</strong> Grown on (001) Si by CVD 96<br />
8.3. TEM <strong>Analysis</strong> <strong>of</strong> β-<strong>SiC</strong> Proton Bombarded and Annealed 97<br />
β-<strong>SiC</strong><br />
References 99
<strong>Analysis</strong> <strong>of</strong> <strong>the</strong> Extended Defects<br />
<strong>in</strong> <strong>3C</strong>-<strong>SiC</strong><br />
by<br />
Ezra Jacobus Olivier<br />
Submitted <strong>in</strong> partial fulfilment <strong>of</strong> <strong>the</strong><br />
requirements for <strong>the</strong> degree<br />
Magister Scientiae<br />
<strong>in</strong> <strong>the</strong> Faculty <strong>of</strong> Science at <strong>the</strong><br />
<strong>Nelson</strong> <strong>Mandela</strong> Metropolitan University<br />
Supervisor: Pr<strong>of</strong> J.H. Neethl<strong>in</strong>g February 2008
Dedicated to my wife and parents
My s<strong>in</strong>cere thanks to <strong>the</strong> follow<strong>in</strong>g people:<br />
ACKNOWLEDGEMENTS<br />
- My supervisor, Pr<strong>of</strong>essor J.H. Neethl<strong>in</strong>g for his guidance and encouragement<br />
throughout this project.<br />
- Any <strong>of</strong> <strong>the</strong> staff or students from <strong>the</strong> NMMU Physics Department whom<br />
played a contribut<strong>in</strong>g role <strong>in</strong> any way dur<strong>in</strong>g my research<br />
- SANHARP for <strong>the</strong>ir f<strong>in</strong>ancial assistance<br />
- My parents, bro<strong>the</strong>r and sister for encouragement and support throughout<br />
- My wife for her understand<strong>in</strong>g, support and love<br />
The assistance <strong>of</strong> <strong>the</strong> follow<strong>in</strong>g people is also gratefully acknowledged:<br />
- Dr Anna Carlsson at FEI Company <strong>in</strong> E<strong>in</strong>dhoven for <strong>the</strong> HRTEM results<br />
- Dr Chris Franklyn at NECSA for do<strong>in</strong>g <strong>the</strong> proton implantation<br />
- Nova<strong>SiC</strong> for supply<strong>in</strong>g <strong>the</strong> <strong>SiC</strong> used <strong>in</strong> <strong>the</strong> study<br />
- Johan Westraadt for <strong>the</strong> anneal<strong>in</strong>g <strong>of</strong> <strong>the</strong> samples at <strong>the</strong> University <strong>of</strong> <strong>the</strong><br />
Witwatersrand
SUMMARY<br />
The dissertation focuses on <strong>the</strong> analysis <strong>of</strong> <strong>the</strong> <strong>extended</strong> <strong>defects</strong> present <strong>in</strong> as-grown<br />
and proton bombarded β-<strong>SiC</strong> (annealed and unannealed) grown by chemical vapour<br />
deposition (CVD) on (001) Si. The proton irradiation was done to a dose <strong>of</strong> 2.8 × 10 16<br />
protons/cm 2 and <strong>the</strong> anneal<strong>in</strong>g took place at 1300°C and 1600°C for 1hr. The ma<strong>in</strong><br />
techniques used for <strong>the</strong> analysis were transmission electron microscopy (TEM) and<br />
high resolution TEM (HRTEM).<br />
From <strong>the</strong> diffraction study <strong>of</strong> <strong>the</strong> material <strong>the</strong> phase <strong>of</strong> <strong>the</strong> <strong>SiC</strong> was confirmed to be<br />
<strong>the</strong> cubic beta phase with <strong>the</strong> z<strong>in</strong>c-blende structure. The ma<strong>in</strong> <strong>defects</strong> found <strong>in</strong> <strong>the</strong> β-<br />
<strong>SiC</strong> were stack<strong>in</strong>g faults (SFs) with <strong>the</strong>ir associated partial dislocations and<br />
microtw<strong>in</strong>s. The SFs were uniformly distributed throughout <strong>the</strong> foil. The SFs were<br />
identified as hav<strong>in</strong>g a fault vector <strong>of</strong> <strong>the</strong> type 1/3 with bond<strong>in</strong>g partial<br />
dislocations <strong>of</strong> <strong>the</strong> type 1/6 by us<strong>in</strong>g image simulation. The SFs were also<br />
found to be predom<strong>in</strong>antly extr<strong>in</strong>sic <strong>in</strong> nature by us<strong>in</strong>g HRTEM analysis <strong>of</strong> SFs<br />
viewed edge-on. Also both bright and dar-field images <strong>of</strong> SFs on <strong>in</strong>cl<strong>in</strong>ed planes<br />
exhibited symmetrical and complementary fr<strong>in</strong>ge contrast images. This is a result <strong>of</strong><br />
<strong>the</strong> anomalous absorption ratio <strong>of</strong> <strong>SiC</strong> ly<strong>in</strong>g between that <strong>of</strong> Si and diamond.<br />
The analysis <strong>of</strong> <strong>the</strong> annealed and unannealed irradiated β-<strong>SiC</strong> yielded no evidence <strong>of</strong><br />
radiation damage or change <strong>in</strong> <strong>the</strong> crystal structure <strong>of</strong> <strong>the</strong> β-<strong>SiC</strong>. This confirmed that<br />
β-<strong>SiC</strong> is a radiation resistant material. The critical proton dose for <strong>the</strong> creation <strong>of</strong><br />
small dislocation loops seems to be higher than for o<strong>the</strong>r compound semiconductors<br />
with <strong>the</strong> z<strong>in</strong>c-blende structure.
OPSOMMING<br />
Hierdie dissertasie handel oor die uitgebreide defekte <strong>in</strong> β-<strong>SiC</strong> gegroei deur middel<br />
van chemiese damp neerslag op (001) silikon substrate. Transmissie<br />
elektronmikroskopie is ook gebruik om die defekte <strong>in</strong> proton bestraalde beta<br />
silikonkarbied (nie uitgegloei en uitgegloei) te ondersoek. Die beta silikonkarbied is<br />
bestraal met 400 keV protone tot ’n dosis van 2.8 × 10 16 protone/cm 2 .<br />
Die uitgloei<strong>in</strong>g van die materiaal het plaasgev<strong>in</strong>d by 1300°C en 1600°C vir een<br />
uur. Hoë resolusie transmissie elektronmikroskopie is ook aangewend om die<br />
stapelfoute <strong>in</strong> <strong>SiC</strong> te analiseer.<br />
Met behulp van elektron diffraksie en hoë resolusie transmissie<br />
elektronmikroskopie is die fase van die silikonkarbied bevestig as die kubiese<br />
beta fase. Die ho<strong>of</strong> defekte wat gev<strong>in</strong>d is <strong>in</strong> die <strong>SiC</strong> was stapelfoute met hul<br />
geassosieerde parsiële ontwrigt<strong>in</strong>gs en mikro-tweel<strong>in</strong>ge. Die stapelfoute was uniform<br />
gespasieer deur die material. Dit is bevestig deur die gebruik van beeldsimulasie dat<br />
die stapelfoute ‘n fout vektor van die tipe 1/3 en<br />
b<strong>in</strong>d<strong>in</strong>gs parsiële ontwrigt<strong>in</strong>gs van die tipe 1/6 het. Verder is ook vasgestel, met<br />
behulp van hoë resolusie transmissie elektronmikroskopie dat die stapelfoute<br />
ho<strong>of</strong>saaklik ekstr<strong>in</strong>siek is. Beeld simulasie van stapelfoute het aangetoon dat die<br />
ligveld en donkerveld beelde van die foute semetries en komplimentêre is. Dit is as<br />
gevolg van die waarde van die anomale absorpsie koefficient van die silikonkarbied<br />
wat tussen die van silikon en diamand lê.<br />
Die analise van die uitgegloeide en nie-uitgegloeide bestraalde beta silikonkarbied het<br />
geen tekens van stral<strong>in</strong>gskade <strong>of</strong> fase verander<strong>in</strong>g getoon nie. Dit het net weereens<br />
bewys dat beta silikonkarbied ‘n bestral<strong>in</strong>gs weerstandige materiaal is en dat die dosis<br />
van protone wat nodig is vir die vorm<strong>in</strong>g van kle<strong>in</strong> ontwrigt<strong>in</strong>gs lussies heelwat hoër<br />
is vir die materiaal as vir ander saamgestelde halfgeleiers met die s<strong>in</strong>k-blende<br />
struktuur.
CONTENTS<br />
Chapter 1: INTRODUCTION 1<br />
Chapter 2: OVERVIEW OF CRYSTAL DEFECTS IN β-SIC 3<br />
Page<br />
2.1. Introduction 3<br />
2.2. Cubic and Z<strong>in</strong>c-Blende Crystal Structures 3<br />
2.2.1. The Face-Centred Cubic Structure 3<br />
2.2.2. The Z<strong>in</strong>c-Blende Structure 5<br />
2.3. Po<strong>in</strong>t Defects, Defect Migration and Anneal<strong>in</strong>g Behaviour 7<br />
2.3.1. Introduction 7<br />
2.3.2. Po<strong>in</strong>t Defects 7<br />
2.3.3. Defect Migration and Anneal<strong>in</strong>g Behaviour 8<br />
2.4. Dislocations 10<br />
2.4.1. Introduction 10<br />
2.4.2. The Dislocation L<strong>in</strong>e and Burgers Vector 11<br />
2.4.3. Motion <strong>of</strong> Dislocations 13<br />
2.5. Partial Dislocations and Slip 15<br />
2.6. Dislocations <strong>in</strong> <strong>the</strong> Z<strong>in</strong>c-Blende Structure 16<br />
2.7. Stack<strong>in</strong>g Faults 18<br />
2.8. Tw<strong>in</strong>n<strong>in</strong>g 21<br />
2.9. Vacancy and Interstitial Loops 22<br />
Chapter 3: ION IMPLANTATION AND RADIATION DAMAGE 24<br />
3.1. Introduction 24<br />
3.2. Ion Ranges 24<br />
3.2.1. Introduction 24<br />
3.2.2. Range Calculations 25<br />
3.2.3. The L<strong>in</strong>hard, Scharff and Schiott (LSS) Theory 26<br />
3.3. Defects Created by Ion Implantation 30<br />
3.3.1. Introduction 30<br />
3.3.2. Theory <strong>of</strong> Displacement Damage 30<br />
3.3.3. Defects 34<br />
3.3.3.1. Defects Created Dur<strong>in</strong>g Irradiation 34<br />
3.3.3.2. Aggregation <strong>of</strong> Po<strong>in</strong>t Defects and Growth <strong>of</strong> Defect 34<br />
Clusters dur<strong>in</strong>g Heat Treatment<br />
Chapter 4: DYNAMICAL THEORY OF ELECTRON DIFFRACTION 36<br />
AND IMAGE SIMULATION<br />
4.1. Introduction 36<br />
4.2. Theory <strong>of</strong> Electron Diffraction 36<br />
4.2.1. The Direct Space Bragg Equation 36<br />
4.2.2. The Bragg Equation <strong>in</strong> Reciprocal Space 38<br />
4.2.3. Darw<strong>in</strong>-Howie-Whelan Equations 40<br />
4.2.4. DHW Equations: Two-Beam Case 42
4.2.5. Two-Beam Scatter<strong>in</strong>g Matrix 44<br />
4.2.6. Crystal Defects and Displacement Fields 45<br />
4.3. Image Contrast for Selected Defects 47<br />
4.3.1. L<strong>in</strong>e Defects 47<br />
4.3.2. Stack<strong>in</strong>g Faults 47<br />
4.3.3. Fr<strong>in</strong>ge Contrast Observed from Tw<strong>in</strong>n<strong>in</strong>g 51<br />
Chapter 5: LITERATURE REVIEW OF <strong>SiC</strong> 53<br />
5.1. Introduction 53<br />
5.2. CVD Growth <strong>of</strong> β-<strong>SiC</strong> on Si 53<br />
5.3. Crystal Defects <strong>in</strong> β-<strong>SiC</strong> 56<br />
5.3.1. Introduction 56<br />
5.3.2. Accommodation <strong>of</strong> Misfit and Interfacial Tw<strong>in</strong>n<strong>in</strong>g 57<br />
5.3.3. Mechanisms for <strong>the</strong> Formation <strong>of</strong> Stack<strong>in</strong>g Faults 58<br />
5.3.4. Stack<strong>in</strong>g Fault Energy <strong>in</strong> β-<strong>SiC</strong> 61<br />
5.4. Ion Implantation and Radiation Damage <strong>in</strong> β-<strong>SiC</strong> 61<br />
Chapter 6: EXPERIMENTAL DETAILS 64<br />
6.1. Introduction 64<br />
6.2. CVD Growth <strong>of</strong> β-<strong>SiC</strong> 64<br />
6.3. Proton Implantation <strong>in</strong> β-<strong>SiC</strong> 64<br />
6.3.1. Ion Ranges and Defect Production 65<br />
6.4. Anneal<strong>in</strong>g Procedure 67<br />
6.5. Sample Preparation 68<br />
Chapter 7: RESULTS AND DISCUSSION 69<br />
7.1. Introduction 69<br />
7.2. Extended Defects <strong>in</strong> β-<strong>SiC</strong> Grown by CVD on (001) Si 69<br />
7.2.1. Crystal Structure and <strong>the</strong> <strong>SiC</strong>/Si Interface 70<br />
7.2.2. Stack<strong>in</strong>g Faults and Partial Dislocations 74<br />
7.2.3. Tw<strong>in</strong>n<strong>in</strong>g 82<br />
7.3. Hydrogen Implanted β-<strong>SiC</strong> 83<br />
7.3.1. Unannealed 83<br />
7.3.2. Annealed at 1300°C 84<br />
7.3.3. Annealed at 1600°C 85<br />
7.4. Reassessment <strong>of</strong> Unimplanted Sample 94<br />
7.4.1. Overview <strong>of</strong> Investigation 94<br />
7.4.2. Results 95<br />
Chapter 8: CONCLUSION 96<br />
8.1. Introduction 96<br />
8.2. TEM <strong>Analysis</strong> <strong>of</strong> β-<strong>SiC</strong> Grown on (001) Si by CVD 96<br />
8.3. TEM <strong>Analysis</strong> <strong>of</strong> β-<strong>SiC</strong> Proton Bombarded and Annealed 97<br />
β-<strong>SiC</strong><br />
References 99
1<br />
CHAPTER ONE<br />
INTRODUCTION<br />
Silicon carbide (<strong>SiC</strong>) is a wide band gap semiconduct<strong>in</strong>g ceramic with excellent<br />
<strong>the</strong>rmal, mechanical and electrical properties. It has a wide range <strong>of</strong> applications,<br />
which <strong>in</strong>clude its use as structural material for high temperature applications <strong>in</strong><br />
aggressive environments (Eveno et al. (1992)). It is proposed as an encapsulat<strong>in</strong>g<br />
material for nuclear fuel <strong>in</strong> light water and gas-cooled fission reactors due to its good<br />
resistance to neutron radiation damage and also as material to be used <strong>in</strong> fusion<br />
environments due to its excellent <strong>the</strong>rmal stability. Hence <strong>the</strong> study <strong>of</strong> this material<br />
and its structural <strong>in</strong>tegrity under different conditions similar to that experienced <strong>in</strong> <strong>the</strong><br />
reactor environment is important. A detailed knowledge <strong>of</strong> <strong>the</strong> <strong>extended</strong> <strong>defects</strong><br />
present <strong>in</strong> <strong>the</strong> <strong>SiC</strong> after <strong>the</strong> growth process is required toge<strong>the</strong>r with <strong>the</strong> study <strong>of</strong> <strong>the</strong><br />
evolution <strong>of</strong> <strong>the</strong> micro and nano-structure <strong>of</strong> <strong>SiC</strong> at high temperatures and irradiation<br />
conditions.<br />
This dissertation focuses on a transmission electron microscopy (TEM) study <strong>of</strong> <strong>the</strong><br />
<strong>extended</strong> <strong>defects</strong> <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> prepared by chemical vapour deposition on silicon (Si)<br />
substrates, as well as <strong>the</strong> <strong>in</strong>vestigation <strong>of</strong> <strong>the</strong> <strong>defects</strong> produced <strong>in</strong> this material by 400<br />
keV hydrogen implanted to a dose <strong>of</strong> 2.8 × 10 16 protons/cm 2 . The effects <strong>of</strong> post<br />
irradiation anneal<strong>in</strong>g on <strong>the</strong> implanted material was also <strong>in</strong>vestigated for samples<br />
annealed at temperatures <strong>of</strong> 1300°C and 1600°C for one hour. Silicon is <strong>the</strong> preferred<br />
substrate for <strong>the</strong> CVD growth <strong>of</strong> s<strong>in</strong>gle crystall<strong>in</strong>e <strong>3C</strong>-<strong>SiC</strong> and due to <strong>the</strong> lattice<br />
mismatch between <strong>SiC</strong> and Si, <strong>the</strong> tw<strong>in</strong>s, stack<strong>in</strong>g faults, partial dislocations and<br />
misfit dislocations present <strong>in</strong> <strong>the</strong> <strong>SiC</strong> will be similar to <strong>the</strong> lattice <strong>defects</strong> present <strong>in</strong><br />
<strong>SiC</strong> grown on o<strong>the</strong>r substrates. In a nuclear reactor such as <strong>the</strong> pebble bed modular<br />
reactor, <strong>the</strong> <strong>SiC</strong> layers <strong>in</strong> <strong>the</strong> nuclear fuel particles are irradiated with high energy<br />
neutrons. The po<strong>in</strong>t <strong>defects</strong> produced by <strong>the</strong> neutrons will be <strong>the</strong> same type as those<br />
produced by high energy protons. In this <strong>in</strong>vestigation <strong>SiC</strong> were bombarded with<br />
protons and <strong>the</strong>n annealed to determ<strong>in</strong>e <strong>the</strong> critical proton dose needed to produce<br />
<strong>extended</strong> defect clusters dur<strong>in</strong>g post-implantation anneal<strong>in</strong>g. The anneal<strong>in</strong>g at<br />
elevated temperatures also allowed <strong>the</strong> <strong>in</strong>vestigation <strong>of</strong> possible phase transformations<br />
<strong>in</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong>.
2<br />
The characteristics <strong>of</strong> two-beam diffraction contrast TEM images <strong>of</strong> stack<strong>in</strong>g faults<br />
(SFs) and th<strong>in</strong> tw<strong>in</strong> platelets <strong>in</strong> <strong>SiC</strong> were <strong>in</strong>vestigated by us<strong>in</strong>g image simulation.<br />
S<strong>in</strong>ce it is expected that <strong>the</strong> anomalous absorption ratio <strong>of</strong> <strong>SiC</strong> should lie between that<br />
<strong>of</strong> diamond and Si, it is important to <strong>in</strong>vestigate how <strong>the</strong> value <strong>of</strong> this ratio would<br />
affect <strong>the</strong> symmetry <strong>of</strong> bright and dark-field TEM images <strong>of</strong> SFs <strong>in</strong> <strong>SiC</strong>. Image<br />
simulation was carried out us<strong>in</strong>g <strong>the</strong> s<strong>of</strong>tware developed by Head et al. (1973)<br />
published by Marc De Graef (2003).<br />
The outl<strong>in</strong>e <strong>of</strong> <strong>the</strong> dissertation is as follows. In Chapter 2 a brief discussion <strong>of</strong> <strong>the</strong><br />
crystal <strong>defects</strong> found <strong>in</strong> face-centred cubic (fcc) and z<strong>in</strong>c-blende crystal structures is<br />
given. Emphasis is placed on <strong>the</strong> <strong>extended</strong> <strong>defects</strong> found <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong>. Defects created<br />
dur<strong>in</strong>g ion bombardment and <strong>the</strong>ir migration dur<strong>in</strong>g heat treatment is also discussed <strong>in</strong><br />
general. Chapter 3 discusses <strong>the</strong> <strong>the</strong>ory <strong>of</strong> ion implantation and radiation damage <strong>in</strong><br />
materials. Fundamental <strong>the</strong>ories used <strong>in</strong> calculat<strong>in</strong>g <strong>the</strong> ion ranges and displacement<br />
damage are given and <strong>the</strong> coalescence <strong>of</strong> po<strong>in</strong>t <strong>defects</strong> lead<strong>in</strong>g to <strong>the</strong> formation <strong>of</strong><br />
defect clusters dur<strong>in</strong>g heat treatment is discussed. Chapter 4 deals with <strong>the</strong> <strong>the</strong>ory<br />
relat<strong>in</strong>g to <strong>the</strong> method <strong>of</strong> image simulation. This is done by briefly discuss<strong>in</strong>g <strong>the</strong><br />
dynamical <strong>the</strong>ory <strong>of</strong> electron diffraction and <strong>the</strong> derivation <strong>of</strong> <strong>the</strong> Darw<strong>in</strong>-Howie-<br />
Whelan (DHW) equations. A simple method for calculat<strong>in</strong>g a l<strong>in</strong>e pr<strong>of</strong>ile across <strong>the</strong><br />
fr<strong>in</strong>ges <strong>of</strong> a stack<strong>in</strong>g fault is also given. Chapter 5 is a literature review <strong>of</strong> previous<br />
research done on <strong>3C</strong>-<strong>SiC</strong> focus<strong>in</strong>g on crystal growth, crystal <strong>defects</strong> and behaviour<br />
under irradiation conditions. Chapter 6 gives <strong>the</strong> experimental details <strong>of</strong> <strong>the</strong> current<br />
<strong>in</strong>vestigation on <strong>3C</strong>-<strong>SiC</strong>. Chapter 7 conta<strong>in</strong>s <strong>the</strong> results <strong>of</strong> a transmission electron<br />
microscopy (TEM) study <strong>of</strong> <strong>the</strong> as-grown and implanted <strong>3C</strong>-<strong>SiC</strong> along with <strong>the</strong><br />
discussion <strong>of</strong> <strong>the</strong> results. Chapter 8 lists <strong>the</strong> conclusions derived from <strong>the</strong> results <strong>of</strong><br />
this dissertation.
2.1. Introduction<br />
3<br />
CHAPTER TWO<br />
OVERVIEW OF CRYSTAL DEFECTS IN <strong>3C</strong>-<strong>SiC</strong><br />
In this chapter a short overview <strong>of</strong> <strong>the</strong> types <strong>of</strong> crystal <strong>defects</strong> found <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> is<br />
presented. The content presented is by no means complete but focuses on <strong>the</strong><br />
important aspects needed to understand <strong>the</strong> analysis <strong>of</strong> <strong>the</strong> experimental results later<br />
on. The reader is advised to consult o<strong>the</strong>r references written specifically for this<br />
purpose if a deeper understand<strong>in</strong>g is required (Hull et al. (1984)). This chapter<br />
focuses ma<strong>in</strong>ly on <strong>the</strong> <strong>defects</strong> found <strong>in</strong> <strong>the</strong> z<strong>in</strong>c-blende structure s<strong>in</strong>ce this is <strong>the</strong><br />
crystal structure <strong>of</strong> <strong>3C</strong>-<strong>SiC</strong>. The structure itself is expla<strong>in</strong>ed firstly by referr<strong>in</strong>g to a<br />
more simple structure i.e. <strong>the</strong> fcc structure with special attention given to <strong>the</strong> atomic<br />
stack<strong>in</strong>g sequences <strong>in</strong> <strong>the</strong> pr<strong>in</strong>cipal crystallographic directions. Fur<strong>the</strong>rmore <strong>the</strong> types<br />
<strong>of</strong> po<strong>in</strong>t <strong>defects</strong> found <strong>in</strong> <strong>SiC</strong> are discussed and <strong>the</strong>ir diffusion characteristics are<br />
reviewed. F<strong>in</strong>ally <strong>the</strong> nature <strong>of</strong> <strong>the</strong> <strong>extended</strong> <strong>defects</strong> present <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> is discussed.<br />
2.2. Cubic and Z<strong>in</strong>c-Blende Crystal Structures<br />
2.2.1 The Face-Centred Cubic Structure<br />
The unit cell <strong>of</strong> <strong>the</strong> face-centred cubic (fcc) structure is a cube with atoms situated at<br />
<strong>the</strong> corners and centres <strong>of</strong> <strong>the</strong> cube faces as illustrated <strong>in</strong> Fig. 2.1. It conta<strong>in</strong>s four<br />
1<br />
atoms <strong>in</strong> positions 000 ; 0<br />
1 1<br />
2 2 ; 2 2<br />
1 1 1<br />
0 and 0 2 2 , with <strong>the</strong> smallest translation vectors<br />
be<strong>in</strong>g <strong>of</strong> <strong>the</strong> type (1/2). This corresponds to half a surface diagonal <strong>of</strong> <strong>the</strong> unit<br />
cell.
Fig. 2.1. The face-centred cubic structure with shortest translation vector ½<br />
4<br />
Fur<strong>the</strong>rmore all next larger translational vectors such as that <strong>of</strong> <strong>the</strong> and <br />
type can be decomposed <strong>in</strong>to vectors <strong>of</strong> <strong>the</strong> type (1/2) for example:<br />
(1/2)[110] + (1/2) 110]<br />
= [100]<br />
[ _<br />
(1/2)[110] + (1/2) [110] = [110]<br />
The stack<strong>in</strong>g sequence <strong>of</strong> <strong>the</strong> {100} and {110} planes are ABABAB…and that <strong>of</strong> <strong>the</strong><br />
{111} planes ABCABC…. (termed close-packed planes). The two stack<strong>in</strong>g sequences<br />
are shown <strong>in</strong> Fig. 2.2.<br />
(a) (b)<br />
Fig. 2.2. The stack<strong>in</strong>g sequences <strong>of</strong> (a) <strong>the</strong> {100} and (b) <strong>the</strong> {111} planes <strong>in</strong> a fcc<br />
structure.<br />
A<br />
B<br />
A<br />
B<br />
A<br />
B<br />
A<br />
B<br />
A<br />
B<br />
C<br />
A<br />
B<br />
C
2.2.2 The Z<strong>in</strong>c-Blende Structure<br />
5<br />
The z<strong>in</strong>c-blende structure can be seen as composed <strong>of</strong> a basic fcc framework with<br />
atoms situated at positions as discussed above, with <strong>the</strong> addition <strong>of</strong> a fur<strong>the</strong>r set <strong>of</strong><br />
1 1<br />
atoms situated at positions 4 4 4<br />
1 3 1 1 3 1 3<br />
3 3<br />
; 4 4 4 ; 4 4 4 and 4 4 4<br />
3 . The two sets <strong>of</strong> atoms may be<br />
<strong>of</strong> <strong>the</strong> same type as for example <strong>in</strong> diamond (carbon), silicon, germanium or it may<br />
consist <strong>of</strong> a composition <strong>of</strong> different elements such as for GaAs, ZnS and <strong>SiC</strong>. The<br />
unit cell consists <strong>of</strong> 8 atoms and can be seen as two <strong>in</strong>terpenetrat<strong>in</strong>g fcc lattices<br />
a<br />
displaced by a translation <strong>of</strong> 111 as shown <strong>in</strong> Fig. 2.3.<br />
4<br />
Fig. 2.3. The <strong>SiC</strong> z<strong>in</strong>c-blende unit cell with <strong>the</strong> smaller atoms be<strong>in</strong>g <strong>the</strong> carbon and<br />
<strong>the</strong> larger silicon.<br />
The stack<strong>in</strong>g sequence <strong>of</strong> <strong>the</strong> z<strong>in</strong>c-blende structure for <strong>the</strong> {100}, {110} and {111}<br />
planes are analogous to that <strong>of</strong> <strong>the</strong> fcc-structure with <strong>the</strong> only difference be<strong>in</strong>g that <strong>the</strong><br />
two sets <strong>of</strong> atoms create a doubl<strong>in</strong>g <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g sequence <strong>in</strong> <strong>the</strong> sense that <strong>in</strong> <strong>the</strong><br />
{100} and {110} cases one obta<strong>in</strong>s a sequence AA'BB'AA'… and <strong>in</strong> <strong>the</strong> {111} a<br />
sequence AA'BB'CC'… is seen (where <strong>the</strong> ' denotes <strong>the</strong> alternate set). This is shown<br />
<strong>in</strong> Fig. 2.4 for <strong>the</strong> stack<strong>in</strong>g <strong>of</strong> <strong>the</strong> {100} and {111} planes.
(a) (b)<br />
6<br />
Fig. 2.4. Stack<strong>in</strong>g sequences <strong>of</strong> <strong>the</strong> z<strong>in</strong>c-blende structure for <strong>the</strong> (a) {100} and (b)<br />
{111} planes<br />
A<br />
A'<br />
B<br />
B'<br />
A<br />
A'<br />
B<br />
B'<br />
A<br />
A'<br />
B<br />
B'<br />
The two consecutive layers <strong>of</strong> <strong>the</strong> constituent atoms <strong>in</strong> <strong>the</strong> same type <strong>of</strong> configuration<br />
are considered as a s<strong>in</strong>gle plane consist<strong>in</strong>g <strong>of</strong> a double layer <strong>of</strong> Si and C atoms. Thus a<br />
ABABAB… or ABCABC… stack<strong>in</strong>g sequence as <strong>in</strong> <strong>the</strong> case for <strong>the</strong> fcc structure<br />
results with <strong>the</strong> only difference be<strong>in</strong>g that a layer conta<strong>in</strong>s both <strong>the</strong> constituent atoms.<br />
-<strong>SiC</strong> is a cubic phase <strong>of</strong> <strong>SiC</strong> with a z<strong>in</strong>c-blende structure as shown <strong>in</strong> Fig. 2.3. The<br />
{110} planes consist <strong>of</strong> double layers <strong>of</strong> Si and C atoms. Also <strong>the</strong> stack<strong>in</strong>g sequence<br />
for <strong>the</strong> {100} and {110} planes are AA'BB'AA'… and AA'BB'CC' for <strong>the</strong> {111}<br />
planes (where <strong>the</strong> ' denotes <strong>the</strong> stack<strong>in</strong>g sequence <strong>of</strong> <strong>the</strong> alternate set <strong>of</strong> atoms). Note<br />
also that <strong>the</strong> consecutive layers <strong>of</strong> <strong>the</strong> silicon and carbon atoms <strong>in</strong> <strong>the</strong> sequence for<br />
e.g., AA' is closely spaced and thus <strong>the</strong> stack<strong>in</strong>g sequence can be generalized to be<br />
ABABAB… and ABCABC… <strong>in</strong> <strong>the</strong> {100}, {110} and {111} case respectively by<br />
comb<strong>in</strong><strong>in</strong>g <strong>the</strong> two closely spaced layers to form one plane.<br />
A<br />
B<br />
A<br />
B<br />
A<br />
B<br />
A<br />
A'<br />
B<br />
B'<br />
C<br />
C'<br />
A<br />
A'<br />
B<br />
B'<br />
C<br />
C'<br />
A<br />
B<br />
C<br />
A<br />
B<br />
C
2.3 Po<strong>in</strong>t Defects, Defect Migration and Anneal<strong>in</strong>g Behaviour<br />
2.3.1 Introduction<br />
7<br />
In this section <strong>the</strong> different types <strong>of</strong> po<strong>in</strong>t <strong>defects</strong> <strong>in</strong> a crystal lattice are discussed.<br />
Fundamental statistics on <strong>the</strong>ir concentration <strong>in</strong> <strong>the</strong> lattice are presented and <strong>the</strong>ir<br />
diffusion mechanisms are considered. In general <strong>the</strong> diffusion behaviour <strong>of</strong> po<strong>in</strong>t<br />
<strong>defects</strong> <strong>in</strong> <strong>SiC</strong> cannot be fully expla<strong>in</strong>ed yet, s<strong>in</strong>ce it is a complex process that is still<br />
be<strong>in</strong>g researched.<br />
3.2 Po<strong>in</strong>t Defects<br />
A po<strong>in</strong>t defect is def<strong>in</strong>ed as <strong>the</strong> <strong>in</strong>sertion or absence <strong>of</strong> an atom <strong>in</strong> a crystal lattice<br />
uncharacteristic to that <strong>of</strong> <strong>the</strong> normal atomic sites. When <strong>the</strong> defect is a result <strong>of</strong> atoms<br />
compris<strong>in</strong>g <strong>of</strong> <strong>the</strong> parent lattice it is termed an <strong>in</strong>tr<strong>in</strong>sic defect and when <strong>the</strong> atom is an<br />
impurity it is termed extr<strong>in</strong>sic. Intr<strong>in</strong>sic po<strong>in</strong>t <strong>defects</strong> may fur<strong>the</strong>r be characterised as<br />
be<strong>in</strong>g vacancy or <strong>in</strong>terstitial, vacancy be<strong>in</strong>g <strong>the</strong> absence <strong>of</strong> an atom <strong>in</strong> a regular atomic<br />
site and an <strong>in</strong>terstitial <strong>the</strong> <strong>in</strong>sertion <strong>of</strong> an atom <strong>in</strong>to a non-lattice site. This is depicted<br />
<strong>in</strong> Fig. 2.5.<br />
Fig. 2.5. The <strong>in</strong>tr<strong>in</strong>sic <strong>defects</strong>, vacancy and <strong>in</strong>terstitial present <strong>in</strong> a lattice (from Hull<br />
et al. (1984))<br />
Extr<strong>in</strong>sic po<strong>in</strong>t <strong>defects</strong> may be characterised as substitutional, where an atom <strong>of</strong> <strong>the</strong><br />
parent lattice ly<strong>in</strong>g <strong>in</strong> a lattice site is replaced by an impurity atom, or <strong>in</strong>terstitial,<br />
where <strong>the</strong> impurity atom is at a non-lattice site (Fig. 2.6).
8<br />
Fig. 2.6. The extr<strong>in</strong>sic <strong>defects</strong>, substitutional and <strong>in</strong>terstitial present <strong>in</strong> a lattice (from<br />
Hull et al. (1984))<br />
Vacancies and <strong>in</strong>terstitials can be produced by plastic deformation and also due to<br />
high-energy particle irradiation. Also <strong>in</strong>tr<strong>in</strong>sic <strong>defects</strong> are produced by virtue <strong>of</strong><br />
temperature, a <strong>the</strong>rmodynamically stable concentration <strong>of</strong> <strong>the</strong>se <strong>defects</strong> are always<br />
present above 0 K. The equilibrium fraction <strong>of</strong> po<strong>in</strong>t <strong>defects</strong> is given by <strong>the</strong> relation,<br />
n<br />
eq<br />
E f<br />
nt<br />
exp( )<br />
(2.1)<br />
kT<br />
where nt is <strong>the</strong> total number <strong>of</strong> atomic sites, k Boltzmann’s constant and Ef <strong>the</strong> energy<br />
<strong>of</strong> formation <strong>of</strong> one defect. For a vacancy, <strong>the</strong> energy is that required to remove one<br />
atom from <strong>the</strong> lattice and place it on <strong>the</strong> surface <strong>of</strong> <strong>the</strong> crystal, and for <strong>the</strong> <strong>in</strong>terstitial it<br />
is <strong>the</strong> energy required to remove one atom from <strong>the</strong> surface and <strong>in</strong>sert it <strong>in</strong>to an<br />
<strong>in</strong>terstitial site. Typically <strong>the</strong> vacancy formation energy is two to four times less than<br />
that <strong>of</strong> <strong>the</strong> <strong>in</strong>terstitial and consequently <strong>the</strong> concentration <strong>of</strong> <strong>in</strong>terstitials is far lower<br />
than that <strong>of</strong> vacancies at a certa<strong>in</strong> temperature T.<br />
2.3.3 Defect Migration and Anneal<strong>in</strong>g Behaviour<br />
In general po<strong>in</strong>t <strong>defects</strong> are cont<strong>in</strong>uously attempt<strong>in</strong>g to migrate, <strong>the</strong> success <strong>of</strong> which<br />
critically depends on <strong>the</strong> defect migration energy Em and is largely determ<strong>in</strong>ed by <strong>the</strong><br />
specimen temperature. Stra<strong>in</strong> <strong>in</strong>duced motion is also possible but it is <strong>the</strong> aim <strong>of</strong> this<br />
section to discuss <strong>the</strong> former. The atomic jump frequency at temperature T <strong>of</strong> a po<strong>in</strong>t<br />
defect is given by,
v<br />
m<br />
Em<br />
vm0<br />
exp( )<br />
(2.2)<br />
kT<br />
9<br />
where Em is <strong>the</strong> migration energy <strong>of</strong> <strong>the</strong> po<strong>in</strong>t defect, ei<strong>the</strong>r vacancy or <strong>in</strong>terstitial and<br />
υm0 is <strong>the</strong> atomic vibration frequency. This atomic jump frequency is a measure <strong>of</strong><br />
how <strong>of</strong>ten it is expected for atoms to leave <strong>the</strong>ir normal positions but it does not say<br />
anyth<strong>in</strong>g about <strong>the</strong> direction <strong>the</strong> atom will take for any particular jump. The atom<br />
accord<strong>in</strong>gly takes a very haphazard path as it wanders through <strong>the</strong> crystal and this<br />
path is termed a random walk and could be correlated or uncorrelated depend<strong>in</strong>g on<br />
<strong>the</strong> type <strong>of</strong> defect <strong>in</strong> question s<strong>in</strong>ce <strong>the</strong> probabilities <strong>of</strong> motion for each are different.<br />
A substitutional impurity jump<strong>in</strong>g through a vacancy mechanism for example is said<br />
to have a correlated random walk because <strong>the</strong>re is not an equal probability <strong>of</strong> its<br />
nearest neighbours be<strong>in</strong>g vacant.<br />
Thus if a region <strong>in</strong> a crystal conta<strong>in</strong>s a concentration <strong>of</strong> po<strong>in</strong>t <strong>defects</strong> all <strong>of</strong> <strong>the</strong>se<br />
<strong>defects</strong> will perform random walks but will also eventually disperse and spread<br />
throughout <strong>the</strong> crystal caus<strong>in</strong>g a net transport <strong>of</strong> matter. This happens s<strong>in</strong>ce <strong>the</strong>re is a<br />
difference <strong>in</strong> concentration <strong>of</strong> <strong>defects</strong> <strong>in</strong> <strong>the</strong> crystal caus<strong>in</strong>g a concentration gradient.<br />
This net transport <strong>of</strong> matter is described by Fick’s first law,<br />
dc<br />
J D<br />
(2.3)<br />
dx<br />
dc<br />
where J is <strong>the</strong> flux, <strong>the</strong> concentration gradient and D <strong>the</strong> diffusion coefficient,<br />
dx<br />
which depends on <strong>the</strong> mechanism <strong>of</strong> diffusion, type <strong>of</strong> crystal structure and jump<br />
frequency. This equation can also be given <strong>in</strong> a more useful form known as Fick’s<br />
second law,<br />
2<br />
dC d C<br />
D<br />
(2.4)<br />
2<br />
dt<br />
dx
10<br />
<strong>in</strong> which <strong>the</strong> rate <strong>of</strong> change <strong>of</strong> <strong>the</strong> concentration with time is described. This is useful<br />
s<strong>in</strong>ce it enables <strong>the</strong> change <strong>in</strong> concentration to be determ<strong>in</strong>ed at different time<br />
<strong>in</strong>tervals. Fur<strong>the</strong>rmore <strong>the</strong> diffusion coefficient may be given as,<br />
D<br />
Q<br />
/ kT<br />
D0e<br />
(2.5)<br />
where D0 is called <strong>the</strong> pre-exponential factor and is temperature <strong>in</strong>dependent and Q is<br />
<strong>the</strong> energy <strong>of</strong> activation for diffusion which depends on <strong>the</strong> diffus<strong>in</strong>g species.<br />
In general <strong>the</strong> formation energy <strong>of</strong> <strong>in</strong>terstitials is considerably higher than that <strong>of</strong><br />
vacancies but <strong>the</strong>ir migration energy is far lower. This is due to <strong>the</strong> considerable<br />
lattice stra<strong>in</strong> created by <strong>the</strong> <strong>in</strong>terstitial and thus reliev<strong>in</strong>g this stra<strong>in</strong> by means <strong>of</strong><br />
migration should be highly probable. If <strong>in</strong>terstitials are formed <strong>the</strong>rmally or by o<strong>the</strong>r<br />
means <strong>the</strong>y will rapidly migrate to s<strong>in</strong>ks. Such s<strong>in</strong>ks will <strong>in</strong>clude <strong>the</strong> surface, o<strong>the</strong>r<br />
<strong>in</strong>terstitials or vacancies and <strong>extended</strong> <strong>defects</strong>. If <strong>the</strong> <strong>in</strong>terstitials come <strong>in</strong>to contact<br />
<strong>the</strong>y will form stable clusters and if <strong>the</strong>y come <strong>in</strong>to contact with vacancies mutual<br />
annihilation will occur. If two <strong>in</strong>terstitials form a cluster it is termed a di-<strong>in</strong>terstitial<br />
and if three form a cluster it is a tri-<strong>in</strong>terstitial etc. The probability <strong>of</strong> migration <strong>of</strong><br />
<strong>the</strong>se clusters with<strong>in</strong> <strong>the</strong> lattice become <strong>in</strong>creas<strong>in</strong>gly less probable as <strong>the</strong>y <strong>in</strong>crease <strong>in</strong><br />
size and will when reach<strong>in</strong>g a sufficient size act as a s<strong>in</strong>k for <strong>the</strong> migration <strong>of</strong> o<strong>the</strong>r<br />
po<strong>in</strong>t <strong>defects</strong>. The same process holds for vacancy migration with <strong>the</strong> only difference<br />
be<strong>in</strong>g that <strong>the</strong> migration energy for vacancies are much higher than that <strong>of</strong> <strong>in</strong>terstitials.<br />
In general <strong>the</strong> migration energy <strong>of</strong> substitutional impurities <strong>in</strong> <strong>the</strong> lattice is also higher<br />
than that <strong>of</strong> <strong>the</strong> <strong>in</strong>terstitial.<br />
2.4. Dislocations<br />
2.4.1 Introduction<br />
In this section <strong>the</strong> <strong>the</strong>ory <strong>of</strong> dislocations is presented. Concepts <strong>of</strong> dislocation l<strong>in</strong>e and<br />
Burgers vector are expla<strong>in</strong>ed and also different types <strong>of</strong> dislocations <strong>in</strong> a crystal are<br />
discussed. The motion <strong>of</strong> dislocations through <strong>the</strong> processes <strong>of</strong> glide and climb is
11<br />
expla<strong>in</strong>ed and <strong>the</strong> concept <strong>of</strong> partial dislocations and slip discussed s<strong>in</strong>ce it is<br />
important <strong>in</strong> <strong>the</strong> understand<strong>in</strong>g <strong>of</strong> <strong>the</strong> formation <strong>of</strong> stack<strong>in</strong>g faults.<br />
2.4.2. The Dislocation L<strong>in</strong>e and Burgers Vector<br />
To understand <strong>the</strong> concept <strong>of</strong> <strong>the</strong> dislocation l<strong>in</strong>e and Burgers vector <strong>the</strong> follow<strong>in</strong>g<br />
geometrical analogy may be followed and is shown <strong>in</strong> Fig. 2.7. Imag<strong>in</strong>e a dislocation<br />
be<strong>in</strong>g produced by mak<strong>in</strong>g a cut <strong>in</strong> a perfect crystal. Follow<strong>in</strong>g this <strong>the</strong> two sides <strong>of</strong><br />
<strong>the</strong> crystal are shifted with respect to one ano<strong>the</strong>r by a translation vector <strong>of</strong> <strong>the</strong> lattice.<br />
Then <strong>the</strong> miss<strong>in</strong>g material is added or surplus material removed. The shifted surface<br />
<strong>of</strong> <strong>the</strong> cut may grow toge<strong>the</strong>r <strong>in</strong> such a way that <strong>the</strong> cut is no longer discernable. Thus<br />
at <strong>the</strong> <strong>in</strong>side <strong>of</strong> <strong>the</strong> crystal, at <strong>the</strong> edge <strong>of</strong> <strong>the</strong> former cut surface, a l<strong>in</strong>e <strong>of</strong> strongly<br />
disturbed material rema<strong>in</strong>s. This l<strong>in</strong>e is called <strong>the</strong> “dislocation l<strong>in</strong>e”.<br />
(a) (b)<br />
(c)<br />
Fig 2.7. The geometrical analogy <strong>of</strong> a Volterra-cut describ<strong>in</strong>g a dislocation present<br />
<strong>in</strong> a crystal (from Bollmann (1970)). (a) A cut is made <strong>in</strong> a perfect crystal. (b) The two<br />
sides <strong>of</strong> <strong>the</strong> crystal are shifted with respect to one ano<strong>the</strong>r by a translation vector <strong>of</strong><br />
<strong>the</strong> lattice. (c) The shifted surface <strong>of</strong> <strong>the</strong> cut may grow toge<strong>the</strong>r <strong>in</strong> such a way that <strong>the</strong><br />
cut is no longer discernable.
12<br />
The dislocation is surrounded by perfect but elastically deformed material and <strong>the</strong> cut<br />
is called a “Volterra-cut”. With<strong>in</strong> a crystal, dislocations are characterised by two<br />
parameters:<br />
1. <strong>the</strong> path <strong>of</strong> <strong>the</strong> l<strong>in</strong>e <strong>in</strong> <strong>the</strong> crystal,<br />
2. <strong>the</strong> Burgers vector (a measure <strong>of</strong> <strong>the</strong> strength <strong>of</strong> <strong>the</strong> dislocation).<br />
Also <strong>the</strong> material <strong>in</strong> <strong>the</strong> immediate surround<strong>in</strong>gs <strong>of</strong> <strong>the</strong> dislocation is called <strong>the</strong><br />
“dislocation core”. The Burgers vector is def<strong>in</strong>ed by compar<strong>in</strong>g <strong>the</strong> disturbed crystal<br />
with <strong>the</strong> perfect. An arbitrary l<strong>in</strong>e sense is attributed to <strong>the</strong> dislocation l<strong>in</strong>e. A closed<br />
circuit <strong>in</strong> <strong>the</strong> sense <strong>of</strong> a right-handed screw is <strong>the</strong>n made around <strong>the</strong> dislocation l<strong>in</strong>e <strong>in</strong><br />
an area <strong>of</strong> unperturbed material. This closed circuit is <strong>the</strong>n repeated step by step with<br />
reference to <strong>the</strong> perfect crystal <strong>in</strong> which <strong>the</strong> circuit will rema<strong>in</strong> open. The vector<br />
po<strong>in</strong>t<strong>in</strong>g from <strong>the</strong> end to <strong>the</strong> beg<strong>in</strong>n<strong>in</strong>g <strong>in</strong> <strong>the</strong> reference crystal is def<strong>in</strong>ed as <strong>the</strong><br />
Burgers vector. The procedure is shown <strong>in</strong> Fig. 2.8.<br />
(a)<br />
(c)<br />
(b)<br />
(d)<br />
Fig. 2.8. The Burgers circuit used <strong>in</strong> def<strong>in</strong><strong>in</strong>g <strong>the</strong> Burgers vector <strong>of</strong> a dislocation<br />
(from Bollmann (1970)). (a) A closed circuit <strong>in</strong> <strong>the</strong> sense <strong>of</strong> a right-handed screw is<br />
made around <strong>the</strong> dislocation l<strong>in</strong>e <strong>in</strong> an unperturbed material. (b) The closed circuit is<br />
repeated step by step with reference to <strong>the</strong> perfect crystal <strong>in</strong> which <strong>the</strong> circuit will<br />
rema<strong>in</strong> open. The vector po<strong>in</strong>t<strong>in</strong>g from <strong>the</strong> end to <strong>the</strong> beg<strong>in</strong>n<strong>in</strong>g <strong>in</strong> <strong>the</strong> reference<br />
crystal is def<strong>in</strong>ed as <strong>the</strong> Burgers vector. (c) and (d) The same procedure is followed<br />
with <strong>the</strong> circuit be<strong>in</strong>g closed <strong>in</strong> <strong>the</strong> distorted crystal.
13<br />
Straight dislocations fur<strong>the</strong>rmore can be categorised <strong>in</strong>to edge and screw dislocations.<br />
An edge dislocation may be seen as a result <strong>of</strong> an extra plane <strong>in</strong>serted <strong>in</strong>to <strong>the</strong> material<br />
and a screw as a shear <strong>of</strong> two consecutive planes such that it seems that two orig<strong>in</strong>al<br />
parallel crystal planes are now jo<strong>in</strong>ed as a helical surface (shown <strong>in</strong> Fig. 2.9.).<br />
Fig. 2.9. A geometrical description <strong>of</strong> a screw dislocation present <strong>in</strong> a crystal (from<br />
Bollmann (1970)). (a) A cut is made <strong>in</strong> <strong>the</strong> perfect crystal. (b) The two sides <strong>of</strong> <strong>the</strong><br />
crystal is sheared by a translation vector <strong>of</strong> <strong>the</strong> lattice.<br />
The dislocation l<strong>in</strong>e and Burgers vectors <strong>of</strong> an edge dislocation are always normal to<br />
one o<strong>the</strong>r and those <strong>of</strong> a screw dislocation are parallel. In general a dislocation may be<br />
seen as a comb<strong>in</strong>ation <strong>of</strong> edge and screw dislocations with associated edge and screw<br />
Burgers vector components given by,<br />
a) screw component bscrew = b·cos(α)<br />
b) edge component bedge = b·s<strong>in</strong>(α)<br />
where α is <strong>the</strong> angle between <strong>the</strong> dislocation l<strong>in</strong>e l and <strong>the</strong> Burgers vector b <strong>of</strong> <strong>the</strong><br />
dislocation.<br />
(a) (b)<br />
2.4.3. Motion <strong>of</strong> Dislocations<br />
It is possible for a dislocation to move with<strong>in</strong> a material under plastic deformation.<br />
This may be understood by <strong>the</strong> concepts <strong>of</strong> glide and climb <strong>of</strong> a dislocation.
14<br />
Dislocation glide takes place on <strong>the</strong> slip plane which is <strong>the</strong> {111} for fcc and z<strong>in</strong>c<br />
blende structures and is a result <strong>of</strong> atoms above and below <strong>the</strong> plane be<strong>in</strong>g displaced<br />
by b with respect to one ano<strong>the</strong>r.<br />
Fig. 2.10. A representation <strong>of</strong> <strong>the</strong> process <strong>of</strong> glide <strong>in</strong> a crystal (from Bollmann<br />
(1970))<br />
If a dislocation climbs it moves perpendicular to <strong>the</strong> glide plane. It is only possible if<br />
<strong>the</strong> dislocation has an edge component and if <strong>the</strong> temperature and stress is sufficient<br />
for diffusion to be possible.<br />
Fig. 2.11. A representation <strong>of</strong> <strong>the</strong> process <strong>of</strong> climb <strong>in</strong> a crystal (from Bollmann<br />
(1970))<br />
In general glide is <strong>the</strong> only mechanism present at low temperatures dur<strong>in</strong>g plastic<br />
deformation. Also from <strong>the</strong>se mechanisms <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> slip plane may be<br />
made. It is def<strong>in</strong>ed as <strong>the</strong> plane which conta<strong>in</strong>s both <strong>the</strong> dislocation l<strong>in</strong>e and Burgers<br />
vector <strong>of</strong> <strong>the</strong> dislocation.
2.5 Partial Dislocations and Slip<br />
15<br />
In <strong>the</strong> fcc structure slip occurs on <strong>the</strong> close-packed planes {111} with <strong>the</strong> slip<br />
direction . Slip <strong>in</strong>volves <strong>the</strong> slid<strong>in</strong>g <strong>of</strong> close-packed planes <strong>of</strong> atoms over each<br />
o<strong>the</strong>r. To see how this can occur <strong>the</strong> follow<strong>in</strong>g may be done. Consider that <strong>the</strong> close-<br />
packed planes can be simulated by a set <strong>of</strong> hard spheres.<br />
Fig. 2.12. The simulation <strong>of</strong> close-packed planes by a set <strong>of</strong> hard spheres to describe<br />
<strong>the</strong> concept <strong>of</strong> slip <strong>in</strong> a crystal lattice (from Hull et al. (1984))<br />
The full circles represents <strong>the</strong> A layer, <strong>the</strong> sites marked B <strong>the</strong> B layer and C <strong>the</strong> C<br />
layer. Consider next <strong>the</strong> movement <strong>of</strong> <strong>the</strong> layers when <strong>the</strong>y are sheared over each<br />
o<strong>the</strong>r. It will be found that that <strong>the</strong> B layer <strong>of</strong> atoms will first move to <strong>the</strong> nearby C<br />
site and <strong>the</strong>n to <strong>the</strong> next B site <strong>in</strong>stead <strong>of</strong> directly mov<strong>in</strong>g from B site to B site. Thus<br />
<strong>the</strong> unit lattice displacement b1 is achieved by two movements about <strong>the</strong> C positions<br />
represented by b2 and b3. If one considers <strong>the</strong> movement <strong>of</strong> <strong>the</strong> unit 1/2 <br />
dislocation <strong>in</strong> <strong>the</strong> fcc lattice it implies that <strong>the</strong> motion will take place via two<br />
dislocations termed partial dislocations with Burgers vectors b2 and b3 <strong>in</strong> a reaction <strong>of</strong><br />
<strong>the</strong> type,<br />
1<br />
2<br />
110 <br />
1 1 <br />
211 121<br />
<br />
6 6<br />
(2.6)<br />
This occurs s<strong>in</strong>ce it is energetically favourable for <strong>the</strong> edge dislocation to dissociate<br />
accord<strong>in</strong>g to Frank’s rule with b 2 edge = a 2 /2 which is greater than bp1 2 + bp2 2 = a 2 /3.
16<br />
2.6 Dislocations <strong>in</strong> <strong>the</strong> Z<strong>in</strong>c-Blende Structure<br />
As <strong>in</strong> fcc metals, slip occurs <strong>in</strong> <strong>the</strong> z<strong>in</strong>c blende structure on close packed planes along<br />
close packed directions. The predom<strong>in</strong>ant slip system is {111} with <strong>the</strong><br />
majority <strong>of</strong> dislocations ly<strong>in</strong>g along and directions.<br />
The asymmetry <strong>in</strong> <strong>the</strong> z<strong>in</strong>c-blende lattice causes two possible dislocations <strong>in</strong> <strong>the</strong><br />
lattice for identical Burgers vector and dislocation l<strong>in</strong>e direction. As discussed earlier<br />
<strong>the</strong> {111} layers consist <strong>of</strong> sets <strong>of</strong> closely spaced double layers <strong>of</strong> <strong>the</strong> constituent<br />
atoms with small <strong>in</strong>ternal spac<strong>in</strong>g<br />
a 3a<br />
and large external spac<strong>in</strong>g <strong>of</strong> . The glide<br />
4 3<br />
4<br />
<strong>of</strong> a dislocation can <strong>the</strong>n ei<strong>the</strong>r cut through <strong>the</strong> small <strong>in</strong>ternal spac<strong>in</strong>g or large external<br />
spac<strong>in</strong>g. Two sets <strong>of</strong> dislocations are thus generated depend<strong>in</strong>g on <strong>the</strong> glide system.<br />
The set <strong>of</strong> dislocations caus<strong>in</strong>g slip between <strong>the</strong> closest spaced layers is called <strong>the</strong><br />
glide set while <strong>the</strong> o<strong>the</strong>r called <strong>the</strong> shuffle set. The shuffle set is commonly assumed<br />
to be <strong>the</strong> correct description s<strong>in</strong>ce fewer bonds have to be broken for <strong>the</strong> dislocations<br />
to move across <strong>the</strong> {111} planes.<br />
The properties <strong>of</strong> dislocations <strong>in</strong> <strong>the</strong> z<strong>in</strong>c-blende structure may be expla<strong>in</strong>ed <strong>in</strong> terms<br />
<strong>of</strong> <strong>the</strong> concept <strong>of</strong> a 60° dislocation. In this configuration <strong>the</strong> slip plane is <strong>of</strong> <strong>the</strong> {111}<br />
type with <strong>the</strong> dislocation l<strong>in</strong>e and Burgers vector mak<strong>in</strong>g an angle <strong>of</strong> 60° with one<br />
ano<strong>the</strong>r. This is illustrated <strong>in</strong> Fig. 2.13.
17<br />
Fig. 2.13. The geometrical setup <strong>of</strong> <strong>the</strong> sixty degree dislocation <strong>in</strong> a z<strong>in</strong>c-blende<br />
crystal lattice (from Stirland et al. (1976))<br />
There are three possible sixty degree dislocations that could lie on one {111} plane.<br />
The core <strong>of</strong> <strong>the</strong> dislocation conta<strong>in</strong>s atoms with broken bonds and <strong>the</strong> axis <strong>of</strong><br />
<strong>the</strong> dislocation lies along a set <strong>of</strong> identical type atoms. If GaAs is used as an example,<br />
<strong>the</strong> broken bonds may belong to ei<strong>the</strong>r Ga or As atoms. The dislocations are called an<br />
α and β dislocation respectively depend<strong>in</strong>g on whe<strong>the</strong>r <strong>the</strong> broken bonds belong to Ga<br />
or As atoms respectively. This situation is illustrated <strong>in</strong> Fig. 2.14.<br />
Fig. 2.14. The (a) α and (b) β type sixty degree dislocations <strong>in</strong> a z<strong>in</strong>c-blende crystal<br />
lattice (from Stirland et al. (1976))
18<br />
The polarity <strong>of</strong> <strong>the</strong> broken bonds <strong>in</strong> both situations differ and thus <strong>the</strong> α dislocation is<br />
positive and <strong>the</strong> β negative. Accord<strong>in</strong>gly <strong>the</strong>y will have different mobilities and<br />
activation energies. The dotted l<strong>in</strong>es <strong>in</strong> <strong>the</strong> figure <strong>in</strong>dicate <strong>the</strong> extra (double) half-<br />
planes <strong>of</strong> Ga-As atoms which generate <strong>the</strong> edge component <strong>of</strong> <strong>the</strong> two types <strong>of</strong><br />
dislocations.<br />
2.7 Stack<strong>in</strong>g Faults<br />
A stack<strong>in</strong>g fault is a planar defect <strong>in</strong> which <strong>the</strong> regular stack<strong>in</strong>g sequence <strong>of</strong> a local<br />
region with<strong>in</strong> <strong>the</strong> crystal has been disrupted. These faults are ma<strong>in</strong>ly <strong>in</strong>troduced due to<br />
plastic deformation through mechanical or <strong>the</strong>rmal stra<strong>in</strong> but may also be <strong>in</strong>troduced<br />
<strong>in</strong>to a crystal dur<strong>in</strong>g a CVD growth process. Stack<strong>in</strong>g faults are not expected <strong>in</strong> an<br />
ABABAB… type stack<strong>in</strong>g sequence s<strong>in</strong>ce no alternative configuration for an A layer<br />
rest<strong>in</strong>g on a B exists. However stack<strong>in</strong>g faults are possible <strong>in</strong> <strong>the</strong> ABCABC… type<br />
stack<strong>in</strong>g sequence s<strong>in</strong>ce alternate stack<strong>in</strong>g configurations do exist. This is frequently<br />
seen <strong>in</strong> <strong>the</strong> close-packed {111} planes for <strong>the</strong> fcc and z<strong>in</strong>c-blende structures.<br />
Fig. 2.15. The (a) <strong>in</strong>tr<strong>in</strong>sic and (b) extr<strong>in</strong>sic stack<strong>in</strong>g faults <strong>in</strong> an fcc lattice (from Hull<br />
et al. (1984))<br />
(a)<br />
(b)<br />
Two types <strong>of</strong> stack<strong>in</strong>g faults are possible, termed <strong>in</strong>tr<strong>in</strong>sic and extr<strong>in</strong>sic. Intr<strong>in</strong>sic<br />
stack<strong>in</strong>g faults are formed as a result <strong>of</strong> <strong>the</strong> removal <strong>of</strong> a layer <strong>in</strong> <strong>the</strong> stack<strong>in</strong>g<br />
sequence, as shown <strong>in</strong> Fig. 2.15(a) where part <strong>of</strong> a C layer has been removed, and
19<br />
extr<strong>in</strong>sic stack<strong>in</strong>g faults are <strong>the</strong> <strong>in</strong>verse <strong>of</strong> <strong>in</strong>tr<strong>in</strong>sic and relates to <strong>the</strong> <strong>in</strong>sertion <strong>of</strong> an<br />
extra layer between two exist<strong>in</strong>g layers as <strong>in</strong> Fig. 2.15(b) where an A layer has been<br />
<strong>in</strong>serted between a C and B.<br />
Fig. 2.16. A description <strong>of</strong> <strong>the</strong> modification <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g sequence <strong>in</strong> a lattice due to<br />
<strong>the</strong> presence <strong>of</strong> <strong>in</strong>tr<strong>in</strong>sic and extr<strong>in</strong>sic stack<strong>in</strong>g faults (from Käckell et al. (1998))<br />
Fur<strong>the</strong>rmore if an <strong>in</strong>tr<strong>in</strong>sic stack<strong>in</strong>g fault is <strong>in</strong>troduced by <strong>the</strong> removal <strong>of</strong> a B layer for<br />
example, it creates a stack<strong>in</strong>g sequence ABCA/CABC which correlates to only one<br />
modified layer (Fig. 2.16.). When a C layer for example is <strong>in</strong>serted between an A and<br />
B such as <strong>in</strong> <strong>the</strong> case <strong>of</strong> an extr<strong>in</strong>sic stack<strong>in</strong>g fault it creates a stack<strong>in</strong>g sequence<br />
ABCA/C/BCABC which implies <strong>the</strong> modification <strong>of</strong> two layers.
20<br />
(a) (b)<br />
Fig. 2.17. The dissociation <strong>of</strong> (a) an edge dislocation <strong>in</strong> an fcc lattice <strong>in</strong>to (b) two<br />
partial dislocations bound<strong>in</strong>g between <strong>the</strong>m a ribbon <strong>of</strong> stack<strong>in</strong>g fault (from Hull et<br />
al. (1984))<br />
A stack<strong>in</strong>g fault is generated by <strong>the</strong> dissociation <strong>of</strong> an edge dislocation <strong>in</strong>to two partial<br />
dislocations by a reaction <strong>of</strong> <strong>the</strong> type,<br />
_ _ _<br />
b edge b p b 1 p2<br />
(2.7)<br />
An edge dislocation present <strong>in</strong> <strong>the</strong> crystal with Burgers vector <strong>of</strong> <strong>the</strong> type 1/2 <br />
(Fig. 2.17(a)) dissociates <strong>in</strong>to two partials with Burgers vector <strong>of</strong> <strong>the</strong> type 1/6 <br />
type through a reaction shown <strong>in</strong> equation 2.7 (Fig. 2.17(b)). With <strong>the</strong> dissociation <strong>the</strong><br />
two partial dislocations leave between <strong>the</strong>m a stack<strong>in</strong>g fault and are <strong>in</strong> turn referred to<br />
as <strong>the</strong> bond<strong>in</strong>g partials. The associated energy per unit area γ <strong>of</strong> a stack<strong>in</strong>g fault is<br />
known as <strong>the</strong> stack<strong>in</strong>g-fault energy and is given by <strong>the</strong> relation,<br />
[ E ( faulted)<br />
E(<br />
perfect)]<br />
/ A<br />
(2.8)<br />
with E be<strong>in</strong>g <strong>the</strong> energy and A <strong>the</strong> area. Values typically lie <strong>in</strong> <strong>the</strong> range 1-1000<br />
mJ/m 2 .
2.8 Tw<strong>in</strong>n<strong>in</strong>g<br />
21<br />
Tw<strong>in</strong>n<strong>in</strong>g is a process <strong>in</strong> which a region <strong>of</strong> <strong>the</strong> crystal is deformed through a<br />
homogenous shear process. This produces <strong>the</strong> orig<strong>in</strong>al crystal structure but <strong>in</strong> a<br />
different orientation. The simplest case is <strong>the</strong> result <strong>in</strong> which <strong>the</strong> tw<strong>in</strong>ned area is a<br />
mirror image <strong>of</strong> <strong>the</strong> orig<strong>in</strong>al crystal. This is depicted <strong>in</strong> Fig. 2.18.<br />
Fig. 2.18. A geometrical description <strong>of</strong> tw<strong>in</strong>n<strong>in</strong>g <strong>in</strong> a crystal lattice (from Hull et al.<br />
(1984))<br />
The open circles represent <strong>the</strong> position <strong>of</strong> <strong>the</strong> atoms before tw<strong>in</strong>n<strong>in</strong>g and <strong>the</strong> filled<br />
circles after. The l<strong>in</strong>e x-y is termed <strong>the</strong> tw<strong>in</strong> boundary with atoms above and below it<br />
mirror images <strong>of</strong> one ano<strong>the</strong>r. Tw<strong>in</strong>n<strong>in</strong>g may occur due to plastic deformation or<br />
<strong>in</strong>troduced <strong>in</strong>to <strong>the</strong> material dur<strong>in</strong>g <strong>the</strong> CVD growth process. It differs from <strong>the</strong><br />
process <strong>of</strong> slip s<strong>in</strong>ce no rotation <strong>of</strong> <strong>the</strong> lattice is tak<strong>in</strong>g place. In <strong>the</strong> fcc and z<strong>in</strong>c-<br />
blende structures tw<strong>in</strong>n<strong>in</strong>g takes place on <strong>the</strong> {111} system. It is as a result <strong>of</strong><br />
a shear caus<strong>in</strong>g a displacement <strong>of</strong> 1/6 on each consecutive plane and can be<br />
seen as <strong>the</strong> motion <strong>of</strong> partial dislocations with b = 1/6 on consecutive {111}<br />
planes ly<strong>in</strong>g parallel to x-y.<br />
Fur<strong>the</strong>rmore a modification to <strong>the</strong> electron diffraction pattern obta<strong>in</strong>ed from a tw<strong>in</strong>ned<br />
area <strong>in</strong> a crystal also occurs. The tw<strong>in</strong>ned sections generate a different set <strong>of</strong><br />
diffractions spots accord<strong>in</strong>g to its crystallographic orientation which are superimposed<br />
onto <strong>the</strong> diffraction pattern from <strong>the</strong> perfect crystal. Such a situation is depicted <strong>in</strong><br />
Fig. 2.19.
22<br />
Fig. 2.19. The modification <strong>of</strong> <strong>the</strong> diffraction pattern due to tw<strong>in</strong>n<strong>in</strong>g <strong>in</strong> a crystal with<br />
<strong>the</strong> electron beam along <strong>the</strong> (from Hirsch et al. (1965))<br />
The f<strong>in</strong>al orientation <strong>of</strong> a tw<strong>in</strong> platelet and correspond<strong>in</strong>g tw<strong>in</strong> spots may be obta<strong>in</strong>ed,<br />
<strong>in</strong> <strong>the</strong> case <strong>of</strong> fcc crystals, by a 180° rotation <strong>of</strong> <strong>the</strong> lattice about <strong>the</strong> {111} planes.<br />
This is seen by rotat<strong>in</strong>g <strong>the</strong> diffraction pattern from <strong>the</strong> perfect crystal by 180° and<br />
superimpos<strong>in</strong>g it onto <strong>the</strong> orig<strong>in</strong>al (Fig. 2.17). This example is called primary<br />
tw<strong>in</strong>n<strong>in</strong>g <strong>in</strong> fcc materials.<br />
2.9 Vacancy and Interstitial Loops<br />
If a material conta<strong>in</strong>s a concentration <strong>of</strong> vacancies and <strong>in</strong>terstitials <strong>in</strong> excess <strong>of</strong> <strong>the</strong><br />
equilibrium concentration <strong>the</strong>se po<strong>in</strong>t <strong>defects</strong> may diffuse and coalesce to form<br />
clusters and precipitate as platelets which will collapse to form a dislocation loop as<br />
<strong>the</strong>y grow larger as shown <strong>in</strong> Fig. 2.20.
23<br />
Fig. 2.20. The process <strong>of</strong> vacancy loop formation <strong>in</strong> a lattice (from Hull et al. (1984))<br />
This figure depicts <strong>the</strong> process for vacancy loop formation. The vacancies cluster to<br />
form a platelet with <strong>the</strong> subsequent collapse <strong>of</strong> <strong>the</strong> planes <strong>in</strong>ward to form a dislocation<br />
loop. The dislocation loop may under certa<strong>in</strong> conditions enclose an area constitut<strong>in</strong>g a<br />
stack<strong>in</strong>g fault. The stack<strong>in</strong>g fault may be removed by <strong>the</strong> motion <strong>of</strong> a partial<br />
dislocation across <strong>the</strong> loop. Fur<strong>the</strong>rmore, loops <strong>in</strong> general lie on close-packed planes.
3.1. Introduction<br />
24<br />
CHAPTER THREE<br />
ION IMPLANTATION AND RADIATION DAMAGE<br />
In this chapter a model <strong>of</strong> <strong>the</strong> implantation <strong>of</strong> ions <strong>in</strong>to matter and <strong>the</strong> subsequent<br />
radiation damage is presented. Theories for <strong>the</strong> calculation <strong>of</strong> ion ranges are discussed<br />
and reference is made to <strong>the</strong> most popular approach, that <strong>of</strong> <strong>the</strong> LSS <strong>the</strong>ory, to<br />
illustrate <strong>the</strong> procedure taken <strong>in</strong> do<strong>in</strong>g such a calculation. The result shown <strong>in</strong> Chapter<br />
6 was however obta<strong>in</strong>ed us<strong>in</strong>g s<strong>of</strong>tware developed by Ziegler et al. (1985) us<strong>in</strong>g<br />
Monte Carlo methods. Follow<strong>in</strong>g this <strong>the</strong>ory <strong>of</strong> displacement damage and anneal<strong>in</strong>g<br />
<strong>of</strong> <strong>defects</strong> are presented.<br />
3.2. Ion ranges<br />
3.2.1 Introduction<br />
Dur<strong>in</strong>g ion implantation, <strong>the</strong> implanted species can lose energy to <strong>the</strong> target atoms <strong>in</strong><br />
two ways, namely electronic excitations (<strong>in</strong>elastic) and nuclear collisions (elastic). At<br />
higher energies <strong>the</strong> ion is stripped <strong>of</strong> some or all <strong>of</strong> its electrons and multiply ionized<br />
at which stage <strong>the</strong> energy loss is ma<strong>in</strong>ly due to <strong>in</strong>elastic processes, i.e. electronic<br />
excitation. Occasionally <strong>the</strong> mov<strong>in</strong>g atom will <strong>in</strong>teract directly with lattice atoms and<br />
collisions <strong>of</strong> <strong>the</strong> Ru<strong>the</strong>rford type results. As <strong>the</strong> k<strong>in</strong>etic energy <strong>of</strong> <strong>the</strong> mov<strong>in</strong>g atom<br />
decreases its degree <strong>of</strong> ionization does as well until <strong>the</strong> atom essentially becomes<br />
neutral at which stage collisions <strong>of</strong> <strong>the</strong> hard-sphere type will dom<strong>in</strong>ate. The rate at<br />
which energy is lost is given <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g form,<br />
dE<br />
N[<br />
S n ( E)<br />
Se<br />
( E)]<br />
(3.1)<br />
dx<br />
where N is <strong>the</strong> number <strong>of</strong> target atoms per unit volume, Sn(E) and Se(E) def<strong>in</strong>ed as <strong>the</strong><br />
nuclear and electronic stopp<strong>in</strong>g powers respectively. The nuclear stopp<strong>in</strong>g power<br />
Sn(E) is given by,
S n Td<br />
(3.2)<br />
25<br />
and is also referred to as <strong>the</strong> stopp<strong>in</strong>g cross-section. T is <strong>the</strong> energy transferred and dσ<br />
<strong>the</strong> differential cross-section. The electronic stopp<strong>in</strong>g power is proportional to <strong>the</strong><br />
velocity <strong>of</strong> <strong>the</strong> <strong>in</strong>com<strong>in</strong>g ion and thus is proportional to E k<strong>in</strong> , <strong>the</strong> k<strong>in</strong>etic energy <strong>of</strong><br />
<strong>the</strong> particle. Us<strong>in</strong>g equation 3.1 <strong>the</strong> implanted range R is given by,<br />
0<br />
1<br />
( ) ( )<br />
E<br />
dE<br />
R (3.3)<br />
N S E S E<br />
0<br />
n<br />
e<br />
with E0 be<strong>in</strong>g <strong>the</strong> energy <strong>of</strong> implantation.<br />
3.2.2 Range Calculations<br />
The first step taken <strong>in</strong> f<strong>in</strong>d<strong>in</strong>g <strong>the</strong> equation relat<strong>in</strong>g <strong>the</strong> nuclear stopp<strong>in</strong>g power is by<br />
solv<strong>in</strong>g <strong>the</strong> scatter<strong>in</strong>g <strong>in</strong>tegral (equation 3.4) for <strong>the</strong> centre <strong>of</strong> mass system scatter<strong>in</strong>g<br />
angle θ given by,<br />
U m<br />
du<br />
2 p <br />
(3.4)<br />
0 V ( u)<br />
2 2<br />
1<br />
p u<br />
E<br />
r<br />
where p is <strong>the</strong> impact parameter, u = 1/r with r <strong>the</strong> distance between collid<strong>in</strong>g atoms.<br />
V(u) <strong>the</strong> <strong>in</strong>teratomic potential and<br />
energy. Also Um satisfies <strong>the</strong> follow<strong>in</strong>g,<br />
1<br />
V ( U<br />
)<br />
p<br />
m 2 2<br />
U m<br />
E<br />
r<br />
0<br />
E r<br />
M 1M<br />
2<br />
E<br />
with E <strong>the</strong> projectile’s<br />
M M M )<br />
1(<br />
1 2<br />
which represents <strong>the</strong> reciprocal <strong>of</strong> <strong>the</strong> m<strong>in</strong>imum distance <strong>of</strong> approach.<br />
(3.5)<br />
Solv<strong>in</strong>g this <strong>in</strong>tegral <strong>the</strong> scatter<strong>in</strong>g angle θ is <strong>the</strong>n used to obta<strong>in</strong> <strong>the</strong> energy<br />
transferred to a struck atom by <strong>the</strong> relation,
4M 1M<br />
2 2 <br />
2 <br />
T E<br />
s<strong>in</strong> T s<strong>in</strong><br />
2<br />
m<br />
(3.6)<br />
( M M ) 2 2<br />
1<br />
2<br />
26<br />
From this <strong>the</strong> differential cross-section d 2pdp<br />
may be obta<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong><br />
impact parameter p which <strong>in</strong> turn is determ<strong>in</strong>ed by us<strong>in</strong>g equation 3.6. Follow<strong>in</strong>g this<br />
<strong>the</strong> nuclear stopp<strong>in</strong>g power Sn(E) may be determ<strong>in</strong>ed by solv<strong>in</strong>g equation 3.2 between<br />
<strong>the</strong> limits 0 to <strong>the</strong> maximum energy transferred<br />
4M<br />
M<br />
E<br />
1 2<br />
Tm (3.7)<br />
2<br />
( M 1 M 2 )<br />
In f<strong>in</strong>d<strong>in</strong>g Sn(E) <strong>the</strong> differential cross-section dσ is found which <strong>in</strong> turn depends<br />
heavily on <strong>the</strong> choice <strong>of</strong> <strong>the</strong> <strong>in</strong>teratomic potential V(r) used. Also <strong>in</strong> most cases <strong>the</strong><br />
scatter<strong>in</strong>g <strong>in</strong>tegral cannot be solved analytically and numerical methods should be<br />
employed which br<strong>in</strong>gs with it an added complexity. The most widely used <strong>the</strong>ory <strong>in</strong><br />
<strong>the</strong> prediction <strong>of</strong> ion ranges <strong>in</strong> a solid is <strong>the</strong> LSS <strong>the</strong>ory developed by L<strong>in</strong>hard, Scharf<br />
and Schiott. In <strong>the</strong> follow<strong>in</strong>g section <strong>the</strong> ma<strong>in</strong> results <strong>of</strong> this <strong>the</strong>ory will be discussed<br />
but it is left to <strong>the</strong> reader to consult fur<strong>the</strong>r references for a more detailed<br />
understand<strong>in</strong>g.<br />
3.2.3. The L<strong>in</strong>hard, Scharff and Schiott (LSS) Theory<br />
The LSS <strong>the</strong>ory uses a Thomas-Fermi model <strong>of</strong> <strong>the</strong> <strong>in</strong>teraction between heavy ions to<br />
derive a nuclear stropp<strong>in</strong>g power Sn and electronic stopp<strong>in</strong>g power Se. The electronic<br />
stopp<strong>in</strong>g power is proportional to <strong>the</strong> velocity <strong>of</strong> <strong>the</strong> mov<strong>in</strong>g atom s<strong>in</strong>ce <strong>the</strong> process <strong>of</strong><br />
energy loss is dependent on <strong>the</strong> k<strong>in</strong>etic energy <strong>of</strong> <strong>the</strong> mov<strong>in</strong>g atom and <strong>the</strong> nuclear<br />
stopp<strong>in</strong>g power is obta<strong>in</strong>ed through <strong>the</strong> procedure shown <strong>in</strong> <strong>the</strong> previous section.<br />
The <strong>in</strong>teratomic potential used has <strong>the</strong> form,<br />
2<br />
Z1Z<br />
2e<br />
r <br />
V ( r)<br />
<br />
<br />
(3.8)<br />
4 r a <br />
0
27<br />
where a = a0(Z1 2/3 + Z2 2/3 ) -½ and is <strong>the</strong> Thomas-Fermi screen<strong>in</strong>g function which is<br />
tabulated numerically. An approximation to <strong>the</strong> function is also given by,<br />
r<br />
r <br />
<br />
a<br />
<br />
(3.9)<br />
2 1<br />
a r 2 2 [ c ]<br />
a<br />
with c 3 result<strong>in</strong>g <strong>in</strong> <strong>the</strong> best average fit to <strong>the</strong> potential.<br />
Us<strong>in</strong>g approximation methods <strong>in</strong> f<strong>in</strong>d<strong>in</strong>g <strong>the</strong> solution <strong>the</strong> LSS <strong>the</strong>ory predicts a<br />
nuclear stopp<strong>in</strong>g power Sn <strong>of</strong> <strong>the</strong> form shown <strong>in</strong> Fig. 3.1.<br />
Fig. 3.1. Nuclear and electronic stopp<strong>in</strong>g powers <strong>in</strong> reduced units. Full-drawn curve<br />
represents <strong>the</strong> Thomas-Fermi nuclear stopp<strong>in</strong>g power, <strong>the</strong> dot and dash l<strong>in</strong>es <strong>the</strong><br />
electronic stopp<strong>in</strong>g for k=0.15 and k=1.5. The dashed l<strong>in</strong>e gives <strong>the</strong> nuclear stopp<strong>in</strong>g<br />
power for <strong>the</strong> r -2 potential (from Carter et al. (1976))<br />
The energies and distances are expressed <strong>in</strong> terms <strong>of</strong> dimensionless parameters ε and<br />
ρ given by,<br />
aM 2<br />
E<br />
(3.10)<br />
2<br />
Z Z e M M )<br />
1<br />
2<br />
( 1 2<br />
M M<br />
RN4a<br />
(3.11)<br />
2 1 2<br />
and 2<br />
( M 1 M 2 )
28<br />
The nuclear stopp<strong>in</strong>g power Sn is thus given by <strong>the</strong>se dimensionless parameters as<br />
d<br />
<br />
which is a function <strong>of</strong> ε only. The electronic stopp<strong>in</strong>g power Se is given by,<br />
d<br />
<br />
d<br />
<br />
<br />
d<br />
<br />
n<br />
e<br />
k<br />
1<br />
2<br />
1 2 1 2<br />
3 2<br />
1 0.<br />
0793Z1<br />
Z 2 ( M 1 M 2 )<br />
6<br />
with k Z1<br />
2 3 2 3 3 4 3 2 1 2<br />
( Z1<br />
Z 2 ) M 1 M 2<br />
(3.12)<br />
(3.13)<br />
A universal curve cannot be obta<strong>in</strong>ed for <strong>the</strong> electronic stopp<strong>in</strong>g power s<strong>in</strong>ce k<br />
depends on <strong>the</strong> collid<strong>in</strong>g atoms. Fig. 3.1 shows <strong>the</strong> electronic stopp<strong>in</strong>g power plotted<br />
for two values <strong>of</strong> k be<strong>in</strong>g 1.5 and 0.15. The straight l<strong>in</strong>e <strong>in</strong>dicates <strong>the</strong> dependence on<br />
velocity.<br />
Fig. 3.2. Reduced range-energy plots for various values <strong>of</strong> <strong>the</strong> electronic stopp<strong>in</strong>g<br />
parameter k (from Carter et al. (1976))<br />
The average total path length ρ is <strong>the</strong>n given by,<br />
<br />
<br />
<br />
0<br />
d { S n ( )<br />
S e ( )}<br />
(3.14)
29<br />
Us<strong>in</strong>g this expression, curves <strong>of</strong> ρ versus ε can be plotted by use <strong>of</strong> numerical<br />
<strong>in</strong>tegration for specific k values as illustrated <strong>in</strong> Fig. 3.2.<br />
If R is expressed <strong>in</strong> µg/cm 2 and E <strong>in</strong> keV <strong>the</strong>n equations 3.11 and 3.10 become,<br />
166<br />
M<br />
1<br />
R<br />
(3.15)<br />
2 / 3 2 3<br />
2<br />
( Z<br />
(<br />
1 Z 2 ) M 1 M 2 )<br />
and<br />
32.<br />
5<br />
M 2<br />
E<br />
(3.16)<br />
2 3 2 3 1 2<br />
Z Z ( Z Z ) ( M M )<br />
1<br />
2<br />
1<br />
2<br />
1<br />
2<br />
The range R is <strong>the</strong>n calculated for a specific Z1, Z2 and E comb<strong>in</strong>ation by calculat<strong>in</strong>g ε<br />
and k and simply read<strong>in</strong>g ρ from <strong>the</strong> graph us<strong>in</strong>g equation 3.11 for <strong>the</strong> conversion.<br />
Fig. 3.3. The geometrical description <strong>of</strong> <strong>the</strong> projected and lateral ranges Rp and R <br />
(from Neethl<strong>in</strong>g (1980))<br />
In practice <strong>the</strong> average projected range Rp is usually measured. It is def<strong>in</strong>ed as <strong>the</strong><br />
component <strong>of</strong> <strong>the</strong> range R parallel to <strong>the</strong> ion beam direction. The lateral range R is<br />
<strong>the</strong> component <strong>of</strong> R perpendicular to <strong>the</strong> ion beam direction. This is depicted <strong>in</strong> Fig.<br />
3.3. At energies sufficiently high <strong>the</strong> ion tends to move <strong>in</strong> a straight l<strong>in</strong>e and only<br />
undergoes appreciable angular deflection when <strong>the</strong> energy is reduced such that <strong>the</strong><br />
dom<strong>in</strong>at<strong>in</strong>g process <strong>of</strong> energy loss is due to hard-sphere collisions. Under <strong>the</strong>se<br />
circumstances <strong>the</strong> average total path length R and average projected range are<br />
approximately <strong>the</strong> same.
3.3 Defects Created by Ion Implantation<br />
3.3.1 Introduction<br />
30<br />
This section discusses <strong>in</strong> short <strong>the</strong> ma<strong>in</strong> aspect <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> displacement damage.<br />
The concept <strong>of</strong> threshold displacement energies (TDE) is discussed and <strong>the</strong> different<br />
processes <strong>in</strong>volved <strong>in</strong> <strong>the</strong> slow<strong>in</strong>g down <strong>of</strong> <strong>the</strong> ion <strong>in</strong> <strong>the</strong> lattice are expla<strong>in</strong>ed. A<br />
simple calculation is used to determ<strong>in</strong>e <strong>the</strong> number <strong>of</strong> displaced atoms per <strong>in</strong>com<strong>in</strong>g<br />
ion. The type <strong>of</strong> <strong>defects</strong> created <strong>in</strong> <strong>the</strong> damage process is discussed and <strong>the</strong>ir evolution<br />
with anneal<strong>in</strong>g is expla<strong>in</strong>ed.<br />
3.3.2 Theory <strong>of</strong> Displacement Damage<br />
The effect <strong>of</strong> atoms be<strong>in</strong>g displaced dur<strong>in</strong>g irradiation with high energy ions is<br />
perhaps <strong>the</strong> most important process tak<strong>in</strong>g place. The displacement <strong>of</strong> an atom from<br />
its regular lattice site depends on a value Ed which is <strong>the</strong> m<strong>in</strong>imum energy that is<br />
required to displace <strong>the</strong> atom from its site. If <strong>the</strong> atom receives energy <strong>in</strong> excess <strong>of</strong> Ed<br />
it is possible that fur<strong>the</strong>r displacements may result from <strong>the</strong> movement <strong>of</strong> <strong>the</strong> primary<br />
displaced atom. If a hard-sphere approximation and classical non-relativistic<br />
mechanics is used, <strong>the</strong> maximum energy transferred to a stationary atom by a particle<br />
with mass M1 and energy E is given by equation 3.7,<br />
Thus <strong>the</strong> m<strong>in</strong>imum <strong>in</strong>cident energy required to displace an atom is given by,<br />
E<br />
m<br />
2<br />
( M 1 M 2 )<br />
Ed<br />
(3.17)<br />
4M<br />
M<br />
1<br />
2<br />
To determ<strong>in</strong>e <strong>the</strong> number <strong>of</strong> displacements produced by an <strong>in</strong>com<strong>in</strong>g ion Nd it is<br />
important to understand <strong>the</strong> amount <strong>of</strong> energy transferred to <strong>the</strong> atom through<br />
collisions <strong>in</strong> <strong>the</strong> process <strong>of</strong> slow<strong>in</strong>g down.
31<br />
As stated previously hard-sphere collisions at <strong>the</strong> lower end <strong>of</strong> <strong>the</strong> energy range<br />
dom<strong>in</strong>ate as <strong>the</strong> process for atomic displacements. This occurs s<strong>in</strong>ce <strong>the</strong> k<strong>in</strong>etic<br />
energy <strong>of</strong> <strong>the</strong> ion is not sufficient to penetrate <strong>the</strong> electron cloud <strong>of</strong> <strong>the</strong> stationary<br />
atom. If <strong>the</strong> k<strong>in</strong>etic energy is sufficient to penetrate <strong>the</strong> electron cloud collisions <strong>of</strong> <strong>the</strong><br />
Ru<strong>the</strong>rford type dom<strong>in</strong>ate. To approximate <strong>the</strong> transition from hard-sphere scatter<strong>in</strong>g<br />
to Ru<strong>the</strong>rford scatter<strong>in</strong>g a <strong>the</strong>ory developed by K<strong>in</strong>ch<strong>in</strong> and Pease (1955) is<br />
considered.<br />
The <strong>the</strong>ory assumes that <strong>the</strong> presence <strong>of</strong> <strong>the</strong> electron cloud cuts <strong>of</strong>f <strong>the</strong> Coulomb<br />
<strong>in</strong>teraction between <strong>the</strong> nuclei <strong>of</strong> <strong>the</strong> mov<strong>in</strong>g and stationary atoms at a distance r0<br />
given by,<br />
r<br />
0<br />
a<br />
(3.18)<br />
3 3 1 2<br />
( Z Z )<br />
2<br />
1<br />
0<br />
2<br />
2<br />
Thus if <strong>the</strong> k<strong>in</strong>etic energy <strong>of</strong> <strong>the</strong> ion is less than <strong>the</strong> Coulomb potential between <strong>the</strong><br />
mov<strong>in</strong>g and stationary nuclei it is assumed that <strong>the</strong> collisions are <strong>of</strong> <strong>the</strong> hard-sphere<br />
type. If <strong>the</strong> k<strong>in</strong>etic energy is greater than <strong>the</strong> Coulomb potential <strong>the</strong>n <strong>the</strong> collisions are<br />
<strong>of</strong> <strong>the</strong> Ru<strong>the</strong>rford type. A k<strong>in</strong>etic energy LA at which this transition occurs is def<strong>in</strong>ed<br />
and given by,<br />
L<br />
2 3 2 3 1 2<br />
2E<br />
RZ<br />
1 Z 2 ( Z1<br />
Z 2 ) ( M 1 M 2 )<br />
A (3.19)<br />
M 1<br />
where ER is <strong>the</strong> Rydberg energy and M1 <strong>the</strong> mass <strong>of</strong> <strong>the</strong> stationary atom. Seitz and<br />
Koehler (1956) also showed that <strong>the</strong> assumption <strong>of</strong> Ru<strong>the</strong>rford scatter<strong>in</strong>g is only valid<br />
for scatter<strong>in</strong>g angles <strong>of</strong> <strong>the</strong> order b/a, where b is <strong>the</strong> distance <strong>of</strong> closest approach.<br />
Thus at smaller angles <strong>the</strong> effect <strong>of</strong> screen<strong>in</strong>g electrons is still felt. Therefore<br />
collisions only occur for impact parameters less than r0, at which <strong>the</strong> m<strong>in</strong>imum energy<br />
which can be transferred is,<br />
E<br />
*<br />
2<br />
2<br />
2<br />
2<br />
3<br />
2<br />
3<br />
4E<br />
R Z1<br />
Z 2 ( Z1<br />
Z 2 ) M 2<br />
( )<br />
(3.20)<br />
E<br />
M<br />
1
32<br />
Thus if E * >Ed all Ru<strong>the</strong>rford collisions displace atoms, but an energy LB exist above<br />
which only some <strong>of</strong> <strong>the</strong> Ru<strong>the</strong>rford collisions do so. LB is given by,<br />
L<br />
B<br />
2<br />
ER<br />
2 2 2 3 2 3 M 2<br />
4 Z1<br />
Z 2 ( Z1<br />
Z 2 )( )<br />
(3.21)<br />
E<br />
M<br />
d<br />
1<br />
Thus to summarize at higher energies <strong>the</strong> ion undergoes Ru<strong>the</strong>rford collisions and <strong>in</strong><br />
some cases atoms are displaced. If <strong>the</strong> energy falls below LB all Ru<strong>the</strong>rford collisions<br />
displace atoms. The reason why only some Ru<strong>the</strong>rford collisions displace atoms at<br />
energies E>>LB is s<strong>in</strong>ce half <strong>the</strong> energy lost to atoms which receive energy less than<br />
Ed and are thus not displaced. This energy loss is dissipated through lattice vibrations<br />
termed <strong>the</strong>rmal spikes.<br />
As <strong>the</strong> energy <strong>of</strong> <strong>the</strong> ion decreases fur<strong>the</strong>r <strong>the</strong> importance <strong>of</strong> energy loss due to hard-<br />
sphere collisions become more important than that <strong>of</strong> electronic energy loss and for<br />
this reason K<strong>in</strong>ch<strong>in</strong> and Pease (1955) def<strong>in</strong>ed a transition energy LC. It is a def<strong>in</strong>ed as<br />
<strong>the</strong> energy above which all energy loss is due to electronic excitation and below<br />
which is due to hard-sphere collisions. The value <strong>of</strong> LC is difficult to obta<strong>in</strong> <strong>in</strong> general<br />
but is approximated for <strong>in</strong>sulators as,<br />
1 M <br />
LC <br />
<br />
E<br />
m <br />
<br />
4 e <br />
G<br />
(3.22)<br />
where ΔEG is <strong>the</strong> band gap between valence and conduction band, me electron mass.<br />
For metals <strong>the</strong> relation is,<br />
1 M <br />
LC <br />
<br />
<br />
m <br />
<br />
4 e <br />
f<br />
where εf is <strong>the</strong> Fermi level but <strong>in</strong> general LC may be approximated by,<br />
(3.23)<br />
LC A keV (3.24)
33<br />
where A is <strong>the</strong> atomic weight. Thus <strong>in</strong> general it is found that <strong>the</strong> number <strong>of</strong> atoms<br />
displaced by an <strong>in</strong>com<strong>in</strong>g ion Nd depends firstly on its energy. If its energy is less than<br />
LC <strong>the</strong> number <strong>of</strong> displaced atoms is determ<strong>in</strong>ed as follows. Consider an atom with<br />
<strong>in</strong>itial energy T < LC mov<strong>in</strong>g <strong>in</strong> a solid. The first collision <strong>the</strong>n results <strong>in</strong> a shar<strong>in</strong>g <strong>of</strong><br />
<strong>the</strong> energy T with <strong>the</strong> struck atom which on average is half that <strong>of</strong> T. Both atoms will<br />
<strong>the</strong>n cont<strong>in</strong>ue with energy 1/2T and collide with one atom each shar<strong>in</strong>g <strong>the</strong>ir energy<br />
with <strong>the</strong> struck atom with a factor <strong>of</strong> an half on average. This process will cont<strong>in</strong>ue<br />
until <strong>the</strong> average energy <strong>of</strong> <strong>the</strong> struck atoms falls below 2Ed and if this occurs after q<br />
groups <strong>of</strong> collisions <strong>the</strong> total amount <strong>of</strong> displaced atoms will be given by,<br />
q<br />
N d 2 ( T 2Ed<br />
)<br />
(3.25)<br />
This expression is valid if T is greater than 2Ed, if T is between Ed and 2Ed only one<br />
atom is displaced and below Ed obviously no atom is displaced. If T is bigger than LC<br />
<strong>the</strong> expression becomes LC/2Ed.<br />
If <strong>the</strong> energy <strong>of</strong> <strong>the</strong> atom is bigger than LC and LB K<strong>in</strong>ch<strong>in</strong> and Pease showed that <strong>the</strong><br />
mov<strong>in</strong>g atom will displace ΔN atoms <strong>in</strong> los<strong>in</strong>g energy ΔE, where<br />
N P E<br />
E<br />
(3.26)<br />
and P is given by,<br />
d<br />
meZ<br />
1[<br />
1<br />
ln( E 2Ed<br />
]<br />
P (3.27)<br />
2<br />
4M<br />
( Z / Z)<br />
ln( 4m<br />
E / M I )<br />
1<br />
effective<br />
4M<br />
M<br />
1 2<br />
with <br />
and 2<br />
2<br />
( M 1 M 2 )<br />
ion is light <strong>the</strong> expression may be simplified to,<br />
N<br />
d<br />
P E LC<br />
) bL<br />
<br />
Ed<br />
<br />
<br />
<br />
e<br />
2<br />
I 8. 8Z<br />
<strong>the</strong> mean ionization potential. If <strong>the</strong> implanted<br />
( 0 C<br />
(3.28)
34<br />
where b = 0.5 if LB > LC and b = 0.25 <strong>of</strong> LB
35<br />
a vacancy dislocation loop if <strong>the</strong> cluster growth takes place <strong>in</strong> <strong>the</strong> presence <strong>of</strong> a gas.<br />
Instead gas bubbles or voids may form, if <strong>the</strong> gas eventually diffuses out <strong>of</strong> <strong>the</strong><br />
bubble.
36<br />
CHAPTER FOUR<br />
DYNAMICAL THEORY OF ELECTRON DIFFRACTION AND IMAGE<br />
SIMULATION<br />
4.1 Introduction<br />
This chapter discusses <strong>in</strong> basic terms <strong>the</strong> technique <strong>of</strong> image simulation <strong>of</strong> two-beam<br />
TEM images <strong>of</strong> <strong>defects</strong>. The <strong>the</strong>ory presented is taken from <strong>the</strong> book “An<br />
Introduction to Conventional Electron Microscopy” written by Marc de Graef (2003)<br />
and it is once aga<strong>in</strong> left to <strong>the</strong> reader to consult this book for any fur<strong>the</strong>r <strong>in</strong>formation.<br />
The <strong>the</strong>ory <strong>of</strong> electron diffraction is discussed from a dynamical po<strong>in</strong>t <strong>of</strong> view<br />
<strong>in</strong>troduc<strong>in</strong>g <strong>the</strong> concepts <strong>of</strong> <strong>the</strong> Bragg equation, reciprocal space and <strong>the</strong> Ewald<br />
sphere. This is <strong>the</strong>n used to derive <strong>the</strong> Darw<strong>in</strong> Howie Whelan equation which <strong>in</strong> turn<br />
is used for image simulations. Only <strong>the</strong> two-beam case is considered and <strong>the</strong><br />
scatter<strong>in</strong>g matrix approach is expla<strong>in</strong>ed. F<strong>in</strong>ally a simple method to obta<strong>in</strong> l<strong>in</strong>e<br />
<strong>in</strong>tensity pr<strong>of</strong>iles from stack<strong>in</strong>g faults and tw<strong>in</strong>ned areas on <strong>in</strong>cl<strong>in</strong>ed planes is shown.<br />
4.2. Theory <strong>of</strong> Electron Diffraction<br />
4.2.1 The Direct Space Bragg Equation<br />
Although <strong>the</strong> process <strong>of</strong> electron scatter<strong>in</strong>g with<strong>in</strong> a crystal is <strong>in</strong> general a complicated<br />
process it may be expla<strong>in</strong>ed <strong>in</strong> a simple way through <strong>the</strong> Bragg relation. Consider Fig.<br />
4.1.
37<br />
Fig. 4.1. (a) A wave <strong>in</strong>cident on a set <strong>of</strong> hkl planes at an angle θ, (b) <strong>the</strong> <strong>in</strong>cident and<br />
diffracted directions and <strong>the</strong> plane normal must lie <strong>in</strong> a planar section through a<br />
conical surface with its top <strong>in</strong> <strong>the</strong> plane (from De Graef (2003))<br />
A wave is <strong>in</strong>cident on a set <strong>of</strong> planes with <strong>in</strong>dices (hkl) at an angle θ. Assum<strong>in</strong>g <strong>the</strong><br />
planes to be semi-transparent mirrors, part <strong>of</strong> <strong>the</strong> <strong>in</strong>tensity <strong>of</strong> <strong>the</strong> wave will be<br />
transmitted through <strong>the</strong> first plane and part <strong>of</strong> it reflected. If Snell’s law is applied<br />
s<strong>in</strong>ce an ideal mirror was assumed, <strong>the</strong> <strong>in</strong>cident and reflected angles must be <strong>the</strong> same,<br />
and both directions <strong>of</strong> <strong>the</strong> wave must be coplanar about a normal n to <strong>the</strong> (hkl) planes.<br />
Next, if <strong>the</strong> reflected waves from <strong>the</strong> first and second planes are considered it is seen<br />
that <strong>the</strong>re is a difference <strong>in</strong> path length between <strong>the</strong> two which relate to <strong>the</strong> waves<br />
be<strong>in</strong>g <strong>in</strong> or out <strong>of</strong> phase. This path length difference is given by <strong>the</strong> sum <strong>of</strong> <strong>the</strong><br />
distance PO’ and O’Q which <strong>in</strong> turn is related to <strong>the</strong> <strong>in</strong>terplanar spac<strong>in</strong>g dhkl and <strong>the</strong><br />
angle θ as follows:<br />
PO'O'Q d<br />
hkl<br />
2d<br />
s<strong>in</strong> d<br />
hkl<br />
s<strong>in</strong> <br />
hkl<br />
s<strong>in</strong> <br />
(4.1)<br />
For <strong>in</strong>-phase propagation <strong>of</strong> <strong>the</strong> two reflected waves, known as constructive<br />
<strong>in</strong>terference, <strong>the</strong> path length difference should be an <strong>in</strong>tegral number <strong>of</strong> wavelengths<br />
and thus for constructive <strong>in</strong>terference <strong>the</strong> relation is,<br />
2 s<strong>in</strong> n<br />
(4.2)<br />
d hkl
38<br />
which reduces to, 2d hkl s<strong>in</strong> for use <strong>in</strong> <strong>the</strong> electron microscope s<strong>in</strong>ce <strong>the</strong> n is<br />
<strong>in</strong>corporated <strong>in</strong>to <strong>the</strong> value <strong>of</strong> dhkl us<strong>in</strong>g <strong>the</strong> Miller <strong>in</strong>dex notation . It should be noted<br />
that <strong>the</strong> Bragg equation only describes <strong>the</strong> geometric representation <strong>of</strong> diffraction and<br />
does not give any <strong>in</strong>formation on <strong>the</strong> <strong>in</strong>tensity <strong>of</strong> <strong>the</strong> diffracted wave. The equation is<br />
derived with respect to a specific plane and states that <strong>the</strong> <strong>in</strong>cident and diffracted<br />
waves must travel <strong>in</strong> directions which lie on a conical surface with top <strong>in</strong> <strong>the</strong><br />
diffract<strong>in</strong>g plane and open<strong>in</strong>g angle π/2-θ as illustrated <strong>in</strong> Fig. 4.1(b).<br />
4.2.2 The Bragg Equation <strong>in</strong> Reciprocal Space<br />
The direct space Bragg equation however elegant and simple is not useful when <strong>the</strong><br />
absolute direction <strong>of</strong> a diffracted wave needs to be determ<strong>in</strong>ed. To do this <strong>the</strong> Bragg<br />
equation should be reformulated <strong>in</strong> terms <strong>of</strong> reciprocal space which <strong>in</strong> turn leads to<br />
<strong>the</strong> Ewald sphere construction.<br />
From <strong>the</strong> de Broglie relation it is known that a plane wave may be represented by its<br />
momentum vector p = hk where k has units <strong>of</strong> reciprocal length and that λ = 1/|k|. k<br />
fully characterises <strong>the</strong> direction and wavelength <strong>of</strong> <strong>the</strong> plane wave with respect to <strong>the</strong><br />
crystal reference frame and thus a reciprocal space equivalent <strong>of</strong> <strong>the</strong> Bragg equation<br />
can be derived s<strong>in</strong>ce <strong>the</strong> plane (hkl) may be represented by <strong>the</strong> vector ghkl <strong>in</strong><br />
reciprocal space.<br />
Fig. 4.2. (a) The <strong>in</strong>cident and diffracted wave vectors with reciprocal lattice vector g,<br />
(b) The redrawn vector construction (from De Graef (2003))
39<br />
Consider Fig. 4.2 which illustrates <strong>the</strong> wave vectors <strong>of</strong> <strong>the</strong> <strong>in</strong>cident and diffracted<br />
waves <strong>in</strong> Fig. 4.1 with <strong>the</strong> reciprocal lattice vector g. By redraw<strong>in</strong>g <strong>the</strong> vector<br />
construction such that k' starts at po<strong>in</strong>t C Fig. 4.2 (b) is obta<strong>in</strong>ed which shows <strong>the</strong><br />
relation,<br />
k' k<br />
g<br />
(4.3)<br />
This is equivalent to <strong>the</strong> direct space Bragg condition and is shown by project<strong>in</strong>g <strong>the</strong><br />
above equation onto <strong>the</strong> vector g,<br />
k'<br />
g k<br />
g<br />
g<br />
g;<br />
k g s<strong>in</strong> k g s<strong>in</strong> g<br />
s<strong>in</strong> s<strong>in</strong> 1<br />
<br />
d<br />
hkl<br />
2<br />
(4.4)<br />
from which <strong>the</strong> Bragg equation follows. The Bragg condition is thus satisfied<br />
whenever <strong>the</strong> po<strong>in</strong>t C falls on <strong>the</strong> perpendicular bisector plane <strong>of</strong> <strong>the</strong> vector g. The<br />
perpendicular bisector plane to <strong>the</strong> vector ghkl consists <strong>of</strong> all po<strong>in</strong>ts that are at <strong>the</strong><br />
same distance from <strong>the</strong> orig<strong>in</strong> <strong>of</strong> reciprocal space and <strong>the</strong> reciprocal lattice po<strong>in</strong>t hkl.<br />
This leads to a simple geometric construction for <strong>the</strong> direction <strong>of</strong> a diffracted wave<br />
when <strong>the</strong> <strong>in</strong>cident wave vector and crystal orientation are given. This is known as <strong>the</strong><br />
Ewald sphere and is depicted <strong>in</strong> Fig 4.3.<br />
Fig. 4.3. The Ewald sphere construction (from De Graef (2003))
40<br />
The follow<strong>in</strong>g procedure is followed. First draw <strong>the</strong> reciprocal lattice with orig<strong>in</strong> O.<br />
Then draw <strong>the</strong> <strong>in</strong>cident wave vector k such that its end po<strong>in</strong>t co<strong>in</strong>cides with O. The<br />
po<strong>in</strong>t C is <strong>the</strong>n taken as <strong>the</strong> center <strong>of</strong> a sphere with radius |k|. Whenever a reciprocal<br />
lattice po<strong>in</strong>t falls on this sphere, <strong>the</strong> Bragg condition is satisfied and a diffracted<br />
vector k+g may occur. Note also that more than one reciprocal lattice po<strong>in</strong>t may lie on<br />
<strong>the</strong> Ewald sphere which <strong>in</strong>dicates that multiple beam diffraction may occur. This is<br />
due to a relaxation <strong>of</strong> <strong>the</strong> Bragg condition due to <strong>the</strong> sample geometry caus<strong>in</strong>g <strong>the</strong><br />
reciprocal po<strong>in</strong>ts to extend <strong>in</strong>to rods (see Hirsch et al. (1965)).<br />
4.2.3 Darw<strong>in</strong>-Howie-Whelan Equations<br />
Bragg’s law shows that electrons are diffracted <strong>in</strong> directions k' given by k+g for all<br />
reciprocal lattice po<strong>in</strong>ts g ly<strong>in</strong>g on <strong>the</strong> Ewald sphere. Hence it may be anticipated that<br />
<strong>the</strong> total wave function at <strong>the</strong> exit surface <strong>of</strong> <strong>the</strong> crystal will be a superposition <strong>of</strong><br />
plane waves, one <strong>in</strong> each <strong>of</strong> <strong>the</strong> directions predicted by <strong>the</strong> Bragg equation. All that<br />
rema<strong>in</strong>s is <strong>the</strong> computation <strong>of</strong> <strong>the</strong> complex amplitudes <strong>of</strong> each <strong>of</strong> <strong>the</strong> diffracted waves<br />
which are given by <strong>the</strong> Darw<strong>in</strong>-Howie-Whelan (DHW) equations:<br />
d<br />
dz<br />
g<br />
2is<br />
<br />
g<br />
g<br />
i<br />
i<br />
g g<br />
'<br />
e<br />
' g q '<br />
g g<br />
<br />
'<br />
g<br />
(4.5)<br />
These DHW equations are derived from <strong>the</strong> Schröd<strong>in</strong>ger equation for dynamical<br />
electron scatter<strong>in</strong>g,<br />
2<br />
2 2<br />
2 8<br />
me<br />
4 K 0 4<br />
U ( r)<br />
V ( r)<br />
<br />
2 c<br />
(4.6)<br />
h<br />
where Vc is a complex electrostatic lattice potential Vc=V + iW. The Fourier<br />
transform <strong>of</strong> which is given by,<br />
V ( r)<br />
V V<br />
'(<br />
r)<br />
iW ( r)<br />
V<br />
c<br />
0<br />
0<br />
<br />
2igr<br />
Vg e i<br />
g0 g<br />
W<br />
g<br />
e<br />
2igr<br />
(4.7)
41<br />
V0 is <strong>the</strong> mean <strong>in</strong>ner potential <strong>of</strong> <strong>the</strong> crystal and is positive and causes acceleration <strong>of</strong><br />
<strong>the</strong> electrons and hence <strong>the</strong> relativistic acceleration <strong>of</strong> <strong>the</strong> electron <strong>in</strong>creases to,<br />
^ ^<br />
C <br />
<br />
V<br />
(4.8)<br />
0<br />
from which <strong>the</strong> wave number k0 is given by,<br />
k<br />
0<br />
^<br />
2me <br />
(4.9)<br />
h<br />
which <strong>in</strong> turn is used to simplify <strong>the</strong> Schröd<strong>in</strong>ger equation to,<br />
2 2<br />
2<br />
4<br />
k 4<br />
[ U iU ] <br />
(4.10)<br />
with,<br />
0<br />
2me<br />
U ( r)<br />
2<br />
h<br />
<br />
g 0<br />
V<br />
2 igr<br />
ge <br />
; (4.11)<br />
igr<br />
g e W<br />
2me<br />
2<br />
U ' ( r)<br />
; (4.12)<br />
2<br />
h<br />
g<br />
To solve equation 4.10 an empty crystal approximation (i.e. U+iU’= 0) is used to<br />
obta<strong>in</strong> a general solution <strong>of</strong> <strong>the</strong> form,<br />
r ( )<br />
<br />
g<br />
(4.13)<br />
i k g r<br />
g e<br />
2 ( 0 )<br />
which upon substitution <strong>in</strong>to equation 4.10 and us<strong>in</strong>g a high energy approximation <strong>the</strong><br />
DHW equations are obta<strong>in</strong>ed. To follow <strong>the</strong> complete derivation <strong>the</strong> reader is referred<br />
to de Graef (2003). The symbol θg <strong>in</strong> equation 4.5 is known as <strong>the</strong> phase factor, sg is<br />
<strong>the</strong> deviation from <strong>the</strong> Bragg condition and,<br />
1 1 e<br />
i<br />
q <br />
g<br />
g '<br />
g<br />
i<br />
g<br />
'<br />
g<br />
(4.14)
'<br />
with ξg <strong>the</strong> ext<strong>in</strong>ction distance, g <strong>the</strong> absorption length and g g <br />
g<br />
'<br />
.<br />
The ext<strong>in</strong>ction distance and absorption length is given by <strong>the</strong> follow<strong>in</strong>g,<br />
1 U g<br />
<br />
k g cos<br />
g<br />
'<br />
1 U g<br />
'<br />
k g cos<br />
g<br />
42<br />
The ext<strong>in</strong>ction distances as calculated by Electron Diffraction (2003) is given <strong>in</strong> Table<br />
4.1. for commonly used reflections for <strong>3C</strong>-<strong>SiC</strong>.<br />
Table 4.1. – The ext<strong>in</strong>ction distances <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> at 200 kV as calculated by Electron<br />
Diffraction (2003).<br />
4.2.4 DHW Equations: Two Beam Case<br />
Index ξg (<strong>in</strong> Å)<br />
111 622<br />
220 825<br />
200 1915<br />
400 1383<br />
113 1431<br />
When <strong>the</strong> special case <strong>in</strong> which <strong>the</strong> <strong>in</strong>tensity <strong>of</strong> <strong>the</strong> <strong>in</strong>cident beam is shared only<br />
between <strong>the</strong> transmitted ψ0 and one diffracted beam ψg is considered, <strong>the</strong> DHW<br />
equations are given by two coupled differential equations. They are,
43<br />
beam left-hand side g' = 0 g' = g<br />
g<br />
d 0<br />
<br />
e<br />
0: 2 is 0<br />
0 0 i <br />
'<br />
g (4.15)<br />
dz<br />
<br />
q<br />
0<br />
i g<br />
d g<br />
e<br />
<br />
g: 2 is g<br />
g i 0 ' g<br />
(4.16)<br />
dz<br />
q <br />
These equations may be written <strong>in</strong> <strong>the</strong> form,<br />
dT<br />
dz<br />
dS<br />
dz<br />
i<br />
isgT<br />
S ; (4.17)<br />
q<br />
g<br />
i<br />
is g S T ; (4.18)<br />
q<br />
g<br />
by means <strong>of</strong> <strong>the</strong> follow<strong>in</strong>g substitution,<br />
)<br />
z<br />
0 T ( z e ; (4.19)<br />
i<br />
g z<br />
g S(<br />
z e e ; (4.20)<br />
)<br />
i<br />
with i(<br />
s ) ' g . This is done to remove <strong>the</strong> dependence on <strong>the</strong> normal<br />
<br />
0<br />
absorption length and to render <strong>the</strong> equations more symmetric. Thus follow<strong>in</strong>g this<br />
procedure <strong>the</strong> solution to <strong>the</strong> problem is derived as follows. Equation 4.17 is written<br />
as,<br />
q<br />
g dT<br />
S s gq<br />
gT<br />
; (4.21)<br />
i<br />
dz<br />
and<br />
dS<br />
dz<br />
2 q<br />
g d T<br />
sg<br />
q<br />
i<br />
dz<br />
2<br />
g<br />
dT<br />
dz<br />
g<br />
i <br />
g<br />
0<br />
(4.22)
44<br />
Substituted <strong>in</strong>to equation 4.18 and rearranged to give,<br />
2<br />
d T 2 2<br />
T 0<br />
(4.23)<br />
2<br />
dz<br />
with <br />
s<br />
2<br />
g<br />
1<br />
<br />
q q<br />
g<br />
g<br />
This is <strong>the</strong> differential equation for <strong>the</strong> harmonic oscillator with general solution,<br />
T ( z)<br />
Acos(<br />
z) B s<strong>in</strong>( z)<br />
(4.24)<br />
From <strong>the</strong> boundary condition at <strong>the</strong> entrance surface T(z = 0) = 1 <strong>the</strong> value <strong>of</strong> A is<br />
found to be 1. The value for B is found by comput<strong>in</strong>g dT/dz and substitut<strong>in</strong>g <strong>in</strong>to <strong>the</strong><br />
equation for S above and apply<strong>in</strong>g <strong>the</strong> boundary condition S(z = 0) = 0 which leads to<br />
B = -isg/σ and thus,<br />
is g<br />
T ( s,<br />
z)<br />
cos( z) s<strong>in</strong>( z)<br />
; (4.25)<br />
<br />
i<br />
S(<br />
s,<br />
z)<br />
s<strong>in</strong>( z)<br />
; (4.26)<br />
q <br />
g<br />
These are <strong>the</strong> general solutions to <strong>the</strong> dynamical two-beam equations <strong>in</strong>clud<strong>in</strong>g<br />
absorption. Note also that σ is complex.<br />
4.2.5 The Two-Beam Scatter<strong>in</strong>g Matrix<br />
In Section 4.2.4 <strong>the</strong> solutions to equations 4.17 and 4.18 was found by us<strong>in</strong>g <strong>the</strong><br />
follow<strong>in</strong>g <strong>in</strong>itial conditions:<br />
T ( s,<br />
z 0)<br />
1;<br />
S ( s,<br />
z 0)<br />
0 ; (4.27)
45<br />
In addition ano<strong>the</strong>r set <strong>of</strong> <strong>in</strong>dependent <strong>in</strong>itial conditions can be used to obta<strong>in</strong> a special<br />
solution:<br />
T '( s,<br />
z 0)<br />
0 ;<br />
S '( s,<br />
z 0)<br />
1;<br />
(4.28)<br />
which shows that it is assumed that all <strong>the</strong> <strong>in</strong>cident amplitude is <strong>in</strong> <strong>the</strong> direction <strong>of</strong> <strong>the</strong><br />
scattered beam ra<strong>the</strong>r than <strong>the</strong> transmitted beam. Us<strong>in</strong>g this it is found that <strong>the</strong><br />
solutions to T' and S' is given by,<br />
'<br />
T ( s,<br />
z)<br />
S(<br />
s,<br />
z)<br />
;<br />
'<br />
S ( s,<br />
z)<br />
T(<br />
s,<br />
z)<br />
; (4.29)<br />
Thus <strong>the</strong> expressions for transmitted and scatter<strong>in</strong>g amplitudes are swapped and <strong>the</strong><br />
sign <strong>of</strong> <strong>the</strong> excitation error changed. From equation 4.26 it is also seen that S(-s,z) =<br />
S(s,z). The amplitude at <strong>the</strong> exit plane <strong>of</strong> <strong>the</strong> crystal with thickness z0 is <strong>the</strong>n given by,<br />
<br />
0 <br />
<br />
<br />
<br />
g <br />
z<br />
z<br />
0<br />
e<br />
(<br />
<br />
'<br />
0<br />
) z<br />
( )<br />
where T T(<br />
s,<br />
z)<br />
.<br />
S<br />
( s,<br />
z ) e<br />
0<br />
<br />
<br />
T<br />
<br />
Se<br />
i<br />
<br />
<br />
T<br />
<br />
Se<br />
(<br />
0<br />
) z0<br />
scat 0<br />
i<br />
'<br />
g<br />
g<br />
Se<br />
T<br />
Se<br />
T<br />
i<br />
( )<br />
g<br />
i<br />
( )<br />
g<br />
<br />
0 <br />
<br />
<br />
<br />
<br />
<br />
g <br />
<br />
<br />
<br />
<br />
z0<br />
(4.30)<br />
(4.31)<br />
Sscat is known as <strong>the</strong> scatter<strong>in</strong>g matrix and is obta<strong>in</strong>ed by <strong>the</strong> knowledge that <strong>the</strong><br />
general solution to a system <strong>of</strong> N first-order differential equations may be written as a<br />
l<strong>in</strong>ear comb<strong>in</strong>ation <strong>of</strong> special solutions, where <strong>the</strong> special solutions can be obta<strong>in</strong>ed by<br />
stat<strong>in</strong>g N <strong>in</strong>dependent boundary conditions. Thus <strong>in</strong> <strong>the</strong> case for N = 2 two sets <strong>of</strong><br />
<strong>in</strong>dependent boundary conditions given by equations 4.27 and 4.28 are used to f<strong>in</strong>d
46<br />
solutions which <strong>in</strong> turn are comb<strong>in</strong>ed to give a general solution from which <strong>the</strong><br />
scatter<strong>in</strong>g matrix is obta<strong>in</strong>ed. The general solution may <strong>the</strong>n be written as follows,<br />
z ) S scat ( s,<br />
z ) (<br />
0)<br />
(4.32)<br />
( 0<br />
0<br />
It should also be noted that <strong>the</strong> scatter<strong>in</strong>g matrix is a complex matrix <strong>of</strong> <strong>the</strong> form,<br />
S S iS<br />
(4.33)<br />
scat<br />
r<br />
i<br />
4.2.6 Crystal Defects and Displacement Fields<br />
The presence <strong>of</strong> a defect <strong>in</strong> a crystal causes a disruption <strong>in</strong> <strong>the</strong> positions <strong>of</strong> <strong>the</strong> atoms<br />
as predicted by <strong>the</strong> Bravais lattice translation vectors. The atoms are displaced from<br />
<strong>the</strong>ir positions by a vector R which is a function <strong>of</strong> position r. This vector field R(r) is<br />
known as <strong>the</strong> displacement field and relates <strong>the</strong> displaced position r’ <strong>of</strong> <strong>the</strong> atom to<br />
<strong>the</strong> ideal position r by,<br />
r '<br />
r R(r)<br />
(4.34)<br />
Also <strong>the</strong> presence <strong>of</strong> this displacement field causes a deviation <strong>in</strong> <strong>the</strong> potential <strong>of</strong> <strong>the</strong><br />
crystal from <strong>the</strong> perfect crystal at <strong>the</strong> deformed position given by,<br />
'<br />
V ( r) V ( r R)<br />
V<br />
0<br />
V<br />
0<br />
<br />
<br />
<br />
g<br />
<br />
g<br />
V<br />
V<br />
g<br />
g<br />
e<br />
e<br />
2ig(<br />
rR<br />
)<br />
2igR<br />
e<br />
2igr<br />
(4.35)<br />
which shows that <strong>the</strong> electrostatic potential is modified by a phase factor αg as<br />
follows,<br />
V<br />
g<br />
i<br />
(r)<br />
g<br />
V e with ( r) 2g<br />
R(r)<br />
g<br />
g (4.36)
47<br />
When this is <strong>in</strong>corporated <strong>in</strong>to <strong>the</strong> Darw<strong>in</strong>-Howie-Whelan equations it modifies <strong>the</strong><br />
equations to be,<br />
d<br />
dz<br />
g<br />
2i(<br />
s<br />
g<br />
i<br />
g g<br />
'<br />
e<br />
' g q '<br />
g<br />
g<br />
dR<br />
g ) g<br />
i<br />
'<br />
(4.37)<br />
g<br />
dz<br />
which only constitutes a change <strong>in</strong> <strong>the</strong> excitation error to an effective excitation error<br />
given by,<br />
dR(r)<br />
( r) s g <br />
(4.38)<br />
dz<br />
eff<br />
sg g<br />
4.3. Image Contrast for Selected Defects<br />
4.3.1 L<strong>in</strong>e Defects<br />
As discussed <strong>in</strong> section 2.4 a dislocation is a l<strong>in</strong>ear lattice defect which may be<br />
characterised by a Burgers vector b and a dislocation l<strong>in</strong>e direction u. It may be<br />
characterised as be<strong>in</strong>g <strong>of</strong> edge character with b·u = 0 or screw character with b || u or<br />
a mixture <strong>of</strong> both. In general <strong>the</strong> displacement field <strong>of</strong> a dislocation is given by,<br />
1 s<strong>in</strong> 2<br />
1<br />
2<br />
cos<br />
<br />
R b<br />
be<br />
b u<br />
ln r <br />
(4.39)<br />
2<br />
4(<br />
1<br />
)<br />
2(<br />
1<br />
)<br />
4(<br />
1<br />
)<br />
<br />
where υ is Poissons’s ratio, be <strong>the</strong> edge component <strong>of</strong> <strong>the</strong> Burgers vector and (r,θ)<br />
polar coord<strong>in</strong>ates <strong>in</strong> a plane perpendicular to <strong>the</strong> dislocation l<strong>in</strong>e direction.<br />
4.3.2 Stack<strong>in</strong>g Faults<br />
The aim <strong>of</strong> this section is not to describe <strong>the</strong> full process <strong>of</strong> simulat<strong>in</strong>g an <strong>in</strong>cl<strong>in</strong>ed<br />
stack<strong>in</strong>g fault with its associated partial dislocations but ra<strong>the</strong>r to describe a relatively<br />
easy way <strong>of</strong> obta<strong>in</strong><strong>in</strong>g one-dimensional <strong>in</strong>tensity pr<strong>of</strong>iles <strong>of</strong> stack<strong>in</strong>g fault fr<strong>in</strong>ges. The<br />
procedure uses a scatter<strong>in</strong>g matrix approach discussed <strong>in</strong> Section 4.2.5 with <strong>the</strong>
48<br />
associated displacement field <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g fault to obta<strong>in</strong> a l<strong>in</strong>e <strong>in</strong>tensity pr<strong>of</strong>ile <strong>of</strong><br />
<strong>the</strong> fr<strong>in</strong>ges.<br />
As discussed <strong>in</strong> Section 2.6 stack<strong>in</strong>g faults are produced by a disruption <strong>of</strong> <strong>the</strong> regular<br />
stack<strong>in</strong>g sequence <strong>of</strong> a material. It can also be seen as two regions <strong>of</strong> <strong>the</strong> crystal<br />
hav<strong>in</strong>g been shifted relative to one ano<strong>the</strong>r by a distance given by a displacement<br />
vector R which may be a non-lattice translation as shown by Fig. 4.4. The<br />
displacement vector has a component parallel to <strong>the</strong> fault plane R|| and normal to it<br />
R . Thus <strong>the</strong> presence <strong>of</strong> this fault will <strong>in</strong>troduce a phase shift α = 2πg·R which<br />
modifies <strong>the</strong> solution to <strong>the</strong> DHW equations for <strong>the</strong> bottom section (section II) <strong>of</strong> <strong>the</strong><br />
crystal. If α is an <strong>in</strong>teger multiple <strong>of</strong> π no defect contrast will be seen which shows<br />
that R is a lattice translation vector and effectively no modification <strong>of</strong> <strong>the</strong> solutions to<br />
<strong>the</strong> DHW equations result. If R is a non-lattice translation, α is not an <strong>in</strong>teger multiple<br />
<strong>of</strong> π and image contrast <strong>of</strong> <strong>the</strong> form <strong>of</strong> alternat<strong>in</strong>g light and dark fr<strong>in</strong>ges will be<br />
formed.<br />
Fig. 4.4. (a) Schematic illustration <strong>of</strong> a planar defect separat<strong>in</strong>g crystal sections I<br />
and II; (b) <strong>in</strong>cl<strong>in</strong>ed planar defect comb<strong>in</strong>ed with <strong>the</strong> column approximation (from De<br />
Graef (2003))<br />
To simulate such an <strong>in</strong>tensity pr<strong>of</strong>ile <strong>the</strong> follow<strong>in</strong>g procedure is used. Consider a<br />
crystal with thickness z0 conta<strong>in</strong><strong>in</strong>g a stack<strong>in</strong>g fault as shown <strong>in</strong> Fig. 4.4 (b). Region 1<br />
has been shifted with respect to region II by a vector R. The top section <strong>of</strong> <strong>the</strong> crystal<br />
(region 1) is defect free and is described by a scatter<strong>in</strong>g matrix Sscat(s1.z1) where z1 is<br />
<strong>the</strong> distance from <strong>the</strong> top surface to <strong>the</strong> fault. The bottom section is shifted with<br />
respect to <strong>the</strong> top and is described by a different scatter<strong>in</strong>g matrix M(s2.z2) s<strong>in</strong>ce it lies<br />
<strong>in</strong> a different orientation with respect to <strong>the</strong> top. The two scatter<strong>in</strong>g matrices are as<br />
follows,
S<br />
( s , z ) e<br />
<br />
<br />
T<br />
i<br />
Se<br />
(<br />
/ 0<br />
) z1<br />
scat 1 1<br />
<br />
M ( s , z ) e<br />
2<br />
2<br />
'<br />
0<br />
(<br />
/ ) z<br />
'<br />
2<br />
<br />
<br />
T<br />
i<br />
Se<br />
<br />
g<br />
g<br />
Se<br />
T<br />
Se<br />
T<br />
i<br />
( )<br />
i<br />
g<br />
( )<br />
M can fur<strong>the</strong>r be decomposed <strong>in</strong>to,<br />
i<br />
g<br />
1<br />
0 <br />
<br />
<br />
<br />
T S e 1 0<br />
2 2<br />
M M P(<br />
<br />
<br />
<br />
g ) S 2P(<br />
g )<br />
i g i<br />
g ( )<br />
<br />
i<br />
g <br />
0<br />
e S<br />
e T 0<br />
e<br />
2<br />
2<br />
<br />
g<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
49<br />
(4.40)<br />
(4.41)<br />
2 (4.42)<br />
with P be<strong>in</strong>g <strong>the</strong> defect phase shift matrix,<br />
1<br />
0 <br />
P(<br />
) <br />
<br />
<br />
<br />
(4.43)<br />
i<br />
0<br />
e <br />
Us<strong>in</strong>g this, <strong>the</strong> transmitted and scattered amplitudes at <strong>the</strong> exit plane <strong>of</strong> <strong>the</strong> crystal are<br />
obta<strong>in</strong>ed by <strong>the</strong> product <strong>of</strong> <strong>the</strong> scatter<strong>in</strong>g matrices for each section us<strong>in</strong>g <strong>the</strong> <strong>in</strong>itial<br />
condition (1,0) T ,<br />
<br />
0 <br />
<br />
<br />
<br />
<br />
g <br />
1<br />
<br />
( <br />
g ) S 2P(<br />
) S1<br />
<br />
<br />
<br />
0<br />
<br />
P g<br />
From this <strong>the</strong> <strong>in</strong>tensities are obta<strong>in</strong>ed by squar<strong>in</strong>g <strong>the</strong> wave functions.<br />
(4.44)<br />
Note that <strong>the</strong> values <strong>of</strong> z1 and z2 depend on <strong>the</strong> <strong>in</strong>com<strong>in</strong>g beam direction B, foil<br />
normal N and <strong>the</strong> plane normal P <strong>of</strong> <strong>the</strong> plane <strong>in</strong> which <strong>the</strong> fault lies, note also that <strong>the</strong><br />
ext<strong>in</strong>ction distance used depends on <strong>the</strong> diffraction vector g. Thus to solve <strong>the</strong><br />
problem <strong>the</strong> follow<strong>in</strong>g geometrical setup is used and shown <strong>in</strong> Fig. 4.5.
50<br />
Fig. 4.5. The geometrical setup used <strong>in</strong> simulat<strong>in</strong>g a l<strong>in</strong>e <strong>in</strong>tensity pr<strong>of</strong>ile <strong>of</strong> a<br />
stack<strong>in</strong>g fault<br />
Firstly <strong>the</strong> projected foil thickness zproj and range Dproj <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g fault <strong>in</strong> <strong>the</strong><br />
direction <strong>of</strong> <strong>the</strong> beam is determ<strong>in</strong>ed. This is done by f<strong>in</strong>d<strong>in</strong>g <strong>the</strong> angle between <strong>the</strong><br />
foil normal N and <strong>the</strong> beam direction B, θ, from which zproj is determ<strong>in</strong>ed by <strong>the</strong><br />
trigonometric relation,<br />
z0<br />
z proj (4.45)<br />
cos<br />
and <strong>the</strong> projected lateral range Dproj by <strong>the</strong> relation,<br />
D proj<br />
with<br />
N.<br />
D tan. z<br />
(4.46)<br />
z0<br />
0<br />
D where α is <strong>the</strong> angle between <strong>the</strong> plane normal P and <strong>the</strong> foil normal<br />
tan<br />
Follow<strong>in</strong>g this, <strong>the</strong> lateral range is divided <strong>in</strong>to n equally spaced po<strong>in</strong>ts as shown <strong>in</strong><br />
Fig. 4.5 (b). The number n determ<strong>in</strong>es <strong>the</strong> resolution <strong>of</strong> <strong>the</strong> solution to <strong>the</strong> problem but<br />
also <strong>in</strong>fluences <strong>the</strong> comput<strong>in</strong>g time. Next <strong>the</strong> values <strong>of</strong> z1 and z2 are determ<strong>in</strong>ed<br />
through <strong>the</strong> relations for each po<strong>in</strong>t,
z0<br />
z2 . xn<br />
cos<br />
and (4.47)<br />
D<br />
1<br />
proj<br />
z z proj z<br />
(4.48)<br />
2<br />
51<br />
Hence z1 and z2 are used with <strong>the</strong> values <strong>of</strong> s1 and s2 and <strong>the</strong> ext<strong>in</strong>ction distance <strong>in</strong> <strong>the</strong><br />
scatter<strong>in</strong>g matrices to determ<strong>in</strong>e <strong>the</strong> exit plane waves which <strong>in</strong> turn is used to<br />
determ<strong>in</strong>e <strong>the</strong> <strong>in</strong>tensity at each po<strong>in</strong>t xn used. Also <strong>in</strong> general <strong>the</strong> excitation errors s1<br />
and s2 for <strong>the</strong> two regions do not differ significantly and may be assumed to be equal.<br />
The foil thickness for this derivation was also assumed to stay constant but is not<br />
necessarily true which <strong>in</strong> turn adds ano<strong>the</strong>r element <strong>of</strong> complexity to <strong>the</strong> problem.<br />
4.3.3 Fr<strong>in</strong>ge Contrast Observed from Tw<strong>in</strong>n<strong>in</strong>g<br />
Tw<strong>in</strong>n<strong>in</strong>g as expla<strong>in</strong>ed <strong>in</strong> Section 2.7 is a region <strong>in</strong> which part <strong>of</strong> <strong>the</strong> crystal is<br />
displaced through a homogenous shear process to produce <strong>the</strong> orig<strong>in</strong>al crystal<br />
structure but <strong>in</strong> a different orientation. Thus a similar geometrical situation as for<br />
stack<strong>in</strong>g faults may occur but with <strong>the</strong> difference that one part <strong>of</strong> <strong>the</strong> crystal (region 1<br />
or 2) will be tw<strong>in</strong>ned with respect to <strong>the</strong> orig<strong>in</strong>al crystal. This causes a change <strong>in</strong> that<br />
<strong>the</strong> excitation error <strong>of</strong> <strong>the</strong> tw<strong>in</strong>ned area and will be different to that <strong>of</strong> <strong>the</strong> normal<br />
crystal and should be <strong>in</strong>corporated <strong>in</strong>to <strong>the</strong> scatter<strong>in</strong>g matrix for that region. To<br />
simulate such a situation <strong>the</strong> same procedure is used as for a stack<strong>in</strong>g fault with <strong>the</strong><br />
only difference be<strong>in</strong>g that <strong>the</strong> excitation error changes.
52<br />
Figure 4.6 – The geometrical considerations for simulation <strong>of</strong> a l<strong>in</strong>e <strong>in</strong>tensity pr<strong>of</strong>ile<br />
<strong>of</strong> a narrow tw<strong>in</strong><br />
Fur<strong>the</strong>rmore a different situation (depicted figure 4.6) may occur <strong>in</strong> which a narrow<br />
tw<strong>in</strong>ned section is present with<strong>in</strong> <strong>the</strong> material. In this case <strong>the</strong> crystal has three regions<br />
<strong>in</strong> which a scatter<strong>in</strong>g matrix should be calculated and also three sections where <strong>the</strong><br />
plane wave propagates through <strong>the</strong> orig<strong>in</strong>al crystal and <strong>the</strong>n tw<strong>in</strong>ned crystal (section<br />
1), orig<strong>in</strong>al crystal <strong>the</strong>n tw<strong>in</strong>ned crystal and <strong>the</strong>n orig<strong>in</strong>al crystal aga<strong>in</strong> (section 2) and<br />
f<strong>in</strong>ally through a tw<strong>in</strong>ned crystal and <strong>the</strong>n through <strong>the</strong> orig<strong>in</strong>al crystal (section 3). This<br />
will produce an <strong>in</strong>terest<strong>in</strong>g fr<strong>in</strong>ge contrast image divided <strong>in</strong>to three sections. To<br />
simulate such a situation is similar to that <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g fault with <strong>the</strong> only<br />
difference be<strong>in</strong>g that <strong>the</strong> calculations are done for each section and that for section<br />
two, three thicknesses <strong>of</strong> <strong>the</strong> regions z1, z2 and z3 should be determ<strong>in</strong>ed. Also <strong>the</strong><br />
excitation error <strong>of</strong> <strong>the</strong> tw<strong>in</strong>ned region should be taken <strong>in</strong>to account. In general such a<br />
calculation is not as trivial as discussed above but <strong>the</strong> section is given to aid <strong>in</strong> <strong>the</strong><br />
understand<strong>in</strong>g <strong>of</strong> what such a calculation might entail. Factors such as <strong>the</strong> calculation<br />
<strong>of</strong> <strong>the</strong> deviation parameter <strong>of</strong> <strong>the</strong> tw<strong>in</strong>ned section with reference to <strong>the</strong> bulk crystal<br />
structure must be done. In addition, care must be taken that <strong>the</strong> correct two-beam<br />
condition is used to form <strong>the</strong> image.
5.1 Introduction<br />
53<br />
CHAPTER FIVE<br />
LITERATURE REVIEW OF <strong>SiC</strong><br />
The follow<strong>in</strong>g chapter consists <strong>of</strong> a literature review <strong>of</strong> earlier work done on <strong>3C</strong>-<strong>SiC</strong><br />
with regards to growth by CVD on Si substrates, crystal <strong>defects</strong> and defect generation<br />
by ion irradiation. A literature review was also done with regard to diffusion<br />
characteristics but it is presented toge<strong>the</strong>r with <strong>the</strong> discussion <strong>of</strong> <strong>the</strong> results <strong>of</strong> proton<br />
bombarded <strong>3C</strong>-<strong>SiC</strong> <strong>in</strong> Chapter 7 s<strong>in</strong>ce it makes more sense to do so.<br />
5.2 CVD Growth <strong>of</strong> <strong>SiC</strong> on Si<br />
The CVD growth <strong>of</strong> <strong>3C</strong>-<strong>SiC</strong> {001} on Si {001} takes place through <strong>the</strong> classical<br />
mechanism <strong>of</strong> nucleation and growth. Dur<strong>in</strong>g <strong>the</strong> first step <strong>of</strong> <strong>the</strong> process only<br />
propane is <strong>in</strong>troduced at 1100°C leav<strong>in</strong>g only <strong>the</strong> substrate as a source <strong>of</strong> Si after<br />
which <strong>the</strong> substrate is ramped up to <strong>the</strong> growth temperature <strong>of</strong> 1380°C and <strong>the</strong> silane<br />
<strong>in</strong>troduced.<br />
Pirouz et al. (1987) expla<strong>in</strong>ed that this two-step process causes <strong>the</strong> formation <strong>of</strong> two-<br />
dimensional nuclei on <strong>the</strong> substrate. Assum<strong>in</strong>g <strong>the</strong>n a “pill box” shape with height h<br />
(<strong>of</strong> <strong>the</strong> order <strong>of</strong> one or a few monolayers), <strong>the</strong> critical radius r <strong>of</strong> <strong>the</strong> nuclei is given<br />
by,<br />
r h<br />
hH<br />
( )]<br />
(5.1)<br />
f / g /[ f / g f / s s / g<br />
with γf/g, γf/s, γs/g <strong>the</strong> film/gas, film/substrate, substrate/gas <strong>in</strong>terfacial energies per unit<br />
area respectively and ΔH <strong>the</strong> enthalpy change per unit volume between <strong>the</strong> reactants<br />
and <strong>the</strong> product. The energy barrier for <strong>the</strong> formation <strong>of</strong> a stable nucleus ΔG is given<br />
by,<br />
(5.2)<br />
G hr<br />
f / g
54<br />
The critical radius and energy barrier for <strong>the</strong> formation <strong>of</strong> stable nuclei both depend<br />
on γf/s. The film/substrate <strong>in</strong>terfacial energy <strong>in</strong> turn depends on <strong>the</strong> bond<strong>in</strong>g across <strong>the</strong><br />
<strong>in</strong>terface where <strong>the</strong> existence <strong>of</strong> elastic stra<strong>in</strong>s and dangl<strong>in</strong>g bonds <strong>in</strong>creases its value<br />
and thus <strong>in</strong>creases r and ΔG. Consequently epitaxially oriented nuclei will have a<br />
lower <strong>in</strong>terfacial energy and a smaller critical radius and energy <strong>of</strong> formation barrier.<br />
Misoriented nuclei will have a larger <strong>in</strong>terfacial energy with a larger critical radius<br />
and energy <strong>of</strong> formation barrier than oriented nuclei and will be subcritical and<br />
dissolve away. The rate <strong>of</strong> nucleation, N, is proportional to exp(-ΔG/kT) and thus<br />
many more oriented than misoriented nuclei will form.<br />
However <strong>the</strong> film/substrate <strong>in</strong>terfacial energy is expected to be a function <strong>of</strong><br />
temperature and its contribution to <strong>the</strong> energy <strong>of</strong> formation barrier decreases with<br />
<strong>in</strong>creas<strong>in</strong>g temperature. This <strong>in</strong> turn reduces <strong>the</strong> anisotropy between oriented and<br />
misoriented nuclei formation and it is expected that at higher temperatures <strong>the</strong>ir<br />
deposition rates become more equal. This is evidenced by <strong>the</strong> fact that <strong>the</strong> growth <strong>of</strong><br />
<strong>SiC</strong> at higher temperatures results <strong>in</strong> a polycrystall<strong>in</strong>e structure. If <strong>the</strong> film/substrate<br />
energy is considered <strong>the</strong> energy <strong>of</strong> oriented nuclei will be <strong>in</strong> a global m<strong>in</strong>ima <strong>of</strong> <strong>the</strong><br />
energy function with an orientation relationship between <strong>the</strong> substrate and nucleus <strong>of</strong><br />
(001)Si||(001)<strong>SiC</strong> ; 110]<br />
Si|| 110]<br />
<strong>SiC</strong>. However Pirouz et al. (1987) mentioned that local<br />
[ _<br />
[ _<br />
m<strong>in</strong>ima <strong>in</strong> <strong>the</strong> energy function also exists for o<strong>the</strong>r orientations. For example a nuclei<br />
tw<strong>in</strong>ned on a 111)<br />
plane relative to an epitaxial one will have <strong>the</strong> orientation<br />
( _<br />
relationship with respect to <strong>the</strong> substrate as (001)Si|| 221)<br />
<strong>SiC</strong> ; 110]<br />
Si|| 114]<br />
( _<br />
[ _<br />
[ _<br />
<strong>SiC</strong> .<br />
These types <strong>of</strong> nuclei are expected to have a lower film/substrate <strong>in</strong>terfacial energy<br />
than misoriented nuclei but higher than oriented. Thus a fraction <strong>of</strong> <strong>the</strong> stable nuclei<br />
will have a tw<strong>in</strong>ned orientation relationship. Fur<strong>the</strong>r, <strong>the</strong> lateral growth <strong>of</strong> <strong>the</strong> tw<strong>in</strong>ned<br />
nuclei will be slower than that <strong>of</strong> <strong>the</strong> oriented ones and will grow easier <strong>in</strong> directions<br />
<strong>in</strong>cl<strong>in</strong>ed to <strong>the</strong> <strong>in</strong>terface. As a result <strong>the</strong>y tend to growth th<strong>in</strong> and long with<strong>in</strong> <strong>the</strong> film.<br />
Growth at 1380°C results <strong>in</strong> migration <strong>of</strong> <strong>the</strong> tw<strong>in</strong>n<strong>in</strong>g boundaries and <strong>the</strong><br />
consumption or decrease <strong>in</strong> <strong>the</strong> volume <strong>of</strong> <strong>the</strong> tw<strong>in</strong>s.<br />
Powell et al. (1987) <strong>in</strong> a different publication expla<strong>in</strong>s that dur<strong>in</strong>g <strong>the</strong> <strong>in</strong>itial stages <strong>of</strong><br />
deposition a large number <strong>of</strong> supracritical nuclei form on <strong>the</strong> Si substrate with
55<br />
different orientations. The <strong>in</strong>terfacial energy <strong>of</strong> <strong>the</strong> epitaxially oriented nuclei is lower<br />
than that <strong>of</strong> <strong>the</strong> misoriented ones and as a result <strong>the</strong> former nuclei will have a<br />
relatively faster lateral growth whereas <strong>the</strong> latter grows preferentially <strong>in</strong> vertical<br />
directions. Once <strong>the</strong> different nuclei meet a gra<strong>in</strong> boundary stress results between <strong>the</strong><br />
nuclei which tends to cause <strong>the</strong> epitaxial nuclei to grow at <strong>the</strong> expense <strong>of</strong> <strong>the</strong><br />
misoriented. This is proportional to <strong>the</strong> difference <strong>in</strong> <strong>in</strong>terfacial energies between <strong>the</strong><br />
two nuclei, <strong>the</strong> surface areas <strong>of</strong> <strong>the</strong> gra<strong>in</strong>s and <strong>the</strong> gra<strong>in</strong> boundary energy.<br />
If <strong>the</strong> substrate is heated to <strong>the</strong> growth temperature and <strong>the</strong> silane and propane are<br />
<strong>in</strong>troduced, <strong>the</strong> nuclei will grow vertically and laterally <strong>in</strong> a three dimensional manner<br />
called one-step deposition. This causes <strong>the</strong> vertical growth <strong>of</strong> <strong>the</strong> gra<strong>in</strong>s to occur faster<br />
than lateral growth <strong>of</strong> <strong>the</strong> epitaxially oriented ones caus<strong>in</strong>g a polycrystall<strong>in</strong>e<br />
aggregate.<br />
In <strong>the</strong> two-step process, <strong>the</strong> island nuclei start form<strong>in</strong>g dur<strong>in</strong>g <strong>the</strong> ramp<strong>in</strong>g process at<br />
much lower temperatures with <strong>the</strong>ir growth be<strong>in</strong>g predom<strong>in</strong>antly two-dimensional<br />
with Si com<strong>in</strong>g from <strong>the</strong> substrate. In <strong>the</strong> extreme case where only propane is<br />
admitted <strong>in</strong> <strong>the</strong> ramp<strong>in</strong>g stage with no Si present <strong>in</strong> <strong>the</strong> vapour phase vertical growth<br />
is <strong>in</strong>hibited. Due to <strong>the</strong> predom<strong>in</strong>ant lateral growth, epitaxially oriented nuclei grow<br />
much faster than misoriented ones which rapidly consume misoriented gra<strong>in</strong>s before<br />
bulk deposition takes place.<br />
Fig. 5.1. Atomic force microscopy images <strong>of</strong> <strong>3C</strong>-<strong>SiC</strong> epilayers hav<strong>in</strong>g different<br />
thicknesses: (a) 0.08; (b) 0.13; (c) 0.7 and (d) 3.7 µm (from Yun et al. (2006)).
56<br />
Yun et al. (2006) used AFM to show <strong>the</strong> process <strong>of</strong> nucleation and growth at different<br />
stages dur<strong>in</strong>g growth. The images show isolated nuclei form<strong>in</strong>g on <strong>the</strong> substrate at an<br />
epilayer thickness <strong>of</strong> 0.08 µm dur<strong>in</strong>g <strong>the</strong> <strong>in</strong>itial stages <strong>of</strong> deposition. The nuclei merge<br />
at a thickness <strong>of</strong> 0.13 µm and develop fur<strong>the</strong>r to form a cont<strong>in</strong>uous {100} faced<br />
surface at 0.7 µm. They expla<strong>in</strong>ed that <strong>the</strong> driv<strong>in</strong>g force for <strong>the</strong> growth to take place<br />
through this process is a result <strong>of</strong> <strong>the</strong> lattice mismatch between <strong>the</strong> <strong>SiC</strong> and Si as well<br />
as <strong>the</strong>rmal mismatch.<br />
F<strong>in</strong>ally <strong>the</strong> researchers Sudarshan et al. (2006), Nagasawa et al. (2002), and Jacob et<br />
al. (2000) all presented results that are consistent with what is discussed above.<br />
5.3 Crystal Defects <strong>in</strong> <strong>SiC</strong><br />
5.3.1. Introduction<br />
The ma<strong>in</strong> crystal <strong>defects</strong> found <strong>in</strong> epitaxially grown <strong>3C</strong>-<strong>SiC</strong> are stack<strong>in</strong>g faults with<br />
<strong>the</strong>ir accompany<strong>in</strong>g partial dislocations, tw<strong>in</strong>s and <strong>in</strong>version doma<strong>in</strong> boundaries.<br />
In a TEM study by Jacob et al. (2000) on epitaxially grown <strong>3C</strong>-<strong>SiC</strong> on Si(001) a high<br />
density <strong>of</strong> planar <strong>defects</strong> ly<strong>in</strong>g on {111} planes was found. Defects <strong>in</strong>cluded stack<strong>in</strong>g<br />
faults and microtw<strong>in</strong>s and <strong>the</strong> authors suggested that <strong>the</strong> <strong>defects</strong> were <strong>in</strong>troduced<br />
dur<strong>in</strong>g <strong>the</strong> growth process to accommodate <strong>the</strong> large lattice and <strong>the</strong>rmal mismatches<br />
between <strong>the</strong> crystals.<br />
Similar f<strong>in</strong>d<strong>in</strong>gs were reported by Stoemenos et al. (2004) <strong>in</strong> a TEM study on 300µm<br />
thick freestand<strong>in</strong>g <strong>3C</strong>-<strong>SiC</strong>. Microtw<strong>in</strong>s, stack<strong>in</strong>g faults and <strong>in</strong>version doma<strong>in</strong><br />
boundaries were observed and <strong>the</strong>ir presence attributed to <strong>the</strong> same reasons as above.<br />
Powell et al. (1987) <strong>in</strong> an extensive microscopy study on <strong>3C</strong>-<strong>SiC</strong> found microtw<strong>in</strong>s,<br />
stack<strong>in</strong>g faults and also misfit dislocations accommodat<strong>in</strong>g <strong>the</strong> lattice mismatch. They<br />
presented possible mechanisms for <strong>the</strong> nucleation and growth <strong>of</strong> <strong>the</strong> <strong>defects</strong> and <strong>the</strong>se<br />
will be discussed <strong>in</strong> Sections 5.3.2 and 5.3.3.
57<br />
Fur<strong>the</strong>rmore Pirouz et al. (1987), Ho et al. (1999) and Nagasawa et al. (2002) all<br />
study<strong>in</strong>g epitaxially grown <strong>3C</strong>-<strong>SiC</strong> on Si (001) found microtw<strong>in</strong>s and stack<strong>in</strong>g faults<br />
as <strong>the</strong> predom<strong>in</strong>ant <strong>defects</strong> <strong>in</strong> <strong>the</strong> <strong>SiC</strong> and attributed <strong>the</strong>ir presence to lattice and<br />
<strong>the</strong>rmal mismatch.<br />
5.3.2 Accomodation <strong>of</strong> Misfit and Interfacial Tw<strong>in</strong>n<strong>in</strong>g<br />
The mechanism for <strong>the</strong> accommodation <strong>of</strong> misfit dur<strong>in</strong>g growth and its relationship to<br />
<strong>in</strong>terfacial tw<strong>in</strong>n<strong>in</strong>g has been thoroughly expla<strong>in</strong>ed by Powell et al. (1987). The total<br />
misfit, f, between an epilayer and substrate is accommodated by <strong>the</strong> elastic stra<strong>in</strong>, ε,<br />
<strong>of</strong> planes <strong>in</strong> <strong>the</strong> regions <strong>of</strong> good register and by <strong>in</strong>terfacial dislocations, δ. The spac<strong>in</strong>g<br />
<strong>of</strong> misfit dislocations with Burgers vector b would be S = b/δ. The Si/<strong>SiC</strong> system has<br />
a lattice mismatch <strong>of</strong> about 20% (f ≈ 0.2). Assum<strong>in</strong>g that <strong>the</strong> entire misfit is<br />
accommodated by misfit dislocations, f = δ and S ≈ 5b. Thus an array <strong>of</strong> edge misfit<br />
dislocations <strong>in</strong> <strong>SiC</strong> parallel to <strong>the</strong> <strong>in</strong>terface with b = a/2 [110] at a spac<strong>in</strong>g <strong>of</strong> S ≈ 5d110<br />
would accommodate all <strong>the</strong> misfit.<br />
Fur<strong>the</strong>rmore it is also frequently observed that where misfit dislocations become<br />
irregular, tw<strong>in</strong>n<strong>in</strong>g occurs. The tw<strong>in</strong>s nucleate at <strong>the</strong> <strong>in</strong>terface due to coherency<br />
stresses result<strong>in</strong>g from <strong>the</strong> lattice mismatch. Tw<strong>in</strong>n<strong>in</strong>g may be thought <strong>of</strong> as <strong>the</strong><br />
motion <strong>of</strong> partial dislocations with b = a/6 on consecutive {111} planes.<br />
Consider<strong>in</strong>g <strong>the</strong> substrate plane to be (001), a pure misfit edge dislocation with<br />
Burgers vector a/2 110]<br />
[ _<br />
would be <strong>the</strong> most efficient type <strong>of</strong> dislocation to<br />
accommodate <strong>the</strong> lattice mismatch hav<strong>in</strong>g strength 2a / 2 accord<strong>in</strong>g to Frank’s rule.<br />
A shear type dislocation glid<strong>in</strong>g on <strong>the</strong> (111) plane for example with b = a/2 101]<br />
will have an edge component along <strong>the</strong> 110]<br />
direction <strong>of</strong> half <strong>the</strong> strength <strong>of</strong> <strong>the</strong> pure<br />
[ _<br />
edge with 2a / 4 . Thus <strong>the</strong> shear type dislocation is only half as efficient as <strong>the</strong> edge<br />
misfit dislocation <strong>in</strong> accommodat<strong>in</strong>g <strong>the</strong> misfit. On <strong>the</strong> o<strong>the</strong>r hand partial dislocations<br />
<strong>of</strong> <strong>the</strong> type a/6 121]<br />
or a/6 211]<br />
glid<strong>in</strong>g on a (111) plane will also have an edge<br />
[ _<br />
[ _<br />
component with strength 2a / 4 and is just as efficient as <strong>the</strong> shear type dislocation<br />
<strong>in</strong> misfit accommodation. Fur<strong>the</strong>rmore a tw<strong>in</strong> as discussed above may be considered<br />
[ _
58<br />
as <strong>the</strong> propagation <strong>of</strong> a number <strong>of</strong> partials with <strong>the</strong> width <strong>of</strong> <strong>the</strong> tw<strong>in</strong> determ<strong>in</strong>ed by<br />
<strong>the</strong> number n <strong>of</strong> <strong>the</strong> partials. Such a system can be considered hav<strong>in</strong>g Burgers vector<br />
equal to that <strong>of</strong> a superdislocation with b = na/6 121]<br />
[ _<br />
hav<strong>in</strong>g strength n 2a / 4 . Thus<br />
a tw<strong>in</strong> formed by glide <strong>of</strong> <strong>the</strong>se partials is much more efficient <strong>in</strong> mismatch<br />
accommodation than <strong>the</strong> edge misfit and shear dislocations.<br />
It is necessary to consider <strong>the</strong> lattice mismatch <strong>in</strong> <strong>the</strong> 110]<br />
also. A set <strong>of</strong> pure edge<br />
[ _ _<br />
dislocations with b = a/2 110]<br />
at a spac<strong>in</strong>g <strong>of</strong> 5d {110} orthogonal to <strong>the</strong> first array<br />
would satisfy <strong>the</strong> mismatch stra<strong>in</strong> along this direction also. The strength <strong>of</strong> <strong>the</strong> edge<br />
_ _<br />
[ _ _<br />
component <strong>of</strong> an a/6 [ 121]<br />
partial along <strong>the</strong> 110]<br />
[ _ _<br />
direction is 2a / 12 and thus <strong>the</strong><br />
tw<strong>in</strong> needs to be six (111) layers thick to be able to accommodate <strong>the</strong> mismatch along<br />
[ _ _<br />
<strong>the</strong> 110]<br />
direction as efficiently as an edge misfit dislocation hav<strong>in</strong>g b = a/2 110]<br />
. In<br />
ano<strong>the</strong>r publication Powell et al. (1987) presented high resolution TEM results <strong>of</strong><br />
misfit dislocations present at <strong>the</strong> <strong>SiC</strong>/Si <strong>in</strong>terface. They found <strong>the</strong> dislocations occur<br />
on average every four lattice fr<strong>in</strong>ges.<br />
5.3.3. Mechanisms for <strong>the</strong> Formation <strong>of</strong> Stack<strong>in</strong>g Faults<br />
The mechanism <strong>of</strong> formation <strong>of</strong> stack<strong>in</strong>g faults <strong>in</strong> <strong>SiC</strong> is still widely debated. It is<br />
obvious that <strong>the</strong> driv<strong>in</strong>g force beh<strong>in</strong>d <strong>the</strong> process is <strong>the</strong> low stack<strong>in</strong>g fault energy <strong>in</strong><br />
<strong>SiC</strong> but <strong>the</strong> exact process <strong>of</strong> formation <strong>of</strong> <strong>the</strong> SFs dur<strong>in</strong>g growth seems not to be as a<br />
result <strong>of</strong> one th<strong>in</strong>g but ra<strong>the</strong>r a comb<strong>in</strong>ation <strong>of</strong> different processes dur<strong>in</strong>g growth. The<br />
possible mechanisms are discussed by Powell et al. (1987) and Pirouz et al. (1987).<br />
(a) Firstly if it is assumed that <strong>the</strong> process <strong>of</strong> growth <strong>of</strong> <strong>the</strong> material occurs through<br />
<strong>the</strong> formation <strong>of</strong> discrete nuclei on <strong>the</strong> substrate surface, it can be envisioned that<br />
some <strong>of</strong> <strong>the</strong> nuclei may be deposited on <strong>in</strong>correct sites form<strong>in</strong>g surfaces <strong>of</strong> mismatch<br />
boundary. When <strong>the</strong> correctly deposited islands meet <strong>the</strong> <strong>in</strong>correctly deposited islands<br />
an <strong>in</strong>verted pyramid <strong>of</strong> stack<strong>in</strong>g fault could result grow<strong>in</strong>g on <strong>the</strong> <strong>in</strong>cl<strong>in</strong>ed {111} faces<br />
with its {001} base at <strong>the</strong> <strong>in</strong>terface. Us<strong>in</strong>g a simple geometric consideration (depicted<br />
<strong>in</strong> (e)) it can be shown that depend<strong>in</strong>g on <strong>the</strong> nature <strong>of</strong> <strong>the</strong> dislocations at <strong>the</strong><br />
[ _ _
59<br />
<strong>in</strong>tersection <strong>of</strong> <strong>the</strong> fault, <strong>the</strong> faces <strong>of</strong> <strong>the</strong> pyramid would ei<strong>the</strong>r be all <strong>of</strong> <strong>the</strong> same<br />
nature or alternatively <strong>in</strong>tr<strong>in</strong>sic and extr<strong>in</strong>sic faults. It can also be shown that <strong>the</strong><br />
measurement <strong>of</strong> <strong>the</strong> width <strong>of</strong> <strong>the</strong> fault at <strong>the</strong> surface <strong>of</strong> <strong>the</strong> foil enables one to measure<br />
<strong>the</strong> po<strong>in</strong>t <strong>of</strong> nucleation.<br />
(b) It is possible for <strong>in</strong>terfacial stack<strong>in</strong>g faults to have formed dur<strong>in</strong>g <strong>the</strong> cool<strong>in</strong>g<br />
stages due to <strong>the</strong> lattice mismatch between <strong>the</strong> substrate and film. In this case, <strong>the</strong><br />
stack<strong>in</strong>g fault would nucleate at <strong>the</strong> <strong>in</strong>terface dur<strong>in</strong>g <strong>the</strong> cool<strong>in</strong>g stages and<br />
subsequently propagate <strong>in</strong>to <strong>the</strong> bulk by glide <strong>of</strong> partial dislocations. However <strong>the</strong><br />
activation energy <strong>of</strong> dislocation motion <strong>in</strong> <strong>SiC</strong> is very high and it is unlikely that <strong>the</strong><br />
stack<strong>in</strong>g fault nucleated would propagate throughout <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> layer. Thus <strong>the</strong><br />
faults are limited to a region close to <strong>the</strong> <strong>in</strong>terface with <strong>the</strong> bond<strong>in</strong>g partial<br />
dislocations parallel to <strong>the</strong> <strong>in</strong>terface.<br />
(c) The next possibility is <strong>the</strong> spontaneous generation <strong>of</strong> dislocations at <strong>the</strong> <strong>in</strong>terface<br />
as soon as <strong>the</strong> epitaxial layer beg<strong>in</strong>s to grow. Conditions on <strong>the</strong> <strong>in</strong>terface such as<br />
contam<strong>in</strong>ation, roughness, moderate stresses would enhance <strong>the</strong> generation at lower<br />
temperatures <strong>of</strong> growth. Accord<strong>in</strong>g to Booker et al. (1978) <strong>the</strong> dislocations always<br />
form <strong>in</strong> pairs and once <strong>the</strong> dislocations are generated <strong>the</strong>y split <strong>in</strong>to two Shockley<br />
partials bound<strong>in</strong>g a ribbon <strong>of</strong> stack<strong>in</strong>g fault between <strong>the</strong>m.<br />
(d) The fourth possibility is that dislocation loops are generated with<strong>in</strong> <strong>the</strong> bulk <strong>of</strong> <strong>the</strong><br />
material presumably dur<strong>in</strong>g <strong>the</strong> cool<strong>in</strong>g stages by <strong>the</strong> stresses caused by <strong>the</strong> <strong>the</strong>rmal<br />
mismatch between <strong>the</strong> substrate and film. Follow<strong>in</strong>g this, <strong>the</strong> glide dislocations would<br />
split up <strong>in</strong>to Shockley partials bound<strong>in</strong>g wide ribbons <strong>of</strong> stack<strong>in</strong>g fault between <strong>the</strong>m<br />
s<strong>in</strong>ce <strong>the</strong> stack<strong>in</strong>g fault energy <strong>in</strong> <strong>SiC</strong> is very low. The <strong>in</strong>teraction between partials<br />
glid<strong>in</strong>g on different {111} planes would give rise to <strong>in</strong>tersect<strong>in</strong>g faults.
60<br />
Fig. 5.2 A schematic diagram <strong>of</strong> <strong>the</strong> fault on <strong>the</strong> (111) plann<strong>in</strong>g assum<strong>in</strong>g that it has<br />
nucleated at po<strong>in</strong>t O. W1 and W2 are <strong>the</strong> widths <strong>of</strong> <strong>the</strong> fault on <strong>the</strong> top and bottom <strong>of</strong><br />
<strong>the</strong> foil respectively.<br />
(e) The last possibility is that stack<strong>in</strong>g faults arise dur<strong>in</strong>g <strong>the</strong> growth <strong>of</strong> <strong>the</strong> film due to<br />
<strong>in</strong>correct deposition <strong>of</strong> some nuclei on <strong>the</strong> surface <strong>of</strong> <strong>the</strong> grow<strong>in</strong>g film. This could be<br />
as a result <strong>of</strong> a rapid deposition rate, <strong>in</strong>homogeneities <strong>in</strong> <strong>the</strong> film or impurities <strong>in</strong> <strong>the</strong><br />
carrier gas. Fur<strong>the</strong>rmore diffusion rates <strong>in</strong> <strong>SiC</strong> are very low at <strong>the</strong> temperature <strong>of</strong><br />
deposition and thus <strong>the</strong> nuclei would not have enough time to readjust <strong>the</strong>mselves to<br />
<strong>the</strong> correct position. Once a mismatch boundary has formed stack<strong>in</strong>g faults will grow<br />
on <strong>the</strong> {111} faces with fur<strong>the</strong>r growth <strong>of</strong> <strong>the</strong> film. A schematic illustration <strong>of</strong> this is<br />
given <strong>in</strong> Fig. 5.2. It is assumed <strong>the</strong> stack<strong>in</strong>g fault nucleated at po<strong>in</strong>t O due to some<br />
<strong>in</strong>homogeneity and grow wider as <strong>the</strong> layer thickened <strong>in</strong> <strong>the</strong> [001] direction. S<strong>in</strong>ce <strong>the</strong><br />
directions are parallel to <strong>the</strong> Peierls valleys <strong>in</strong> crystals with sphalerite structure,<br />
and <strong>the</strong> Peierls energy is expected to be very high <strong>in</strong> <strong>SiC</strong>, <strong>the</strong> growth occurs such that<br />
<strong>the</strong> partials bound<strong>in</strong>g <strong>the</strong> stack<strong>in</strong>g fault lie along <strong>the</strong>se directions. Measur<strong>in</strong>g <strong>the</strong> width<br />
<strong>of</strong> <strong>the</strong> fault at <strong>the</strong> top <strong>of</strong> <strong>the</strong> foil, W1 enables one to determ<strong>in</strong>e <strong>the</strong> depth below <strong>the</strong><br />
surface, h, at which <strong>the</strong> stack<strong>in</strong>g fault nucleated,<br />
h W1<br />
/<br />
2<br />
Yun et al. (2006) expla<strong>in</strong>ed that <strong>the</strong> stack<strong>in</strong>g fault formation is a result <strong>of</strong> a large<br />
number <strong>of</strong> tw<strong>in</strong>s form<strong>in</strong>g on {111} planes dur<strong>in</strong>g <strong>the</strong> early stages <strong>of</strong> growth prior to<br />
<strong>the</strong> coalescence <strong>of</strong> nuclei. The atomic stack<strong>in</strong>g errors may form on {111} planes <strong>of</strong>
61<br />
each nuclei because <strong>of</strong> <strong>the</strong> low surface energy <strong>of</strong> <strong>the</strong> plane. The atomic errors lead to<br />
tw<strong>in</strong> formation and stack<strong>in</strong>g faults appear at tw<strong>in</strong> boundaries.<br />
5.3.4. Stack<strong>in</strong>g Fault Energy <strong>in</strong> <strong>SiC</strong><br />
The stack<strong>in</strong>g fault energy <strong>in</strong> <strong>SiC</strong> has been found to be very low and this is one <strong>of</strong> <strong>the</strong><br />
ma<strong>in</strong> reasons for <strong>the</strong> large number <strong>of</strong> hexagonal polytypes that are found <strong>in</strong> <strong>SiC</strong>. For<br />
<strong>3C</strong>-<strong>SiC</strong> <strong>the</strong> stack<strong>in</strong>g fault energies have been <strong>the</strong>oretically calculated by Denteneer et<br />
al. (1987), Iwata et al. (2002), Käckell et al. (1998) and L<strong>in</strong>defelt et al. (2003).<br />
Denteneer et al. (1987) used an adaptation <strong>of</strong> <strong>the</strong> ANNNI (axial next-nearest-neigbour<br />
Is<strong>in</strong>g) and found <strong>the</strong> stack<strong>in</strong>g fault energy γesf for <strong>the</strong> extr<strong>in</strong>sic fault as -15 mJ.m -2 and<br />
for <strong>the</strong> <strong>in</strong>tr<strong>in</strong>sic γisf = 12 mJ.m -2 . This is <strong>in</strong>terest<strong>in</strong>g s<strong>in</strong>ce <strong>the</strong> value for <strong>the</strong> extr<strong>in</strong>sic<br />
fault is negative which shows that <strong>the</strong> formation <strong>of</strong> <strong>the</strong> fault is energy efficient.<br />
Denteneer expla<strong>in</strong>s that this is an <strong>in</strong>dicator <strong>of</strong> <strong>the</strong> occurrence <strong>of</strong> polytypism <strong>in</strong> <strong>SiC</strong>.<br />
Iwata et al. (2002) also found a negative stack<strong>in</strong>g fault energy <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> <strong>of</strong> -2.70<br />
mJ.m -2 and -6.27 mJ.m -2 us<strong>in</strong>g two different methods which are <strong>the</strong> supercell and<br />
ANNNI method respectively. Käckell et al. (1998) us<strong>in</strong>g a scheme based on density-<br />
functional <strong>the</strong>ory and <strong>the</strong> local-density approximation found values <strong>of</strong> -28 mJ.m -2 and<br />
-3.4 mJ.m -2 for extr<strong>in</strong>sic and <strong>in</strong>tr<strong>in</strong>sic faults respectively.<br />
Fur<strong>the</strong>rmore Denteneer et al. (1989), Karch et al. (1994) and Cheng et al. (1990)<br />
found values <strong>of</strong> 13.8 mJ.m -2 , 11.1 mJ.m -2 , -7.1 mJ.m -2 for <strong>in</strong>tr<strong>in</strong>sic and -6.1 mJ.m -2 ,<br />
-15.4 mJ.m -2 and -32.3 mJ.m -2 for extr<strong>in</strong>sic faults respectively. The negative SF<br />
energy values for <strong>the</strong> extr<strong>in</strong>sic fault is suggested to be due to <strong>the</strong> fact that <strong>the</strong> stack<strong>in</strong>g<br />
sequence <strong>of</strong> <strong>the</strong> extr<strong>in</strong>sic fault is close to <strong>the</strong> bulk stack<strong>in</strong>g sequence <strong>of</strong> <strong>the</strong> more<br />
stable hexagonal polytypes <strong>of</strong> <strong>SiC</strong>.<br />
5.4 Ion Implantation and Radiation Damage <strong>in</strong> <strong>SiC</strong><br />
The process <strong>of</strong> energetic displacement cascades <strong>in</strong> <strong>SiC</strong> is not a trivial process. There<br />
is anisotropy between <strong>the</strong> threshold displacement energies (TDE) <strong>of</strong> <strong>the</strong> two<br />
subsystems and also a difference <strong>in</strong> recovery <strong>of</strong> <strong>the</strong> subsystems after <strong>the</strong><br />
displacements has taken place. The TDE is also highly directional vary<strong>in</strong>g greatly
62<br />
with direction. Values published by Perlado et al. (2000) range from 42 – 112 eV for<br />
Si and 26 – 58 eV for C depend<strong>in</strong>g on <strong>the</strong> direction. He<strong>in</strong>isch et al. (2004), <strong>in</strong> a study<br />
on displacement damage <strong>in</strong> <strong>SiC</strong> <strong>in</strong> fission reactors, stated that <strong>the</strong> displacement value<br />
for Si could be taken as 35 eV and for C as 20 eV s<strong>in</strong>ce <strong>the</strong>se are <strong>the</strong> average values<br />
taken over all directions <strong>in</strong> <strong>the</strong> <strong>SiC</strong> <strong>the</strong> also stated that <strong>the</strong>re are actually four<br />
m<strong>in</strong>imum recoil damage energies that should be considered to create displacements <strong>in</strong><br />
<strong>SiC</strong>, depend<strong>in</strong>g on <strong>the</strong> projectile/target comb<strong>in</strong>ations: 41 eV (C/Si), 35 eV (Si/Si), 24<br />
eV (Si/C) and 20 eV (C/C). Vladimirov et al. (1998) used values <strong>of</strong> 93 eV for Si and<br />
16.3 eV for C <strong>in</strong> <strong>the</strong>ir calculations as well as Ryazanov et al. (2002).<br />
Thus it is obvious that <strong>the</strong>re is still a lot <strong>of</strong> uncerta<strong>in</strong>ty about <strong>the</strong> correct values for<br />
TDE to be used <strong>in</strong> calculations for <strong>SiC</strong> and is as mentioned before, a result <strong>of</strong> <strong>the</strong><br />
anisotropy with<strong>in</strong> <strong>the</strong> material. Hence <strong>the</strong> method <strong>of</strong> computer simulation is used to<br />
determ<strong>in</strong>e <strong>the</strong> displacement mechanisms and TDE <strong>in</strong> <strong>the</strong> material.<br />
Gao et al. (2001) used molecular dynamics (MD) s<strong>of</strong>tware to simulate Si<br />
displacement cascades <strong>in</strong> <strong>SiC</strong>. Their simulation showed that dur<strong>in</strong>g a high energy Si<br />
primary knock on atom (PKA), multiple sub-cascades were generated. This prevented<br />
both a compact structure <strong>of</strong> displaced atoms to form dur<strong>in</strong>g <strong>the</strong> collision phase and a<br />
high-energy density dur<strong>in</strong>g <strong>the</strong> <strong>the</strong>rmal spike. They attributed this to <strong>the</strong> light masses<br />
<strong>of</strong> <strong>the</strong> <strong>SiC</strong> system and small scatter<strong>in</strong>g cross-sections <strong>of</strong> <strong>the</strong> Si and C atoms.<br />
Fur<strong>the</strong>rmore <strong>the</strong> high melt<strong>in</strong>g temperature <strong>of</strong> <strong>the</strong> <strong>SiC</strong> also <strong>in</strong>hibits <strong>the</strong> development <strong>of</strong><br />
a liquid like cascade which results <strong>in</strong> a very short lifetime <strong>of</strong> <strong>the</strong> <strong>the</strong>rmal spike. This<br />
along with <strong>the</strong> formation energies <strong>of</strong> <strong>defects</strong> provide a physical basis for defect<br />
production and cluster<strong>in</strong>g <strong>in</strong> <strong>the</strong> material. They found that at <strong>the</strong> peak <strong>of</strong> <strong>the</strong> <strong>the</strong>rmal<br />
spikes <strong>the</strong> number <strong>of</strong> displaced Si and C atoms are essentially <strong>the</strong> same but that <strong>the</strong> Si<br />
sublattice partially recovers as <strong>the</strong> energy <strong>of</strong> <strong>the</strong> cascades dissipate <strong>in</strong>to <strong>the</strong> lattice,<br />
while <strong>the</strong> damage <strong>in</strong> <strong>the</strong> C sublattice does not. The number <strong>of</strong> Si <strong>in</strong>terstitials surviv<strong>in</strong>g<br />
after <strong>the</strong> cascade was on average three times smaller than for C. This is possible s<strong>in</strong>ce<br />
most silicon atoms are only displaced a short distance from <strong>the</strong>ir lattice positions and<br />
recomb<strong>in</strong>e dur<strong>in</strong>g <strong>the</strong> <strong>the</strong>rmal spike phase. They expla<strong>in</strong> that as a consequence <strong>of</strong> <strong>the</strong><br />
above facts, <strong>the</strong> formation <strong>of</strong> a dispersed cascade is seen with <strong>defects</strong> spread over <strong>the</strong><br />
simulation box, and most <strong>in</strong>terstitials rema<strong>in</strong> as s<strong>in</strong>gle <strong>defects</strong>. Only a small fraction<br />
<strong>of</strong> <strong>in</strong>terstitials form clusters, <strong>the</strong> biggest be<strong>in</strong>g four atoms. Fur<strong>the</strong>rmore Perlado et al.
63<br />
(2000) found that subsequent anneal<strong>in</strong>g <strong>of</strong> <strong>the</strong>ir simulated cascades yielded no<br />
considerable change <strong>in</strong> <strong>the</strong> defect configuration. Thus it may be concluded that almost<br />
all <strong>in</strong>terstitials are generated dur<strong>in</strong>g <strong>the</strong> early stages <strong>of</strong> <strong>the</strong> cascade and <strong>the</strong>n directly<br />
quenched <strong>in</strong>to <strong>the</strong>ir f<strong>in</strong>al arrangements. When consider<strong>in</strong>g vacancies <strong>the</strong> same results<br />
are seen as for <strong>in</strong>terstitials.<br />
Gao et al. (2001) also simulated high energy cascades <strong>in</strong> <strong>the</strong> <strong>SiC</strong> created by Au ions.<br />
They found a disordered region with a very high defect concentration but also with a<br />
large anisotropy regard<strong>in</strong>g <strong>the</strong> amount <strong>of</strong> <strong>in</strong>terstitials <strong>of</strong> Si and C. It clearly<br />
demonstrated that a heavy energetic ion can create a large disordered cluster <strong>in</strong> <strong>SiC</strong>.
6.1 Introduction<br />
64<br />
CHAPTER SIX<br />
EXPERIMENTAL DETAILS<br />
This chapter deals with <strong>the</strong> experimental procedures used <strong>in</strong> this <strong>in</strong>vestigation.<br />
Specifics on <strong>the</strong> growth process, ion implantation, anneal<strong>in</strong>g procedure and <strong>the</strong> TEM<br />
sample preparation are given.<br />
6.2 CVD Growth <strong>of</strong> <strong>SiC</strong><br />
The <strong>3C</strong>-<strong>SiC</strong> used <strong>in</strong> this study was grown by Nova<strong>SiC</strong> <strong>in</strong> France us<strong>in</strong>g H2 as carrier<br />
gas and C3H8 (propane) and SiH4 (silane) as precursors on a silicon substrate. The<br />
layer deposition was carried out at 200 mbar. The first step <strong>of</strong> <strong>the</strong> process consisted <strong>of</strong><br />
a carbonisation <strong>of</strong> Si at 1100 °C under a mixture <strong>of</strong> H2 and C3H8 and <strong>the</strong> second step<br />
was <strong>the</strong> epitaxial growth, performed at 1380 °C with a growth rate <strong>of</strong> 2 µm/h.<br />
6.3 Proton Implantation <strong>in</strong> <strong>SiC</strong><br />
To <strong>in</strong>vestigate <strong>the</strong> damage created by light ion bombardment <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> <strong>the</strong> material<br />
was implanted with 400keV protons to a total dose <strong>of</strong> 2.2 × 10 16 over a 1cm diameter<br />
region. The 400 keV protons were generated by implant<strong>in</strong>g <strong>the</strong> material with 800 keV<br />
H2 + ions which upon enter<strong>in</strong>g <strong>the</strong> material splits up <strong>in</strong>to 400 keV H + . The implantation<br />
was done at NECSA (National Energy Corporation <strong>of</strong> South Africa). SRIM<br />
(Stopp<strong>in</strong>g Ranges <strong>of</strong> Ions <strong>in</strong> Matter) developed by Ziegler et al. (1985) was used to<br />
calculate <strong>the</strong> proton range. This s<strong>of</strong>tware uses Monte Carlo methods to determ<strong>in</strong>e <strong>the</strong><br />
ion range and defect production with<strong>in</strong> <strong>the</strong> material.
6.3.1 Ion Range and Defect Production<br />
65<br />
Fig. 6.1. A plot <strong>of</strong> <strong>the</strong> ion distribution <strong>of</strong> 400 keV protons implanted <strong>in</strong>to <strong>3C</strong>-<strong>SiC</strong><br />
obta<strong>in</strong>ed by us<strong>in</strong>g SRIM<br />
Fig. 6.1. shows a SRIM plot <strong>of</strong> <strong>the</strong> hydrogen ion distribution <strong>of</strong> 400 keV protons<br />
implanted <strong>in</strong>to <strong>3C</strong>-<strong>SiC</strong>. The plot shows a maximum concentration <strong>of</strong> ions at 2.87 µm.<br />
This is with<strong>in</strong> <strong>the</strong> thickness <strong>of</strong> <strong>the</strong> <strong>SiC</strong> layer used for this study. Fig. 6.2. shows SRIM<br />
plots <strong>of</strong> <strong>the</strong> numbers <strong>of</strong> Si and C vacancies produced for displacement energies given<br />
by He<strong>in</strong>isch et al. (2004) and Vladimirov et al. (1998).
(a)<br />
(b)<br />
66<br />
Fig. 6.2. (a) Si and C vacancies produced per unit distance <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> us<strong>in</strong>g<br />
displacement energies <strong>of</strong> 35 eV and 20 eV respectively (He<strong>in</strong>isch et al.); (b) Si and C<br />
vacancies produced per unit distance <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> us<strong>in</strong>g displacement energies <strong>of</strong> 93 eV<br />
and 16.3 eV respectively (Vladimirov et al.)<br />
SRIM gives <strong>the</strong> number <strong>of</strong> vacancies produced per ion for <strong>the</strong> two sets <strong>of</strong><br />
displacement energies to be 13.2 and 10.1 respectively. It is also evident from Fig.<br />
6.2. that <strong>the</strong> number <strong>of</strong> Si vacancies is always less than that <strong>of</strong> <strong>the</strong> C due to its<br />
displacement energy be<strong>in</strong>g higher. In <strong>the</strong> case <strong>of</strong> Fig. 6.2 (b) <strong>the</strong> anisotropy is very<br />
high with an order <strong>of</strong> magnitude difference <strong>in</strong> <strong>the</strong> numbers produced. Fig. 6.3. shows<br />
a plot <strong>of</strong> <strong>the</strong> number <strong>of</strong> displaced atoms per unit distance per <strong>in</strong>com<strong>in</strong>g ion for <strong>the</strong><br />
displacement energies used by He<strong>in</strong>sich et al. (2004).
67<br />
Fig. 6.3. Total number <strong>of</strong> displacements produced <strong>in</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> per <strong>in</strong>com<strong>in</strong>g ion per<br />
Angstrom.<br />
If an area around <strong>the</strong> peak <strong>of</strong> <strong>the</strong> projected range is selected, as <strong>in</strong>dicated on <strong>the</strong> above<br />
figure, an estimate <strong>of</strong> <strong>the</strong> total number <strong>of</strong> displacements per unit volume around <strong>the</strong><br />
damage region can be obta<strong>in</strong>ed for <strong>the</strong> dose used by assum<strong>in</strong>g that <strong>the</strong> number <strong>of</strong><br />
displacements per <strong>in</strong>com<strong>in</strong>g ion per angstrom to be constant over this region.<br />
Multiply<strong>in</strong>g it by <strong>the</strong> dose and divid<strong>in</strong>g it by <strong>the</strong> volume which <strong>in</strong> this case is a disc<br />
with thickness <strong>of</strong> <strong>the</strong> area selected and diameter <strong>of</strong> 1 cm. The number <strong>of</strong><br />
displacements per cm 3 is <strong>the</strong>n found to be 9.5×10 21 <strong>in</strong> <strong>the</strong> damage region. This is<br />
about an order <strong>of</strong> magnitude lower than <strong>the</strong> concentration <strong>of</strong> atoms per unit volume <strong>in</strong><br />
<strong>SiC</strong> which is 9.6×10 22 .<br />
6.4 Anneal<strong>in</strong>g Procedure<br />
After <strong>the</strong> proton implantation two sets <strong>of</strong> samples were sent to <strong>the</strong> University <strong>of</strong> <strong>the</strong><br />
Witwatersrand for anneal<strong>in</strong>g at 1300 °C and 1600 °C for 1 hr. The anneal<strong>in</strong>g was<br />
done <strong>in</strong> vacuum.
6.5 Sample Preparation<br />
68<br />
For <strong>the</strong> analysis <strong>in</strong> <strong>the</strong> TEM, samples were prepared <strong>in</strong> cross-section as illustrated <strong>in</strong><br />
Fig.6.4. Two sections with dimensions 3mm by 0.5 mm were cut from <strong>the</strong> irradiated<br />
area <strong>of</strong> <strong>the</strong> <strong>SiC</strong> samples. The two sections were <strong>the</strong>n glued toge<strong>the</strong>r us<strong>in</strong>g an epoxy<br />
solution with <strong>the</strong> faces <strong>of</strong> <strong>the</strong> irradiated surfaces fac<strong>in</strong>g towards one ano<strong>the</strong>r.<br />
Follow<strong>in</strong>g this two sections <strong>of</strong> <strong>the</strong> same dimensions as before <strong>of</strong> Si were cut and glued<br />
to ei<strong>the</strong>r side <strong>of</strong> <strong>the</strong> <strong>SiC</strong> sections for extra support. The samples were <strong>the</strong>n<br />
mechanically th<strong>in</strong>ned to approximately 250 µm by means <strong>of</strong> wet sandpaper<strong>in</strong>g<br />
followed by polish<strong>in</strong>g on a rotat<strong>in</strong>g pad us<strong>in</strong>g a 3 µm diamond grit solution to remove<br />
any residual surface damage from <strong>the</strong> sandpaper<strong>in</strong>g. This was <strong>the</strong>n mounted on a<br />
copper microscope grid us<strong>in</strong>g an epoxy solution with <strong>the</strong> polished face on <strong>the</strong> grid.<br />
The o<strong>the</strong>r side <strong>of</strong> <strong>the</strong> sample was polished us<strong>in</strong>g <strong>the</strong> same procedure as before to a<br />
total thickness <strong>of</strong> approximately 50 µm. The samples were f<strong>in</strong>ally milled us<strong>in</strong>g a<br />
Gatan PIPS (Precision Ion Polish<strong>in</strong>g System) by means <strong>of</strong> <strong>the</strong> simultaneous<br />
bombardment <strong>of</strong> <strong>the</strong> top and bottom surfaces with 5 kV argon-ion beams at an<br />
<strong>in</strong>cident angle <strong>of</strong> 6°. The mill<strong>in</strong>g cont<strong>in</strong>ued until a small hole appeared <strong>in</strong> <strong>the</strong> sample.<br />
Fig. 6.4. Schematic representation <strong>of</strong> <strong>the</strong> steps <strong>in</strong> <strong>the</strong> preparation <strong>of</strong> a (110) cross-<br />
sectional sample (Neethl<strong>in</strong>g (1980)).
7.1 Introduction<br />
69<br />
CHAPTER SEVEN<br />
RESULTS AND DISCUSSION<br />
The follow<strong>in</strong>g chapter conta<strong>in</strong>s <strong>the</strong> results obta<strong>in</strong>ed from a TEM <strong>in</strong>vestigation <strong>of</strong> <strong>the</strong><br />
as-grown and ion bombarded <strong>3C</strong>-<strong>SiC</strong>. The results <strong>of</strong> this <strong>in</strong>vestigation are also<br />
compared with f<strong>in</strong>d<strong>in</strong>gs <strong>of</strong> o<strong>the</strong>r researchers.<br />
7.2 Extended Defects <strong>in</strong> -<strong>SiC</strong> Grown by CDV on (001) Si<br />
A cross-sectional sample <strong>of</strong> <strong>3C</strong>-<strong>SiC</strong> grown on Si by CVD was prepared us<strong>in</strong>g <strong>the</strong><br />
technique described <strong>in</strong> Section 6.5. Follow<strong>in</strong>g this an <strong>in</strong>vestigation was done by us<strong>in</strong>g<br />
transmission electron microscopy (TEM) as well as high-resolution TEM (HRTEM).<br />
From <strong>the</strong>se results <strong>the</strong> phase <strong>of</strong> <strong>the</strong> <strong>SiC</strong> was determ<strong>in</strong>ed to be <strong>the</strong> cubic (<strong>3C</strong>-<strong>SiC</strong>) z<strong>in</strong>c-<br />
blende structure with a lattice parameter <strong>of</strong> 4.35Å . The <strong>in</strong>terface between <strong>the</strong> <strong>SiC</strong> and<br />
Si was also <strong>in</strong>vestigated. It was found from diffraction data that parallel epitaxial<br />
orientation between <strong>the</strong> two materials was obta<strong>in</strong>ed dur<strong>in</strong>g growth. The consistency<br />
along <strong>the</strong> <strong>in</strong>terface was also <strong>in</strong>vestigated and it was seen to conta<strong>in</strong> a reasonable<br />
amount <strong>of</strong> <strong>in</strong>homogeneities. Examples <strong>of</strong> very th<strong>in</strong> tw<strong>in</strong>ned sections orig<strong>in</strong>at<strong>in</strong>g at <strong>the</strong><br />
<strong>in</strong>terface were also found. In <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> material a large density <strong>of</strong> stack<strong>in</strong>g faults<br />
was observed ly<strong>in</strong>g on <strong>the</strong> {111} close-packed planes with bond<strong>in</strong>g partials <strong>of</strong> <strong>the</strong> 1/6<br />
type. The stack<strong>in</strong>g faults observed <strong>in</strong> an edge-on orientation showed an<br />
<strong>in</strong>terest<strong>in</strong>g feature <strong>in</strong> that <strong>the</strong>y generate satellite streaks <strong>in</strong> a diffraction pattern. This is<br />
consistent with what is normally seen <strong>in</strong> materials conta<strong>in</strong><strong>in</strong>g narrow planar<br />
precipitates. This is due to <strong>the</strong> shape <strong>of</strong> <strong>the</strong> precipitate be<strong>in</strong>g similar to that <strong>of</strong> a plate<br />
like crystal (see p.97 Hirsch et al. (1965)). Fur<strong>the</strong>rmore <strong>the</strong>se stack<strong>in</strong>g faults were<br />
<strong>in</strong>vestigated us<strong>in</strong>g HRTEM and found to be extr<strong>in</strong>sic <strong>in</strong> nature. These f<strong>in</strong>d<strong>in</strong>gs agree<br />
with that found by previous researchers (Sections 5.2-5.3).
70<br />
7.2.1 Crystal Structure and <strong>the</strong> <strong>SiC</strong>/Si Interface<br />
Fig. 7.1. Bright field TEM micrograph <strong>of</strong> <strong>the</strong> Si/<strong>SiC</strong> <strong>in</strong>terfac. B = [011], g =<br />
≥ 0. Inserted are SAD patterns taken from <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> <strong>SiC</strong> and over <strong>the</strong> <strong>in</strong>terface<br />
_<br />
11 1,<br />
s<br />
Fig. 7.1. shows a bright-field TEM micrograph <strong>of</strong> <strong>the</strong> <strong>SiC</strong>/Si <strong>in</strong>terface (<strong>in</strong> figure<br />
caption). The beam direction B is [011] with diffraction vector g = [ 111]<br />
deviation parameter s ≥ 0. The <strong>in</strong>serts are selected area diffraction (SAD) patterns<br />
taken <strong>of</strong> <strong>the</strong> areas <strong>in</strong>dicated and are rotated to reflect <strong>the</strong> correct directions as <strong>in</strong> <strong>the</strong><br />
micrograph s<strong>in</strong>ce <strong>the</strong>re is a rotational relationship between diffraction and image<br />
planes <strong>in</strong> <strong>the</strong> TEM. This rotation relationship has been tabulated for <strong>the</strong> TEM for<br />
various magnifications and is used for <strong>the</strong> correction. In this particular case <strong>the</strong><br />
rotation could also be found easily due to <strong>the</strong> direction <strong>of</strong> growth <strong>of</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> be<strong>in</strong>g<br />
known. The left hand SAD pattern is consistent with <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> phase with beam<br />
direction B = [011] and lattice parameter 4.35 Å. The camera length <strong>of</strong> <strong>the</strong> TEM was<br />
calibrated us<strong>in</strong>g <strong>the</strong> Si substrate as reference. This diffraction pattern (right hand)<br />
corresponds to <strong>the</strong> superposition <strong>of</strong> <strong>the</strong> two sets <strong>of</strong> diffraction patterns <strong>of</strong> <strong>the</strong> Si and<br />
<strong>SiC</strong>, with <strong>the</strong> pattern conta<strong>in</strong><strong>in</strong>g <strong>the</strong> shorter set <strong>of</strong> reciprocal lattice vectors be<strong>in</strong>g that<br />
<strong>of</strong> <strong>the</strong> <strong>SiC</strong> and <strong>the</strong> o<strong>the</strong>r that <strong>of</strong> <strong>the</strong> Si. The follow<strong>in</strong>g procedure was followed.<br />
_<br />
and
Fig. 7.2. Simulated diffraction pattern <strong>of</strong> <strong>the</strong> [011] zone axis <strong>of</strong> <strong>3C</strong>-<strong>SiC</strong>.<br />
71<br />
Fig. 7.2 shows an <strong>in</strong>dexed simulated diffraction pattern <strong>of</strong> <strong>the</strong> [011] zone axis <strong>of</strong> <strong>3C</strong>-<br />
<strong>SiC</strong>. From diffraction <strong>the</strong>ory it is known that <strong>the</strong> diffraction patterns obey <strong>the</strong><br />
follow<strong>in</strong>g relation (Hirsch et al. (1965)),<br />
L Rd hkl<br />
where λ is <strong>the</strong> electron wavelength, L <strong>the</strong> camera length, R <strong>the</strong> distance from <strong>the</strong><br />
transmitted or central diffraction spot to <strong>the</strong> diffraction spot <strong>in</strong> question and dhkl <strong>the</strong><br />
<strong>in</strong>terplanar spac<strong>in</strong>g given by (Hirsch et al. (1965)),<br />
d hkl<br />
<br />
h<br />
2<br />
a<br />
k<br />
2<br />
l<br />
2<br />
with a <strong>the</strong> lattice parameter <strong>of</strong> <strong>the</strong> material and h,k,l <strong>the</strong> Miller <strong>in</strong>dices <strong>of</strong> <strong>the</strong><br />
diffraction vector used. Us<strong>in</strong>g <strong>the</strong>se relations <strong>the</strong> lattice parameter <strong>of</strong> a crystal can be<br />
determ<strong>in</strong>ed from a diffraction pattern provided <strong>the</strong> camera constant λL is known.<br />
R
72<br />
Table 7.1. The values used <strong>in</strong> calculat<strong>in</strong>g <strong>the</strong> camera length. Rm is <strong>the</strong> measured<br />
values <strong>in</strong> arbitrary units us<strong>in</strong>g measur<strong>in</strong>g s<strong>of</strong>tware and R is <strong>the</strong> actual values <strong>in</strong><br />
meters obta<strong>in</strong>ed by us<strong>in</strong>g <strong>the</strong> scal<strong>in</strong>g factor. A value <strong>of</strong> 5.43Å was used for <strong>the</strong> lattice<br />
parameter <strong>of</strong> Si.<br />
Index Rm R (<strong>in</strong> m) λL (<strong>in</strong> m 2 )<br />
93.49 0.007805 2.45E-12<br />
93.49 0.007805 2.45E-12<br />
91.39 0.00763 2.39E-12<br />
91.39 0.00763 2.39E-12<br />
107.27 0.008956 2.43E-12<br />
107.27 0.008956 2.43E-12<br />
Average 2.42E-12<br />
Table 7.2. The values used <strong>in</strong> calculat<strong>in</strong>g <strong>the</strong> lattice parameter <strong>of</strong> <strong>SiC</strong>. Rm is <strong>the</strong><br />
measured values <strong>in</strong> arbitrary units us<strong>in</strong>g measur<strong>in</strong>g s<strong>of</strong>tware and R is <strong>the</strong> actual<br />
values <strong>in</strong> meters obta<strong>in</strong>ed by us<strong>in</strong>g <strong>the</strong> scal<strong>in</strong>g factor.<br />
Index Rm R (<strong>in</strong> m) dhkl (<strong>in</strong> m) Lattice parameter (<strong>in</strong> m)<br />
116.37 0.009715 2.49E-10 4.32E-10<br />
115.86 0.009673 2.51E-10 4.34E-10<br />
115.90 0.009676 2.50E-10 4.34E-10<br />
114.67 0.009573 2.53E-10 4.38E-10<br />
133.69 0.011161 2.17E-10 4.34E-10<br />
132.43 0.011056 2.19E-10 4.38E-10<br />
Average 4.35E-10 ± 0.02E-10<br />
Us<strong>in</strong>g this procedure, <strong>the</strong> lattice parameter <strong>of</strong> <strong>the</strong> <strong>SiC</strong> was determ<strong>in</strong>ed by tak<strong>in</strong>g an<br />
average <strong>of</strong> <strong>the</strong> parameters obta<strong>in</strong>ed us<strong>in</strong>g all <strong>the</strong> and diffraction spots<br />
present <strong>in</strong> <strong>the</strong> pattern. The lattice parameter was found to be 4.35Å ± 0.02Å.<br />
Fur<strong>the</strong>rmore from <strong>the</strong> orientation relationship between <strong>the</strong> two sets <strong>of</strong> diffractions<br />
spots it is clear that exact epitaxial orientation between <strong>the</strong> two materials were<br />
obta<strong>in</strong>ed dur<strong>in</strong>g growth with <strong>the</strong> follow<strong>in</strong>g orientation relationship {001}Si || {001}<strong>SiC</strong><br />
and Si || <strong>SiC</strong>.<br />
Fig. 7.1 also shows evidence <strong>of</strong> planar <strong>defects</strong> orig<strong>in</strong>at<strong>in</strong>g at <strong>the</strong> <strong>in</strong>terface and<br />
extend<strong>in</strong>g <strong>in</strong>to <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> <strong>SiC</strong>. This was confirmed by a high resolution TEM<br />
study <strong>of</strong> <strong>the</strong> <strong>in</strong>terface shown <strong>in</strong> Fig. 7.3.
73<br />
Fig. 7.3. A HRTEM image <strong>of</strong> <strong>the</strong> <strong>SiC</strong>/Si <strong>in</strong>terface with FFTs taken <strong>of</strong> <strong>the</strong> Si and <strong>SiC</strong><br />
(<strong>in</strong>serted)<br />
The <strong>in</strong>serts are <strong>the</strong> images obta<strong>in</strong>ed from fast Fourier transforms (FFT) taken <strong>of</strong> <strong>the</strong><br />
areas <strong>in</strong>dicated. These FFTs are essentially reconstructed images <strong>of</strong> diffraction<br />
patterns <strong>of</strong> <strong>the</strong> areas result<strong>in</strong>g from <strong>the</strong> fact that <strong>in</strong> a TEM <strong>the</strong> images <strong>in</strong> <strong>the</strong> diffraction<br />
plane are Fourier transforms <strong>of</strong> that <strong>in</strong> <strong>the</strong> image plane and vice versa. Thus <strong>the</strong>se<br />
patterns may also be used to determ<strong>in</strong>e <strong>the</strong> lattice parameter <strong>of</strong> <strong>the</strong> <strong>SiC</strong>. Us<strong>in</strong>g <strong>the</strong><br />
same technique as above and a value <strong>of</strong> 4.35ű 0.01Šwas found.<br />
Evidence <strong>of</strong> <strong>defects</strong> orig<strong>in</strong>at<strong>in</strong>g at <strong>the</strong> <strong>in</strong>terface and propagat<strong>in</strong>g along {111} planes is<br />
seen. The orig<strong>in</strong> <strong>of</strong> <strong>the</strong>se <strong>defects</strong> has been ascribed by Powell et al. (Section 5.3.2) to<br />
<strong>the</strong> accommodation <strong>of</strong> <strong>the</strong> lattice and <strong>the</strong>rmal mismatch between <strong>the</strong> <strong>SiC</strong> layer and Si<br />
substrate. This process takes place through <strong>the</strong> <strong>in</strong>troduction <strong>of</strong> misfit dislocations and<br />
tw<strong>in</strong>s. Due to <strong>the</strong> high density <strong>of</strong> tw<strong>in</strong>s and stack<strong>in</strong>g faults at <strong>the</strong> <strong>SiC</strong>/Si <strong>in</strong>terface <strong>the</strong><br />
misfit dislocations could not be resolved at <strong>the</strong> <strong>in</strong>terface.
74<br />
The absence <strong>of</strong> an <strong>in</strong>termediate polycrystall<strong>in</strong>e layer along <strong>the</strong> <strong>in</strong>terface shows that <strong>the</strong><br />
growth did take place accord<strong>in</strong>g to a two-step process with an <strong>in</strong>itial carbonisation<br />
step achieved by only <strong>in</strong>troduc<strong>in</strong>g <strong>the</strong> propane.<br />
7.2.2 Stack<strong>in</strong>g Faults and Partial Dislocations<br />
Fig. 7.4. Bright field TEM micrographs taken from <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> sample. B<br />
= [011], g =<br />
SF selected<br />
_<br />
11 1,<br />
s ≥ 0<br />
Fig. 7.4. shows a set <strong>of</strong> low magnification bright-field TEM micrographs <strong>of</strong> different<br />
areas <strong>in</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> representative <strong>of</strong> <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> material. A high density <strong>of</strong><br />
stack<strong>in</strong>g faults (SF) are seen ly<strong>in</strong>g on <strong>the</strong> four {111} planes. Fr<strong>in</strong>ge contrast typical <strong>of</strong><br />
stack<strong>in</strong>g faults ly<strong>in</strong>g on <strong>in</strong>cl<strong>in</strong>ed and edge-on planes is seen. The stack<strong>in</strong>g faults are<br />
uniformly distributed throughout <strong>the</strong> foil.<br />
The prelim<strong>in</strong>ary results <strong>of</strong> tilt<strong>in</strong>g experiments <strong>in</strong> <strong>the</strong> electron microscope showed <strong>the</strong><br />
fault vector <strong>of</strong> <strong>the</strong> SF to be <strong>of</strong> <strong>the</strong> type 1/3 with bond<strong>in</strong>g partials <strong>of</strong> <strong>the</strong> type<br />
1/6 i.e. <strong>the</strong> partials are <strong>of</strong> <strong>the</strong> Shockley type. This <strong>in</strong>dicates that <strong>the</strong> fault was<br />
not formed as a result <strong>of</strong> <strong>the</strong> aggregation <strong>of</strong> po<strong>in</strong>t <strong>defects</strong>, which would form a Frank<br />
loop with Burgers vector <strong>of</strong> <strong>the</strong> type 1/3.
75<br />
Fig. 7.5. The stack<strong>in</strong>g fault selected for simulation with arrows po<strong>in</strong>t<strong>in</strong>g to <strong>the</strong><br />
thickness fr<strong>in</strong>ges used for foil thickness determ<strong>in</strong>ation<br />
To verify <strong>the</strong> suggested nature <strong>of</strong> <strong>the</strong> fault image, simulation and match<strong>in</strong>g was<br />
performed. Fig. 7.5. shows a magnified bright field TEM micrograph <strong>of</strong> <strong>the</strong> SF shown<br />
<strong>in</strong> Fig. 7.4. This fault was selected for <strong>the</strong> simulation and <strong>the</strong> assumed fault vector and<br />
partial dislocation burgers vectors were used. Follow<strong>in</strong>g this, <strong>the</strong> unknowns <strong>of</strong> <strong>the</strong><br />
problem were determ<strong>in</strong>ed start<strong>in</strong>g with foil thickness. This was done us<strong>in</strong>g <strong>the</strong><br />
thickness fr<strong>in</strong>ges present <strong>in</strong> <strong>the</strong> micrograph as well as <strong>the</strong> technique <strong>of</strong> convergent<br />
beam electron diffraction (CBED) (De Graef (2003)).<br />
Thickness fr<strong>in</strong>ge method<br />
Consider Fig. 7.5. It is known that <strong>the</strong> thickness fr<strong>in</strong>ges present <strong>in</strong> a bright field (or<br />
dark field) micrograph follow <strong>in</strong> <strong>in</strong>crements <strong>of</strong> 1/2ξ (ξ is ext<strong>in</strong>ction distance) for each<br />
successive bright to dark fr<strong>in</strong>ge start<strong>in</strong>g from <strong>the</strong> edge <strong>of</strong> <strong>the</strong> material (Hirsch et al.<br />
(1965)). Thus <strong>in</strong> this case <strong>the</strong> foil thickness was found to be approximately 2.75 ξ at<br />
<strong>the</strong> po<strong>in</strong>t <strong>of</strong> <strong>the</strong> fault us<strong>in</strong>g <strong>the</strong> thickness fr<strong>in</strong>ges as <strong>in</strong>dicated. Us<strong>in</strong>g this and <strong>the</strong><br />
ext<strong>in</strong>ction distance <strong>of</strong> <strong>the</strong> {111} reflection <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> <strong>the</strong> thickness was determ<strong>in</strong>ed to<br />
be ~ 188 nm. A value <strong>of</strong> 622 Å was used for <strong>the</strong> ext<strong>in</strong>ction distance and was found<br />
us<strong>in</strong>g Electron Diffraction (2003).<br />
3 ξ<br />
~2.75 ξ<br />
2 ξ<br />
5/2 ξ<br />
3/2 ξ
CBED<br />
76<br />
Fig. 7.6. shows an annotated micrograph <strong>of</strong> a CBED pattern recorded us<strong>in</strong>g a 400<br />
reflection at <strong>the</strong> area conta<strong>in</strong><strong>in</strong>g <strong>the</strong> stack<strong>in</strong>g fault.<br />
Fig. 7.6. An annotated CBED micrograph used <strong>in</strong> <strong>the</strong> determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> foil<br />
thickness<br />
The distances measured on <strong>the</strong> orig<strong>in</strong>al negative are <strong>in</strong>dicated on <strong>the</strong> figure. The<br />
values calculated accord<strong>in</strong>g to <strong>the</strong> technique outl<strong>in</strong>ed <strong>in</strong> De Graef (2003) are given <strong>in</strong><br />
Table 7.3 with correspond<strong>in</strong>g plotted values and <strong>the</strong> associated l<strong>in</strong>ear trend l<strong>in</strong>e given<br />
<strong>in</strong> Fig. 7.7.<br />
Table 7.3. The measured parameters with calculated variables used <strong>in</strong> obta<strong>in</strong><strong>in</strong>g Fig.<br />
7.7.<br />
ni Δθi Si (nm -2 ) 1/ni 2 (Si/ni) 2 (nm -2 )<br />
2 0.36 0.0050 0.250 6.31667E-06<br />
3 0.80 0.0111 0.111 1.38637E-05<br />
4 1.13 0.0157 0.063 1.55589E-05<br />
5 1.47 0.0205 0.040 1.68515E-05
(Si/ni) 2<br />
0.000018<br />
0.000016<br />
0.000014<br />
0.000012<br />
0.00001<br />
0.000008<br />
0.000006<br />
0.000004<br />
0.000002<br />
0<br />
77<br />
y = -4.99E-05x + 1.89E-05<br />
R 2 = 9.95E-01<br />
0 0.05 0.1 0.15 0.2 0.25 0.3<br />
(1/ni) 2<br />
Fig. 7.7. The plotted values <strong>of</strong> Table 7.1 with <strong>the</strong> l<strong>in</strong>ear trend l<strong>in</strong>e used to determ<strong>in</strong>e<br />
<strong>the</strong> foil thickness and ext<strong>in</strong>ction distance<br />
From <strong>the</strong> trend l<strong>in</strong>e <strong>the</strong> values <strong>of</strong> ext<strong>in</strong>ction distance and foil thickness can be<br />
determ<strong>in</strong>ed from <strong>the</strong> slope and y-<strong>in</strong>tercept <strong>of</strong> <strong>the</strong> graph respectively by tak<strong>in</strong>g <strong>the</strong><br />
square root <strong>of</strong> <strong>the</strong> <strong>in</strong>verse <strong>of</strong> <strong>the</strong> associated value. The ext<strong>in</strong>ction distance was found<br />
to be 142 nm and foil thickness 230 nm. Electron Diffraction (2003) calculates <strong>the</strong><br />
ext<strong>in</strong>ction distance <strong>of</strong> <strong>the</strong> 400 reflection <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> to be 138 nm and is thus <strong>in</strong> good<br />
agreement with <strong>the</strong> obta<strong>in</strong>ed value. The value found for foil thickness us<strong>in</strong>g <strong>the</strong><br />
thickness fr<strong>in</strong>ge method agrees well with that <strong>of</strong> <strong>the</strong> CBED technique but <strong>the</strong> value<br />
found by <strong>the</strong> latter should be taken as <strong>the</strong> more accurate one.<br />
Follow<strong>in</strong>g this <strong>the</strong> anomalous absorption ratio <strong>of</strong> <strong>the</strong> material was chosen. This ratio<br />
affects <strong>the</strong> image contrast characteristics <strong>in</strong> <strong>the</strong> bright and dark-field image and may<br />
lead to <strong>in</strong>correct conclusions if <strong>the</strong> wrong value is used for <strong>the</strong> image simulations. A<br />
value between that <strong>of</strong> diamond and silicon was assumed s<strong>in</strong>ce no <strong>the</strong>oretical value<br />
could be found <strong>in</strong> literature. A value <strong>of</strong> 0.055 was used. The values <strong>of</strong> diamond and<br />
silicon are ~ 0.01 and ~ 0.1 respectively and can be calculated by f<strong>in</strong>d<strong>in</strong>g <strong>the</strong> ratio <strong>of</strong><br />
<strong>the</strong> absorption length to that <strong>of</strong> <strong>the</strong> ext<strong>in</strong>ction distance which <strong>in</strong> turn could be found<br />
by various s<strong>of</strong>tware packages such as JEMS - Electron Microscopy S<strong>of</strong>tware.
78<br />
After determ<strong>in</strong><strong>in</strong>g all <strong>the</strong> variables <strong>of</strong> <strong>the</strong> problem <strong>the</strong> fault was simulated us<strong>in</strong>g<br />
s<strong>of</strong>tware developed by Head et al. (1955) us<strong>in</strong>g a technique analogous to that<br />
discussed <strong>in</strong> Section 4.3.3.<br />
Fig. 7.8. The bright field and dark field TEM micrographs with <strong>the</strong>ir correspond<strong>in</strong>g<br />
simulated images <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g fault selected.<br />
Fig. 7.8. shows both bright and dark field TEM micrographs <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g fault with<br />
<strong>the</strong> correspond<strong>in</strong>g simulated images. Excellent agreement is seen <strong>in</strong> both cases and it
79<br />
may be concluded that <strong>the</strong> <strong>in</strong>itial identification <strong>of</strong> <strong>the</strong> fault was correct. Also <strong>the</strong><br />
fr<strong>in</strong>ges <strong>in</strong> <strong>the</strong> bright and dark-field images are complementary and symmetrical <strong>in</strong><br />
both cases. This be<strong>in</strong>g a direct result <strong>of</strong> <strong>the</strong> magnitude <strong>of</strong> <strong>the</strong> anomalous absorption<br />
ratio for <strong>SiC</strong> which differs from that <strong>of</strong> heavier elements. The complementary and<br />
symmetrical nature <strong>of</strong> bright and dark-field SF images <strong>of</strong> <strong>SiC</strong> is not true <strong>in</strong> general for<br />
<strong>the</strong> heavier elements (Hirsch et al. (1965)).<br />
SF viewed edge on<br />
Fig. 7.9. Bright-field TEM micrograph <strong>of</strong> narrow planar <strong>defects</strong> viewed edge-on. B =<br />
[011], g =<br />
_<br />
11 1,<br />
s ≥ 0. Inserted is <strong>the</strong> SAD pattern taken over <strong>the</strong> area <strong>in</strong>dicated and<br />
below <strong>the</strong> dark field image obta<strong>in</strong>ed by us<strong>in</strong>g <strong>the</strong> spot <strong>in</strong>dicated.
80<br />
Fig. 7.9. shows a bright-field TEM image obta<strong>in</strong>ed from an area with<strong>in</strong> <strong>the</strong> material<br />
conta<strong>in</strong><strong>in</strong>g narrow planar <strong>defects</strong>. A SAD pattern <strong>of</strong> this area showed additional<br />
streaks between <strong>the</strong> diffraction spots. Us<strong>in</strong>g one <strong>of</strong> <strong>the</strong>se streaks to form a dark field<br />
image resulted <strong>in</strong> an <strong>in</strong>version <strong>of</strong> <strong>the</strong> contrast <strong>in</strong>dicat<strong>in</strong>g that <strong>the</strong> <strong>defects</strong> are <strong>the</strong> orig<strong>in</strong><br />
<strong>of</strong> <strong>the</strong> streaks. This is similar to streaks produced by th<strong>in</strong> precipitates viewed edge-on.<br />
A fur<strong>the</strong>r <strong>in</strong>vestigation was conducted by means <strong>of</strong> HRTEM and it was found that <strong>the</strong><br />
streaks are a result <strong>of</strong> extr<strong>in</strong>sic stack<strong>in</strong>g faults present <strong>in</strong> <strong>the</strong> <strong>SiC</strong> viewed edge-on.<br />
Fig. 7.10. HRTEM image and magnified view <strong>of</strong> a fault present <strong>in</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> sample<br />
with a FFT taken over <strong>the</strong> area <strong>in</strong>dicated<br />
{111}<br />
Fig. 7.10. shows a high resolution TEM image <strong>of</strong> <strong>the</strong> sample with a planar defect<br />
viewed edge-on. The {111} planes run <strong>in</strong> directions <strong>in</strong>dicated on <strong>the</strong> figure. The <strong>in</strong>sert<br />
is a magnified view <strong>of</strong> <strong>the</strong> area <strong>in</strong>dicated with its associated FFT.<br />
From <strong>the</strong> magnified view it is clear that <strong>the</strong> defect is an extr<strong>in</strong>sic stack<strong>in</strong>g fault. Two<br />
displaced {111} layers are seen as described <strong>in</strong> Section 2.7. Fur<strong>the</strong>rmore streaks<br />
similar to that seen previously <strong>in</strong> <strong>the</strong> SAD pattern are found <strong>in</strong> <strong>the</strong> FFT image,<br />
<strong>in</strong>dicat<strong>in</strong>g that <strong>the</strong> defect is <strong>the</strong> same type as that observed <strong>in</strong> Fig. 7.9.
81<br />
Fig. 7.11. Bright field TEM micrograph <strong>of</strong> <strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> sample. B = [011], g = _<br />
11 1,<br />
s ≥ 0<br />
The high density <strong>of</strong> stack<strong>in</strong>g faults <strong>in</strong> <strong>the</strong> crystal is attributed to <strong>the</strong> very low stack<strong>in</strong>g<br />
fault energy <strong>in</strong> <strong>SiC</strong>. The fact that <strong>the</strong> faults shows different characteristics <strong>in</strong> terms <strong>of</strong><br />
lengths and comb<strong>in</strong>ations <strong>of</strong> partials, along with its uniform distribution throughout<br />
<strong>the</strong> crystal suggests that <strong>the</strong>ir formation is not a result <strong>of</strong> only one process (e.g. misfit<br />
stra<strong>in</strong>) but ra<strong>the</strong>r a comb<strong>in</strong>ation <strong>of</strong> processes dur<strong>in</strong>g growth and possibly post growth.<br />
Firstly pyramidal stack<strong>in</strong>g faults are seen (<strong>in</strong>dicated by <strong>the</strong> arrow <strong>in</strong> Fig. 7.11.) <strong>in</strong> <strong>the</strong><br />
sample support<strong>in</strong>g <strong>the</strong> <strong>the</strong>ory <strong>of</strong> formation by <strong>the</strong> <strong>in</strong>correct deposition <strong>of</strong> nuclei dur<strong>in</strong>g<br />
bulk growth as expla<strong>in</strong>ed by Powell et al. (1987). Secondly wide ribbons <strong>of</strong> stack<strong>in</strong>g<br />
fault along with <strong>in</strong>tersect<strong>in</strong>g faults <strong>in</strong> <strong>the</strong> form <strong>of</strong> stair-rod structures (<strong>in</strong>dicated by <strong>the</strong><br />
arrow <strong>in</strong> Fig. 7.11.) are also seen support<strong>in</strong>g <strong>the</strong> <strong>the</strong>ory <strong>of</strong> spontaneous generation <strong>of</strong><br />
dislocation loops dur<strong>in</strong>g <strong>the</strong> cool<strong>in</strong>g stages <strong>of</strong> growth due to <strong>the</strong>rmal mismatch.The<br />
subsequent splitt<strong>in</strong>g <strong>of</strong> glide dislocations <strong>in</strong>to partials leads to <strong>the</strong> formation <strong>of</strong> wide<br />
ribbons <strong>of</strong> stack<strong>in</strong>g fault and <strong>the</strong> <strong>in</strong>teractions <strong>of</strong> mov<strong>in</strong>g partials results <strong>in</strong> <strong>in</strong>tersect<strong>in</strong>g<br />
faults (expla<strong>in</strong>ed by Powell et al. (1987)).<br />
Fur<strong>the</strong>rmore, <strong>the</strong> extr<strong>in</strong>sic faults seen <strong>in</strong> <strong>the</strong> high resolution TEM images support <strong>the</strong><br />
<strong>the</strong>ory that extr<strong>in</strong>sic faults have a much lower energy than <strong>in</strong>tr<strong>in</strong>sic SFs. This may be<br />
due to <strong>the</strong> fact that <strong>the</strong> stack<strong>in</strong>g sequence <strong>of</strong> <strong>the</strong> extr<strong>in</strong>sic fault is close to <strong>the</strong> bulk<br />
stack<strong>in</strong>g sequence <strong>of</strong> <strong>the</strong> more stable hexagonal phases <strong>of</strong> <strong>SiC</strong>.
7.2.3 Tw<strong>in</strong>n<strong>in</strong>g<br />
Fig. 7.12. Bright-field TEM micrograph <strong>of</strong> a tw<strong>in</strong>ned area <strong>in</strong> <strong>the</strong> <strong>3C</strong>- <strong>SiC</strong> lattice.<br />
B = [011], g =<br />
82<br />
_<br />
11 1,<br />
s ≥ 0. The <strong>in</strong>sert is a SAD pattern <strong>of</strong> <strong>the</strong> area with <strong>the</strong><br />
correspond<strong>in</strong>g dark-field micrograph obta<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong> tw<strong>in</strong> spot <strong>in</strong>dicated<br />
Fig. 7.12. shows a bright-field TEM image <strong>of</strong> an area close to <strong>the</strong> <strong>SiC</strong>/Si <strong>in</strong>terface <strong>in</strong><br />
<strong>the</strong> <strong>SiC</strong> where tw<strong>in</strong>s are present. The tw<strong>in</strong>s were identified by analys<strong>in</strong>g <strong>the</strong> extra<br />
spots <strong>in</strong> <strong>the</strong> SAD pattern shown <strong>in</strong> Fig. 7.12. The extra spots are consistent with a<br />
180° rotation <strong>of</strong> <strong>the</strong> lattice about a {111} plane. Also <strong>the</strong> dark-field image shown <strong>in</strong>
83<br />
Fig. 7.12 which was obta<strong>in</strong>ed by us<strong>in</strong>g one <strong>of</strong> <strong>the</strong> extra spots showed an <strong>in</strong>version <strong>of</strong><br />
<strong>the</strong> contrast over <strong>the</strong> area <strong>in</strong>dicat<strong>in</strong>g <strong>the</strong> tw<strong>in</strong>ned regions. The tw<strong>in</strong>s are long, very<br />
narrow, orig<strong>in</strong>ate at <strong>the</strong> <strong>in</strong>terface and extent only a short distance <strong>in</strong>to <strong>the</strong> bulk <strong>of</strong> <strong>the</strong><br />
material from <strong>the</strong> <strong>in</strong>terface.<br />
The presence <strong>of</strong> tw<strong>in</strong>s is consistent with <strong>the</strong> <strong>the</strong>ories <strong>of</strong> nucleation and growth and<br />
accommodation <strong>of</strong> lattice misfit presented <strong>in</strong> Sections 5.2. and 5.3.2. The tw<strong>in</strong>s are<br />
most probably <strong>in</strong>troduced dur<strong>in</strong>g <strong>the</strong> growth process. S<strong>in</strong>ce <strong>the</strong> film/substrate<br />
<strong>in</strong>terfacial energies <strong>of</strong> <strong>the</strong> tw<strong>in</strong>ned nuclei is higher than that <strong>of</strong> <strong>the</strong> epitaxial ones <strong>the</strong><br />
epitaxial ones will have a faster lateral growth than <strong>the</strong> tw<strong>in</strong>ned nuclei and will also<br />
cause <strong>the</strong> tw<strong>in</strong>s to grow along <strong>in</strong>cl<strong>in</strong>ed {111} planes. This is exactly what is seen <strong>in</strong><br />
results and supports <strong>the</strong> <strong>the</strong>ory presented <strong>in</strong> Sections 5.2 and 5.3.2.<br />
7.3 Hydrogen Implanted -<strong>SiC</strong><br />
In this section <strong>the</strong> results <strong>of</strong> <strong>the</strong> <strong>in</strong>vestigation <strong>of</strong> proton irradiated <strong>3C</strong>-<strong>SiC</strong> to a dose <strong>of</strong><br />
2.8 × 10 16 protons/cm 2 at 400 keV are presented. Implanted samples that were<br />
unannealed and annealed at 1300 °C and 1600°C for one hour were prepared <strong>in</strong> cross-<br />
section and analysed by TEM. In <strong>the</strong> unannealed sample no noticeable change <strong>in</strong> <strong>the</strong><br />
microstructure <strong>of</strong> <strong>the</strong> <strong>SiC</strong> was seen show<strong>in</strong>g <strong>the</strong> same characteristics <strong>of</strong> <strong>the</strong><br />
unimplanted <strong>SiC</strong> sample. Also no effect due to implantation was seen <strong>in</strong> any <strong>of</strong> <strong>the</strong><br />
annealed samples. No evidence <strong>of</strong> a damage region was seen or <strong>of</strong> <strong>defects</strong> hav<strong>in</strong>g<br />
formed as a result <strong>of</strong> <strong>the</strong> irradiation process. The structure <strong>of</strong> <strong>the</strong> <strong>SiC</strong> rema<strong>in</strong>ed<br />
crystall<strong>in</strong>e with <strong>the</strong> cubic <strong>3C</strong> phase for every sample even those annealed at 1600°C.<br />
7.3.1 Unannealed<br />
The unannealed sample yielded no difference from <strong>the</strong> characteristics seen <strong>in</strong> <strong>the</strong><br />
unimplanted sample. No evidence <strong>of</strong> radiation damage or a damage zone was seen<br />
with <strong>the</strong> <strong>SiC</strong> rema<strong>in</strong><strong>in</strong>g crystall<strong>in</strong>e throughout. The ma<strong>in</strong> <strong>defects</strong> seen were stack<strong>in</strong>g<br />
faults with <strong>the</strong>ir associated partial dislocations and tw<strong>in</strong>s orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong><br />
<strong>in</strong>terface and extend<strong>in</strong>g only a limited distance <strong>in</strong>to <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> <strong>SiC</strong> layer.
84<br />
These results agree well with <strong>the</strong> <strong>the</strong>ories proposed by Gao et al. (2001) and Perlado<br />
et al.(2000) from <strong>the</strong>ir molecular dynamics simulations <strong>of</strong> displacement cascades <strong>in</strong><br />
<strong>SiC</strong>. It shows that <strong>the</strong> proton does not impart to <strong>the</strong> lattice enough energy to create<br />
vast displacement cascades capable <strong>of</strong> creat<strong>in</strong>g an amorphous region. This is due to its<br />
small mass compared to <strong>the</strong> constituent atoms and also due to <strong>the</strong> high threshold<br />
displacement energies <strong>of</strong> <strong>the</strong> atoms, especially <strong>the</strong> Si. The anisotropy <strong>of</strong> <strong>the</strong><br />
displacement energies between <strong>the</strong> constituent atoms also play a big role and <strong>the</strong>ir<br />
difference <strong>in</strong> recovery after <strong>the</strong> <strong>the</strong>rmal spike prevents large defect clusters to form<br />
dur<strong>in</strong>g <strong>the</strong> radiation process. It also causes <strong>the</strong> cascade to be dispersed, conta<strong>in</strong><strong>in</strong>g<br />
only s<strong>in</strong>gle po<strong>in</strong>t <strong>defects</strong> and small po<strong>in</strong>t defect clusters. F<strong>in</strong>ally <strong>the</strong> results also show<br />
that a significantly higher proton dose should be used if any form <strong>of</strong> visible radiation<br />
damage is to be produced.<br />
7.3.2 Annealed at 1300°C<br />
No evidence <strong>of</strong> radiation damage was observed after a 1hr anneal at 1300°C. Bright-<br />
field TEM images us<strong>in</strong>g all <strong>the</strong> reflections 200, 111, 220 and 311 showed no<br />
significant changes <strong>in</strong> <strong>the</strong> characteristics observed <strong>in</strong> <strong>the</strong> unimplanted or implanted<br />
unannealed samples. Fig. 7.13. shows a bright field TEM image <strong>of</strong> <strong>the</strong> sample us<strong>in</strong>g a<br />
311 two-beam condition. This image is similar to that obta<strong>in</strong>ed with o<strong>the</strong>r two-beam<br />
conditions. Only th<strong>in</strong> tw<strong>in</strong>s, edge-on stack<strong>in</strong>g faults and <strong>in</strong>cl<strong>in</strong>ed stack<strong>in</strong>g faults were<br />
seen and no damage region or dislocation loops due to <strong>the</strong> cluster<strong>in</strong>g <strong>of</strong> po<strong>in</strong>t <strong>defects</strong><br />
were observed.
85<br />
Fig. 7.13. Bright-field TEM micrograph <strong>of</strong> <strong>3C</strong>-<strong>SiC</strong> sample annealed at 1300°C for 1<br />
hr. B = [011], g = 311, s ≥ 0. The <strong>in</strong>sert is a SAD pattern <strong>of</strong> <strong>the</strong> two beam condition<br />
used.<br />
7.3.3 Annealed at 1600°C<br />
For <strong>the</strong> analysis <strong>of</strong> this sample a beam direction was used s<strong>in</strong>ce <strong>the</strong> sample was<br />
most electron transparent <strong>in</strong> this direction. Also it was assumed that if <strong>the</strong> effects <strong>of</strong><br />
radiation damage was to be seen <strong>in</strong> <strong>the</strong> sample it would be <strong>in</strong> <strong>the</strong> form <strong>of</strong> dislocation<br />
loops which would lie on <strong>the</strong> {111} close packed planes. These loops will be strongly<br />
visible us<strong>in</strong>g 220 type reflections which <strong>in</strong> turn are easily accessible us<strong>in</strong>g a <br />
beam direction. However extensive exam<strong>in</strong>ation <strong>of</strong> <strong>the</strong> sample us<strong>in</strong>g all <strong>of</strong> <strong>the</strong> 220<br />
reflections showed no evidence <strong>of</strong> a damage region or dislocation loops hav<strong>in</strong>g<br />
formed due to <strong>the</strong> cluster<strong>in</strong>g <strong>of</strong> <strong>in</strong>tr<strong>in</strong>sic po<strong>in</strong>t <strong>defects</strong>. The only <strong>defects</strong> found were<br />
stack<strong>in</strong>g faults with <strong>the</strong>ir associated partial dislocations and <strong>in</strong>terest<strong>in</strong>g dislocation<br />
complexes such as shown <strong>in</strong> Fig. 7.14.
86<br />
(a) (b)<br />
Fig. 7.14. (a) Bright-field TEM micrograph <strong>of</strong> <strong>the</strong> bulk <strong>of</strong> <strong>the</strong> <strong>3C</strong>- <strong>SiC</strong> sample<br />
annealed at 1600°C for 1 hr. (b) Bright field micrograph show<strong>in</strong>g dislocation<br />
complexes <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> sample annealed at 1600°C for 1 hr. B = [111], g = 220, s ≥0.<br />
The dislocation complexes seen <strong>in</strong> Fig. 7.14 (b) were <strong>in</strong>itially not observed <strong>in</strong> <strong>the</strong><br />
unimplanted sample and it was <strong>in</strong>itially thought that <strong>the</strong>y might have formed due to<br />
ei<strong>the</strong>r <strong>the</strong> irradiation or anneal<strong>in</strong>g process. This was found not to be <strong>the</strong> case s<strong>in</strong>ce a<br />
reassessment (Section 7.3) <strong>of</strong> <strong>the</strong> unimplanted sample <strong>in</strong> <strong>the</strong> {111} beam direction<br />
showed similar defect complexes.<br />
Fur<strong>the</strong>rmore an electron diffraction <strong>in</strong>vestigation was done to confirm that no phase<br />
change occurred dur<strong>in</strong>g <strong>the</strong> anneal<strong>in</strong>g <strong>of</strong> <strong>the</strong> sample at 1600°C. A diffraction pattern<br />
<strong>of</strong> <strong>the</strong> zone axis was taken and matched with simulated patterns. This was done<br />
for <strong>the</strong> s<strong>in</strong>ce this direction allows an unambiguous differentiation between<br />
cubic and hexagonal <strong>SiC</strong> phases. Excellent agreement was found and it was<br />
concluded that <strong>the</strong> crystal structure rema<strong>in</strong>ed that <strong>of</strong> <strong>the</strong> cubic <strong>3C</strong> phase. The results<br />
are shown <strong>in</strong> Fig. 7.15.
87<br />
(a) (b)<br />
Fig. 7.15. The (a) experimental (b) simulated SAD pattern <strong>of</strong> <strong>the</strong> [110] zone axis <strong>in</strong><br />
<strong>the</strong> <strong>3C</strong>-<strong>SiC</strong> sample annealed at 1600°C for 1 hr<br />
To expla<strong>in</strong> <strong>the</strong> results obta<strong>in</strong>ed for <strong>the</strong> implanted <strong>SiC</strong> <strong>the</strong> follow<strong>in</strong>g is proposed.<br />
Dur<strong>in</strong>g proton implantation po<strong>in</strong>t <strong>defects</strong> (<strong>in</strong>terstitial atoms, vacancies and hydrogen<br />
atoms) are generated simultaneously <strong>in</strong> <strong>the</strong> specimen. In <strong>the</strong> case <strong>of</strong> <strong>SiC</strong> <strong>the</strong> <strong>in</strong>terstitial<br />
atoms and vacancies are from both sublattices, i.e. silicon and carbon. S<strong>in</strong>ce<br />
<strong>in</strong>terstitials and vacancies can have a charge, <strong>the</strong>y can trap electrons or holes. Po<strong>in</strong>t<br />
<strong>defects</strong> produced by irradiation can be trapped by a number <strong>of</strong> s<strong>in</strong>ks <strong>of</strong> which <strong>the</strong><br />
follow<strong>in</strong>g are <strong>the</strong> most important:<br />
1. Interstitial and vacancy recomb<strong>in</strong>ation will result <strong>in</strong> mutual annihilation.<br />
2. Some vacancies may trap hydrogen atoms lead<strong>in</strong>g to <strong>the</strong> formation <strong>of</strong> gas filled<br />
bubbles or two dimensional platelets.<br />
3. Interstitials and vacancies can be trapped by dislocations, dislocation loops and<br />
voids.<br />
In order for <strong>the</strong> displaced atoms and hydrogen to <strong>in</strong>teract dur<strong>in</strong>g post irradiation<br />
anneal<strong>in</strong>g, <strong>the</strong>y must be mobile enough to migrate to <strong>the</strong> various s<strong>in</strong>ks.<br />
S<strong>in</strong>ce dislocation loops on {111} and {110} planes consist <strong>of</strong> double layers <strong>of</strong> silicon<br />
and carbon atoms charge neutrality is ma<strong>in</strong>ta<strong>in</strong>ed. This is a result <strong>of</strong> <strong>the</strong> Coulomb<br />
repulsion <strong>in</strong> dielectric ionic materials which ensures that <strong>in</strong>terstitial coalescence <strong>in</strong>to a<br />
dislocation loop rema<strong>in</strong>s stoichiometric (Ryazanov et al. 2002). The same argument<br />
would apply to vacancy loops.
88<br />
Therefore, for dislocation loops to grow, it is required that both Si and C <strong>in</strong>terstitial<br />
atoms be present at <strong>the</strong> loop nucleation site. For <strong>SiC</strong>, <strong>the</strong>re is a big difference <strong>in</strong> <strong>the</strong><br />
generation rates for Si and C atoms due to <strong>the</strong> large difference <strong>in</strong> threshold<br />
displacement energies <strong>in</strong> <strong>SiC</strong> (one source gives Ed = 93 eV for Si and 16 eV for C,<br />
Vladimirov et al. (1998)). About 5 times more carbon than silicon <strong>in</strong>terstitials are<br />
produced dur<strong>in</strong>g irradiation and <strong>the</strong> carbon <strong>in</strong>terstitials are also more mobile than <strong>the</strong><br />
silicon <strong>in</strong>terstitials as can be seen from <strong>the</strong> follow<strong>in</strong>g data.<br />
Table 7.4. Diffusion coefficients for Si and C <strong>in</strong>terstitials <strong>in</strong> <strong>SiC</strong><br />
Si <strong>in</strong>terstitials C <strong>in</strong>terstitials Temperature Reference<br />
D = 9 x 10 -6 cm 2 /s<br />
(Em = 0.51 eV)<br />
D = 10 -4 exp (-0.3/kT)<br />
= 3 x 10 -6 cm 2 /s<br />
D = 1 x 10 -5 cm 2 /s<br />
(Em = 0.55 eV)<br />
D = 10 -4 exp (-0.2/kT)<br />
= 9.8 x 10 -6 cm 2 /s<br />
1500 K Salvador et al. (2004)<br />
1000 K Ryazanov et al. (2002)<br />
Us<strong>in</strong>g <strong>the</strong> diffusion coefficients for <strong>in</strong>terstitial Si and C from Ryazanov et al. (2002).,<br />
<strong>the</strong> average diffusion distance for Si <strong>in</strong> <strong>SiC</strong> at 1000 K for 30 m<strong>in</strong>utes is found to be<br />
x Dt ~ 700 m.<br />
For carbon <strong>in</strong>terstitials <strong>the</strong> average diffusion distance is found to be approximately<br />
1300 m.<br />
From <strong>the</strong> above calculation is it clear that Si and C <strong>in</strong>terstitials are already mobile<br />
enough at 1000 K for <strong>the</strong> nucleation and growth <strong>of</strong> dislocation loops. The fact that no<br />
dislocation loops were observed <strong>in</strong> <strong>the</strong> samples analysed <strong>in</strong> this <strong>in</strong>vestigation, most<br />
likely <strong>in</strong>dicates that <strong>the</strong> proton dose was below <strong>the</strong> critical dose required for<br />
<strong>in</strong>terstitial coalescence. <strong>SiC</strong> is a very radiation resistant material which makes it ideal<br />
for use <strong>in</strong> fission end fusion reactors.
89<br />
In <strong>the</strong> follow<strong>in</strong>g paragraphs <strong>the</strong> analysis and attempted identification <strong>of</strong> an <strong>in</strong>terest<strong>in</strong>g<br />
defect <strong>in</strong> <strong>the</strong> <strong>SiC</strong> sample is presented. The identification is <strong>in</strong>terest<strong>in</strong>g s<strong>in</strong>ce it shows<br />
that <strong>the</strong> defect is an edge dislocation with Burgers vector and dislocation l<strong>in</strong>e<br />
comb<strong>in</strong>ation uncharacteristic <strong>of</strong> that usually found <strong>in</strong> <strong>3C</strong>-<strong>SiC</strong> and <strong>in</strong> z<strong>in</strong>c-blende<br />
structures. The simulated image <strong>of</strong> <strong>the</strong> defect agrees closely with <strong>the</strong> experimental one<br />
but <strong>the</strong> nature <strong>of</strong> <strong>the</strong> defect could not be proved conclusively through experimental<br />
techniques <strong>in</strong> <strong>the</strong> microscope due to <strong>the</strong> thickness <strong>of</strong> <strong>the</strong> sample prohibit<strong>in</strong>g it from<br />
be<strong>in</strong>g studied <strong>in</strong> different beam directions. A fur<strong>the</strong>r <strong>in</strong>vestigation on this type <strong>of</strong><br />
defect will be conducted <strong>in</strong> future.<br />
Fig. 7.16. Bright-field TEM micrograph <strong>of</strong> an area conta<strong>in</strong><strong>in</strong>g a defect <strong>in</strong>dicated by<br />
<strong>the</strong> black arrow. B = [111], g = 220, s≥0. The white arrow <strong>in</strong>dicates and area<br />
conta<strong>in</strong><strong>in</strong>g residual contrast possibly show<strong>in</strong>g a nucleation po<strong>in</strong>t.<br />
Fig. 7.16 shows a bright-field TEM micrograph <strong>of</strong> <strong>the</strong> area <strong>in</strong> <strong>the</strong> material where <strong>the</strong><br />
dislocation is present. The dislocation is <strong>in</strong>dicated by <strong>the</strong> black arrow. The beam<br />
direction is [111] and <strong>the</strong> reflection 220. The bright field image was taken such that<br />
<strong>the</strong> Kukuchi l<strong>in</strong>e was present on <strong>the</strong> outside edges <strong>of</strong> <strong>the</strong> diffraction spots to obta<strong>in</strong> <strong>the</strong><br />
best contrast. Also present <strong>in</strong> <strong>the</strong> image is a stack<strong>in</strong>g fault which may be used to
90<br />
determ<strong>in</strong>e <strong>the</strong> orientation <strong>of</strong> <strong>the</strong> micrograph accurately s<strong>in</strong>ce it is known that <strong>the</strong><br />
stack<strong>in</strong>g fault is ly<strong>in</strong>g on <strong>the</strong> {111} planes.<br />
The dislocation has a zig zag image character <strong>in</strong>dicat<strong>in</strong>g that <strong>the</strong> dislocation l<strong>in</strong>e is<br />
<strong>in</strong>cl<strong>in</strong>ed extend<strong>in</strong>g most probably from top to bottom <strong>of</strong> <strong>the</strong> foil. Also <strong>the</strong> projected<br />
image <strong>of</strong> <strong>the</strong> dislocation l<strong>in</strong>e is perpendicular to <strong>the</strong> fr<strong>in</strong>ges <strong>of</strong> <strong>the</strong> stack<strong>in</strong>g fault. All<br />
<strong>of</strong> <strong>the</strong>se facts along with <strong>the</strong> visibility and <strong>in</strong>visibility <strong>of</strong> <strong>the</strong> defect us<strong>in</strong>g <strong>the</strong> different<br />
220 reflections po<strong>in</strong>ted towards <strong>the</strong> dislocation be<strong>in</strong>g a 60° type commonly found <strong>in</strong><br />
z<strong>in</strong>c-blende structures with <strong>the</strong> Burgers vector <strong>of</strong> <strong>the</strong> type ½ and <strong>the</strong> dislocation<br />
l<strong>in</strong>e <strong>in</strong> a direction mak<strong>in</strong>g an angle <strong>of</strong> 60° with <strong>the</strong> Burgers vector.<br />
However <strong>the</strong> projected length <strong>of</strong> <strong>the</strong> dislocation is 397 nm, which gives a foil<br />
thickness <strong>of</strong> 561 nm by us<strong>in</strong>g <strong>the</strong> beam direction [111] with <strong>the</strong> dislocation l<strong>in</strong>e [110].<br />
This foil thickness is too thick to be electron transparent. The geometrical setup used<br />
is shown <strong>in</strong> Fig. 7.17.<br />
Fig. 7.17. The geometrical setup used to <strong>in</strong>fer <strong>the</strong> foil thickness from an assumed<br />
dislocation l<strong>in</strong>e direction<br />
Next <strong>the</strong> foil thickness was determ<strong>in</strong>ed by us<strong>in</strong>g CBED. The CBED micrograph taken<br />
<strong>of</strong> <strong>the</strong> area <strong>of</strong> <strong>the</strong> dislocation is shown <strong>in</strong> Fig. 7.18.
Fig. 7.18. CBED pattern taken <strong>of</strong> <strong>the</strong> area <strong>of</strong> <strong>in</strong>terest us<strong>in</strong>g a 220 reflection<br />
91<br />
From <strong>the</strong> figure a l<strong>in</strong>e pr<strong>of</strong>ile was obta<strong>in</strong>ed us<strong>in</strong>g a Matlab method developed for this<br />
purpose along <strong>the</strong> l<strong>in</strong>e <strong>in</strong>dicated. This was done to aid <strong>in</strong> <strong>the</strong> identification <strong>of</strong> <strong>the</strong><br />
positions <strong>of</strong> <strong>the</strong> m<strong>in</strong>ima’s and maximums used <strong>in</strong> <strong>the</strong> calculations. The pr<strong>of</strong>ile is<br />
shown <strong>in</strong> Fig 7.19 with identifications <strong>of</strong> <strong>the</strong> maximums and m<strong>in</strong>ima’s shown.
Intensity<br />
92<br />
Fig. 7.19. The <strong>in</strong>tensity pr<strong>of</strong>ile <strong>of</strong> <strong>the</strong> CBED pattern along <strong>the</strong> l<strong>in</strong>e <strong>in</strong>dicated on Fig.<br />
7.18.<br />
The measurements and subsequent calculated variables <strong>of</strong> <strong>the</strong> technique are given <strong>in</strong><br />
Table 7.3 (<strong>the</strong> units and scale used <strong>in</strong> determ<strong>in</strong><strong>in</strong>g <strong>the</strong> θi’s are irrelevant s<strong>in</strong>ce it is <strong>the</strong><br />
ratio <strong>of</strong> that to <strong>the</strong> Bragg angle that is important). The plotted values are shown <strong>in</strong> Fig.<br />
7.20.<br />
2θB<br />
Position<br />
etc.<br />
θ1<br />
θ2<br />
θ3
93<br />
Table 7.5. The measured parameters with calculated variables used <strong>in</strong> obta<strong>in</strong><strong>in</strong>g Fig.<br />
7.20<br />
(Si/ni) 2<br />
ni Δθi Si (nm -2 ) 1/ni 2 (Si/ni) 2 (nm -2 )<br />
4 13 0.0044 0.063 1.22975E-06<br />
5 28 0.0096 0.040 3.65112E-06<br />
6 38 0.0130 0.028 4.66997E-06<br />
7 49 0.0167 0.020 5.70487E-06<br />
8 58 0.0198 0.016 6.11964E-06<br />
9 67 0.0229 0.012 6.45229E-06<br />
10 76 0.0259 0.010 6.72476E-06<br />
11 86 0.0293 0.008 7.11641E-06<br />
12 95 0.0324 0.007 7.29683E-06<br />
13 105 0.0358 0.006 7.59524E-06<br />
0.000008<br />
0.000007<br />
0.000006<br />
0.000005<br />
0.000004<br />
0.000003<br />
0.000002<br />
0.000001<br />
0<br />
y = -1.09E-04x + 7.94E-06<br />
R 2 = 9.93E-01<br />
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07<br />
(1/ni) 2<br />
Fig. 7.20. The plotted values <strong>of</strong> Table 7.3 with <strong>the</strong> l<strong>in</strong>ear trend l<strong>in</strong>e used to determ<strong>in</strong>e<br />
<strong>the</strong> foil thickness and ext<strong>in</strong>ction distance<br />
Once aga<strong>in</strong> from <strong>the</strong> slope and y-<strong>in</strong>tercept <strong>of</strong> <strong>the</strong> l<strong>in</strong>ear trend l<strong>in</strong>e <strong>the</strong> ext<strong>in</strong>ction<br />
distance and foil thickness were obta<strong>in</strong>ed by tak<strong>in</strong>g <strong>the</strong> square roots <strong>of</strong> <strong>the</strong>ir <strong>in</strong>verses.<br />
The value obta<strong>in</strong>ed for <strong>the</strong> ext<strong>in</strong>ction distance is 96 nm and <strong>the</strong> foil thickness is 355<br />
nm.<br />
The foil thickness determ<strong>in</strong>ed is too th<strong>in</strong> for <strong>the</strong> dislocation l<strong>in</strong>e direction to be that <strong>of</strong><br />
a type s<strong>in</strong>ce <strong>the</strong> projected length <strong>of</strong> <strong>the</strong> dislocation is too long. The only option
94<br />
left is that <strong>the</strong> l<strong>in</strong>e direction is <strong>of</strong> <strong>the</strong> type. The [100] makes an angle <strong>of</strong> 54.74°<br />
with <strong>the</strong> [111] direction and if one <strong>in</strong>fers <strong>the</strong> foil thickness from <strong>the</strong> projected<br />
dislocation length, a thickness <strong>of</strong> ~280 nm is found. Thus it is possible that <strong>the</strong><br />
dislocation l<strong>in</strong>e direction is <strong>of</strong> <strong>the</strong> 100 type with <strong>the</strong> dislocation not runn<strong>in</strong>g<br />
completely from top to bottom <strong>in</strong> <strong>the</strong> foil but ra<strong>the</strong>r hav<strong>in</strong>g been nucleated at a defect<br />
somewhere with<strong>in</strong> <strong>the</strong> foil. A possible nucleation po<strong>in</strong>t is shown on <strong>the</strong> micrograph <strong>in</strong><br />
Fig. 7.16. by <strong>the</strong> white arrow. Some form <strong>of</strong> residual contrast is visible which may<br />
<strong>in</strong>dicate a defect which acted as nucleation po<strong>in</strong>t.<br />
Fig. 7.21. The simulated image <strong>of</strong> <strong>the</strong> dislocation <strong>in</strong>dicated on Fig. 7.16.<br />
None <strong>the</strong> less, <strong>the</strong> identification <strong>of</strong> <strong>the</strong> dislocation was done by simulation and <strong>the</strong><br />
results are shown <strong>in</strong> Fig. 7.21. Excellent agreement is seen but <strong>the</strong> true nature <strong>of</strong> <strong>the</strong><br />
defect will have to be established <strong>in</strong> a future study.<br />
7.4 Reassessment <strong>of</strong> Unimplanted Sample<br />
7.4.1 Overview <strong>of</strong> Investigation<br />
The unimplanted <strong>SiC</strong> sample was <strong>in</strong>vestigated aga<strong>in</strong>, us<strong>in</strong>g a beam direction,<br />
to determ<strong>in</strong>e whe<strong>the</strong>r similar dislocation complexes to those observed <strong>in</strong> <strong>the</strong> hydrogen<br />
implanted <strong>SiC</strong> sample annealed at 1600°C was present. It was found that <strong>the</strong> same<br />
dislocation structures are present <strong>in</strong> <strong>the</strong> unimplanted sample and it was concluded that<br />
dislocation complexes was not <strong>in</strong>troduced as a result <strong>of</strong> <strong>the</strong> irradiation or anneal<strong>in</strong>g<br />
process.
7.4.2 Results<br />
95<br />
Fig. 7.22. Bright-field TEM micrograph <strong>of</strong> <strong>the</strong> unimplanted <strong>3C</strong>-<strong>SiC</strong> sample viewed<br />
under a beam direction with g = 220, s≥0.<br />
Fig. 7.22 shows a bright field TEM micrograph <strong>of</strong> <strong>the</strong> unimplanted sample recorded<br />
us<strong>in</strong>g a beam direction with a g = 220 two-beam condition. The quality <strong>of</strong> <strong>the</strong><br />
micrograph is poor s<strong>in</strong>ce <strong>the</strong> sample was very thick <strong>in</strong> this direction but none <strong>the</strong> less<br />
it is sufficient to show that <strong>the</strong> same dislocation complexes are present <strong>in</strong> this sample<br />
as was observed <strong>in</strong> <strong>the</strong> annealed one. Their presence is most likely due to <strong>the</strong> growth<br />
process and <strong>the</strong>y are identified as predom<strong>in</strong>antly partial and 60 degree type<br />
dislocations.
8.1 Introduction<br />
96<br />
CHAPTER EIGHT<br />
CONCLUSION<br />
In this chapter <strong>the</strong> conclusions that are reached from <strong>the</strong> study are dealt with <strong>in</strong> two<br />
sections. Section 8.2 lists <strong>the</strong> ma<strong>in</strong> results and conclusions for <strong>the</strong> unirradiated and<br />
unannealed samples and Section 8.3 <strong>the</strong> ma<strong>in</strong> results and conclusions for <strong>the</strong> proton<br />
bombarded <strong>3C</strong>-<strong>SiC</strong>. All <strong>the</strong> results listed and conclusions are from a TEM study <strong>of</strong><br />
<strong>the</strong> samples.<br />
8.2 TEM <strong>Analysis</strong> <strong>of</strong> ß-<strong>SiC</strong> Grown <strong>in</strong> (001) Si by CVD<br />
The phase <strong>of</strong> <strong>the</strong> <strong>SiC</strong> was confirmed by electron diffraction and HRTEM to be<br />
<strong>the</strong> cubic ß phase with a lattice parameter <strong>of</strong> 4.35 0.01 Å.<br />
The <strong>SiC</strong> layer was s<strong>in</strong>gle crystall<strong>in</strong>e with <strong>the</strong> same orientation as <strong>the</strong> Si<br />
substrate.<br />
HRTEM <strong>in</strong>dicated that planar <strong>defects</strong> on <strong>in</strong>cl<strong>in</strong>ed {111} planes were generated<br />
at <strong>the</strong> Si/<strong>SiC</strong> <strong>in</strong>terface and propagate through <strong>the</strong> <strong>SiC</strong> layer.<br />
The <strong>SiC</strong> layer conta<strong>in</strong>ed a high density <strong>of</strong> SFs which were uniformly<br />
distributed through <strong>the</strong> foil.<br />
By simulat<strong>in</strong>g SFs and match<strong>in</strong>g it with observed SFs, it was found that <strong>the</strong><br />
SFs have a fault vector <strong>of</strong> <strong>the</strong> type 1/3 with bond<strong>in</strong>g partial dislocations<br />
<strong>of</strong> <strong>the</strong> type 1/6 .<br />
Incl<strong>in</strong>ed SFs exhibited symmetrical and complementary fr<strong>in</strong>ge contrast images<br />
<strong>in</strong> both bright and dark field two-beam imag<strong>in</strong>g conditions.<br />
SFs viewed edge-on generated streaks <strong>in</strong> <strong>the</strong> diffraction pattern similar to that<br />
generated by a th<strong>in</strong> precipitate viewed edge-on.<br />
HRTEM imag<strong>in</strong>g <strong>of</strong> SFs viewed edge-on <strong>in</strong>dicated that <strong>the</strong> SF were extr<strong>in</strong>sic<br />
<strong>in</strong> nature. FFTs obta<strong>in</strong>ed <strong>of</strong> <strong>the</strong> HRTEM image <strong>of</strong> <strong>the</strong>se SFs produced streaks<br />
similar to those observed <strong>in</strong> <strong>the</strong> selected area electron diffraction patterns <strong>of</strong><br />
edge-on SFs.
97<br />
Tw<strong>in</strong> platelets on {111} planes were observed <strong>in</strong> <strong>the</strong> <strong>SiC</strong> layer. These narrow<br />
tw<strong>in</strong>s nucleate at <strong>the</strong> <strong>in</strong>terface and extend only a short distance <strong>in</strong>to <strong>the</strong> <strong>SiC</strong><br />
layer.<br />
The <strong>extended</strong> <strong>defects</strong> observed <strong>in</strong> <strong>the</strong> <strong>SiC</strong> layers were most probably <strong>in</strong>troduced<br />
dur<strong>in</strong>g <strong>the</strong> CVD growth process as a result <strong>of</strong> <strong>the</strong> lattice mismatch between <strong>the</strong> <strong>SiC</strong><br />
epitaxial layer and <strong>the</strong> Si substrate. This <strong>in</strong>ference is streng<strong>the</strong>ned by <strong>the</strong> fact that<br />
hexagonal 6H-<strong>SiC</strong>, which are grown by a different technique, can be prepared with an<br />
extremely low <strong>extended</strong> defect concentration. Fur<strong>the</strong>rmore, it is known that stack<strong>in</strong>g<br />
faults form readily <strong>in</strong> ß-<strong>SiC</strong> due to <strong>the</strong> very low (negative) stack<strong>in</strong>g fault energy <strong>of</strong><br />
this compound. Tw<strong>in</strong>s on <strong>the</strong> o<strong>the</strong>r hand can also relieve misfit stress and it is<br />
<strong>the</strong>refore expected that tw<strong>in</strong>s would be generated at <strong>the</strong> Si/<strong>SiC</strong> <strong>in</strong>terface as was<br />
observed <strong>in</strong> this <strong>in</strong>vestigation.<br />
High resolution transmission electron microscopy analyses <strong>of</strong> stack<strong>in</strong>g faults viewed<br />
edge-on <strong>in</strong>dicate that <strong>the</strong> SFs are extr<strong>in</strong>sic <strong>in</strong> nature, i.e. it consists <strong>of</strong> two displaced<br />
layers yield<strong>in</strong>g a stack<strong>in</strong>g sequence <strong>in</strong> <strong>the</strong> direction <strong>of</strong> …ABCA/C/BCAB…<br />
The symmetrical and complementary nature <strong>of</strong> two-beam bright and dark-field<br />
diffraction contrast images <strong>of</strong> <strong>in</strong>cl<strong>in</strong>ed SFs <strong>in</strong>dicate that <strong>SiC</strong> has an anomalous<br />
absorption ratio between that <strong>of</strong> Si and diamond. This conclusion was confirmed by<br />
match<strong>in</strong>g simulated SF images with experimental SF images, where <strong>the</strong> best match<br />
was obta<strong>in</strong>ed for an anomalous absorption ratio <strong>of</strong> 0.016 for <strong>SiC</strong>. The values for Si<br />
and diamond are 0.1 and 0.01 respectively.<br />
8.3 TEM <strong>Analysis</strong> <strong>of</strong> Proton Bombarded and Annealed ß-<strong>SiC</strong><br />
ß-<strong>SiC</strong> samples were implanted with 400 keV protons to a total dose <strong>of</strong> 2.8 ×<br />
10 16 protons.cm -2 . Post-implantation anneal<strong>in</strong>g was carried for 1 hour at 1300<br />
and 1600 ºC.<br />
In all <strong>the</strong> implanted samples, annealed an unannealed, no evidence <strong>of</strong> radiation<br />
damage was observed. The radiation damage that one would expect to f<strong>in</strong>d <strong>in</strong> proton
98<br />
implanted and annealed materials could <strong>in</strong>clude self <strong>in</strong>terstitial dislocation loops,<br />
hydrogen platelets or bubbles, vacancy loops and voids. The possibility that <strong>the</strong> high<br />
density <strong>of</strong> stack<strong>in</strong>g faults and tw<strong>in</strong> platelets could have acted as s<strong>in</strong>ks for <strong>the</strong> radiation<br />
<strong>in</strong>duced <strong>in</strong>terstitials and hydrogen and thus prevent<strong>in</strong>g self <strong>in</strong>terstitial coalescence and<br />
loop formation is ruled out by <strong>the</strong> fact that <strong>the</strong> same proton bombardment and<br />
anneal<strong>in</strong>g treatment was also performed on semiconductor grade 6H-<strong>SiC</strong> (hexagonal<br />
phase with no visible lattice <strong>defects</strong>) and no radiation damage was detected <strong>in</strong> this<br />
case also. The TEM results <strong>of</strong> <strong>the</strong> 6H-<strong>SiC</strong> were not <strong>in</strong>cluded <strong>in</strong> this dissertation s<strong>in</strong>ce<br />
no lattice <strong>defects</strong> were observed <strong>in</strong> <strong>the</strong>se samples.<br />
The calculation <strong>of</strong> <strong>the</strong> average migration distance for silicon and carbon <strong>in</strong>terstitial<br />
atoms at a 1000°C revealed that <strong>the</strong> <strong>in</strong>terstitial atoms are mobile enough to migrate<br />
over large distances and thus could form dislocation loops if present <strong>in</strong> large enough<br />
concentrations after annihilation by vacancy <strong>in</strong>terstitial recomb<strong>in</strong>ation.<br />
The proton implanted and annealed ß-<strong>SiC</strong> samples also exhibited no evidence <strong>of</strong><br />
phase transformation or an <strong>in</strong>crease <strong>in</strong> lattice defect concentration compared to <strong>the</strong><br />
unimplanted samples.<br />
The results <strong>of</strong> this <strong>in</strong>vestigation confirm <strong>the</strong> notion that <strong>SiC</strong> is a radiation resistant<br />
material. The critical proton dose for <strong>the</strong> creation <strong>of</strong> small dislocation loops seems to<br />
be higher than for o<strong>the</strong>r compound semiconductors with <strong>the</strong> z<strong>in</strong>c-blende structure.<br />
This is most probably due to <strong>the</strong> very high displacement energies <strong>of</strong> Si and C atoms.<br />
The large difference <strong>in</strong> displacement energies for Si and C would also retard <strong>the</strong><br />
formation <strong>of</strong> dislocation loops s<strong>in</strong>ce equal numbers <strong>of</strong> Si and C are required for a loop<br />
to nucleate and grow. The difference <strong>in</strong> lattice recovery for <strong>the</strong> Si and C sub-lattices<br />
after <strong>the</strong> <strong>the</strong>rmal spike produced dur<strong>in</strong>g ion implantation is ano<strong>the</strong>r factor which<br />
would retard dislocation formation and hence contribute to <strong>the</strong> radiation resistance <strong>of</strong><br />
<strong>SiC</strong>.<br />
Future work on <strong>SiC</strong> will <strong>in</strong>clude <strong>the</strong> use <strong>of</strong> higher proton doses to establish <strong>the</strong> critical<br />
dose required for dislocation loop formation. The proton <strong>in</strong>duced radiation damage at<br />
higher doses will also be compared with radiation damage produced by high energy<br />
neutrons <strong>in</strong> <strong>SiC</strong>.
99<br />
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