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Ion Channelling and Electronic Excitations in Silicon

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Ion Channelling and Electronic Excitations in Silicon Ion Channelling and Electronic Excitations in Silicon Anthony Craig Lim MSc Theory and Simulation of Materials, Imperial College London (2011), MSci Physics, Imperial College London (2010) CDT Theory and Simulations of Materials Physics Imperial College London South Kensington Campus, London, SW7 2AZ, United Kingdom THESIS Submitted for the degree of Doctor of Philosophy, Imperial College London 2014 2 The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work I, Anthony Lim, confirm that this thesis is my own work, unless stated otherwise. Where information is from other sources, I confirm that it has been indicated appropriately. 3 Abstract When a high-energy (MeV) atom or neutron collides with an atom in a solid, a region of radiation damage is formed. The intruding atom may cause a large number of atoms to leave their lattice sites with low energies (keV) in a collision cascade. Alternatively, it may cause a single lattice atom to be removed from its crystal lattice site and travel large distances without undergoing any further atomic collisions/scatterings in a process known as channelling. Any moving atom may lose energy via collisions with other atoms or by the excitation of electrons. The microstructural evolution of the irradiated material depends on the rate at which the damaged region cools, which in turn depends on the rate at which electrons are excited and carry energy away. Since silicon is a semiconductor with a band gap of 1.1 eV, it is normally assumed that the rate of excitation of electrons by atoms with low kinetic energies (< 100 eV) is negligible. However, the atomic kinetic energy threshold required for electronic excitation is not understood. This thesis uses large-scale quantum mechanical simulations to investigate how a moving atom loses energy to the electrons in a crystal of silicon. It is possible to calculate the energy transfer from a channelling atom to the host mater- ial’s electrons using time-dependent density functional theory (TDDFT) and we have done so. However, these highly accurate calculations are computationally expensive and cannot be used to study very large systems. Hence, we also use a less accurate method called time-dependent tight binding (TDTB). We show that TDTB and TDDFT simulations of channelling in small silicon systems are in good qualitative agreement and use the cheaper TDTB method to investigate finite-size effects. We also investigate how an atom oscillating around its lattice site transfers energy to the crystal’s electrons. To understand the complex behaviour of the electronic energy transfer, we utilise non-adiabatic perturbation theory. Our simulations and the perturbative analysis both show that the presence of a gap state with a time-dependent energy eigenvalue allows electronic excitations for very low energy (eV) channelling atoms. 4 Acknowledgments First I would like to start with the formalities. The work in this thesis work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Labor- atory under Contract DE-AC52-07NA27344. Computing support for this work came from the Lawrence Livermore National Laboratory Institutional Computing Grand Challenge pro- gram. Anthony Lim was supported by the CDT in Theory and Simulation of Materials at Imperial College funded by EPSRC grant EP/G036888/1. Thank you to my supervisors for their guidance during my time at Imperial. I would also like to thank Dr. Erik Draeger, Dr. Andre Schleife and Dr Daniel Mason for useful discussions and help with the computational side of the project. I am particularly grateful to Dr. Alfredo Correa who has taught me a considerable amount over the past few years. This would not have been possible without the love and support of my girlfriend Kim Birkett. Thank you for being so patient with me, especially over the last few months. I would not be where I am today if it was not for my Nan, Rita Wright, she always encouraged me and supported my interests. This thesis is dedicated to her. Contents Front matter Declaration of authorship . 2 Abstract.......................................... 3 Acknowledgments..................................... 4 Contents.......................................... 5 ListofFigures ...................................... 7 1Introduction 12 1.1 Radiation Damage . 13 1.2 TheVelocityThreshold .............................. 15 2 Literature Review 17 2.1 Theory........................................ 17 2.2 Experiments..................................... 24 2.3 Simulations ..................................... 28 2.4 Artacho’s Simple Perturbation Theory . 33 3TheoreticalBackground 35 3.1 Density Functional Theory (DFT) . 35 3.2 TightBinding(TB) ................................ 39 3.3 Dynamics of Electronic Systems . 42 3.4 Time-Dependent Density Functional Theory (TDDFT) . 45 3.5 Time-Dependent Tight Binding (TDTB) . 48 3.6 Theory and Simulation . 50 4 Time-Dependent Density Functional Theory: A Silicon Ion Channelling in a Silicon Crystal 51 4.1 SimulationMethod................................. 51 4.2 AnalysisTechnique................................. 54 4.3 Results........................................ 55 4.4 Pre-thresholdRegime ............................... 60 4.5 Below Threshold Stopping . 65 5TightBindingModelsofSilicon 66 5.1 Vogl Tight Binding Model . 66 5.2 Goodwin, Skinner and Pettifor (GSP) Tight Binding Model . 67 5.3 TheKwonTBModel................................ 69 5 Contents 6 5.4 Tight Binding and The Electronic Stopping Power . 73 6ModificationofTightBindingModel 74 6.1 Modifications to Tight Binding . 74 6.2 The Smooth Switch Function . 74 6.3 Finite Electronic Bands (FEB) . 75 6.4 Linear Approximation of Pair Potentials (LAPP) . 78 6.5 Dynamical Simulations . 81 7Time-DependentTightBindingResults 82 7.1 The Gradient Extraction Method (GEM) . 82 7.2 The Generalised Gradient Extraction Method (G-GEM) . 83 7.3 IonChannelling................................... 84 7.4 TheOscillatingAtom ............................... 88 7.5 Below Threshold Stopping In Large Systems . 92 8Perturbationtheory 93 8.1 Non-Adiabatic Perturbation Theory . 93 8.2 TheMatrixElement ................................ 96 8.3 Verifying Eq. (8.25) . 97 8.4 The Weak Coupling Limit . 98 8.5 Independent Particle Approximation . 100 8.6 Conditions On The Form of W . 103 8.7 Rewriting the Electronic Energy Transfer . 104 8.8 The Electronic Stopping Power . 105 8.9 The Electronic Stopping Power and Perturbation Theory . 105 9 Perturbative Analysis of Channelling Ions in Silicon 106 9.1 Time Dependent Density Functional Theory (TDDFT) . 106 9.2 Perturbation Theory . 110 10 Summary and Further Work 128 10.1 Further Work . 129 References 132 Appendices A LiteratureReview ................................. 139 B Theoretical Background . 140 C The Kwon Force . 142 List of Figures 1.1 Top is a simple 2D density of states (DOS) for a metal and the bottom panel contains a simple 2D DOS for an insulator. Slow moving atoms will create low energy excitations; these excitations are possible for a metal as there is no separation between the occupied (blue) and unoccupied (red) states. In an insulator there is no low-energy excitation because there is no empty state for theelectrontooccupy................................ 13 1.2 The total (ST ), ionic (SI ) and electronic (S) stopping powers as a function of the channelling ion’s kinetic energy. The data presented here is for silicon (Si) in Si and has been calculated using SRIM. 15 2.1 A cartoon of the analysis used in TOF measurements. A channelling ion is measured by the light gates to have an initial kinetic energy of Ek and after passing through the foil it produces a signal response of E in the detector. By finding the kinetic energy of the ion which produced the same signal response without passing through the foil (Ek! ), we can determine the kinetic energy of the ion after passing through the foil. 26 4.1 The variation in the binding energy per atom (Ea) as a function of cubic lattice parameter (a). The bulk modulus is given by the second derivative and is in close agreement to experiment. The factor of eight in the bulk modulus calculation is because there are eight atoms in the cubic unit cell. 53 4.2 The yellow atoms represent the frozen crystal lattice containing 216 atoms, while the pink atom is the channelling ion. The x and y axes are parallel to the white box, while the zaxis is perpendicular to the page. The channelling ion travels at a constant velocity in the 001 direction (straight out of the ! " page). ........................................ 54 4.3 The raw TDDFT data (blue) minus the Born Oppenheimer data (green) pro- duces the red curve. To produce a flatter line we then subtract Eq. (4.5) from the red curve to get the black line. By fitting a straight line to the black line we obtain the magenta line. The stopping power is given by the gradient of themagentaline. ................................. 56 7 List of Figures 8 4.4 The electronic stopping power calculated using TDDFT is about a factor two smaller than experiment. This can be accounted for because the simulations, unlike experiments, consider
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