STRUCTURE, STABILITY AND DEFECTS OF SINGLE LAYER h-BN Master Thesis by: guus slotman Supervisor: External Advisor: Prof. dr. A. Fasolino dr. ir. B.L.M. Hendriksen October 2012 Theory of Condensed Matter Institute for Molecules and Materials Radboud University ACKNOWLEDGMENTS The making of this thesis was greatly helped by the support of many people, who I will not all name, but here are a few: First of all my friends who always were interested to talk about something else than physics. Second all my co-workers at the Theory of Condensed Matter department who made the working days much more pleasurable. All the various teachers who, at least most of them, always were motivated to teach us students more. In particular I would like to thank my thesis supervisor prof. A. Fasolino for all her help and support during my year at the department. Of course my family needs to be thanked because without them I wouldn’t be here! Last but not least I would like to thank Resie Wijnen for all her support and love during the years. iii CONTENTS 1 introduction1 1.1 Crystal Structures . 1 1.2 Computational Physics . 2 1.3 Structure of this thesis . 3 1.4 Article . 3 2 molecular dynamics in various ensembles5 2.1 Basics of Molecular Dynamics . 5 2.1.1 Numerical Integration of the equations of motion . 5 2.1.2 Simulation of Boron Nitride in the Microcanonical ensemble . 9 2.2 The canonical ensemble . 11 2.2.1 Berendsen thermostat . 11 2.2.2 Nosé-Hoover chain thermostat . 12 2.2.3 Simulation of Boron Nitride in the canonical ensemble . 17 2.3 The Isothermal-Isobaric ensemble . 20 2.3.1 Berendsen barostat . 20 2.3.2 Nosé-Hoover chain barostat . 21 2.3.3 Simulation of Boron Nitride in the Isothermal-Isobaric ensemble . 22 2.4 Conclusion . 24 3 advanced techniques 27 3.1 Steered Molecular Dynamics . 27 3.2 Nudged Elastic Band . 28 4 simulation of hexagonal boron-nitride 31 4.1 Empirical potentials . 31 4.1.1 A potential for BN . 33 4.2 Testing the potential . 35 4.2.1 Lattice constant . 35 4.2.2 Bending rigidity . 38 4.2.3 Nosé-Hoover . 41 4.3 Conclusion . 41 5 defects 43 5.1 Point defects and vacancies . 43 5.1.1 Stone-Wales defects . 44 5.1.2 Other point defects . 45 5.1.3 Vacancies . 46 5.2 Energy barriers . 47 6 summary and conclusion 51 a appendices 53 a.1 Character of the Verlet algorithm . 53 a.2 Coefficients for the BN Tersoff potential . 54 a.3 lammps files . 55 a.4 Melting process . 58 bibliography 61 v INTRODUCTION 1 The topic of this thesis is a single sheet of hexagonal boron nitride, in short: h-BN. At first sight this seems to be quite a simple and random topic, however it is part of one of the hot topics in solid state physics: the study of two-dimensional crystals (2d). Since the discovery of the first 2d crystal made out of carbon, called graphene, the interest in 2d crystals has rapidly extended to other materials, often combined to form man-made heterostructures such as transistors[1]. The reason of interest in h-BN is that is has similar structural properties to graphene, but the electronic properties are different: BN is an insulator with a gap of ∼ 5 - 6 eV[2] instead of a conductor. It is this property that makes it one of the promising dielectric materials for integration in hybrid graphene devices[3]. Although carbon and its two dimensional compound have been extensively studied by means of computer simulation, this is less the case for BN. In this thesis, we try to fill this gap by studying the structural stability, thermal expansion and defect formation in h-BN by means of Molecular Dynamics (MD) based on a classical description of interatomic interactions. The success of this approach for graphene, particularly for temperature dependent properties, has been due to the existence of several accurate models of interactions[4,5,6,7]. These so called bond-order potentials for carbon, pioneered by Tersoff[4], have the im- portant feature of being reactive, namely to allow change of coordination. For BN, Albe, Möller and Heinig[8] have developed a Tersoff potential that is supposed to describe the bulk BN phases and the defect formation energy and has been used to study the effect of irradiation in single layer h-BN[9]. To our knowledge, however, no comprehen- sive study of the structural stability and defect formation energies based on this ap- proach exists up to date. Validation of the results given by this potential opens the way to the development of a reliable potential capable to deal with hybrid BN-graphene structures. 1.1 crystal structures At school we learn that the world around us is divided into three categories: solid, liquid and gas. Although there exist many more states, such as liquid-crystal, super- liquid and plasma states, we will focus all our attention to the solid state. More specific, we talk about the crystalline solid state, meaning that the structure exhibits a long ranged order. Examples of some basic crystal structures are simple body-centered and face-centered cubic crystals. Some elements have more than one stable crystal structure in normal conditions. For example, carbon can exist as ordinary graphite or as the less ordinary diamond. These different flavours of a material are called allotropes. In the last 30 years three new different allotropes of carbon have been discovered: 0d buckeyballs by Kroto et al.[10] in 1985, 1d nanotubes (although observed before) became popular after 1991 and finally in 2004 2d graphene[11]. These discoveries have always been followed by the search 1 2 introduction Figure 1.1: The typical honeycomb structure of both h-BN and graphene. Depicted are the lat- tice parameter a and the nearest neighbour distance R, two quantities we will study in chapter 4. for equivalent structures in different materials. For example in 1995 BN nanotubes were created by Chopra et al.[12] and in 2005 the first experiments were done on two- dimensional BN[13]. There are a couple of remarkable things about 2d materials. First and foremost that they exist at all. It was predicted by Peierls, Landau, Mermin and Wagner that 2d materials could not exist[14]. The way stability is achieved is by rippling of the surface: the 2d crystal becomes somewhat 3-dimensional. More on that in chapter 4. So far we have not mentioned what the material looks like. In figure 1.1 we show the honeycomb-like structure of h-BN, which is the same structure as for graphene. Both 2d materials are existentially a single layer of 3d materials. One of the structural differences between the two materials is the stacking method in the bulk (3d) variants: the way that two layers stack on top of each other. Two layers of graphene (or bulk graphite) is found to have AB-stacking (see figure figure 1.2a), meaning that the carbon atoms do not lie directly on top of each other. Bulk h-BN is found to have AA-stacking, in this case one B atom lies on top of one N atom (see figure 1.2b). The role of defects in materials becomes more important as the devices we create become smaller and smaller. It is therefore necessary to study the effects of these de- fects on the structural and electronic properties of a material. Again, in graphene there is already done extensive research on various point-defects whereas only few results exist for h-BN. 1.2 computational physics Historically there are two flavours of physics that one can distinguish: theory and ex- periment. However it is impossible to analytically solve most models that consider more than two bodies, except for a few special cases (for example the 2d Ising model). 1.3 structureofthisthesis 3 (a) (b) Figure 1.2: Different kinds of stacking: top view of AB-stacking in graphite (left) and side view of AA-stacking in h-BN (right). This is very inconvenient because most interesting problems in condensed matter con- sider of N particles, where N can be anything from a few hundreds to ∼ 1023. There comes in Computational Physics, which can be the bridge between theory and experi- ment. We can make a model of a system and test it against experiments; if the model does not produce the right results the model is wrong. The model can than be tested against theory, or vice versa. Also, computer simulations allow us to perform ’exper- iments’ which are hard to perform in real life without breaking your experimental equipment, for instance in conditions with extreme pressure and temperature. The first steps in computer modelling were done in the USA in the 1950’s where big computers were available in the national research institutes, mainly to help de- velop thermonuclear bombs. At the Los Alamos National Laboratory they had such a machine and this is where Metropolis and colleagues first developed the Metropo- lis Monte Carlo (MC) method[15]. Around the same time Alder and Wainwright per- formed the first Molecular Dynamics (MD) simulations[16], which is the method we use throughout this thesis. Since the beginning of computational physics the basics of the methods have not been changed much, but the methods have been applied to larger and more complex systems, thanks to cheaper and faster computers. 1.3 structure of this thesis This thesis is organised as follows: after this small introductory chapter we give, in chapter 2, a throughout review of the MD techniques we use , including how to per- form simulations in the canonical (NVT) and isobaric-isothermal ensemble (NPT) in sections 2.2 and 2.3.