Structure, Stability and Defects of Single Layer H-BN

STRUCTURE,STABILITYANDDEFECTSOFSINGLELAYER 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.

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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

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 important 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 approach 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 lattice 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 defects 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 consider 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 ’experiments’ 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 performed 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 perform simulations in the canonical (NVT) and isobaric-isothermal ensemble (NPT) in sections 2.2 and 2.3. Later on, in chapter 3 we give a brief overview of two methods to calculate energy barriers. In chapter 4 we give the results of our extensive research on h-BN using the Albe et al.[8] potential. This includes structural properties of a single layer of h-BN in section 4.2. Properties of low energy defects will be discussed in chapter 5.

1.4 article

Parts of chapter 4 and most of chapter 5 were published in Structure, stability and defects of single layer hexagonal BN in comparison to graphene, G.J. Slotman and A. Fasolino, Journal of Physics: Condensed Matter 25-4,045009 (2013)[39].

MOLECULARDYNAMICSINVARIOUSENSEMBLES 2

In this section we ﬁrst review some basics of Molecular Dynamics (MD) and the implementation in the public domain code lammps[17]. Afterwards we explain the basics of performing MD simulations in the canonical ensemble (NVT) and in the isothermal- isobaric ensemble (NPT). The goal is to perform MD simulations on a sheet of hexagonal Boron-Nitride (h-BN). In this section we performed simulations using this system to ﬁnd reasonable parameters to use. We will explain the details of this system later on in chapter 4.

2.1 basics of molecular dynamics

Molecular dynamics is a widely used technique to obtain numerical results for many- body systems that are not analytically solvable. In MD the classical equations of motion are solved numerically for N-particles. From the positions and trajectories that are generated for these particles one can calculate macroscopic quantities such as temperature and pressure as time averages. One can use these results for making educated guesses about experiments that cannot be done in real life, for example experiments with very extreme pressures or experiments with materials that do not exist (yet). A big advantage over the other computational physics method, Monte Carlo, is that from MD the dynamics of the system can be extracted. For example one could follow the path of a particle in a liquid. When doing MD simulations one should always remind that the generated trajectories are very sensitive to the initial conditions (Lyapunov instabil- ity) and thus one should always treat the trajectories with caution: averages should be taken over a large enough time. A basic MD program consists of the following steps: 1. Set up the initial values of your sample (positions, velocities) 2. Calculate the forces on each particle 3. Update the velocities and positions of the particles. Repeat steps 2 and 3 to get a simulation on a relevant timescale. The forces are calculated using empirical (many body) potentials. The best known of such potential is the Lennard-Jones potential. We will discuss such kinds of potentials in section 4.1.

2.1.1 Numerical Integration of the equations of motion

In MD the coordinates of each atom are updated each time step using classical (Newto- nian) dynamics. Therefore we need an efﬁcient algorithm to integrate the equations of motion. On of the most used is the velocity Verlet algorithm. To obtain the algorithm we start with writing the position r of a particle at a time t + ∆t as a Taylor series (while neglecting the higher order terms):

1 r (t + ∆t) ≈ r (t) + r˙ (t)∆t + r¨ (t)∆t2 + ... (2.1) i i i 2 i

5 6 molecular dynamics in various ensembles

Listing 2.1: An implementation of the verlocity Verlet algorithm

initialization for N steps: %perform half time step upgrade of the velocities according to: v (t + 1 ∆t) = v (t) + 1 F (t)∆t i 2 i 2mi i %perform full time step update of the coordinates according to: ∆t ri(t + ∆t) = ri(t) + vi(t + 2 )∆t %recalculate the forces Fi(t + ∆t) %perform half time step update of the velocities: v (t + ∆t) = v (t + ∆t )∆t + 1 F (t + ∆t)∆t i i 2 2mi i end of run

where r˙i and r¨i are the ﬁrst and second derivatives with respect to time, which are of course the velocity v and the acceleration a of a particle. Using Newton’s second law we write:

1 2 ri(t + ∆t) ≈ ri(t) + vi(t)∆t + Fi(t)∆t (2.2) 2mi

where Fi is the force on a particle with mass mi. Notice that ri(t + ∆t) is written in terms of quantities calculated at time t. From this equation we can see that to obtain the new position we need to calculate the forces on each particle. To also get an expression for v we write the equation for ri(t) going backwards in time.

1 2 ri(t) ≈ ri(t + ∆t) − vi(t + ∆t)∆t + Fi(t + ∆t)∆t (2.3) 2mi

Plugging in the previous obtained formula for ri(t + ∆t) we get for vi(t + ∆t):

Fi(t + ∆t) + Fi(t) vi(t + ∆t) = vi(t) + ∆t (2.4) 2mi

Notice that vi is written in terms of the forces at time t and t + ∆t. Therefore we can ﬁrst advance the positions to calculate Fi(t + ∆t). If the forces also depend on velocities, like in a damped system, the scheme has to be changed. This can be done by updating the velocities by half a time step. A algorithm to perform such calculations can be found in listing 2.1. One can do the numerical integration scheme in a more formal way. In this way we can show that the Verlet algorithm is symplectic, namely it preserves the Hamilto- nian dynamics. We will not try to prove everything but we shall explain some basics which we will use later on. For a more rigorous approach please refer to Tuckerman[18] or[19,20]. Let us ﬁrst introduce the Hamiltonian, which is the total energy of a system:

2 pi H(r1...rN, p1...pN) = + U(r1...rN) (2.5) 2mi i X where U is the potential energy. The Hamiltonian equations of motion of a system are:

∂H ∂H q˙ α = p˙ α = − (2.6) ∂pα ∂qα 2.1 basics of molecular dynamics 7 where q and p are the generalized coordinates and momenta. It is easy to see that the Hamilton equations are equivalent to Newton’s second law of motion if you plug in eq. 2.5 into eq. 2.6. We can introduce the matrix: ! q x = .(2.7) p

Now we can write the Hamilton equations even more compactly: ! 0 1 ∂H ∂H x˙ = = M (2.8) −1 0 ∂x ∂x which is the ’symplectic’ notation of the Hamilton equations. It can be shown[18,20] that a transformation is canonical if M and the Jacobian matrix J obey:

M = JT MJ (2.9) which is known as the symplectic property. If this property is obeyed, the numerical integrator exhibits the same properties as the exact result. The integrator is then said to by symplectic. We want to look at the time evolution of any of the phase space vectors. The evolution of such a phase space vector a(q, p), where q and p depend on time, follows

3N 3N da(p, q) ∂a ∂a ∂a ∂H ∂a ∂H = q˙α + p˙α = − dt ∂qα ∂pα ∂qα ∂pα ∂pα ∂qα α=1 α=1 (2.10) X X = {a, H} ≡ iLa where L is the Liouville operator deﬁned by:

3N ∂ ∂H ∂ ∂H iL ≡ − .(2.11) ∂qα ∂pα ∂pα ∂qα α=1 X Integrating equation 2.10 we obtain for the time evolution of the phase space vectors:

iLt a(xt) = e a(x0) (2.12) or: " # " # q(t) q = eiLt 0 .(2.13) p(t) p0

With this formula we can evolve our system over time using the Liouville operator. The goal is now to get a propagation scheme with a discrete time step ∆t instead of t. We can write the Liouville operator as the sum of two independent operators:

" N # " N # ∂H ∂ ∂H ∂ iL = − = iL1 + iL2 (2.14) ∂pα ∂qα ∂qα ∂pα α=1 α=1 X X 8 molecular dynamics in various ensembles

If we now consider the Hamiltonian from 2.5 and the Hamilton equations from 2.6 we get:

N N pi ∂ ∂ iL1 = iL2 = Fi (2.15) mi ∂ri ∂pi i=1 i=1 X X Note that in general iL1 and iL2 do not commute. This means that in general:

eiLt = eiL1t+iL2t 6= eiL1teiL2t (2.16)

which is a bit inconvenient if we want to use the Liouville operator for the propagation of our vector. However, we can use the Trotter theorem: h iP eA+B = lim eA/2PeBeA/2P .(2.17) P→

If we apply this theorem∞ to equation 2.16 we get:

h iP eiLt = lim eiL2t/2PeiL1t/PeiL2t/2P .(2.18) P→

Now we introduce∞ the discrete time step ∆t = t/P and write our equation for ﬁnite P and ﬁnite ∆t:

eiL∆t ≈ eiL2∆t/2eiL1∆teiL2∆t/2 + O(∆t3) (2.19)

which we can use for the propagation of our vector. Let us now apply it to recover the velocity Verlet algorithm. Starting from equation 2.13 plug equations 2.15 into eq. 2.19 you get for a single particle in one dimension: " # " # q(∆t) q 2=.13 eiL∆t 0 p(∆t) p0 " # 2.19 q ≈ eiL2∆t/2eiL1∆teiL2∆t/2 0 (2.20) p0 " # 2.15 ∂ p ∂ ∂ q = exp F ∆t/2 exp ∆t exp F ∆t/2 0 . ∂p m ∂q ∂p p0

We now have exponential operators with derivatives. To apply those to a function g(x) we can take the Taylor expansion of g(x) around the point x + c and get:

c ∂ e ∂x g(x) = g(x + c).(2.21)

We make use of this fact by applying the operators to the position and momentum: " # " # ∂ q q exp F ∆t/2 0 = 0 (2.22) ∂p ∆t p0 p0 + 2 F

" # " # p ∂ q q + ∆t p0 exp ∆t 0 = 0 m (2.23) m ∂q ∆t ∆t p0 p0 + 2 F p0 + 2 F(q0 + ∆t m ) 2.1 basics of molecular dynamics 9

" # " # ∂ q + ∆t p0 q + ∆t 1 (p + ∆t F(q )) exp F ∆t/2 0 m = 0 m 0 2 0 . ∂p ∆t p0 ∆t ∆t 1 ∆t p0 + 2 F(q0 + ∆t m ) p0 + 2 F(q0) + 2 F(q0 + ∆t m (p0 + 2 F(q0)) (2.24)

p Now if we substitute x for q and use that v = m we can write it in a more familiar way: " # " # x(∆t) x(0) + ∆tv(0) + ∆t2 F(x(0)) = 2m .(2.25) 1 v(∆t) v(0) + ∆t 2m [F(x(0)) + F(x(∆t))]

We notice that we obtained the velocity Verlet formulas 2.2 and 2.4 to update the position and velocities. Although this whole scheme seems to overly complicate things we now have shown that the verlocity Verlet algorithm is, to a second order expansion, symplectic1 and time-reversible. The methods described above can be used for more advanced integration methods, for example a multiple time scale integrator like respa (see for instance Tuckerman et al.[21]). More importantly for us, we will use these methods in the next sections to add thermostats to a system.

2.1.2 Simulation of Boron Nitride in the Microcanonical ensemble

In order to test our system we examine how it behaves in the microcanonical ensemble. Thus we perform a simulation using lammps where the number of particles N, the volume V and the total energy E are supposed to be conserved. Since the Verlet algorithm implies an approximation, we have to check for which time step the total energy is conserved. We will look at the quantity:

Nstep 1 Ek − E(0) ∆E = (2.26) Nstep E(0) k=1 X The goal is to find a time step ∆t where the energy remains constant. In figure 2.1 the kinetic, potential and total energy of a system with N = 1296 BN atoms are plotted for the first few thousand time steps. From figure 2.1a we can see that with a time step ∆t 6 1fs the total energy is constant. When we take ∆t bigger (for example ∆t = 3.5fs) we see that the total energy of the system is not conserved. The initial velocities of all system where picked randomly from a Gaussian distribution to correspond with a temperature of 300K. In the first few time steps a part of this kinetic energy is converted to potential energy resulting in a lower temperature of the system. Another way to determine a suitable value for ∆t is to look at the fluctuations in the potential and the total energy. As a rule of thumb the fluctuations in the total energy should not exceed the fluctuations in the potential energy by more than a few percents. We ran our system for 60.000 time steps where the energy was written every 50 steps. The results of this procedure for different values of ∆t can be found in table 2.1. This table shows, again, that we are safe for time steps ∆t smaller than or equal to 1 fs.

1 In appendixA. 1 we show that equation 2.9 is satisﬁed by the verlocity Verlet algorithm. 10 molecular dynamics in various ensembles

30 40 -6.38 -6.36 30 -6.39 20 -6.38 20 -6.40 pe ∆t = 1 fs ∆ te t = 1 fs 10 ke ∆t = 1 fs -6.40 ∆ pe ∆t = 3.5 fs 10 -6.41 pe t = 0.1 fs ∆ te ∆t = 0.1 fs te t = 3.5 fs ∆ ke ∆t = 3.5 fs ke t = 0.1 fs Kinetic energy per atom (meV) Kinetic energy per atom (meV)

Total/Potential energy per atom (eV) -6.42 0 Total/Potential energy per atom (eV) -6.42 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 Number of time steps Number of time steps (a) (b)

Figure 2.1: The kinetic (ke), potential (pe) and total energy (te) per atom as a function of the number of time steps for different values of ∆t. In 2.1a the total energy remains constant over the run, meaning that the choosen time steps (∆t = 0.1 and 1.0 fs) are ﬁne. In 2.1b for a time step of ∆t = 3.5 fs the energy does not remain constant.

Table 2.1: Energy ﬂuctuations in the NVE ensemble. The given values are for the whole system, consisting of 1296 atoms. As seen the ﬂuctuations become considerably larger for bigger time steps.

Time step (fs) Total energy (eV) Potential energy (eV) Rel. Fluctuations 0.01 -8265.729 ± 0.000 -8291.063 ± 1.559 0.0 % 0.1 -8265.725 ± 0.000 -8291.042 ± 0.924 0.02 % 0.5 -8265.623 ± 0.003 -8290.934 ± 0.431 0.58 % 1.0 -8265.294 ± 0.016 -8290.618 ± 0.853 1.90 % 3.0 -8259.401 ± 0.25259 -8285.427 ± 0.493 51.23 % 2.2 the canonical ensemble 11

2.2 the canonical ensemble

Performing simulations in the microcanonical ensemble is easily done and yield the correct results. However, in real life we are not that much interested in performing constant energy experiments, which describe an isolated system. Simulations at a constant temperature and constant pressure come closer to real life. In this section we therefore look at the canonical ensemble (NVT), where the number of particles, the volume and the temperature are fixed. To fix the temperature ("thermostatting") gen- erally two approaches can be distinguished: ’couple’ the system to a heat bath and rescale the velocities accordingly or make use of the "extended phase space" approach. We will discuss examples of both approaches and also distinguish between local and global thermostats. Later, in section 2.3, we show how to extend these methods to the Isothermal-Isobaric (NPT) ensemble, where the pressure is also fixed. Before we continue let us first define the temperature by using the equipartition theorem. This will lead to an instantaneous temperature T:

2 3 mivi EK = NkbT −→ T = ,(2.27) 2 3NkB i X where N is the number of particles and kB Boltzmann constant. We see that the temperature is directly related to the kinetic energy EK.

2.2.1 Berendsen thermostat

The Berendsen thermostat is an example where the system is (weakly) coupled to a heat bath with a constant temperature T0. Physically the system undergoes collisions with a heat bath and exchange kinetic energy as to end up at a certain temperature. A way to describe this process is provided by the Langevin approach:

mir¨i = Fi − miγir˙i + Ri(t) (2.28) where we have a friction term with a damping constant γi and we have Ri(t) which is a Gaussian stochastic variable with zero mean and amplitude:

< Ri(t)Rj(t + ∆t) >= 2miγikT0δ(∆t)δij.(2.29)

However, drawing N Gaussian random numbers at each step is computational very expensive, leading to longer simulation times. The idea of the Berendsen thermostat is to replace this random term by an extra damped term. Berendsen et al.[22] show that this leads to:

1 T0 mir¨i = Fi + mi ( − 1)r˙i.(2.30) 2τt T

where the random part Ri(t) (local noise) from equation 2.28 is substituted by a global temperature coupling. Equation 2.30 is the modiﬁed equation of motion and results in a velocity rescaling at every step according to: s ∆t T0 v = λv = 1 + − 1 v (2.31) τt T 12 molecular dynamics in various ensembles

1 τt = 2γ is the coupling constant and determines how much the heat bath and the system are coupled. T is the current temperature of the system derived from the kinetic energy using equipartition and T0 the desired temperature (or temperature of the heat bath). The resulting change in temperature can be described by: dT 1 = 2γ(T0 − T) = (T0 − T) (2.32) dt τt If we solve this equation for T the role of the coupling constant becomes more apparent as we get an exponential decay towards the equilibrium temperature with the time scale given by τt:

−t/τt T = T0 − Ce (2.33)

For large τt the extra term in the equation of motion (eq. 2.30) disappears and we are left with just Newton’s equations of motion, which is equivalent to the microcanonical ensemble. When τt becomes small all fluctuations in the kinetic energy disappear. We will discuss a suitable value for τt in section 2.2.3. Notice that the rescaling is applied to the whole system and in that sense the Berend- sen thermostat is a so called global thermostat. The problem with this thermostat is that in general the right fluctuations corresponding to the fluctuations in the canonical ensemble are not obtained (see for instance Morishita[23]).

lammps implementation. The implementation of the Berendsen thermostat is easily done: one needs to rescale the velocities each step according to equation 2.31. The Berendsen velocity rescaling is done in lammps by setting ’fix temp/berendsen’ in the input file. In lammps each procedure that is applied to the system during a time step is called ’fix’. The Berendsen fix needs three parameters: Ti, Tf and τt where Ti and Tf are the desired temperatures at the beginning and the end of the run and τt is the temperature damping parameter which also appears in equation 2.31. At every time step the velocities of the particles are rescaled as v = λv where λ is given by equation 2.31. The target temperature T0 is determined at every time step by: n T0 = Ti + (Tf − Ti) (2.34) Ntotal th with n the n time step and Ntotal the total number of time steps of the run. Note that the velocity rescaling is done at the end of the time step thus after the positions and velocities are updated by the Verlet algorithm. An input file for lammps using the Berendsen thermostat can be found in appendixA. 3.

2.2.2 Nosé-Hoover chain thermostat

In the previous method a constant temperature was achieved by using collisions with a heat bath. Now we will consider a different approach in which the Hamiltonian (or the Langrangian) is extended with additional variables. Those variables try to mimic a heat bath. An example of such an extended Hamiltonian is the Nosé Hamiltonian[24]:

N 2 2 pi N ps HN = 2 + U(r ) + + dNkBT ln s (2.35) 2mis 2Q i=1 X 2.2 the canonical ensemble 13

where d is the dimension of the system, s is our additional variable with corresponding momentum ps, T is the desired temperature of the system and Q can be thought of as an effective mass corresponding to s (but it is not a mass as it has different units). It should be noted that the phase space is now extended and that we have 6N + 2 degrees of freedom thanks to s and ps. The logarithmic dependence is chosen to end up with the canonical ensemble[18,19]. Using the Hamilton equations of motion (eq. 2.6) we can derive the equations of motion for this Hamiltonian:

∂HN pi r˙i = = 2 (2.36) ∂pi mis

N ∂HN ∂U(r ) p˙ i = − = − = Fi (2.37) ∂ri ∂ri

∂HN ps s˙ = = (2.38) ∂ps Q

N 2 " N 2 # ∂HN pi dNkT 1 pi p˙ s = − = 3 − = 2 − dNkBT (2.39) ∂s mis s s mis i=1 i=1 X X From the last equation one can see how the desired thermostat works: the sign and strength ofp ˙ s depends on the difference between a kinetic energy like term and temperature. The velocities are rescaled accordingly with s, and Q determines the strength of the coupling. 0 pi 0 ps 0 dt By transforming pi = s , ps = s and dt = s one can reformulate the above equations of motion in such a way that the ’normal’ kinetic energy term is used. How- ever, this transformation is non-canonical and non-symplectic. A different, also non- canonical, transformation was introduced by Hoover[25] leading to the Nosé-Hoover equations. This change of variables is formulated as:

pi dt 1 ds dη p0 = dt0 = = p = p (2.40) i s s s dt0 dt0 s η We have now shown the basic idea of the extension of the Hamiltonian with additional variables. Taking the above steps will lead to the Nosé-Hoover equations. How- ever, these equations will fail when more than one conservation law (for example, conservation of momentum) has to be obeyed. To solve this problem the equations can be extended even further. Martyna et al.[26] developed the idea of Nosé-Hoover chains: the thermostat is coupled to another thermostat leading to a chain of thermostats. This will result in the Nosé-Hoover chain equations for system with i = 1...N particles and j = 1...M chains2:

pi r˙i = (2.41) mi

2 For a full review see Tuckerman [18] and Frenkel and Smit [19] 14 molecular dynamics in various ensembles

pη1 p˙ i = Fi − pi (2.42) Q1

pηj η˙ j = (2.43) Qj

" N 2 # pi pη2 p˙ η1 = − dNkT − pη1 (2.44) mi Q2 i=1 X

"p2 # ηj−1 pηj+1 p˙ ηj6=1,M = − kT − pηj (2.45) Qj−1 Qj+1

" p2 # ηM−1 p˙ ηM = − kT (2.46) QM−1

The last three equations can be rewritten to correspond better with the notation of Frenkel and Smit[19] and lammps:

" N 2 # p˙ η1 1 pi η¨1 = = − dNkT − η˙ 1η˙ 2 (2.47) Q1 Q1 mi i=1 X

1 2 η¨j = Qj−1η˙ j−1 − kT − η˙ jη˙ j+1 (2.48) Qj

1 2 ηM¨ = QM−1η˙ M−1 − kT (2.49) QM

The coefﬁcients Qj can be determined using the equations:

2 Q1 = dNkTτt (2.50)

2 Qj6=1 = kTτt (2.51)

Now the next step is to integrate these equations of motion. To do this we will go back to the the Liouville formalism we introduced in the previous section. We can distinguish the Liouville operator as:

iL = iLNHC + iL1 + iL2 (2.52)

where iL1 and iL2 are deﬁned in equation 2.15. The LNHC operator which we get from the Nosé-Hoover chain equations is given by:

N M M−1 ∂ ∂ Qj∂ QM∂ iLNHC = − η˙ 1pi + η˙j + Gj − Qjη˙ jη˙ j+1 + GM (2.53) ∂pi ∂ηj ∂η˙ j ∂η˙ M i=1 j=1 j=1 X X X 2.2 the canonical ensemble 15 where we have introduced:

N 2 pi G1 = − dNkT (2.54) mi i=1 X

p2 ηj−1 2 Gj = − kT = Qj−1η˙ j−1 − kT (2.55) Qj−1

We now can write our propagator as:

eiL∆t = eiLNHC∆t/2eiL2∆t/2eiL1∆teiL2∆t/2eiLNHC∆t/2 (2.56)

The beauty of this propagator is that the middle part of the propagator is the same as in the microcanonical ensemble (eq. 2.19). To add a thermostat we just have to apply the ’new’ part of the propagator to v, η andη ˙ , perform the verlocity Verlet algorithm and apply the new propagator again. Of course the hard part is to calculate the new propagator. To do this eiLNHC can be factorised even further. This is done in Martyna et al.[27] and Frenkel and Smit[19]. This lead to the equations given in algorithm 2.2. lammps implementation. To perform NVT calculations in lammps using the Nosé-Hoover chain equations one needs to set the ’fix nvt’ command. All relevant functions can be found in fix_nh.cpp. As with the ’fix temp/berendsen’ command one is able to set values for the initial (Ti) and final (Tf) temperature and the temperature damping factor τt. Other values that are possible to set are the number of thermostat chains (M) and an extra drag coefficient γ to damp oscillations. Default values are m = 3 and γ = 0. An algorithm to perform the eiLNHC update is given in pseudo-code in algorithm 2.2. The total propagation of all coordinates and velocities is than given by algorithm 2.3. 16 molecular dynamics in various ensembles

Listing 2.2: Algorithm for the Nosé-Hoover chain equations or LNHC update of the velocities.

%LNHC update of the velocities(also η and η˙ ): Calculate Qj Calculate Gj %update the η velocities fromM to 1: 1 η˙ M−1 = η˙ M−1 ∗ exp( 8 ∆tη˙ M) 1 η˙ M−1 = η˙ M−1 + 4 ∆t ∗ Gj 1 η˙ M−1 = η˙ M−1 ∗ exp( 8 ∆tη˙ M) %do this forj=M-2 untilj=1 %rescale the velocities according to: 0 1 v = v ∗ exp( 2 ∆tη˙ 1) %update the kinetic energy %update η: 1 ηj = ηj + 2 ∆t ∗ η˙ %update the η velocities from1 toM(see above) 1 η˙ M−1 = η˙ M−1 ∗ exp( 8 ∆tη˙ M) 1 η˙ M−1 = η˙ M−1 + 4 ∆t ∗ Gj 1 η˙ M−1 = η˙ M−1 ∗ exp( 8 ∆tη˙ M)

Listing 2.3: Typical MD code for performing NVT simulations using the NHC thermostat

Propagate the added variables according to algorithm 2.2

Perform first step of the velocity Verlet algorithm on v and r

Calculate the forces

Perform second Verlet step on the velocities

Redo the LNHC update according to algorithm 2.2

¥ 2.2 the canonical ensemble 17

2.2.3 Simulation of Boron Nitride in the canonical ensemble

In section 2.1.2 we performed a simple NVE simulation of two dimensional BN to determine a good choice for the time step ∆t. To do this we checked if the total energy was conserved. However, in the NTV ensemble the total energy (kinetic plus potential energy) is not conserved as the various thermostats add energy to the system. The new conserved quantity can then be described by3:

0 H (r, p) = H(r, p) + Hthermostat (2.57)

Therefore we have to check if H 0 (eq 2.57) is conserved. Of course the other quantity of interest in the NVT ensemble is the temperature. What one wants is a fast and stable way to reach the equilibrium temperature after which the temperature fluctuations are small. We have to find out which of the previously described methods accomplish this the best and what the corresponding value for the parameter τt is. We have performed this test on a system consisting of a single sheet of BN with a hexagonal lattice with N = 1296 atoms and periodic boundary conditions. The atoms are given an initial velocity corresponding to a temperature of 300K. However, as before, some of this kinetic energy is transferred to potential energy in the first couple of time steps resulting in a lower temperature.

Berendsen thermostat Nose-Hoover thermostat -6.35 -6.35

-6.36 -6.36

∆t = 0.1 fs ∆t = 0.1 fs ∆t = 1 fs ∆t = 1 fs -6.37 ∆t = 3 fs -6.37 ∆t = 3 fs Conserved energy H'(eV) Conserved energy H'(eV)

-6.38 -6.38 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Number of time steps Number of time steps

(a) (b)

Figure 2.2: The conserved energy H 0 for three different time steps for a) the Berendsen thermostat and b) the Nosé-Hoover chain thermostat. In total 100.000 steps are taken with τt = 0.1 ps and three values of ∆t as indicated.

Let us first look, again, at the choice for the time step ∆t. From the calculations in the microcanonical ensemble we learned that ∆t 6 1.0 fs was fine for energy conservation. Let us check if this is still the case. To this purpose we choose τt = 100 fs= 0.1 ps for both the Berendsen and the Nosé-Hoover chain thermostat. In figure 2.2 we see that, in both cases, for ∆t = 0.1 fs the quantity H 0 is nicely conserved and therefore we will use this value in further calculations. The first task of the thermostat is to get the system to the equilibrium temperature we want to study. In figure 2.3 we see how heating of the system from 150 K to 300 K occurs for the two thermostats with various values for τt. For both thermostats we see

3 Note that in general H0 is not a Hamiltonian 18 molecular dynamics in various ensembles

Berendsen Nose Hoover 350 350

300 300

250 250

200 τ 200 τ t = 0.01 ps t = 0.01 ps τ = 0.1 ps τ = 0.1 ps

Temperature (K) t Temperature (K) t τ = 1.0 ps τ = 1.0 ps 150 t 150 t

100 100 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Number of time steps Number of time steps

(a) (b)

Figure 2.3: The temperature as a function of time for a) the Berendsen thermostat and b) the Nosé-Hoover thermostat for different values of τt. For all simulations the time step is taken to be ∆t = 0.1 fs. The 100000 time steps shown correspond to 10.0 ps.

4 that for the high value of τt = 1.0 ps it takes more than 60.000 steps before the wanted temperature is reached, where in the Nosé-Hoover case we can also distinguish a oscillating character. If we look at the other values for τt we see that it takes roughly 10 · τt before the system is at its desired temperature. Again, we see oscillating behaviour for the Nosé-Hoover thermostat for τt = 0.1 ps. To study this oscillating more closely, in 2.4 we see the energy added by the thermostat over time for τt = 0.1 ps. Again, it is clear that the ﬂuctuations in energy are much higher for the Nosé-Hoover thermostat.

Berendsen Nose-Hoover -6.33 0 -6.33 0

-6.34 -0.01 -6.34 -0.01

-6.35 -0.02 -6.35 -0.02 te te te+thermostat te+thermostat -6.36 thermostat -0.03 -6.36 thermostat -0.03 Total energy (eV) Total energy (eV) -6.37 -0.04 -6.37 -0.04 Thermostat energy (eV) Thermostat energy (eV)

-6.38 -0.05 -6.38 -0.05 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Number of time steps Number of time steps

(a) (b)

Figure 2.4: The total (potential + kinetic) energy and the added energy by a) the Berendsen and b) Nosé-Hoover chain thermostat as function of the time. For the simulations ∆t was 0.1fs and τt = 0.1ps for both thermostats.

To study the ﬂuctuations better we go back to basic statistical mechanics. Fluctua- tions in the NVT ensemble can be described as follows (see for instance Allen and

4 1.0 ps corresponds to 10000 time steps of 0.1 fs. 2.2 the canonical ensemble 19

[28] Tildesley ), starting from the fact that the speciﬁc heat CV is given by the ﬂuctua- tions in total energy5:

2 2 < δH >NVT = kBT CV (2.58)

We can split the Hamiltonian into a potential and a kinetic contribution and write for the kinetic energy ﬂuctuations: 1 δK2 = N k2 T 2 (2.59) 2 f B

Where Nf is the number of degrees of freedom and

δK2 = K2 − hKi2 .(2.60)

We thus can find an exact result for the kinetic energy fluctuations and compare our simulations with this exact value. In figure 2.5 we can find this comparison for different values of τt. The temperature was set to be 150 K with a sample size of N = 1296 atoms. Initially 100.000 steps where taken with the Berendsen thermostat to get the system to the desired temperature, after which 500.000 steps where taken to determine the fluctuations in the kinetic and potential energy. As we can see in figure 2.5 we get the desired value (which is 1 is these units) for the fluctuations for the Nosé-Hoover thermostat for τt 6 0.1ps. For the Berendsen thermostat we never reach the right values, but one should use τt > 0.1ps to get the best result.

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 kin b 0.3 pot b Energy Fluctuations 0.2 kin nh pot nh 0.1 0 0.001 0.01 0.1 1 10 τ t (ps)

Figure 2.5: Kinetic and potential energy fluctuations as defined by formula 2.60 as function of 1 2 2 τt. Fluctuations are in units of 2 NfkBT , so that the value of 1 corresponds to the analytical result for the kinetic energy fluctuations.

5 We will drop the NVT subscript from here, as all averages are done in the canonical ensemble 20 molecular dynamics in various ensembles

2.3 the isothermal-isobaric ensemble

In the previous section we extended the equations of motion for our system with a coupling to a constant heat bath to perform simulations in the canonical ensemble. In this section we do the same but now to achieve a constant pressure. Combining both the thermostat and the barostat we will be able to perform MD calculations in the isothermal-isobaric ensemble (NPT). In the canonical ensemble the temperature could easily be deﬁned. For the pressure it is a bit more cumbersome. In the isobaric ensemble the volume of the system is adjusted in a way that the (average) internal pressure is the same as the (applied) external pressure, which can be described by the pressure virial theorem: D E P(int) = P,(2.61)

where P is the external pressure and P(int) the internal pressure. The instantaneous pressure in a simulation can be calculated by means of the virial:

NkBT 1 P(int) = + r F ,(2.62) V Vd ij ij i

where d is the dimension of the system, Fij is the force on particle i due to particle j and rij is the distance between the two particles. On the right side of the equation we can distinguish two parts: the ﬁrst part is ideal gas law, while the second part is the work added in the system by interactions between particles.

2.3.1 Berendsen barostat

In section 2.2.1 we discussed the Berendsen thermostat, where the velocities of the particles are rescaled leading to a different temperature. If we look at equation 2.62 we see that a change in pressure can be accomplished by changing the distance rij between particles. The Berendsen barostat applies such a scaling of the coordinates and box length l, according to[22]:

1 3 0 ∆t x = 1 − (P0 − P) x = µx (2.63) τp

l0 = µl.(2.64)

Where we had a damping factor τt for the thermostat we now have a similar time constant τp for the barostat. The pressure P0 is the desired pressure of the system while P is the current instantaneous pressure of the system.

lammps To use the Berendsen barostat in lammps you can use the ’fix press/berendsen’ command. As with its temperature equivalent, one can choose the desired initial (Pi) and final (Pf) pressure and the value for τp. After each time step the barostat will rescale the coordinates of the particles and the values for the box size in the specified 2.3 the isothermal-isobaric ensemble 21

dimensions according to equations 2.63 and 2.64. The target pressure P0 is calculated at each time step as: n P0 = Pi + (Pf − Pi) (2.65) Ntotal

th with n the n time step and Ntotal the total number of time steps of the run. After rescaling the box the coordinates of the atoms are rescaled in a similar manner. It is possible to specify different pressure values for different directions.

2.3.2 Nosé-Hoover chain barostat

Obtaining a constant pressure can be done with the same approach as for the Nosé- Hoover thermostat. In this case equations of motion are added to account for a change [29] in volume , with new variables , p and W. First we deﬁne being:

V = ln ,(2.66) V(t = 0) with V the current volume and V(t = 0) the starting value of the volume. The new equations of motion corresponding to the new variables are:

dVp V˙ = (2.67) W

N 2 (int) 1 pi pξ1 p˙ = dV P − P + − p (2.68) N mi Q1 i X where d is as usual the dimension. These equations are added to the equations of motion for the positions, velocities and thermostat variables. These combined equations are called the MTK equations after Martyna, Tobias, and Klein[29]. Decomposing the corresponding Liouville operator and integration of these equations is done in[18,19,29], while a complete algorithm can be found in Martyna et al.[27]. The advantage over the Berendsen scheme is that the MTK equations lead to the correct NPT phase space distribution. lammps Simulations in the NPT ensemble using the Nosé-Hoover methods is done by setting ’ﬁx npt’ command. As for the Berendsen case before one can set the desired initial (Pi) and ﬁnal (Pf) pressure values. As before a value for the damping parameter τp can be set. The barostat mass W depends on this value according to:

2 W = NkBTτp.(2.69)

As mentioned, a complete algorithm can be found in Martyna et al.[27]. 22 molecular dynamics in various ensembles

Berendsen Nose-Hoover -6.37 -6.355

-6.372 ∆t = 0.1 fs -6.374 ∆t = 0.5 fs -6.36 ∆t = 1 fs -6.376 ∆t = 3 fs -6.365 -6.378 -6.38 -6.37 ∆ -6.382 t = 0.1 fs ∆t = 1 fs ∆ -6.384 -6.375 t = 3 fs -6.386 Conserved energy H'(eV/Atom) -6.388 Conserved energy H'(eV/Atom) -6.38 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Number of time steps Number of time steps

(a) (b)

Figure 2.6: The conserved energy H 0 for three different time steps for both the Berendsen method (a) and the Nosé-Hoover chain method (b). In total 100.000 steps are taken with τt = 0.01ps.τp = 20000 ∗ ∆t for Berendsen and τp = 1000 ∗ ∆t for Nosé Hoover.

2.3.3 Simulation of Boron Nitride in the Isothermal-Isobaric ensemble

The goal of this section is to obtain values for ∆t and τp which we can use to perform simulations in the NPT ensemble. We want to be able to perform simulations at any given temperature with zero external pressure. The first check is, again, to check what a suitable time step ∆t would be. When we first performed simulations we found that the Berendsen damping parameter τp must be at least 20.000 time steps for the system to be stable. Thus for determining the time step we take τp = 20000 ∗ ∆t in the case of the Berendsen barostat and τp = 1000 ∗ ∆t in the case of the Nosé Hoover barostat. In both cases we will use τt = 0.01ps. Again we can introduce a conserved quantity H 0, which is the total energy of the system plus the energy added by the thermostat and barostat. Results for different values of ∆t can be found in figure 2.6. We see that in the Berendsen case (figure 2.6a) we need a relatively small time step (∆t = 0.1fs) to stop the energy from floating away. In the Nosé-Hoover case ( figure 2.6b) the time step can be larger. For our simulations we want to apply zero external pressure. In figure 2.7 we have plotted the pressure over time for P = 0 with different values for τp. We see that the fluctuation in the pressure differs between the two cases by a factor ten or larger. An- other quantity at which we look is the enthalpy. The enthalpy is defined as the internal energy plus the product of the pressure and the volume (H = U + pV)6. In figure 2.8 we have plotted the fluctuations in the enthalpy for both the NH and Berendsen cases. We see for Nosé-Hoover, independent of the choice of τp, the same undesired fluctuations as in the pressure, while in the Berendsen case the enthalpy remains constant. We see that the enthalpy fluctuations in figure 2.8 are the same as the pressure fluctuations in figure 2.7b, meaning that the volume is not changed enough to keep the enthalpy constant.

6 The quantity was introduced by J.W. Gibbs while H. Kamerlingh Onnes was the ﬁrst to call it enthalpy [30]. 2.3 the isothermal-isobaric ensemble 23

Berendsen Nose Hoover 0.02 τ 0.3 τ p = 100.0 ps p = 0.1 ps τ = 10.0 ps τ = 1.0 ps pτ 0.2 τ p p = 2.0 ps p = 10.0 ps 0.01 0.1

0 0

-0.1 Pressure (GPa) -0.01 Pressure (GPa) -0.2

-0.02 -0.3 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Number of time steps Number of time steps

(a) (b)

Figure 2.7: The pressure as function of the time steps for different values of τp for both the Berendsen barostat (2.2a) and the Nosé-Hoover chain barostat (2.2b). In total 100.000 steps are taken with τt = 0.1ps and ∆t = 0.1fs. Ones sees that the ﬂuctuations differ by (at least) a factor 10.

From this we conclude that the Berendsen barostat is more suitable for our simulations than the Nosé-Hoover. We will use a value of τp = 2.0ps in further simulations because the ﬂuctuations are the smallest for this value (see ﬁgure 2.7a). 24 molecular dynamics in various ensembles

Enthalpy fluctuations

NH - τ = 0.1 ps τp NH - p = 1.0 ps -6.25 NH - τ = 10.0 ps τp Berendsen - p = 10.0 ps -6.3

-6.35 Enthalpy (eV/Atom)

-6.4

0 20000 40000 60000 80000 100000 Number of time steps

Figure 2.8: Enthalpy ﬂuctuations in the NPT ensemble for Nosé Hoover. The pink line in the middle is for the Berendsen case. 100.000 steps are taken with ∆t = 0.1fs and τt = 0.01ps.

2.4 conclusion

In this section we have described the basic principles of doing a molecular dynamics simulation and how to extend these simulations to various ensembles. Two different thermostats and barostats were studied using the Tersoff potential for a single sheet of boron-nitride. In section 2.2.3 we showed that the Berendsen thermostat is faster in getting the system to a particular temperature but that the Nosé-Hoover thermostat yields the correct canonical ensemble (for certain values of τt). In section 2.3.3 we saw that the NH-barostat gave rise in serious ﬂuctuations in the enthalpy. The NH stats are considered better, but in our system (a two dimensional crystal) this is not the case. Because it is faster and more stable we will use the Berendsen stats in the next sections, but we will test our results against simulations performed with the NH stats. For reference we report in table 2.2 the optimal values found for NVT and NPT simulations for both methods. Probably a better candidate for our system would be the recently developed thermostat by Bussi, Donadio, and Parrinello[31] and the corresponding barostat developed by Bussi, Zykova-Timan, and Parrinello[32]. These are similar to the Berendsen ver- sions, but add an additional random ’noise’ to the system that leads to the correct phase-space distribution. 2.4 conclusion 25

Table 2.2: Values of the different variables we use in the NPT ensemble. The time step ∆t can be increased up to ∼ 1 fs when simulating in other (NVE or NVT) ensembles.

Berendsen Nosé-Hoover ∆t 0.1 fs 0.1 fs

τt 0.1 ps 0.1 ps

τp 0.2 ps 1.0 ps

ADVANCEDTECHNIQUES 3

In this chapter we will briefly discuss two methods to obtain energy profiles as function of a reaction coordinate. The reaction coordinate can be compared to the order parameter in statistical mechanics. In our case we want to study defects, where the reaction coordinate should follow the path which leads the system from the initial state to a state with a defect. The first method, steered molecular dynamics (SMD), can be used to obtain a free energy profile, while the second, nudged elastic band (NEB), can only be used to obtain a total energy profile. Although the free energy gives more information than the ’regular’ energy, we found from tests that SMD was unsuitable for the study of defects. That is why we, in section 5.2, will only give results from the NEB calculations

3.1 steered molecular dynamics

The discovery of Jarzynski’s equality[33] lead to a method, steered molecular dynamics (SMD), to calculate the free energy in non-equilibrium simulations. The goal is to obtain the potential of mean force (PMF) of the system which is the free energy proﬁle with respect to ξ. This method can be best illustrated to pulling part of a molecule. A string is ’attached’ to a molecule while the string moves with a constant velocity v in a particular direction. The potential of this spring is described by: k V = [ξ(r) − λ(t)]2 ,(3.1) 2 where λ is the position of the spring and ξ is the reaction coordinate. The spring constrains ξ to be close to λ, thus the spring should move in the direction of the desired path that will lead to a defect. The position of the spring can be described by:

λt = λ0 + vt (3.2)

In order to use Jarzyski’s equation:

e−β∆F = e−βW (3.3) we have to calculate the work performed on the system. This work, at a time t, is given by[34,35]:

t 0 0 W0→t = −kv dt ξ(rt0 ) − λ0 − vt .(3.4) Z0 Park et al.[34] show that when the spring constant k is strong enough the PMF Φ and the free energy F are related to each other:

F(λ) ≈ Φ(λ).(3.5)

27 28 advanced techniques

Now if we use Jarzyski’s equation we can get for Φ: 1 Φ(λ ) = Φ(λ ) − log e−βW0→t .(3.6) t 0 β Instead of taking the average over the exponential it can be more convenient to make an approximation. We can write the last part of equation 3.6 as (ignoring the subscripts of W):

β2 β3 log e−βW ≈ −β hWi + ( W2 − hWi2) − ( W3 − 3 W2 hWi + 2 hWi3) + ... 2 3! (3.7)

When using a limited amount of samples the second order expansion gives the most accurate results according to[34,35]. However, one should always check is this is the case for the system being studied.

3.2 nudged elastic band

The Nudged Elastic Band (NEB) method is used to find the minimal energy path leading from one (meta-)stable configuration to another. This is particular useful to find a good reaction coordinate, but NEB calculations also produce an energy barrier. The implementation in lammps is done according to Henkelman and Jónsson[36] and Henkelman et al.[37]. A general introduction to the method can be found in Jónsson et al.[38]. At the start of the calculations M replicas of the system are created and run in parallel on M processors. Replica 1 is the first stable configuration while replica M is the stable state at the other side of the energy barrier. Atom coordinates of the replicas in between the two states are initialised by linear interpolation. The forces of all replicas are minimised to come to the minimum energy path, where the total force on each atom is changed to include inter-replica forces, meaning that each atom in replica i is connected (using a spring) to the same atom in replica i − 1 and replica i + 1. s The total force in replica i, on a particle j, includes the spring force Fi,jk between the replicas and is given by[36]:

s Fi,j = −∇U(Ri,j)⊥ + Fi,jk.(3.8)

Where Ri,j are the coordinates of particle j in replica i. We will drop the subscript j from here. The spring force is given by:

s Fik = k(|Ri+1 − Ri| − |Ri − Ri−1|)tˆi.(3.9)

tˆi is the unit tangent vector of replica i. It depends on the direction of the energies between replicas and it is given by[36]:

t+ = R − R if U > U > U t = i i+1 i i+1 i i−1 (3.10) i − ti = Ri − Ri−1 if Ui+1 < Ui < Ui−1 If the system is at a local maximum or minimum the tangent becomes more complex: + − it will become a weighted average between ti and ti where the weight is depended on Ui+1 and Ui−1. 3.2 nudged elastic band 29

Minimising formula 3.8 for all replicas leads to a the minimum energy path. In order to improve the accuracy another step can be taken after this ﬁrst minimisation (as is done in lammps). A description of this step can be found in Henkelman, Uberuaga, and Jónsson[37].

SIMULATIONOFHEXAGONALBORON-NITRIDE 4

Although MD uses classical mechanics, it is used to study systems that are not classical. This seems strange but in most cases it works. Only in cases of light atoms or molecules, such as H2,He and D2, this approach will fail, or more speciﬁc when the mean nearest neighbour distance a is much smaller than the de Broglie thermal wavelength. The (quantum) physics comes around in the interatomic potential, which are designed to represent reality as close as possible. In this chapter we will discuss such a potential designed for bulk Boron-Nitride (BN) and test it on a single layer of hexagonal-BN (h-BN).

4.1 empirical potentials

When doing any numerical calculations on materials the most important part is the model of interatomic interactions. Although the results may not be exact, you want your model to resemble ’real-life’ as good as possible. In the case of molecular dynamics the potential between particles must result in the right phase of the material at a given temperature or pressure. These so called empirical or classical potentials are at- tempts to mimic the interactions arising from quantum mechanics. They are especially useful in the case of systems that are simply too big to do direct quantum calculations and are used to gain insight into the structural properties of complex systems. Empirical potentials consist of a mathematical form with certain constants that are fitted to experimental (or other theoretical) data. A good empirical potential obeys certain rules (as mentioned by Brenner[40]): first, the potential should be flexible. Second, the potential should be accurate. Third, the potential should be transferable and fourth, the potential should be computational efficient. The second and fourth points speak for themselves: you want accurate results for the data set you used to fit and you do not want the calculations to take too long. The other points need a bit more explanation. You want flexibility as you want to incorporate as many properties (or structures) in the fitting data as possible and you want transferability to obtain (qualitative) results on structures that are not included in the data you used for fitting. Probably the simplest of the interatomic potentials are the pairwise potentials such as the Lennard-Jones and the Morse potential in which the interatomic forces depend only on the distance between two atoms. Although they are widely used they become less accurate when the coordination is low (when atoms have few neighbours). This is seen in 4.1, where the bond energies are plotted for different crystal structures: structures with high coordination are energetically favoured2. To simulate more diffi-

A part of this chapter was published as Structure, stability and defects of single layer hexagonal BN in comparison to graphene, G.J. Slotman and A. Fasolino, Journal of Physics: Condensed Matter 25-4,045009 (2013) [39]

2 For reference: graphite has 3 nearest neighbours, diamond 4, simple (sc) 6, body-centered (bcc) 8 and face-centered (fcc) 12.

31 32 simulation of hexagonal boron-nitride

cult materials with covalent bondings like semi-conductors, more complex potentials have been developed. The so called bond order potentials start from the idea that the strength of each bond depends on the local environment: atoms with many neighbours form weaker bonds with each atom than atoms with few neighbours. Examples of bond order potentials include: Tersoff[4], rebo[5,41], airebo[6] and lcbop[7].

Figure 4.1: Energy for two different pair potentials as function of interatomic distance: in the top panel the Lennard-Jones potential and in the bottom panel the Morse potential. On the right the energy for different crystal structures. It is seen that these potentials tend to favour structures with high coordination. Images courtesy of Jan Los. 4.1 empirical potentials 33

4.1.1 A potential for BN

The interaction between the particles in our system is described by a Tersoff bond order potential[4], which is given by: 1 E = V 2 ij i6=j X Vij = fC(rij) fR(rij) + bijfA(rij)

−λ1r fR(r) = Ae

−λ2r fA(r) = −Be

1, r < R − D  1 1 π(r−R) (4.1) fC(r) = − sin , R − D < r < R + D  2 2 2D  0, r > R + D  n n − 1 bij = (1 + β ζij) 2n m m λ (rij−rik) ζij = fc(rik)g(θijk)e 3 k6=i j X, c2 c2 g(θ) = γijk 1 + 2 − 2 2 d (d + (cos(θ) − cosθ0) )

The potential consists of a repulsive part fR(r) and an attractive part fA(r). The cutoff function fC(r) is there to limit the range of the potential leading to less computing time. Normally the cutoff distance is chosen in such a way to include only the first- neighbours. The attractive and repulsive functions are exponential functions (Morse- [42] like) because this leads to the correct bonding behaviour . The term bij, the bond order, is probably the most important part of the potential: it determines the strength of the attractive term as a function of the coordination and of the angle between different bonds. The specific parameters for the Tersoff potential in the case of BN are given by Albe, Möller, and Heinig[8] and are shown in table 4.1. Due to a difference between the definition of the coefficients used by Albe et al.[8] and the definition here there is a discrepancy between the coefficients. In appendixA. 2 the details of the calculation are given. In table 4.2 we give various quantities for this potential for different molecular structures. It is clear that the values are not simply a pairwise function but also depend on the surrounding atoms. 34 simulation of hexagonal boron-nitride

Table 4.1: Coefficients for the Tersoff potential for BN equivalent to the ones defined by Albe et al. [8]. See appendixA. 2 for the relation between these coefficients and those given by Albe et al. [8].

Coefﬁcient BB-interaction NN-interaction BN-interaction m 3 3 3 γ 1 1 1

λ3 0 0 1.9925 c 0.52629 17.7959 1092.9287 d 0.001587 5.9484 12.38

cos(θ0) 0.5 0 -0.5413 n 3.9929061 0.6184432 0.364153367 β 0.0000016 0.019251 0.000011134

λ2 2.07750 2.627272 2.784247 B 1173.197 2563.560 3613.431 R 2 2 2 D 0.1 0.1 0.1

λ1 2.23726 2.82931 2.99836 A 1404.47520 2978.95279 4460.83397

[8] Table 4.2: Structural properties of the Albe et al. potential. The binding energies EB are in eV, the bond lengths R in Å and the bond angles θ in degrees. Values are taken from [8].

Molecule Molecule

B-N EB 6.36 N-B-N EB 8.88

RBN 1.33 RBN 1.45

B-B EB 3.08 B-B-B EB 5.96

RBB 1.59 RBB 1.594

N-N EB 9.91 N-N-N EB 13.86

RNN 1.11 RNN 1.241

B EB 9.18 B-N-N EB 11.28

B RBB 1.594 RBN 1.44

B θ 60 RNN 1.241

N EB 14.64 N EB 8.42

N RNN 1.361 N RNN 1.336

N θ 60 B RBN 1.761 θ 44.5 (a) (b) 4.2 testingthepotential 35

4.2 testing the potential

In this section we will calculate some properties for single layer h-BN. Unless mentioned otherwise, we use in our simulations a sheet of 9800 BN atoms and we call this sample A. Simulations for this sample are done in the NPT ensemble. In all simulations we use the Berendsen thermostat and barostat with values τt = 0.01 ps and τp = 2.0 ps to keep the system at the desired temperature while keeping 0 external pressure. The external pressure is only applied in the xy-plane. The velocities of the atoms are initialised with a Gaussian distribution around the velocity corresponding to the desired temperature. The ﬁrst 200.000 steps are used to stabilize the temperature and pressure. After this, 2.5 · 106 steps are taken, where the last 2.0 · 106 steps are used for averaging. For these averages the lattice box size and atom coordinates where outputted every 400 steps. The value for the time step is ∆t = 0.1fs.

4.2.1 Lattice constant

The ﬁrst step for us is to determine the lattice constant for a single layer of h-BN. For this we distinguish three different structural parameters, namely the in-plane lattice constant a determined from the equilibrium box size obtained at constant pressure P = 0, the B-N nearest neighbour distance R and the nearest neighbour distance projected√ in the xy-plane Rxy. At T = 0 K, for a completely ﬂat sheet of h-BN R = Rxy = a/ 3.

To obtain a value for the lattice constant at zero temperature one can do a direct energy calculation. We do this by calculating the total energy at different values of a, for a completely flat sample. The equilibrium lattice constant at T = 0 K is than the value of a at the minimum energy. In figure 4.2 we see the results for these calculations. We see that the potential is anharmonic except for a small region around its minimum. From a fit around the minimum we conclude that at zero temperature a = 2.5328Å, which is in agreement with the value of a = 2.532Å given by[8] for bulk h-BN.

The next step is to see how the lattice constant changes with the temperature. In figure 4.3 we see what happens with a sample when the temperature is increased: it leads to out of plane deviations making the surface rippled. Simulations based on the bond order potential LCBOPII for graphene[43,44] have found that the lattice parameter first decreases with increasing temperature up to ∼900 K and increases at higher temperature. Calculations based on the quasi-harmonic approximation predict a contraction at least up to 2500 K[45]. Experimentally, data up to 400 K confirm a contraction[46]. In figure 4.4 and 4.5 we show the calculated temperature dependence of the previously defined three quantities: the in-plane lattice constant a, the B-N nearest neighbour distance R and the nearest neighbour distance projected in the xy-plane Rxy. We find that the BN nearest neighbor distance increases linearly up to about 1500 K with a slope of 1.2 · 10−5 ÅK−1, which is almost twice as large as the value 6.5 · 10−6 Å K−1 found in[44] for freestanding graphene using ab initio MD simulations. For the lattice parameter we find thermal contraction up to 1200 K, similarly to prediction for graphene[43]. For higher temperatures, the lattice parameter grows less rapidly than for graphene and, in the studied temperature range up to 2500 K, it never gets larger 36 simulation of hexagonal boron-nitride

−3.5

−6.400

−4.5 −6.405

−5.5 −6.410

Total Energy per atom (eV) Total Energy per atom (eV) −6.415

−6.5 2.2 2.4 2.6 2.8 3.0 3.2 2.49 2.51 2.53 2.55 2.57 Lattice Parameter (Å) Lattice Parameter (Å)

(a) (b)

Figure 4.2: The total (potential) energy per atom as a function of the lattice parameter. From a harmonic ﬁt (thin line) we can determine that the energy minimum lies at a = 2.5328Å. On the left we see that this approximation does not work away from equilibrium due to anharmonic terms in the potential. On the right we see a zoom around the minimum.

(a) T = 0K (b) T = 1500K

Figure 4.3: On the left a completely ﬂat sample of h-BN at zero temperature. On the right a sample heated up to T = 1500K, resulting in rippling of the material.

√ than the value at T = 0 K. At T 6= 0, the lattice parameter a is not equal to 3Rxy due to ﬂuctuations in the out of plane direction. 4.2 testingthepotential 37

1.50

1.49

1.48

B−N distance R (Å) 1.47

1.46 0 500 1000 1500 2000 2500 Temperature (K)

Figure 4.4: Temperature dependence of the average BN interatomic distance R calculated by MD in the NPT ensemble for sample A consisting of 9800 atoms. A linear ﬁt to the −5 −1 calculated points with R = R0 + αT where R0 =1.462 Å and α = 1.2 · 10 Å K describes well the results up to ∼ 1500K.

a 2.532 Rxy 1.462

1.460 2.528 (Å) xy a (Å) 1.458 R 2.524

1.456

2.520 0 500 1000 1500 2000 2500 Temperature (K)

Figure 4.5: Temperature dependence of lattice parameters calculated by MD in the NPT ensemble for sample A consisting of 9800 atoms. Lattice parameter a (left y-axis) and BN interatomic distance projected√ in the xy-plane Rxy (right y-axis). Notice that left and right y-axis differ by a factor 3 38 simulation of hexagonal boron-nitride

4.2.2 Bending rigidity

T=300K 100

10−1 G(q)/N

2 −2 q H(q) /N 10 G(q)/N G0(q)/N

10−3 0.1 1

Figure 4.6: The two correlation functions G(q) (eq. 4.2) and q2H(q) (eq. 4.4) at T = 300K coin- −1 cide as expected from equation 4.3 for wave vectors q . 1 Å yielding the same value of κ from a best ﬁt to the harmonic approximation (eq. 4.4) (dotted line) in the −1 −1 range q = 0.5 − 0.9 Å . Deviations from power-law√ behaviour occur for q & 1 Å close to the Bragg peak position at q = 4π/ 3a = 2.86 Å−1. In the long wavelength limit, q < q∗ ≈ 0.15 Å−1, the correlation functions have a power-law behaviour with a smaller exponent [47].

In the previous section we saw that the in plane lattice parameter shrinks while the distance between atoms grows when temperature is increased. This is due to the fact that the crystal starts to ripple. This rippling is an intrinsic property of a 2d crystal; without it the crystal will not be stable. To study the out of plane ﬂuctuations we can borrow from the theory of membranes in the continuum (see for instance[48,49,50]). Key quantities in this theory are the correlation functions of the height ﬂuctuations and of the normals. The normal-normal correlation function G(q) in the harmonic approximation is given by[51]:

D 2E 1 kBTN G0(q) = n = (4.2) q S κq2

where S = LxLy/N is the area per atom, κ is the bending rigidity and the subscript zero indicates that averages are taken in the harmonic approximation, namely neglecting the coupling of in-plane and out of plane ﬂuctuations in the stress tensor[50]. In the limit of slowly varying height ﬂuctuations, one can show that G(q) is related to the height-height correlation function H(q) by:

2 2 D 2E G(q) = q H(q) ≡ q hq (4.3) 4.2 testingthepotential 39

104 103 2500K 102 101 100 H(q)/N 10−1 10−2 1500K 10−3 300K 10−4 0.1 1 q (Å−1)

Figure 4.7: H(q)/N for T = 300 K, T = 1500 K and T = 2500 K. yielding, in the harmonic approximation,

1 kBTN H (q) = (4.4) 0 S κq4 It has been shown[52] that 4.3 is well reproduced when using numerical results from atomistic simulations only if the height fluctuations are calculated by averaging over the nearest neighbours heights: 1 1 h = h + (h + h + h ) (4.5) 2 0 3 α β γ where h0 is the z coordinate of one atom and hi the z coordinates of its three nearest neighbours. In figure 4.6, we compare G(q) to q2H(q) calculated by MD at T = 300 −1 K. We notice that these functions indeed coincide for q . 1 Å . Below this value the continuum approximation used in the theory of membranes breaks down and deviations from power-law√ behaviour occur, resulting in a peak at the position of the Bragg peak q = 4π/ 3a = 2.86 Å−1. The theory of membranes[50,53] predicts deviations from harmonic behaviour for wavevectors smaller than r 3TY q∗ = (4.6) 8πκ2 where Y is the bulk modulus. In the longwavelength limit, for q < q∗, the correlation functions bend to a lower exponent. This feature reduces the divergence of out of plane fluctuations and stabilizes the 2d crystal[50,51,54]. In figure 4.6 we see that deviations of ∗ −1 the calculated points from G0(q) start at q ≈ 0.15 Å as found for graphene at the same temperature[47], suggesting a similar ratio Y/κ. In figure 4.7 we show H(q)/N calculated by MD at different temperatures. 40 simulation of hexagonal boron-nitride

determining κ. The bending rigidity κ can be obtained by a ﬁt of the slope of the calculated H(q) to 4.4 in the range of q vectors where the harmonic approximation applies, yielding the temperature behaviour shown in 4.9. The value of κ at T = 0 can be found by direct total energy calculations of nanotubes and extrapolation to the limit of inﬁnite radii[55]. We do this by bending a sheet of h-BN to become a nanotube. To obtain the value of κ we use the fact that the energy to bend such a sheet is given by[56]:

1 E = κH2 (4.7) 2 where H is the curvature of the nanotube, namely the inverse radius and E is the energy of the nanotube per unit area. We did this for nanotubes with radii between 12 and 40 Å. Results of these calculations can be found in figure 4.8. This procedure yields κ = 0.54 eV, a value lower than the value κ = 0.82 eV for graphene[51], meaning that h-BN is easier to bend. Combining both the value for κ at T = 0 K and the values obtained from a fit of the slope of H(q) at T 6= 0 K, we obtain figure 4.9, where we see the temperature dependence of κ. Along with the values for h-BN we give the values for graphene, obtained from atomistic Monte-Carlo simulations in[52]. We see that over the whole temperature range κBN < κgraphene.

−2.3080 Fit with κ = 0.5410 eV ) 2 −2.3085

−2.3090

−2.3095 Energy per area (eV / Å

−2.3100 0 0.001 0.002 0.003 1/2 H2 (Å−2)

Figure 4.8: The energy per unit area as function of the curvature of a nanotube. Using equation 4.7 we obtain from a ﬁt the T = 0 value for κ, namely 0.54 eV. 4.3 conclusion 41

2.0

1.8

1.6

1.4

1.2 (eV) κ 1.0

0.8 BN 0.6 Graphene

0.4 0 500 1000 1500 2000 2500 Temperature (K)

Figure 4.9: The bending rigidity κ as function of temperature for both h-BN and graphene. For h-BN the T 6= 0K values are obtained by ﬁtting equation 4.4 for different temperatures. The values for graphene are taken from [52].

4.2.3 Nosé-Hoover

To be sure that the choice of the barostat does not alter the results we also performed some calculations with the Nosé-Hoover barostat and compared them with those obtained with the Berendsen barostat. In 4.10 we see how the in plane nearest neighbour q 2 2 distance Rxy and its standard deviation < Rxy > − < Rxy > behave over time for T = 300 K. Simulations were done for three different cases. One with a Nosé-Hoover thermostat and two with a Berendsen thermostat. One sees that the values do not converge to an exact number, but that the differences are negligible and fall well into the error margins. Since the standard deviation in the Berendsen cases is smaller, our choice for this barostat is well justified. To once more check that our final results are not dependent on the choice of the barostat we compared H(q) for both. In figure 4.11 we see that the differences between both are negligible. From fitting the slope of the data to obtain values for κ, we got similar results, namely κ = 0.8 for both.

4.3 conclusion

In this chapter we have shown our calculations for both the lattice constant and the bending rigidity of a single sheet of h-BN. All quantities behave similarly to those for graphene, although there are differences: the B-N distance for h-BN has a higher gradient with respect to the temperature (fig. 4.4), the in-plane lattice parameter does not become larger than at T = 0 up to melting (fig. 4.5) and the bending rigidity κ is consequently lower (fig. 4.9). In the last section we saw that the choice of the barostat does not influence the outcome. 42 simulation of hexagonal boron-nitride

1.4590 0.0005

0.0004 1.4588

0.0003

(Å) 1.4586 xy

R 0.0002

NH, τ = 1.0 ps τ 1.4584 p NH, p = 1.0 ps B, τ = 2.0 ps Standard deviation (Å) 0.0001 τ p B, p = 2.0 ps B, τ = 10.0 ps τ p B, p = 10.0 ps 1.4582 0.0000 0.0*106 0.5*106 1.0*106 1.5*106 2.0*106 0.0*106 0.5*106 1.0*106 1.5*106 2.0*106 Number of steps Number of steps

(a) (b)

Figure 4.10: The in plane nearest neighbour distance a) and its standard deviation b) for two different barostats, all at T = 300K and with an applied external pressure of P = 0GPa. We give the values for three different pressure damping factor τp, namely τp = 1.0 ps for Nosé-Hoover and for Berendsen τt = 2.0 ps or τp = 10.0 ps. For all three, the temperature damping factor is τt = 0.1ps

104 103 102 101 100 H(q)/N 10−1 10−2 Berendsen Nose−Hoover 10−3 10−4 0.01 0.1 1 10 q (Å−1)

Figure 4.11: H(q)/N for T = 300K for both the Nosé-Hoover and the Berendsen barostat. Differ- ence between the two simulations are negligible. The two ﬁtted lines (almost) completely overlap. The resulting κ is thus almost the same for Nosé-Hoover (κ = 0.801 eV) and for Berendsen (κ = 0.806 eV). DEFECTS 5

Figure 5.1: Melting of a single sheet of h-BN. Periodic boundary conditions are used. More details about the melting process are given in appendixA. 4.

Classical MD simulations are very suitable to search for possible distortions of the lattice. By raising the temperature we have observed the formation of some defects that arise in the melting process which occurs spontaneously at T ∼ 4000 K (see for instance ﬁgure 5.1 and appendixA. 4). In this chapter we will study the energetics of these defects and of others defects that are known to occur in graphene and in semiconductors.

5.1 point defects and vacancies

We distinguish two kinds of point defects, those where the number of atoms remains the same and those where one or more atoms are removed (vacancies). For the ﬁrst group of defects we can calculate the formation energy as:

EF = Edefect − Eperfect,(5.1) where Edefect and Eperfect are the total energies of the sample with and without the defect. The cohesive energy is Ecoh ≡ Eperfect/n = −6.4166 eV. For these calculations

43 44 defects

Figure 5.2: Stone-Wales defects in BN. A pair of atoms is rotated by 90 degrees to form two pentagons and two heptagons (a). Side view of the two buckled structures with similar formation energy (see text): b) SW1 . The bond lengths are 1) 1.47 Å, 2) 1.71 Å, 3) 1.75 Å and 4) 1.58 Å. c) SW2. The bond lengths 1 and 3 change slightly with respect to SW1: 1) 1.45 Å and 3 1.72 Å.

we use a sample of n = 1296 atoms (sample B) which is cooled down to T = 0 K in the nVT ensemble. In these calculations all defects were artiﬁcially created, thus not always observed in high temperature simulations.

5.1.1 Stone-Wales defects

First of all we consider Stone-Wales (SW) defects[57] where a 90 degrees rotation of a pair of atoms transforms four hexagons into two pentagons and two heptagons (see figure 5.2a)). In graphene, SW defects are known to have the lowest formation energy ∼ 4.7 eV[58]. Although SW defects have been theoretically studied in both nanotubes and single layer h-BN[59,60], they have not been found experimentally[61]. Also in our melting simulations we have not observed their formation. Recently, it has been suggested[60] that SW defects in graphene are further stabilized by a sine-like or cosine-like buckle and that this finding should hold also for other hexagonal single-layer crystals. By starting with a flat layer with a SW defect, and raising the temperature to 10 K we find a spontaneous buckling to one of the two structures SW1 and SW2 shown in figure 5.2b) and figure 5.2c) respectively which resemble the sine-like buckle proposed [60] in . We found the cosine-like buckle to be unstable and deform to the structure SW2. The formation energies (table 5.1) of SW1 is slightly lower and both are almost twice the value of a SW in graphene[58,60]. 5.1 point defects and vacancies 45

Figure 5.3: Top and side view of a) the BB defect, b) the antisite defect and c) the tetrahedron defect. Selected values of the interatomic distances: 1) 1.53 Å 2) 1.90 Å 3) 1.60 Å 4) 1.47 Å 5) 1.59 Å 6) 1.61 Å 7) 1.41 Å. In all three cases the defects only cause a local distortion in the z-direction leaving the rest of the sample relativly ﬂat (see table 5.1).

Table 5.1: Formation energy EF of defects that do not change the number of atoms for which EF = Edefect − Eperfect. The height difference h along the z-axis between the highest and lowest points is also given. In brackets we give the standard deviation q z2 − hzi2 evaluated over the whole sample.

Defect EF (eV) h (Å)

SW1 8.6 2.26 (0.22)

SW2 8.8 2.04 (0.28) BB 4.4 1.12 (0.13) Tetra 4.3 1.43 (0.20) Anti-site 6.4 1.41 (0.10)

5.1.2 Other point defects

While raising the temperature close to melting, we observe first two defects that have no counterpart in graphene, and turn out to have the lowest formation energy. In figure 5.3a) we show the defect that we call BB-defect because it results from a broken BN bond. The B atom moves to a metastable state with a slightly longer BN bond length (1.53 Å instead of 1.47 Å) with the two neighbors and the formation of two loose B-B bonds with a large bond length (1.9 Å). No extended deviations in the z- direction occur for this defect (see table 5.1). The formation energy (4.4 eV) is only 0.1 eV higher than the tetrahedron defect (figure 5.3c)) where a N atom pops up and form a NB3 tetrahedron with the three B atoms in the plane. Also this defect does not lead to out of plane deformations in the surrounding. It has been suggested that a BN3 tetrahedron structure with one N atom on top could explain some features in core level spectroscopy[62,63]. We find that this structure is unstable and that the B atom abandons 46 defects

Figure 5.4: Top and side view of a) VBN b) V3B+N c) VB+3N where rebonding and ring formation is obtained by construction (see text and table5.2). In both a) and b) short N-N bonds are created leading to strong out of plane distortions. Selected values of interatomic distances: 1) 1.01 Å 2) 1.91 Å 3) 1.92 Å 4) 1.47 Å 5) 1.90 Å 6) 1.49 Å 7) 1.53 Å 8) 1.58 Å 9) 1.55 Å 10) 1.52 Å.

the layer if lifted from the plane. In ﬁgure 5.3b) we show also the antisite defect which is commonly found in semiconductors. As shown in table 5.1 its formation energy is intermediate between those of the BB and tetrahedron defects and those of the SW defects. In table 5.1 we also give the maximum out of plane displacement and the average value of out of plane displacements in the whole sample. We can see that, with the exception of the antisite defect, the formation energy seems to be related to the out of plane distortions, as suggested for grain boundaries in graphene[64].

5.1.3 Vacancies

Lastly, we consider the various vacancies occurring when one or more atoms are removed. We deﬁne the vacancy giving as subscript the removed atoms, e.g. VN as a system missing one N atom. Apart from the VBN vacancy that keeps the stoichiometry, for these cases the evaluation of the formation energy requires to know the chemical potential of bulk phases of the constituents[65]. Different formation energies can result from use of different reference systems. Moreover, the chemical potential is often ap- proximated by the cohesive energy, i.e. its value at T = 0[65]. For BN, the formation [66] energy of the VBN, V3B+N and VB+3N vacancies has been calculated ab-initio in by use of the chemical potentials of bulk metallic boron and solid nitrogen. As discussed in Ref.[67], this approach is not reliable for a phenomenological potential like the one we are using. Therefore in table 5.2 we just give the energy difference ∆E between the energy of the perfect sample (Eperfect = nEcoh) and the one of the sample relaxed after creation of the vacancy. We notice that VB and VN have the same ∆E because they both involve three BN broken bonds. Once the vacancy is created, no new bonds are formed. The three atoms surrounding the vacancy remain two-fold coordinated with the same bond angle. The bond length with the two nearest neighbours contracts from 1.46 Å to 1.44 Å. 5.2 energy barriers 47

Table 5.2: The number nv of atoms removed to create the vacanvy and the energy difference ∆E between the perfect sample and the sample with a defect for various vacancies. The quantity A = ∆E − nvEcoh is given for the minimal energy structure without rebonding (see text), while A0 is the same quantity for the defects after ring formation (see ﬁgure 5.4). For the latter defects we also give the height difference h along the z-axis between the highest and lowest points and in brackets the standard deviation [ z2 − hzi2]1/2 evaluated over the whole sample.

0 Defect nv ∆E (eV) A (eV) A (eV) h (Å)

VB 1 11.7 5.3

VN 1 11.7 5.3

VB+N 2 19.7 6.9 8.8 2.06 (0.35)

VB+3N 4 36.0 10.4 10.9 1.42 (0.31)

V3B+N 4 36.2 10.6 16.1 2.63 (0.22)

Since, the single vacancies VN and VB have the same ∆E, we have a qualitative indication of the formation energy by subtracting from ∆E the cohesive energy for each removed atom, irrespective of its nature, namely A ≡ ∆E − nvEcoh, where nv is the number of missing atoms. Also for VBN, V3B+N and VB+3N we find that no new bonds are formed after creation of the vacancy and that the structural changes are negligible. In table 5.2 we give the corresponding values of ∆E and A. To establish whether these are indeed the lowest energy structures, we have brought the atoms around the vacancy closer to each other inducing the formation of new bonds re-establishing three-fold coordination as shown in figure 5.4 for VBN, V3B+N and VB+3N. All these structures remain bonded after relaxation but have higher energy than the ones with no rebonding. This is due to the strong N-N bonds of only ≈ 1.01 Å in VBN and V3B+N that cause strong local out of plane distortions and hinder the ring formation. Indeed, the ring structure of the V3B+N vacancy shown in figure 5.4b) is obtained by construction and does not occur spontaneously whereas the VB+3N, with no N-N bonds, may form spontaneously at finite temperature since it has an energy only marginally higher than the one without rebonding.

5.2 energy barriers

To obtain more information about the formation energy of the BB and tetrahedron defects we applied the nudged elastic band (NEB) method as described in section 3.2. To reiterate: we have M replicas of the system, where replica 1 is the sample without a defect and replica M = 16 is the system with a defect. The force in each replica is changed to include a inter-replica force with a spring constant of k = 10 eV Å−2.2 After minimising the force a reaction coordinate and a energy barrier are obtained. We performed NEB calculations for both the BB and tetrahedron defects. The resulting energy proﬁle is shown in ﬁgure 5.5. We see that the barrier for the BB defect is

2 These are in lammps reduced (metal) units. 48 defects

6 2.9 eV

5 BB

Formation Energy (eV) 3

2 Tetra

0 2 4 6 8 10 12 14 16 M

Figure 5.5: The energy profile calculated using the nudged elastic band method for the tetrahedron and the BB defect. M = 1 represent the perfect sample and M = 16 is the final sample with the defect. An illustration of the corresponding reaction coordinates can be found in figures 5.6 and 5.7. The formation energy is defined to be EF = Edefect − Eperfect. The maximum for the BB defect is 8.0 eV and for the tetrahedron defect is 5.1 eV, giving an energy difference between the two of 2.9 eV.

2.9 eV higher than that of the tetrahedron defect. Illustrations of the corresponding reaction paths can be found in ﬁgure 5.6 and 5.7. We see that the reaction coordinate is relatively easy to identify: in both cases there is one atom that moves while the other atoms adjust their positions only slightly. 5.2 energy barriers 49

(a) M = 1 (b) M = 1

(e) M = 11 (f) M = 11

(g) M = 13 (h) M = 13

(i) M = 16 (j) M = 16

Figure 5.6: The reaction path for the tetrahedron defect with top (left) and side (right) view. A N atom (depicted in black) simply moves in the z-direction from the stable conﬁgu- ration (M = 1) too the (meta)stable conﬁguration (M = 16), while its neighbouring B atoms start to form bonds. 50 defects

(a) M = 1 (b) M = 1

(e) M = 10 (f) M = 10

(g) M = 16 (h) M = 16

Figure 5.7: Top (left) and side (right) view of the ﬁnal states of the various replicas for the BB defect. The reaction coordinate that can be distinguished is the simple path of the B atom (depicted in black) from the ﬁrst replica (M = 1) till the last (M = 16). SUMMARYANDCONCLUSION 6

The goal of this thesis was to study the structure and stability of a single sheet of hexagonal boron nitride by means of MD simulations and an interatomic potential developed by Albe, Möller, and Heinig[8]. In chapter 2 we presented to the reader the necessary theoretical background of performing molecular dynamics in various ensembles. The goal was to find which method to obtain a particular temperature and zero pressure was the best for our two-dimensional system. In the end we favoured the method developed by Berendsen et al.[22] over the Nosé-Hoover method. In future work the Berendsen method should be replaced by the one developed by Bussi et al.[32], because the one by Bussi is able to obtain the correct phase-space distribution for the various ensembles.1 In the chapters 4 and 5 we have presented an overview of our research on a single layer of h-BN and compared our findings to those for graphene. In chapter 4 we presented our results for various quantities: we find that the non-monotonic behaviour of the lattice parameter, the expansion of the interatomic distance and the growth of the bending rigidity with temperature are qualitatively similar to those of graphene. The energetics of point defects is extremely different in graphene and h-BN. In chapter 5 we show that Stone-Wales defects have formation energy twice as large as the two lowest energy defects in h-BN which involve either a broken bond or an out of plane displacement of a N atom to form a tetrahedron with three B atoms in the plane. For the last two defects we obtained a energy profile using nudged elastic band calculations. In future work one could try to obtain a free energy barrier using the reaction coordinates we specified. We tried doing this using steered molecular dynamics, however we found this method not suitable for the study of defects and therefore a different method such as umbrella sampling should be used instead. In summary, we have presented an extended study of the structural properties of single layer h-BN. Although the interatomic potential gives rather good results, some values (such as the lattice constant) differ quite a bit from the experimental value. In future work one could try to improve this potential or create a new one based on one of the many potentials that exist for carbon (see for instance:[4,5,6,7]). Validation of results given by this potential can lead to the next step: to combine the potential for h-BN with one of the potentials for graphene. For this a potential to describe the interlayer interaction between carbon and BN is also needed. For graphite systems such a potential for interlayer interactions was developed by Kolmogorov and Crespi[68]. Some test simulations performed by us prove that it is possible to combine the different potentials, however, for such hybrid structures the coefficients for the Kolmogorov- Crespi potential need to be re-calculated. In conclusion, we have shown that there still are a lot of open problems to solve and we therefore hope that our results will stimulate further research on this topic.

1 Code for the Bussi thermostat to be used with lammps can be found on the TCM source code repository.

APPENDICES A a.1 character of the verlet algorithm

The equations of motion for the verlocity Verlet algorithm are:

F(x(0)) x(∆t) = x(0) + ∆tv(0) + ∆t2 2m (A.1) 1 v(∆t) = v(0) + ∆t [F(x(0)) + F(x(∆t))] 2m In chapter 2 we showed how these equations are obtained from the Hamilton equations. To once more show that the Verlet algorithm is symplectic, we want to show that the symplectic property:

M = JT MJ (A.2) holds. We have:

!  ∂x(∆t) ∂x(∆t)  0 1 M = J = ∂x(0) ∂v(0) ,  ∂v(∆t) ∂v(∆t)  . (A.3) −1 0 ∂x(0) ∂v(0)

Now we calculateA. 2:

 ∂x(∆t) ∂v(∆t)  !  ∂x(∆t) ∂x(∆t)  0 1 JT MJ = ∂x(0) ∂x(0) ∂x(0) ∂v(0)  ∂x(∆t) ∂v(∆t)   ∂v(∆t) ∂v(∆t)  ∂v(0) ∂v(0) −1 0 ∂x(0) ∂v(0)  ∂x(∆t) ∂v(∆t)   ∂v(∆t) ∂v(∆t)  = ∂x(0) ∂x(0) ∂x(0) ∂v(0)  ∂x(∆t) ∂v(∆t)   ∂x(∆t) ∂x(∆t)  (A.4) ∂v(0) ∂v(0) − ∂x(0) − ∂v(0)   0 ∂x(∆t) ∂v(∆t) − ∂x(∆t) ∂v(∆t) = ∂x(0) ∂v(0) ∂v(0) ∂x(0)  ∂x(∆t) ∂v(∆t) ∂x(∆t) ∂v(∆t)  ∂v(0) ∂x(0) − ∂x(0) ∂v(0) 0

Let us calculate the terms of the matrix J for the Verlet algorithm:

∂x(∆t) ∆t2 ∂F(x(0)) = 1 + ∂x(0) 2m ∂x(0) ∂x(∆t) = ∆t ∂v(0) (A.5) ∂v(∆t) ∆t ∂F(x(0)) ∂F(x(t)) = + ∂x(0) 2m ∂x(0) ∂x(0) ∂v(∆t) ∆t ∂F(x(0)) ∂F(x(t)) = 1 + + ∂v(0) 2m ∂v(0) ∂v(0)

53 54 appendices

∂2F Using the fact that F(x) = m ∂x2 and seeing that the derivatives with respect to x(0) and v(0) are then zero, we get:

! 0 1 JT MJ = = M (A.6) −1 0

a.2 coefficients for the bn tersoff potential

As mentioned in the text there is a discrapancy between Albe et al.[8] and lammps in the deﬁnition of the coefﬁcients for the Tersoff potential. Left lammps and right the ones by Albe et. al.

1 E = V 1 2 ij E = Vij i6=j 2 X i6=j  X  Vij = fC(rij) fR(rij) + bijfA(rij) V = f (r ) f (r ) + b f (r )  ij C ij R ij ij A ij  D √  0 −β 2S(r−r0) −λ1r  fR(r) = e fR(r) = Ae  S − 1  √ −λ2r  SD fA(r) = −Be  0 −β 2/S(r−r0)  fA(r) = e  S − 1  1, r < R − D   1, r < R − D π(r−R)  fC(r) =  1 1  2 − 2 sin 2D , R − D < r < R + D   1 1 π(r−R)   fC(r) = − sin , R − D < r < R + D    2 2 2D 0, r > R + D  0, r > R + D  1   n n − 2n  bij = (1 + β ζij)   n n − 1  b = (1 + γ χ ) 2n m m  ij  ij λ (rij−rik)  ζij = fc(rik)g(θijk)e 3  3 3  λ (rij−rik)  χij = fc(rik)g(θijk)e 3 k6=i,j  X  k6=i,j 2 2  X c c  2 2 g(θ) = γijk 1 + −  c c d2 (d2 + (cos(θ) − cosθ )2)  g(θ) = 1 + − 0  2 2 2  d d + (h − cosθijk)   (A.7)

The differences are subtle. To calculate the coefﬁcients this is what we have done (the resulting potential ﬁle can be found in appendixA. 3): A.3 lammps files 55

Coefﬁcients lammps Coefﬁcients Albe et al.[8] m 3 (from formula) γ 1 (from formula)

λ3 λ3 c c d d

cos(θ0) h n n

β γalbe λ β p2/S 2 albe √ B SD0 eβalbe 2/Sr0 S−1 √ λ1 βalbe 2S √ D0 βalbe 2Sr0 A S−1 e

Table A.1: Coefﬁcients for the Tersoff potential for BN equivalent to the ones deﬁned by Albe et al. [8] a.3 lammps files

In this section we present two files that can be used to perform simulations in lammps using the Albe et al.[8] potential. In listingA. 1 one finds the potential file for lammps. In this file the coefficients for the BN potential are given. One can use this file by using the commands: pair_style tersoff pair_coeff ** ./potentials/BN_albe.tersoff B N where ’./potentials/BN_albe.tersoff’ is the name of the file. ¥ In listingA. 2 one finds a typical input file for lammps, for performing simulations in the NPT ensemble using the potential for BN. 56 appendices

Listing A.1: The potential ﬁle for BN used for lammps

# B and N mixture, parameterized for Tersoff potential # values are from Albe et.al - Rad. Eff. Def. Sol. 141 (1997) 85.

#values are calculated from Albe et. al from # m = 3 # gamma = 1 # c = c, d=d # cos(thetat0 = h) # n = n # beta = gamma_albe # lambda1=beta_albe*sqrt(2*S_albe) # lambda2=beta_albe*sqrt(2/S_albe) # lambda3=lambda3 # A=D0/(S-1)*exp(lambda1*r0) # B=S*D0/(S-1)*exp(lambda2*r0) # R = R, D=D

# Tersoff parameters for various elements and mixtures # multiple entries can be added to this file, LAMMPS reads the ones it needs # these entries are in LAMMPS "metal" units: # A,B = eV; lambda1,lambda2,lambda3 = 1/Angstroms; R,D = Angstroms # other quantities are unitless

# format of a single entry (one or more lines): # element 1, element 2, element 3, # m, gamma, lambda3, c, d, costheta0, n, # beta, lambda2, B, R, D, lambda1, A

B B B 3 1 0 0.52629 0.001587 0.5 3.9929061 0.0000016 2.07750 1173.197 2 0.1 2.23726 1404.47520

N N N 3 1 0 17.7959 5.9484 0 0.6184432 0.019251 2.627272 2563.560 2 0.1 2.82931 2978.95279

N N B 3 1 1.9925 1092.9287 12.38 -0.5413 0.364153367 0.000011134 0 0 2 0.1 0 0

N B B 3 1 1.9925 1092.9287 12.38 -0.5413 0.364153367 0.000011134 2.784247 3613.431 2 0.1 2.99836 4460.83397

B N N 3 1 1.9925 1092.9287 12.38 -0.5413 0.364153367 0.000011134 2.784247 3613.431 2 0.1 2.99836 4460.83397

B N B 3 1 0 0.52629 6.28433 0.5 3.9929061 0.0000016 0 0 2 0.1 0 0

B B N 3 1 1.9925 1092.9287 12.38 -0.5413 0.364153367 0.000011134 0 0 2 0.1 0 0

N B N 3 1 0 17.7959 5.9484 0 0.6184432 0.019251 0 0 2 0.1 0 0

¥ A.3 lammps files 57

Listing A.2: A typical example of a lammps input ﬁle

#Setting a log file for this simulation log ./log.txt

#setting the units to metal instead of SI (or others) units metal #setting boundary conditions (x,y,z) to periodic in x and y, and fixed in z boundary p p f

#setting the atom_style: #atomic: standard without charge atom_style atomic

#reading data from an input file containing xyz coordinates of atoms, masses etc read_data bn_sheet_n_1296.lammps

#turning Newton’s 3rd law on or off for pairwise and bonded interactions #needed for pair_style tersoff newton on

#setting the pair style (the interatomic potential to use) pair_style tersoff #setting the pair coefficients specific for this material pair_coeff ** ../../potentials/BN_albe.tersoff B N

#This command sets parameters that affect the building of pairwise neighbor lists neighbor 0.2 bin #setting the delay for re-checking the neighbour list neigh_modify every 20 delay 100 check yes

#velocity: setting the initial velocity to a particular temperature velocity all create 300.0 1234567890 dist gaussian

#setting the time step of the simulation (in fs) timestep 0.0001

#fix nve is needed to update positions and velocities fix 1 all nve #fix temp/berendsen sets the temperature fix 2 all temp/berendsen 300 300 0.01 #fix press/berendsen sets the pressure fix 3 all press/berendsen x 0 0 10 y 0 0 10 couple xy

#defining the output to the screen (every 100 time steps) thermo 100 thermo_style custom step temp pe ke etotal vol press enthalpy pxx pyy pzz

#running the simulation run 100000

¥ 58 appendices

a.4 melting process

In chapter 5 we already briefly discussed the melting of a single sheet of h-BN. In this section we will illustrate this melting process by two images. Both images are the result of simulations in the NPT ensemble with a temperature of T = 4500 K and pressure P = 0. The initial sheet of h-BN consists of N = 9800 atoms and periodic boundary conditions are used. The melting consists of various stages. The first one is the forming of small defects, such as the ones highlighted in figureA. 1. These defects can lead to the second stage of the melting, where the hexagonal crystal around the defect is destroyed. Most B atoms and a few N atoms will create 3d structures around, what are now, edges of the crystal. In figureA. 2 we see that these 3d structures become larger to form big clusters of atoms. The 2d crystal is now almost completely destroyed while many N atoms form N2.

Figure A.1: Snapshot of the melting of h-BN in various stages. Highlighted are some small defects that form the ﬁrst stage of melting, while other parts of the 2d crystal are already destroyed. A.4 melting process 59

Figure A.2: A snapshot of the melting of h-BN in a later stage. Seen is that the initial sheet of BN is almost completely destroyed and big clusters of mostly B atoms are created, while many N atoms form N2.

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