Molecular Dynamics Studies of Low-Energy Atom Impact Phenomena on Metal Surfaces during Crystal Growth

Link¨oping Studies in Science and Technology

Dissertation No. 1028

Molecular Dynamics Studies of Low-Energy Atom

Impact Phenomena on Metal Surfaces during

Crystal Growth

Dragan Adamovi´c

Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden

ISBN 91-85523-56-9 ISSN 0345-7524

Abstract

It is a well-known fact in the materials science community that the use of low-energy atom impacts during thin film deposition is an effective tool for altering the growth behavior and for increasing the crystallinity of the films. However, the manner in which the incident atoms affect the growth kinetics and surface morphology is quite complicated and still not fully understood. This provides a strong incentive for further investigations of the interaction among incident atoms and surface atoms on the atomic scale. These impact-induced energetic events are non-equilibrium, transient processes which complete in picoseconds. The only ac-cessible technique today which permits direct observation of these events is molec-ular dynamics (MD) simulations.

This thesis deals with MD simulations of low-energy atom impact phenomena on metal surfaces during crystal growth. Platinum is chosen as a model system given that it has seen extended use as a model surface over the past few decades, both in experiments and simulations. In MD, the classical equations of motion are solved numerically for a set of interacting atoms. The atomic interactions are cal-culated using the embedded atom method (EAM). The EAM is a semi-empirical, pair-functional interatomic potential based on density functional theory. This po-tential provides a physical picture that includes many-atom effects while retaining computational efficiency needed for larger systems.

Single adatoms residing on a surface constitute the smallest possible clusters and are the fundamental components controlling nucleation kinetics. Small two-dimensional clusters on a surface are the result of nucleation and are present during the early stages of growth. These surface structures are chosen as targets in the simulations (papers I and II) to provide further knowledge of the atomistic processes which occur during deposition, to investigate at which impact energies the different kinetic pathways open up, and how they may affect growth behav-ior. Some of the events observed are adatom scattering, dimer formation, cluster disruption, formation of three-dimensional clusters, and residual vacancy forma-tion. Given the knowledge obtained, papers III and IV deal with growth of several layers with the aim to study the underlying mechanisms responsible for altering

vi

growth behavior and how the overall intra- and interlayer atomic migration can be controlled by low-energy atom impacts.

PREFACE

This thesis is a summary of the work I have carried out in the Theoretical Physics Group and the Thin Film Physics Group at Link¨opings University between June 2000 and June 2006. The thesis is divided in two parts. The first part is intended to give an introduction to the research field dealt with. The second part includes the scientific papers.

Since I started my PhD studies, many people here at the physics department have contributed to the progress in me gaining deeper knowledge in physics in general and to the scientific work I have carried out. First and foremost, I would like to express my deep gratitude to my supervisor Docent Peter M¨unger and co-adviser Docent Valeriu Chirita for their guidance and support over the years. We have had many discussions regarding the simulation results and how to write the scientific papers. It has truly been a learning process. A special thanks goes to Lars Hultman and Joe Greene in providing additional guidance how to become a better writer.

I would like to thank the senior staff of Theoretical and Computational Physics groups who gladly shared their knowledge. To my fellow, former and present, PhD students, diploma workers, and scholars, thank you for putting up with me and providing support when needed. I would also like to thank the people in the Thin Film group for inviting me to your yearly trips and for the discussions we have had regarding experiments vs. computer simulations. Thank you Ingeg¨ard Andersson for all the help with administrative issues.

I acknowledge financial support from the Swedish Research Council (VR), and the Swedish Foundation for Strategic Research (SSF) Strategic Research Center on Materials Science and Nanoscale Surface Engineering.

I would also like to thank my friends outside the university for being there and for all the things we have done together. We have lived the life, no doubt!

Dragan Adamovi´c Link¨opings Universitet, 2006

CONTENTS

1 Introduction. 1

2 Crystal growth. 3

2.1 Film growth from the vapor phase. . . 3

2.2 Growth modes. . . 4

2.3 Film microstructure. . . 5

3 Low-energy atom impacts during crystal growth. 7 3.1 The basics. . . 7

3.2 Surface and sub-surface effects. . . 8

3.3 Bombardment and substrate temperature. . . 9

3.4 Bombardment and growth modes. . . 9

3.5 Aims of the thesis. . . 10

4 Molecular dynamics. 13 5 The embedded atom method. 17 5.1 Fundamentals. . . 18 5.2 Phenomenology. . . 19 5.3 Analytic forms. . . 20 5.4 Applications. . . 22 5.5 Other methods. . . 23 6 Simulation experiments. 25 6.1 Substrate specifics. . . 25 6.2 Paper I. . . 26 6.3 Paper II. . . 26 6.4 Paper III. . . 27 6.5 Paper IV. . . 28 ix

x Contents

Paper I 31

Paper II 37

Paper III 59

CHAPTER

1

Introduction.

Before the invention of computers, more than half a century ago, the progress in science was determined by the interplay of experiment and theory. An experiment is characterized by the set-up of a physical system which is then subjected to mea-surements, and results are obtained in numeric form. A theory is a representation of the physical system, usually in the form of mathematical equations. The model is then authenticated by its ability to reproduce the system behavior for specifi-cally chosen circumstances. In many cases, approximations are required to carry out the calculations.

The advent of ’high-speed’ computers in 1960’s altered the conventional pic-ture of progress in science by introducing a new element in the interplay of theory and experiment, the computer experiment. In a computer experiment, or a com-puter simulation, the model representing a system is provided by theorists and the calculations are carried out by the computers. This often requires the knowledge in transforming the mathematical equations into a form that the computers can handle.

By utilizing simulations, complexity can be introduced and more realistic sys-tems can be studied. This opens a path towards better understanding of theory and experiment. However, it is important to realize that simulations play a much more significant role than validating experiments and or theories. In many cases, simulations are the only alternative. Needless to say, the birth of computer simu-lations is not only a link between theory and experiment it is also a powerful tool to propel progress in new directions.

Today, simulations are indispensable in most research areas. Many branches within the scientific community conduct research in which the time evolution of a system on the atomic scale is of great interest. An extensively used technique for simulating the time evolution of a set of interacting atoms is molecular dynamics (MD). The method finds use, for example, in studying assembly phenomena dur-ing thin film growth.

2 Introduction.

The incentive for investigating growth on the atomic level by means of com-puter simulations is the need for increased fundamental knowledge, which can not be obtained by experiments or theories alone, in order to meet the ever more stringent requirements on the quality of thin films for developing advanced micro-electronics, optical and magnetic devices, and coatings with enhanced mechanical properties.

But even if the knowledge to manufacture sophisticated thin-film applications are accessible, they will only find practical use if satisfying the industrial demand for mass production at low cost. This, in turn, has led to the development of growth techniques away from near-equilibrium deposition toward kinetically limited pro-cesses. One of the approaches being explored includes the use of low-energy, high flux ion irradiation1 _{at low temperatures.}

Given that thermally activated atomistic processes are exponentially depressed at low temperatures, the focus turns to the events induced by single ion impacts. The latter are picosecond-events, and the only accessible technique today which permits direct observation of these non-equilibrium transient mechanisms is MD simulations.

In order to conduct computer experiments in which the effects of single atom impacts are studied, some 103_{− 10}6_{atoms and 10}3_{− 10}9_{time steps are required.}

Typically, several hundreds of simulations are carried out to provide statistics. For this purpose, semi-empirical, pair-functional models which are much less complex than first-principles calculations but incorporates the necessary many-body effects, and are not much more computationally demanding than simple pair potentials, are commonly used for describing the atomic interactions.

This thesis deals with molecular dynamics studies of low-energy atom impact phenomena on metal surfaces during crystal growth from the vapor phase using the embedded atom method (EAM). The chapters are organized as follows. In chapter 2, a brief summary of crystal growth from the vapor phase without bom-bardment is given. Chapter 3 is intended to illustrate the basics of low-energy ion irradiation during deposition and the aims of this study. In chapter 4, the computational method for studying the time evolution of a set of atoms is given. The model describing the atomic interactions is presented in chapter 5. In chapter 6 a summary of the included papers is given.

1_{The term ”ion irradiation” is used to refer to bombardment by fast neutrals as well as} accelerated ions.

CHAPTER

2

Crystal growth.

2.1

Film growth from the vapor phase.

Before discussing the effects of ion bombardment during deposition it is instructive to go over the basics of film growth from the vapor phase without ion irradiation [1-3]. The thermally activated processes involved in the film evolution are governed by characteristic times determined by activation energies, attempt frequency factors, and the substrate temperature. The time scale for thermal events range from picoseconds to miliseconds.

In the initial phase of film formation, two main processes take place, arrival of atoms at the substrate and the migration of atoms (adatoms) on the substrate surface. This is illustrated schematically in Fig. 2.1. The adatoms migrate over the surface via diffusive random hops before meeting other adatoms with the possibility of creating a small cluster, a nucleus. A nucleus may capture additional atoms and in that way increase its size or it may dissociate releasing migrating adatoms. The stability of a cluster increases with increasing size. In nucleation theory, the critical island size is defined as the size at which for the first time the island becomes more stable with the addition of just one more atom. This typically occurs with 3 or 4 atoms, depending on the surface orientation.

As the deposition process continues, clusters grow larger by capturing mi-grating adatoms and subcritical clusters. Larger clusters may also grow at the expense of neighboring smaller clusters by ripening. Eventually, clusters join, ei-ther by growth or migration. This coalescence continues and forms a network of connected clusters which is followed by the filling of the remaining voids resulting in a film with full coverage.

4 Crystal growth. a) b) c) d) e) f) g) h)

Figure 2.1. Typical atomistic processes occurring on a surface during physical vapor deposition. (a) adatom migration, (b) nucleation, (c) cluster migration, (d) cluster dis-sociation, (e) adatom detachment, (f) adatom attachment, (g) cluster coalescence, (h) adatom descend from higher level terrace.

2.2

Growth modes.

There are three primary modes for crystal growth on surfaces [3], as illustrated schematically in Fig. 2.2. In the island, or Volmer-Weber mode, three-dimensional (3-D) clusters are formed on the substrate surface. This often occurs when the bonding between the deposited atoms is stronger than to the substrate. Two-dimensional (2-D) layer-by-layer, or Frank-van der Merwe mode, displays the op-posite characteristics. The bonding between the deop-posited atoms is now equal to or less than that to the substrate atoms. The third growth mode, Stranski-Krastanov, is a combination of the first two in which the growth proceeds first by deposition of one or more monolayers (ML) and is then followed by the formation of 3-D islands. In the case of homoepitaxy (self-growth) one would expect layer-by-layer growth due to the bonding nature between the substrate and deposited atoms. However, the prevailing growth mode is strongly correlated to the substrate temperature and deposition rate. For layer-by-layer mode to occur, deposited atoms on top of 2-D islands must be given the time to descend on to the lower level terrace. If they can not leave the higher level terrace in time they will aggregate with other

2.3 Film microstructure. 5

deposited adatoms on the same level and form clusters. This, in turn, may initiate 3-D growth. 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 000000000000 000000000000 111111111111 111111111111 000000000000 000000000000 111111111111 111111111111 000000000000 000000000000 111111111111 111111111111 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 a) b) c) Deposition

Figure 2.2. Schematic representation of the three primary crystal growth modes. (a) island or Volmer-Weber, (b) layer-by-layer or Frank-van der Merwe, (c) layer-island or Stranski-Krastanov.

2.3

Film microstructure.

When the films grow thicker they develop a microstructure. There are four differ-ent types of microstructures [4], each as a result of the degree of surface and bulk diffusion, compiled in a structure zone model. The structure zone model demon-strates how film growth proceeds as a function of the substrate temperature. At low temperatures the film consists of fibrous grains with a high density of defects. As the temperature is increased, the microstructure of the film becomes rather inhomogeneous with V-shaped grains. Next, the grains grow columnar rather than fibrous. At very high temperatures the crystal growth is characterized by equiaxed grains. In common for all cases is that the density is quite low with large microporosity which exerts a strong negative influence on the thin-film functional properties.

6

References.

[1] K. L. Chopra, Thin Film Phenomena, McGraw-Hill Book Company, Inc. (1969).

[2] Z. Zhang and M. G. Lagally, Science 276 (1997) 377.

[3] J. A. Venables, G. D. T. Spiller, and M. Hanb¨ucken, Rep. Prog. Phys. 47 (1984) 399.

CHAPTER

3

Low-energy atom impacts during crystal growth.

3.1

The basics.

The impingement of energetic atoms or ions on solid surfaces is known to produce a wide variety of effects [1-4]. Many of these effects are beneficial for thin-film deposition while some are detrimental and can be avoided by keeping the energy low.

The actual energy transfer in the ion/surface interactions depend on the inci-dent ion and target atom mass. Considering a head-on binary elastic collision of an incident ion of mass mi and kinetic energy Ei, and a target atom with mass

mtinitially at rest, the transferred energy Tmis given by

Tm= 4mimt

(mi+ mt)2

Ei= γEi. (3.1)

The energy transfer efficiency factor γ is the main feature of ion bombardment as it gives incident ions the ability to displace target (film) atoms. This individual fact is responsible for altering the nucleation and coalescence kinetics during growth.

In the case of mi ∼ mt, Ei of only a few eV is sufficient to move a target

atom from one surface site to another. Incident ions with a few tens of eV can, in addition of activating surface processes, recoil into the sub-surface layers or dislodge surface atoms into the vapor phase. Thus, it appears that there is an energy regime, ranging from a few eV to several tens of eV, for which incident ions can cause rearrangement of the surface but not of the bulk. This is usually referred to as the ’low-energy’ range.

Note that if the mass of the incident ions is greater or smaller than the target atoms, the irradiation energy has to be raised in order to provide similar effects in terms of atomic displacements on the surface or in the sub-surface region compared to the case where mi ∼ mt.

8 Low-energy atom impacts during crystal growth.

3.2

Surface and sub-surface effects.

Some of the impact-induced events that can occur on the surface and in the sub-surface region during low-energetic deposition are rather intuitive and what follows is a brief summary of the basic processes and their effect on the morphological de-velopment. The processes below will be labeled ’wanted’ or ’unwanted’ which are to be interpreted in the context of layer-by-layer growth since this type of growth mode is higly desirable for crystal growers. For simplicity of the discussion, the incident species are considered to be of the same type as the substrate atoms (ho-moepitaxy), however, some remarks on how the identified processes may affect heteroepitaxial growth will be given.

On an atomically flat surface, an incident atom can either be deposited di-rectly on the surface or it can recoil into the substrate. The latter can result in two different reactions, both distinguished by the impact energy. Assuming that the incident energy is rather low, what follows is that a substrate atom is pushed up onto the surface. This particular type of process is usually labeled as an in-terlayer exchange event. In homoepitaxy, inin-terlayer exchange is if no significance. However, in heteroepitaxy, it is usually of interest to keep the mixing probability between the two materials as low as possible in order to keep the interface sharp-ness high.

If the impact energy is raised, an interlayer exchange process can again occur, however, with two (or more) substrate atoms being pushed up onto the surface in-stead of only one, leaving one (or more) surface vacancies behind. These processes are usually identified as adatom/vacancy pair formation events. In heteropitaxy the vacancies can act as sinkholes for the other type of deposited material which results in intermixing. In homoepitaxy, adatom/vacancy pair formation can both have wanted and unwanted effects. Formation of additional adatoms can nucleate with other nearby adatoms. This is typically considered to be a wanted effect if it occurs on the lowest level terrace as it can contribute in keeping the island number density high. However, if adatom/vacancy formation occurs on a higher level terrace 3D growth can be accelerated which is an unwanted effect.

As the deposition process continues small islands will nucleate and some of them will be subjected to energetic impacts. Similar to bombardment of a flat surface, a number of events involving the island atoms can be initiated. There are basically three main types of events that can occur. At rather low irradiation energies, the incident atom will in most cases simply be deposited on top of the island without causing any rearrangement involving the island. This is usually an unwanted effect since it can lead to nucleation on top of the island surface with other deposited adatoms which, in turn, can initiate 3D growth.

If the irradiation energy is increased, the impacts can induce a push-out pro-cess. This simply means that one of the outer island atoms is pushed to a neigh-boring surface site by the incident atom which settles at the formerly occupied surface site. These types of events are wanted since they act to keep the surface topograpy flat which promotes layer-by-layer growth.

At even higher impact energies, scattering of outer island atoms can occur. A reduction in island size during film growth is known to have two effects, a reduced

3.3 Bombardment and substrate temperature. 9

probability for arriving adatoms to be deposited on top of islands, and an increased frequency, for already deposited adatoms atop islands, to reach island edges and subsequently descend onto the lower level terrace. Obviously, this promotes layer-by-layer growth and is a highly wanted feature.

3.3

Bombardment and substrate temperature.

Thermally activated atomic motion is present, in princinple, as long as the tem-perature is above 0 K, while bombardment-induced mass transport only depends on the impact energy (or energy transfer) and stops as soon as the excess ki-netic energy is dissipated through coupling into the surface. At low temperatures, where the thermally activated processes are exponentially depressed, the effects of ion-irradiation become increasingly significant. One could say that ion irradiation provides the means of having high temperature processes (locally) while keeping the substrate temperature low.

3.4

Bombardment and growth modes.

Low-energy (often < 100 eV) ion bombardment is an effective tool for altering the growth mode from 3-D, or island growth, to 2-D, or layer-by-layer growth, and is known to increase the epitaxial thickness of films deposited at low temperatures [1-12]. The proposed model(s) explaining the growth mode transition is that ion bombardment enhances inter- and intralayer mass transport during deposition, leading to changes in initial nucleation and coalescence of islands which affect the overall morphological development and microstructure evolution. For example, it has been suggested that incident ions are responsible for suppressing 3D island formation by removal of atoms from barely stable nuclei during the initial nucle-ation stages of growth. It has also been proposed that ion bombardment enhances the adatom number density via disruption of dimers which, in turn, results in higher island nucleation rates. Another explanation is that during bombardment, adatoms are created via displacement of surface atoms and that these form one or several adatom islands close to the impact site leading to an increased island number density.

The manner in which low-energetic incident atoms affect the growth kinetics and surface morphology is quite complicated and still not fully understood. This provides a strong incentive for further investigations of the interaction among inci-dent ions and surface atoms on the atomic scale. These impact-induced energetic events are non-equilibrium transient processes which complete in picoseconds, and as such, not accessible even to state-of-the-art in situ experimental techniques, which can only detail initial and final states. The only accessible techniques to-day which permit direct observation of these events are computer simulations, in particular molecular dynamics (MD) simulations [13-19].

10 Low-energy atom impacts during crystal growth.

3.5

Aims of the thesis.

The objective of this thesis is to employ MD simulations in order to provide further insight into low-energy atom impact phenomena which occur during growth and how they may affect growth behavior, specifically,

• To study the kinetic pathways as induced by low-energy atom impacts of single adatoms and small 2-D clusters.

• To elucidate how the impact-induced processes affect nucleation, coalescence, and morphological development during growth.

• To demonstrate how the overall intra- and interlayer mass transport can be controlled via low-energy single atom impacts during growth.

With a view to keep things simple, a metal is chosen as a model surface since metal bonds (in contrast to semiconductors or molecular materials) have essentially no directionality that can be used to direct atomic processes. To avoid further unnecessary complications, such as lattice mismatch and differences in surface energy, the incident atoms are chosen to be of the same type as the substrate atoms (homoepitaxy).

In the included papers, Pt(111) is chosen as a model surface due to extensive field ion microscopy [20-23] and scanning tunneling microscopy [24-29] data, and for the simplicity of the existing semi-empirical models describing the cohesion of late transition metals. All papers deal with low-energy (5 - 50 eV) normally incident Pt atom impacts on Pt(111).

The results and conclusions presented in the papers are expected to be vaild for most face-centered cubic (111) metallic planes as inferred by the strong similarities in the surface properties of these systems [29-32].

11

References.

[1] J. E. Greene, D. T. J. Hurle (Ed.), Handbook of Crystal Growth, Fundamentals, Part A, Vol. 1, North-Holland, Inc. (1993), p.639. [2] D. L. Smith, Thin-Film Deposition, Principles & Practice, MacGraw-Hill

Book Company, Inc. (1995).

[3] W. Ensinger, Nucl. Instr. and Meth. in Phys. Res. B 127 (1997) 796. [4] I. Petrov, P. B. Barna, L. Hultman, and J. E. Greene, J. Vac. Sci.

Technol. A 21 (2003) S117.

[5] N. D. Telling, M. D. Crapper, D. R. Lovett, S. J. Guilfoyle, C. C. Tang, M. Petty, Thin Solid Films 317 (1998) 278.

[6] S. J. Guilfoyle, R. J. Pollard, P. J. Grundy, J. Appl. Phys. 79 (1998) 4939.

[7] T. -Y. Lee, S. Kodambaka, J. G. Wen, R. D. Twesten, J. E. Greene, and I. Petrov, Appl. Phys. Lett. 84 (2004) 2796.

[8] L. Hultman, G. H˚akansson, U. Wahlstr¨om, J. -E. Sundgren, I. Petrov, F. Adibi, and J. E. Greene, Thin Solid Films 205 (1991) 153.

[9] N. -E. Lee, G. A. Tomasch, and J. E. Greene, Appl. Phys. Lett. 65 (1994) 3236

[10] M. V. Ramana, H. A. Atwater, A. J. Kellock, J. E. E. Baglin, Appl. Phys. Lett. 62 (1993) 2566.

[11] S. Tungasmita, J. Birch, P. O. ˚A. Persson, K. J¨arrendahl, and L. Hultman, Appl. Phys. Lett. 76, (2000) 170.

[12] J. Y. Tsao, E. Chason, K. Horn, D. Brice, and S. T. Picraux, Nucl. Instrum. Methods Phys. Res., Sect. B 39, (1989) 72.

[13] D. Adamovi´c, E. P. M¨unger, V. Chirita, L. Hultman, and J. E. Greene Appl. Phys. Lett. 86 (2005) 211915.

[14] M. Villarba, H. J´onsson, Surf. Sci. 324 (1995) 35.

[15] J. Jacobsen, B. H. Cooper, J. P. Sethna, Phys. Rev. B 58 (1998) 15847. [16] X. W. Zhou and H. N. G. Wadley, J. Appl. Phys. 84 (1998) 2301. [17] C. M. Gilmore and J. A. Sprague, Phys. Rev. B 44 (1991) 8950. [18] M. Koster and H. M. Urbassek, Surf. Sci. 496 (2002) 196.

[19] M. Kitabatake and J. E. Greene, Thin Solid Films 272 (1996) 271. [20] K. Kyuno and G. Ehrlich, Surf. Sci. 437 (1999) 29.

[21] K. Kyuno and G. Ehrlich, Phys. Rev. Lett. 84 (2000) 2658.

[22] K. Kyuno, A. G¨olzh¨auser, and G. Ehrlich, Surf. Sci. 397 (1998) 191. [23] P. J. Feibelman, J. S. Nelson, and G. L. Kellogg, Phys. Rev. B 49

(1994) 10548.

[24] S. Esch, M. Bott, T. Michely, and G. Comsa, Appl. Phys. Lett. 67 (1995) 3209.

[25] M. Kalff, M. Breeman, M. Morgenstern, T. Michely, and G. Comsa, Appl. Phys. Lett. 70 (1997) 182.

[26] T. Michely and C. Teichert, Phys. Rev. B 50 (1994) 11156. [27] M. Bott, T. Michely, and G. Comsa, Surf. Sci. 272 (1992) 161.

[28] T. Michely, M. Kalff, G. Comsa, M. Strobel, and K. -H. Heinig. Phys. Rev. Lett. 86 (2001) 2589.

12 Low-energy atom impacts during crystal growth.

[29] S. Esch, M. Breeman, M. Morgenstern, T. Michely, and G. Comsa, Surf. Sci. 365 (1996) 187.

[30] S. C. Wang and G. Ehrlich, Surf. Sci. 239 (1990) 301.

[31] J. Naumann, J. Osing, A. J. Quinn, and I. V. Shvets, Surf. Sci. 388 (1997) 212.

[32] W. Wulfhekel, N. N. Lipkin, J. Kliewer, G. Rosenfeld, L. C. Jorritsma, B. Poelsema, and G. Comsa, Surf. Sci. 348 (1996) 227.

CHAPTER

4

Molecular dynamics.

Molecular dynamics is a commonly employed technique for studying the motion of a set of interacting objects in space [1]. The objects can, in principle, be of any kind but are usually atoms or molecules. If the atoms are treated classically, the equations of motion are given by

Fi= miai, (4.1)

where Fi is the force acting on atom i with mass mi giving it an acceleration ai.

Integrating ai once and twice with respect to time yields the velocity and position

vectors vi and ri for atom i, respectively.

The exerted force Fi on atom i is usually caused by the interaction with the

remaining atoms in the system. If the interatomic energy potential Ep is known,

an explicit form of the force Fi can be obtained via

Fi= −∇iEp, (4.2)

where ∇iEp is the gradient of the potential Ep with respect to the position vector

ri.

Usually, the equations of motion (4.1) must be solved numerically. A standard method is the finite difference approach [2]. The basics are as follows. First, the time axis is discretized in time steps, where δt is the time step between consecutive points along the time axis. Then, given the atomic positions, velocities and accel-erations at time t, the same quantities can be obtained at a later time t + δt to a sufficient degree of accuracy. By repeating the procedure, the time evolution of the system can be followed. A finite difference scheme is known as a time integration algorithm and is the engine in molecular dynamics.

Finite difference methods are usually based on Taylor expansions truncated at some term. Hence, the accuracy of the algorithm is related to the truncation

14 Molecular dynamics.

r_i

v_i

a_i

t t_{+ δt} t t_{+ δt} t t+ δt t t+ δt

Figure 4.1. Successive steps of the velocity Verlet algorithm. The stored variables are in grey boxes.

errors. The truncation error can be reduced by decreasing the time step δt. Typ-ically, the time step is significantly smaller than the time it takes for the atom to move its own length.

What defines a successful algorithm can be summarized with the following at-tributes: it should be fast and require little memory, it should permit the use of a long time step δt, it should duplicate the classical trajectory as closely as possible, it should satisfy the known conservation laws for energy and momentum and be time-reversible, and it should be simple in form and easy to program.

The MD results presented in this thesis and the papers included, are based on the very popular velocity Verlet time integration algorithm [3] due to its numeri-cal stability and simplicity. The method uses 9N words of storage, where N is the number of atoms.

The procedure consists of three steps, as illustrated in Fig. 4.1, and begins with knowing all the positions, velocities, and accelerations at time t. The first step is to calculate the new position vectors ri(t + δt) by a truncated Taylor expansion

ri(t + δt) = ri(t) + vi(t)δt +1

2ai(t)δt

2_{, (4.3)}

and the velocities at mid-step by using

vi(t +

1

2δt) = vi(t) + 1

2ai(t)δt . (4.4) The mid-step velocities are not used to calculate any physical properties but are merely there to provide means for storing the accelerations aiat time t. Given the

positions riat time t + δt, the accelerations ai at time t + δt can be calculated via

ai= −

1

m∇iEp. (4.5) Finally, the velocities vi at mid-step are replaced with

vi(t + δt) = vi(t +

1 2δt) +

1

2ai(t + δt) , (4.6) which completes the algorithm. Note that it is never needed to simultaneously store the values of ri, vi, and ai at two different times. The time integration

procedure is usually repeated many times until some requirements are met which ends the molecular dynamics simulation.

15

References.

[1] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Inc. (1987).

[2] L. R˚ade and B. Westergren, Mathematics Handbook for Science and Engineering, Studentlitteratur, Inc. (1989).

[3] W. C. Swoope, H. C. Andersen, P. H. Berens, and K. R. Wilson, J. Chem. Phys. 76 (1982) 637.

CHAPTER

5

The embedded atom method.

The embedded atom method is a semi-empirical, pair-functional model for calcu-lating the cohesive energy of a metallic system. The success of the EAM since its formulation in 1983 by Daw and Baskes [1-2] is explained by the fact that it is much less complex than first-principles calculations but still manages to mimic the coordination-dependent metallic bonding, and is not significantly more com-putationally demanding than pair potential schemes.

In the embedded-atom treatment, each atom is viewed as being embedded in a host lattice consisting of all other atoms. The fundamental concept is that the embedding energy for a particular species of atom is a unique function of the host electron density, independent of the source of that electron density. In addition, there is an energy from two-body electrostatic interactions. The ansatz for the cohesive energy for a set of atoms is

Ecoh = X i Gi(ρi) + 1 2 X i,j,i6=j Uij(Rij) , (5.1) with ρi= X j(6=i) ρj(Rij) , (5.2)

where G is the embedding function, ρiis the electron density at atom i determined

by the densities from all other atoms j, U is the electrostatic two-body interaction, and Rij is the separation distance between atoms i and j. The embedding energy

is defined as the interaction of the atom with the background electron gas. The background gas at that atom is taken to be a linear superposition of spherically averaged tail densities from all other atoms.

18 The embedded atom method.

5.1

Fundamentals.

The EAM format (eq. 5.1) can be derived from the density functional theory (DFT). This approach has been covered in depth by Daw [3] and what follows is a short summary of the process.

The cohesive energy Ecoh of a solid in terms of the electron density ρ within

the DFT [4] is given by Ecoh= G[ρ]+ 1 2 X i,j,i6=j ZiZj Rij − X i Z _Z iρ(r) |r − Ri| dr+1 2 Z Z _ρ(r 1)ρ(r2) |r1− r2| dr1dr2−Eatoms, (5.3) where the sums over i and j are over the nuclei of the solid, Z is the charge of a nucleus, R is the position vector of a nucleus, and r is the position vector for the electron density. The first term in the above equation represents the kinetic, ex-change, and correlation energy functional of the electrons. The second, third, and fourth terms represent the repulsive nucleus-nucleus, attractive electron-nucleus, and Coulombic electron-electron contributions, respectively. The last term is the total energy of the atoms at an infinite distance from each other.

Equation 5.1 can be obtained from 5.3 by making following two assumptions. First, it is assumed that the functional G[ρ] is semilocal

G[ρ] = Z

g(ρ(r), ∇ρ(r), . . .))dr (5.4) which means that G can be divided into a sum of contributions from individual volumes. Second, the electron density of the solid ρs(r) is approximated by a

linear superposition of the densities of the individual atoms

ρs(r) =

X

i

ρai(r − Ri) . (5.5)

The first assumption is motivated by studies of the response function of the nearly uniform electron gas [4]. The second assumption is justified if the covalent bonding in the metal is negligible [5]. The latter implies that the EAM is expected only to be valid for simple metals and for early and late transition metals.

By rearranging eq. 5.3 and substituting ρ(r) with ρs(r) it can be written as

Ecoh= G[ X i ρai] − X i G[ρai] + 1 2 X i,j,i6=j Uija , (5.6)

where the first two terms represent the difference in kinetic, exchange, and corre-lation energies going from the case of isolated atoms to the solid. The last term is the electrostatic energy of the overlaping charge distributions.

By considering a region around atom i, it is reasonable to assume that ρa i

dom-inates and that the background density from the surrounding atoms in that region is relatively small and slowly varying. Therefore, it seems plausible to approximate the background density by a constant ρi.

5.2 Phenomenology. 19

The embedding energy for an atom in an electron gas of some constant density (neutralized by a positive background density) can thus be defined as

Gi(ρi) = G[ρai + ρi] − G[ρai] − G[ρi] . (5.7)

By using the above defined embedding energy, the cohesive energy from eq. 5.6 can be written as Ecoh= X i Gi(ρi) + 1 2 X ij,i6=j Uija + Eerr, (5.8)

where the error is a function of the constant background density. Setting the error to zero gives an equation for optimal ρi. In general, ρi is a functional of the true

background density.

The difference between eqs. 5.3 and 5.8 is that the functional G[ρ] for the solid has been reduced to evaluating the embedding function G(ρi) for each atom. This

becomes an advantage only if the constant background density is approximated by

ρi=

X

j(6=i)

ρaj(r − Rj) (5.9)

which, in turn, leads to that atomistic calculations become very straightforward. Note that since the embedding energy does not depend on the source of the background density, same embedding functions can be used to evaluate the energy for atoms in a homo-nuclear solid as well as in an alloy.

5.2

Phenomenology.

What makes the EAM so appealing is its physical picture of metallic bonding. The perspective where each atom is viewed as being embedded in a host electron gas created by its neighboring atoms provides the means for describing and un-derstanding the very important many-body interactions that are present in solids. In particular, it is simple to demonstrate how bonding is affected by coordination Z.

It has been suggested [6] that the cohesive energy should scale something like −√Z for metallic systems. Given the non-linearity for the cohesive energy and that the two-body electrostatic interactions are not coordination-dependent, the em-bedding function must be non-linear. This non-linearity reflects the bond strength saturation with increasing background density which, in turn, requires that the curvature of the embedding function must be positive.

A way of realizing the presence of many-body interactions within the EAM is to examine the change in cohesive energy when making small perturbations from a reference state. The basic approach is to replace G with a Taylor series expansion about the unperturbed host electron density ρ. By choosing a uniform homonu-clear solid as the reference state, the cohesive energy for small distortions can be

20 The embedded atom method. −1 0 1 2 3 4 5 6 7 8 9 −1 −0.8 −0.6 −0.4 −0.2 0 S ca le d p ot en ti al en er gy E

Scaled separation distance r

Figure 5.1. Potential energy as a function of the separation. Here, E(0) = −1, E0(0) = 0, and E00(0) = 1 written as Ecoh≈ X i [Gi(ρ) − G 0 i(ρ) · ρ] + 1 2 X i,j,i6=j [Uij+ 2G 0 i(ρ)· ρaj+ G 00 i(ρ) · (ρaj)2] . (5.10)

Thus, the effective two-body interactions in the above equation are environment dependent, in that the interaction between two atoms depends on the slopes and curvatures of their embedding functions which, in turn, depend on the background density for each atom. This clearly shows that, for example, atoms in the bulk will be treated differently from atoms at the surface. This is the strength of the EAM, and is directly connected to the non-linearity of the embedding function.

5.3

Analytic forms.

In order to use the EAM in practice, functional forms must be established for the embedding function G, the two-atom interaction U , and the atomic electron density ρ. The equations used in this thesis are derived by Johnson [8-9] from a nearest-neighbor approach for fcc metals both for homonuclear systems and binary alloys (Ni, Cu, Pd, Ag, Pt, Au). A brief summary of the procedure in obtaining the analytic forms is given below.

It has been shown that, for a broad range of materials, the potential (or atomic interaction) energy per atom Ep1in a crystal as a function of the nearest-neighbor

distance r, or any length parameter, is well approximated by [10]

Ep1= −Ee  1 + α r re− 1  e−α(rer−1) (5.11)

5.3 Analytic forms. 21 with α = 3 ΩeBe Ee  , (5.12)

where the subscript e is used to indicate evaluation at equilibrium, reis the

nearest-neighbor distance, Ωeis the atomic volume, Beis the bulk modulus, and Eeis the

cohesive energy. The functional behavior of eq. 5.11 is illustrated in Fig. 5.1. While the EAM does not require the atomic interactions to be limited to nearest-neighbors only, it provides a very simple scheme in obtaining useful ana-lytic forms while retaining the first-order effects. Thus, for a perfect crystal the two-body interaction energy per atom as a function of the separation distance r can be written as

Ua = 6U (r) , (5.13)

where U (r) represents the interaction between two atoms. Similarly, the back-ground electron density for each atom is then

ρa= 12ρ(r) , (5.14)

where ρ(r) is the tail density from one nearest-neighboring atom. By studying the sperically averaged free-atom charge densities calculated from Hartree-Fock theory for late transition metals [11-12], it can be seen that the density is well approximated by a single exponential term in the range of distances of interest in EAM calculations. Thus, the electronic density is taken as

ρ(r) = ρee−β(

r

re−1), r ≤ r

c. (5.15)

The two-body potential is taken as a Born-Mayer repulsion [13] in order to have the same analytic form as ρ(r), i.e.

U (r) = Uee−γ(

r

re−1), r ≤ r

c, (5.16)

where rc is a cutoff parameter and β and γ are model parameters.

The potential energy per atom in a perfect crystal from the EAM (nearest-neighbor approximation) is

Ep1= G(ρa) + Ua (5.17)

which in combination with eqs. 5.11, 5.15, and 5.16 leads to an expression for the embedding function G. Dropping the subscript a, the final form of G becomes

G(ρ) = −Ee  1 − α βln  ρ ρe  ρ ρe α/β − 6Ue  ρ ρe α/γ . (5.18)

As seen, the embedding function is non-linear and when the electron density goes to zero so does G. The model parameters β, γ, ρe, and Ue are determined

via a fitting procedure in which the lattice constant, cohesive energy, unrelaxed vacancy-formation energy, bulk modulus and Voigt-average shear modulus are used

22 The embedded atom method.

as input parameters to obtain the best fit [9]. The parameters given in [9] also yield a positive curvature of the embedding function.

The exact choice of the cutoff radius rc plays no significant role as long as

it is well within the gap between first and second neighbors. In this thesis, the Johnson potential is replaced by a cubic spline in the interval ranging from 1.04 to 1.5 of the nearest neighbor distance. The spline goes to zero at 1.5 of the nearest neighbor distance which is the interaction cutoff.

In the case of a binary alloy with atoms of type A and B, UAA _{and U}BB _are

given by their respective model, while the two-body interaction between type A and B is given by a weighted arithmetic mean

UAB(r) = UBA(r) = 1 2  ρA_(r) ρB_(r)U BB_{(r) +}ρB(r) ρA_(r)U AA_(r)_{. (5.19)}

The above equation also imposes a criterion on the weighting factors, in that the electron density for any of the alloy constituent cannot be zero at any distance for which the two-body potential is non-zero. Therefore, a cutoff distance for the two-body potentials must be shorter than the interaction range for the density functions.

Note that the analytic form is fitted to bulk solid properties and is thus ex-pected to work best for bulk calculations.

5.4

Applications.

The EAM has been applied to a large variety of problems related to bulk properties of late transition metals. A review of the various applications are summarized in [14] and show that calculations of, for example, bulk phonons, thermal expansion coefficients, melting points, and point defects are in quite good agreement with experimental values. In addition, metal surface properties such as surface energies and relaxations have also been examined. Generally, the surface energies are sys-tematically too low, though the ordering with respect to face is correct. The error in the absolute surface energy is due to neglecting the slope of the background electron density experienced by the surface atoms [3]. The interlayer spacings, calculated for relaxed surface geometries relative to the spacing in bulk geome-tries, show contraction between the top surface layer and the first sub-surface layer. Moreover, the rougher (110) surfaces demonstrate larger relaxations than the smoother (100) and (111) surfaces. Both of these general features agree with the trends found in experimental data.

The analytic EAM model derived by Johnson [9] have been tested (by our research group) by evaluating forces of displaced atoms in the bulk and energy barriers for surface diffusion. Comparisons with first-principles calculations and experimental values have been made. For the bulk test, a cell of 108 Pt atoms was used. The atoms were initially at equilibrium positions. One atom was then displaced from the equilibrium position in the [111] direction in increments of 0.3 ˚

A while keeping the remaining atoms fixed. The force acting on that atom was calculated by using both EAM of Johnson and DFT. The DFT calculations were

5.5 Other methods. 23 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 −40 −20 0 20 40 EAM VASP F or ce [e V /˚A ] Distance [˚A]

Figure 5.2. Comparison of the analytic EAM model of Johnson with DFT. The results show the force acting on one atom when displaced from its equilibrium position in the [111] direction in a cell of 108 Pt atoms.

performed with the plane-wave package VASP [15] using the projector augmented wave (PAW) method [16-17] and a generalized gradient approximation (GGA) [18]. The results are illustrated in Fig. 5.2 and demonstrate very good agreement between the two models.

Surface diffusion activation energies for adatoms and small clusters have been calculated with the Johnson parametrization [19-20] and show good agreement with first-principles calculations and experimental values. For example, the calcu-lated energy barrier for adatom diffusion with EAM yields a value of 0.22 eV and experiments demonstrate values of 0.25 eV and 0.26 eV.

5.5

Other methods.

Since the mid 1980’s many semi-empirical, pair-functional models have been de-veloped for the purpose of calculating the potential energy for various materials. For the interested reader, a good starting point is a review by S¸akir Erko¸c [21] which includes some 30 models for metals (Daw-Baskes(EAM), Finnis-Sinclair), semiconductors (Tersoff, Stillinger-Weber), and insulators (Brenner, Erko¸c).

24

References.

[1] M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50 (1983) 1285. [2] M. S. Daw and M. I. Baskes, Phys. Rev. B 29 (1994) 6443. [3] M. S. Daw, Phys. Rev. B 39 (1989) 7441.

[4] P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. [5] I. M. Torres, Interatomic Potentials, Academic Press, Inc. (1972). [6] A. Carlsson, H. Ehrenreich and D. Turnbull (Eds.), Solid State Physics,

Vol. 43, Academin Press, Inc. (1990). [7] S. M. Foiles, Phys. Rev. B 32 (1985) 3409. [8] R. A. Johnson, Phys. Rev. B 37 (1988) 3924. [9] R. A. Johnson, Phys. Rev. B 39 (1989) 12554.

[10] J. H. Rose, J. R. Smith, F. Guinea, and J. Ferrante, Phys. Rev. B 29 (1984) 2963.

[11] E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [12] A. D. McLean and R. S. McLean, At. Data Nucl. Data Tables 26

(1981) 197.

[13] M. Born and J. E. Mayer, Z. Phys. 75 (1932) 1.

[14] M. S. Daw, S. M. Foiles, and M. I. Baskes, Mat. Sci. Rep. 9 (1993) 251. [15] G. Kresse, and J. Furthm¨uller, Phys. Rev. B 54, (1996) 11169.

[16] P. E. Bl¨ochl, Phys. Rev. B 50, (1994) 17953.

[17] G. Kresse and D. Joubert, Phys. Rev. B 59, (1999) 1758.

[18] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys Rev. B 46, (1992) 6671.

[19] E. P. M¨unger, V. Chirita, J. E. Greene, and J. -E. Sundgren, Surf. Sci. 355 (1996) L325.

[20] E. P. M¨unger, V. Chirita, L. Hultman, and J. E. Greene, Surf. Sci. 539 (2003) L567.

CHAPTER

6

Simulation experiments.

The purpose of this chapter is to give a brief summary of the results in the papers included. In common for all papers is that they focus on the intra- and interlayer atomistic patways as induced by normally incident Pt atoms on Pt(111) with ki-netic energies E ranging from 5 to 50 eV.

The MD program used to perform all simulation experiments was developed and written by Peter M¨unger (my supervisor). The program has been used in previous MD studies with great success [1-11].

My contributions to the papers included are as follows. I performed the simu-lations. I conducted the analysis of the MD data. I compiled the results. I wrote the first draft of each manuscript and was involved in all iterations until the final version.

6.1

Substrate specifics.

The Pt(111) substrate used in the MD simulations consists of nine layers, 16x18 atoms each, as illustrated in Fig. 6.1. The atoms in the bottom layer are kept at fixed positions in order to prevent translation and or rotation of the substrate in the simulation box. The heat reservoir consists of two layers which are located directly above the fixed layer. The reservoir temperature is chosen to 1000 K. The number of time steps between rescaling of the atomic velocities in the heat bath is set to 10. The atoms in the remaining layers are allowed to interact freely according to the EAM potential with a time step ≤ 1 femtosecond. Periodic in-plane boundary conditions were employed along orthogonal [110] and [112].

26 Simulation experiments. [110] [111] [110] [112] [111] [132] Static layer Heat reservoir Dynamically relaxed

Figure 6.1. Pt(111) substrate used in the computer experiments.

6.2

Paper I.

This study is initiated by investigating the effects of low-energy Pt irradiation of single Pt adatoms on Pt(111). Single adatoms residing on the surface are chosen as targets in the bombardment process since they constitute the smallest possible clusters and are the fundamental components controlling the nucleation kinetics during growth. It is shown that the introduction of energetic species enhances adatom migration and that 10 ps are sufficient to fully accommodate for the atomic motion initiated the impacts with up to 50 eV. The ion-irradiation-induced mass transport is compared with thermal adatom migration at 1000 K. The underlying mechanisms responsible for the enhanced migration are isolated. Important results are:

• Target adatom and incident atom scattering is observed for all E. • Dimer formation is observed for all E.

• Incident atom and substrate atom exchange occurs at E ≥ 15 eV. • Residual vacancy formation occurs at E ≥ 40 eV.

• Adatom migration is enhanced up to a factor of ∼ 3.

• Total migration including the target adatom, the incident atom, and ejected substrate atoms, is enhanced up to a factor of ∼ 6.

6.3

Paper II.

In this paper, the focus is on low-energy Pt irradiation of 2-D compact Pt3, Pt7,

Pt19, and Pt37clusters on Pt(111). The range in size is chosen to represent clusters

6.4 Paper III. 27

the target clusters (hexagonal starting with Pt7) are chosen since they correspond

to the most energetically stable configurations on (111) planes.

The results are divided in three parts, intralayer effects, interlayer effects, and defect formation. In common for all cases is that a few ps (< 10 ps) are sufficient for the impact-induced processes to complete. The focus in the first part is on the intralayer effects where it is demonstrated that a rich variety of atomistic processes can be triggered during the bombardment process. Based upon visual inspection of the MD simulations, the overall results can be divided into three classes: cluster preservation, cluster reconfiguration, and cluster disruption. Important results are:

• As E increases the island preservation probability decreases for all island sizes.

• The overall preservation probability increases with island size.

• With E < 15 eV cluster reconfiguration is observed only to occur for Pt3

and Pt7.

• With E ≥ 20 eV cluster disruption starts to occur.

The second part deals with interlayer events, specifically, the formation of 3-D is-lands. The results demonstrate four different energy-dependent mechanisms lead-ing to the transition from 2-D to 3-D islands. Important results are:

• Negligible probability for 3-D island formation involving Pt3.

• Decreasing 3-D island formation probability with increasing E for Pt7.

• For Pt19and Pt37, the probability for 3-D island formation is ∼ 0.2 and 0.4,

respectively, irrespective of E.

The third and last part deals with residual point defect formation, i.e. adatom/vacancy pair formation events. Important results are:

• Point defects in clusters are only observed to occur for Pt19 and Pt37.

• A 3-D island is always created when a residual vacancy in a cluster is formed. • Point defect formation is negligible up to E ∼ 30 eV.

6.4

Paper III.

With the knowledge basis obtained with papers I and II, paper III deals with low-energy Pt irradiation on Pt(111) during homoepitaxy. Given that it is known from experiments that low-energy ion irradiation is an effective tool for altering the growth from 3-D to 2-D mode in this energy range, the aim of this paper is to investigate the underlying mechanisms responsible for the transition in growth behavior. In the simulations, 5 ML are deposited for each energy E with an

28 Simulation experiments.

arrival rate of 100 ps−1_{, for a total simulation time of 1 µs. This corresponds}

to a deposition rate of 5x105 _{µs/mm, which is 10}3 _{times higher compared to}

experimental conditions (electron beam-physical vapor deposition). At E = 25 eV, 5 ML are deposited with an arrival rate of 1 ns−1_{, i.e. only 2 orders of magnitude}

higher than experimental rates, for a total simulation time of 1 µs or 1.5 billion time steps. Important results are:

• Ion-irraditiaion-induced mass transport is separated from thermal migration during growth, where the former is completed within 10 ps following the impacts.

• The irradiation-induced mass transport is strongly dependent on the impact energy.

• Thermal migration is essentially independent of deposition energy. • With E ≥ 20 eV layer-by-layer growth is obtained and maintaned.

• The growth mode is determined solely by the atomistic processes in the first 10 ps following the impacts and the energy level of the impacts.

• The same kinetic pathways, leading to an even more clear layer-by-layer growth mode, are identified at fluxes approaching experimental conditions.

The MD simulation with the lower deposition rate took approximately 7 months to finish on a Intel Pentium 4 - 3.40 GHz.

Comment: In order to determine the growth mode, the specular anti-phase intensity I, as would be measured in, for example, reflection high-energy electron diffraction experiments, is calculated from the simulations as a function of time. The expression for the normalized anti-phase intensity is given by

I = ∞ X n=0 (−1)n(Θn− Θn+1) 2 , (6.1)

where Θn is the fractional coverage of the nth layer.

6.5

Paper IV.

This paper is built upon the same simulations as paper III, however, the focus is to give detailed information regarding the intra- and interlayer mass transport during deposition as a function of the energy E. The results are divided in two parts, interlayer and intralayer activity. In order to investigate the mass transport as induced by bombardment, the time between impacts is divided in two intervals, 10 ps and 90 ps. The former represents bombardment-induced activity while the latter corresponds to thermal migration. Important results are:

• Number of impacts generating interlayer events and the total displacement at each event is strongly correlated to incident energy E.

6.5 Paper IV. 29

• With E ≥ 20 eV, the net impact-induced interlayer mass transport is in the upward direction (towards the surface).

• The final distributions of the substrate and deposited atoms are strongly correlated to the impact energy E.

• Interlayer mixing with substrate surface atoms start to occur at E = 15 eV. • Interlayer mixing with substrate sub-surface atoms start to occur at E = 30

eV.

• Sputtering is observed with E ≥ 25 eV, reaching a value of less than 1% at E = 50 eV.

• The mass transport within a layer is strongly correlated to the layer coverage. • Maximum intralayer migration within a layer occurs at a coverage of ∼ 0.05

ML.

30

References.

[1] E. P. M¨unger, V. Chirita, J. E. Greene, and J. -E. Sundgren, Surf. Sci. 355 (1996) L325.

[2] M. H. Carlberg, V. Chirita, and E. P. M¨unger, Phys. Rev. B 54 (1996) 2217.

[3] M. H. Carlberg, V. Chirita, and E. P. M¨unger, Nucl. Instrum. Methods Phys. Res. B 112 (1996) 109.

[4] M. H. Carlberg, V. Chirita, and E. P. M¨unger, Thin Solid Films 317 (1998) 10.

[5] V. Chirita, E. P. M¨unger, J. -E. Sundgren, and J. E. Greene, Appl. Phys. Lett. 72 (1998) 127.

[6] E. P. M¨unger, V. Chirita, J. -E. Sundgren, and J. E. Greene, Thin Solid Films 318 (1998) 57.

[7] V.Chirita, E. P. M¨unger, J. E. Greene, and J. -E. Sundgren, Surf. Sci. 436 (1999) L641.

[8] M. H. Carlberg, E. P. M¨unger, and L. Hultman, J. Phys. Cond. Mat. 11 (1999) 6509.

[9] V. Chirita, E. P. M¨unger, J. E. Greene, and J. -E. Sundgren, Thin Solid Films 370 (2000) 179.

[10] M. H. Carlberg, E. P. M¨unger, and L. Hultman, J. Phys. Cond. Mat. 12 (2000) 79.

[11] E. P. M¨unger, V. Chirita, L. Hultman, and J. E. Greene, Surf. Sci. 539 (2003) L567.